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Cornell University Llbrery 
 
 arV19281 
 
 The academic algebra. 
 
 3 1924 031 286 549 
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€ntan antr §rabfaiirs's Sta%matkal Series. 
 
 THE 
 
 ACADEMIC ALGEBRA. 
 
 BY 
 
 WILLIAM F. BEADBURY, A.M., 
 
 HEAD MASTER OF THE CAMBRIDGE LATIN SCHOOL, 
 
 GRENVILLE C. EMERY, A.M., 
 
 MASTER IS THE BOSTON LATIN SCHOOL. 
 
 BOSTON: 
 THOMPSON, BROWN, AND COMPANY, 
 
 23 Hawlet Street. 
 
Copyright, 1889, 
 By William F. Bradbury and jGrenville C. Embry. 
 
 JoHM Wilson and Son, Cambkidge. 
 
PREFACE. 
 
 The favor with which the Eaton and Bradbury's Mathe- 
 matical Series has been reoeived has encouraged the au- 
 thors to prepare this book. It is <i|signed to meet the 
 demand for a fuller treatment of Factoring, for more 
 numerous examples for practice; -and for the more ad- 
 vanced work now required in our" best High Schools and 
 Academies, and for admission to many of the Colleges. 
 
 In all the subjects treated the exercises have been care- 
 fully graded to lead the student from the simple to the 
 more difficult. 
 
 Special attention is invited to the treatment of Positive 
 and Negative Numbers in Chapter II., of Addition and 
 Subtraction as a single topic, as well as Multiplication 
 and Division, in both Integral and Fractional Numbers; 
 to the arrangement of the equations in Elimination; to 
 the interpretation of negative results, and of the forms, 
 
 — , - , and - ; and to the treatment of Affected Quad- 
 A 
 
 ratic Equations. 
 
 Although a work on Mathematics has a natural order 
 for the development of topics, yet this may not be the 
 best order for the pupil. To awaken the pupil's interest 
 in algebraic operations, a few problems have been intro- 
 duced in Chapter I., and to keep this interest alive, and 
 
IV PREFACE. 
 
 give him some idea of the beauty and utility of Algebra 
 in its application, teachers are recommended, after com- 
 pleting Chapters I. and II., to pass over to Chapter XL, 
 and while going over this and Chapter XII. to take the 
 part omitted. It may also be better for the younger pupils 
 to omit the demonstration in Art. 108, and the Theorems 
 in Arts. 121 and 122, until they become more familiar 
 with algebraic reasoning. 
 
 The adjective. Arithmetic, is used by recent writers on 
 Algebra, and in referring to a series of numbers, as it corre- 
 sponds with the adjectives, Geometric and Harmonic, it 
 has been adopted in this work. 
 
 At the end of the book are the Examination Questions 
 in Algebra for admission to several New England Colleges, 
 from September, 1884, to September, 1888, inclusive. 
 
 W. E. B. 
 
 G. C. E. 
 Cambeidge, Mass., June, 1889. 
 
 Note. Great care has been taken to avoid errors. If any are 
 found, a statement of them to the authors will be thankfully 
 received. 
 
CONTEI^TS. 
 
 Chapter Page 
 
 I. Definitions and Notation. Axioms. Simple Problems 1 
 
 II. Positive and Negative Nhmbbbs 15 
 
 III. Addition and Subteaotion 17 
 
 IV. Multiplication 37 
 
 v. Division 49 
 
 VI. Theorems of Development 61 
 
 VII. Factoring 67 
 
 VIII. Greatest Common Divisor 90 
 
 IX. Least Common Multiple 98 
 
 X. Fractions 103 
 
 XI. Simple Equations 141 
 
 XII. Equations of the First Degree containing Two or 
 
 more Unknown Numbers .... 177 
 
 XIII. Generalization. Discussion of Problems 198 
 
 Negative Results and the Forms ;^, ^> ^ • • ■ • 205 
 
 XIV. Indeterminate Equatioks 211 
 
 Inequalities 214 
 
 XV. Involution 218 
 
 Evolution 225 
 
 XVI. Theory op Indices 241 
 
VI CONTENTS. 
 
 Chapteb Page 
 
 XVII. Radicals 251 
 
 Imaginabies . 266 
 
 Binomial Sukds 272 
 
 XVIII. Radical Equations 274 
 
 Puke Equations 276 
 
 XIX. Affected Quadratic Equations 280 
 
 XX. Simultaneous Quadratic Equations containing 
 
 Two Unknown Numbers 297 
 
 XXI. Peopbbties of Quadratic Equations 309 
 
 XXII. Ratio and Proportion 318 
 
 Variation 327. 
 
 XXIII. Arithmetic Progression 331 
 
 Gbometeio Progression 340 
 
 Harmonic Progression 349 
 
 XXIV. The Binomial Theorem 352 
 
 XXV. Logarithms 357 
 
 Examination Papers, 1884-1889, for Admission to 
 
 Harvard 375 
 
 Yale 383 
 
 Amherst 388 
 
 Dartmouth 395 
 
 Brown 398 
 
 Institute of Technology . . 405 
 
ALGEBRA. 
 
 CHAPTEE I. 
 
 DEFINITIONS AND NOTATION. 
 
 L Algebra is the science which treats of numbers. 
 
 In Arithmetic the numbers are positive, and represented 
 by figures ; while in Algebra numbers may be positive or 
 negative, real or imaginary, and represented by figures 
 or letters. 
 
 In most algebraic operations these numbers have an abstract 
 signification ; that is, they represent the measure, absolute or ap- 
 proximate, of some quantity referred to a unit of its own kind, 
 arbitrarily selected as the standard. 
 
 Through the annexation of the name of the measuring unit they 
 become concrete. 
 
 Note 1. Quantity, the subject of all mathematical investigation, 
 has been variously used to mean not only anything which can be meas- 
 med, — as time, distance, etc., — but also the abstract number arising from 
 its measurement, and the concrete number representing its measurement, 
 together with their various combinations. " It is any symbol which re- 
 sults from the rules of calculation." (De Morgan.) 
 
 In this work the word " quantity " will be used to mean the thing 
 measured, or the concrete number representing its measurement. When 
 the abstract number arising from its measurement is clearly meant, the 
 word "number" will be used. 
 
 1 
 
2 ALGEBRA. 
 
 2. The first letters of the alphabet, a, b, c, etc., gen- 
 erally stand for what are called known numbers, — that is, 
 those whose values are given; and the last letters, x, y, z, etc., 
 for unknown numbers, — that is, those whose values are to 
 be determined. Accented and subscript letters, as r', r", 
 etc. (read, r prime, r second, etc.), or r^, r^, etc. (read, r sub 
 one, r sub two, etc.), are often used to represent numbers of 
 the same kind, but of different values ; as the terms of a 
 continued proportion, or different rates of interest. 
 
 Each letter may represent any number whatever, but 
 throughout the same investigation the same letter is sup- 
 posed to stand for the same number. 
 
 THE SIGNS, -1-, — , X, -=-. 
 
 3. Addition is denoted by the sign + (read, plus), which 
 indicates that the number following is to be added to that 
 which precedes ; thus, 3 + 2 (that is, 3 plus 2) signifies 
 that 2 is to be added to 3. 
 
 4. Subtraction is denoted by the sign — (read, minus), 
 which indicates that the number following is to be sub- 
 tracted from that which. precedes; thus, 7 — 4 (that is, 
 7 minus 4) signifies that 4 is to be subtracted from 7. 
 
 These signs also denote the character, or quality, of 
 numbers. Thus, those before which the plus (+) sign 
 stands are called positive numbers ; and tliose before which 
 the minus (— ) sign stands, negative numbers. This exten- 
 sion of meaning will be fully explained later. 
 
 5. Multiplication is denoted by the sign X (read, into, 
 times, or multiplied hy), which indicates that the number 
 preceding is to be multiplied by that which follows ; 
 thus, 6x5 (that is, 5 times 6) signifies that 6 is to be 
 multiplied by 5. 
 
FACTORS AND POWEKS. 3 
 
 Between a figure and a letter, or between letters, the 
 sign is usually omitted ; thus, 6 a 6 is the same as 
 6 X a X 6. 
 
 Sometimes the sign X is replaced by a point above the 
 line ; thus, 8 • 6 • 4 is the same as 8 X 6 X 4. 
 
 When figures are to be multiplied together, some sign 
 for multiplication must always be employed; thus, 23 
 has already a meaning assigned to it, — namely, the sum 
 of two tens and three units, or twenty-three, — and hence 
 cannot stand for 2 X 3, or 2 ■ 3. 
 
 6. Division is denoted by the sign -i- (read, divided ly), 
 which indicates that the number preceding is to be divided 
 by that which follows ; thus, 9 -r- 3 (that is, 9 divided by 3) 
 signifies that 9 is to be divided by 3. 
 
 Division is also indicated by the sign :, and by the 
 fractional form ; thus, 9 : 3, f , and 9-^3, all have for 
 their value 3. 
 
 FACTORS AND POWERS. 
 
 7. A Factor is any one of several numbers, integral or 
 fractional, which are to be multiplied together to form a 
 product. 
 
 8. Any one or more of the factors which go to make 
 up the product is called the Coefficient of the remaining 
 factors. Thus, in 3 a & c, 3 is the coefficient of a 6 c, or 6 c 
 the coefficient of 3 a, or 3 a 6 the coefficient of c, and so on. 
 
 A coefficient is called numerical, literal, or mixed, 
 according as it is a numeral, a letter or letters, or a 
 numeral and letters combined. The three cases above, 
 taken in their order, illustrate this. 
 
 By coefficient, the numerical coefficient, together with the 
 sitrn of the expression, is usually meant. If no figure is ex- 
 pressed, a unit is understood ; thus, x is the same as 1 x. 
 
4 ALGEBRA. 
 
 9. The Beciprocal of a number is a unit divided by that 
 number; thus, the reciprocal of o is | ; of a;, — . 
 
 10. A Power is the product obtained by repeating a 
 number a given number of times. 
 
 11. An Index, or Exponent, is some number symbol, 
 either positive or negative, integral or fractional, placed to 
 the right, and a little above the number. 
 
 If the index is positive and integral, it indicates how 
 many times the number enters as a factor into the power. 
 
 Thus, 2* = 2x2X2x2 = 16; read, 2 fourth power, or 
 
 the fourth power of 2. 
 
 a' ^ a X a') read, a second power, or a square. 
 
 a^ =^ a X a X a; read, a third power, or a cube. 
 
 a" = aXa-X a . . . to n factors ; read, a nth power. 
 
 Exponents and coefficients must be carefully distin- 
 
 guished. 
 
 
 Thus, 
 
 x* = xXxXxXx 
 
 while 
 
 ix — X + X + X + X 
 
 12. A Root is one of the equal factors into which a 
 number may be resolved. 
 
 A root is indicated by the radical sign ^, the initial 
 letter of the word radix. The root index is written at 
 the top of the sign, though the index denoting the sec- 
 ond, or square, root is generally omitted. Thus, 
 
 \^; read, the second root, or the square root, of a. 
 '\Jx\ read, the third root, or the cube root, of a. 
 -V^a; ; read, tlie wth root of x. 
 
ALGEBRAIC EXPEESSIONS. 5 
 
 ALGEBRAIC EXPRESSIONS. 
 
 13. An Algebraic Expression is a single mimber symbol, 
 or a collection of such symbols, generally connected by 
 algebraic signs. 
 
 14 The Terms of an algebraic expression are the parts 
 which are connected by the signs + or — , the sign gener- 
 ally being considered as part of the term ; thus, 3x + c 
 — 7 y is an algebraic expression of three terms, 3x, + c, 
 and — 7 y. 
 
 15. A Monomial is an algebraic expression which con- 
 tains a single term; as, a, or 3x, or 5bxy. 
 
 16. A Polynomial is an algebraic expression which con- 
 tains two or more terms ; as, x + y, 3 a + ix — 7 aby, 
 or c + 2d — e+5d. 
 
 17. A Binomial is a polynomial of two terms ; as, 
 3x + 3y, OT X — y. 
 
 18. A Trinomial is a polynomial of three terms ; as, 
 3a + x — 6cd. 
 
 19. A Sesidnal is a binomial in which one term is plus 
 and the other minus ; as, .a; — y. 
 
 20. Algebraic expressions are sometimes classed as 
 simple and compound; the former consisting of one term, 
 the latter of two or more. 
 
 21. Like Terms, or Similar Terms, are those which do 
 not differ, or differ only in their signs or coefficients ; 
 as, 4: ax, and —3 ax. Other terms are unlike or dis- 
 similar ; as, bed, and 4 a &. 
 
 22. The Degree of a term is denoted by the sum of the 
 exponents of the literal factors ; thus, 2 a is of the first 
 decree, and 6 a^ a^ and 5 a^ o^ are of the seventh degree. 
 
6 ALGEBRA. 
 
 23. Homogeneous Terms are those of the same degree; 
 as, 5aa^, icthc, and x^z. 
 
 24. A Polynomial is homogeneous if all its terms are 
 of the same degree ; thus, 4 asa^ ■\- b ahc + Zxy^ is & 
 homogeneous polynomial. 
 
 SIGNS OF EQUALITY, INEQUALITY, GROUPING, ETC. 
 
 25. Equality is denoted by the sign = (read, equals, or 
 is equal to), which indicates that the number following 
 it is equal to that which precedes it; thus, $1 = 100 
 cents, signifies that one dollar is equal to one hundred 
 cents. Such a statement is called an equation ; that por- 
 tion which precedes the sign = is called the first mem- 
 ber, and that which follows, the second member. 
 
 26. Inequality is denoted by the sign > or < (read, 
 greater than, less than), which indicates that the number 
 standing at the vertex is less than the number standing 
 at the opening of the two lines ; thus, 6 < 8 < 9 signi- 
 fies that 8 is greater than 6, but less than 9. 
 
 27. The Signs of Inference are •.• (read, because or since), 
 and .'. (read, hence or therefore). 
 
 :■ 2« = 8, .*. W=2. 
 
 28. The Signs of Grouping are the different forms of 
 the Bracket (),[],{}, the "Vinculum — , and the Bar |. 
 They indicate that all the numbers included or connected 
 are to be considered as a single number, and are to be 
 subjected to the same operation; thus, 
 
 a 
 (a + b — c), [a + 6 — c], {a + b — a), a + b — c, 
 
 — c 
 
 indicate that a, b, and — c are to be considered as one 
 whole, and subjected to the same operation. 
 
EXEECISES. 7 
 
 29. The Sign of Continuation is ... , or - - - (read, 
 and so 07i) ; thus, 1 ■ 2 • 3 . . . r signifies the product of 
 the natural numbers from 1 to r inclusive, whatever the 
 value of r. 
 
 The abbreviated form \r (read, factorial r) has the 
 same signification. 
 
 30. The Sign of Infinity is c». In mathematics In- 
 finity means a number which is greater than any assign- 
 able number. 
 
 31. Exercises in Translation from Algebraic into Common 
 Language. 
 
 1. a + b, a — b, a b, a -i- b, a — b, a > b. 
 
 2. 3^ 3», 3» 3", a/3, v'S, \^3, VS. 
 
 3. 5 VW^m, \/l^s> Va+^. 
 
 4. (a + by, (a + by+5Va. 
 
 5. (x + ay — (aJ — ay = 4^ax. 
 
 6. (a + b) + (a — b) = 2 a. 
 
 7. (a + b)-(a-b) = 2b. 
 
 8. (a + b)(a-b) = a^- b\ 
 
 9. (»i + 1) a 4- (w + 1) 6 = 44. 
 
 10- (-+3(^+S<^+i- 
 
 11. 
 
 a + b a — b 
 
 l o+d 
 V m + m' 
 
 x—tj X +y 
 
 12. a" b* c^X aH^d' ^ a* 6» c^ 
 
 13. a' - 5 a^ 6 + 10 a» 6^ - 10 a^b^ + 5 a.b* - &^ 
 
 14. -v^a; — 2/ + VS w — (m + to)''. 
 
 a 
 
 16. _^ X (a; - y + «). 
 
8 ALGEBBA. 
 
 32. Exercises in Translation from Common Language into 
 
 Algebraic Language. 
 
 1. a is equal to b. 
 
 2. a is greater than c. 
 
 3. The sum of a and c is greater than their difference. 
 
 4. The quotient of a divided by c is less than their 
 product. 
 
 5. The second power of a is equal to the cube of c in- 
 creased by one. 
 
 6. The square root of a diminished by one is equal to the 
 third root of c. 
 
 7. a prime, x sub naught, c «th power, b second sub two. 
 
 8. The sum of a, b, c, d, divided by their product equals 
 what? 
 
 9. Five times the square root of a added to the cube of 
 the sum of a and b equals zero. 
 
 10. Six times the cube of the sum of a and b divided by 
 the sum of a and b equals six times the square of a added 
 to twelve times the product of a and b, increased by six times 
 the square of b. 
 
 33. Exercises in finding the Numerical Values of Literal 
 
 Expressions. 
 
 To find the numerical value of an algebraic expression 
 when the values of the letters are known, we must sub- 
 stitute the given values for the letters, and perform the 
 operations indicated by the signs. 
 
 The numerical value of 8 a — 5* + e^, when a := 4, J = 2, 
 and c = 5, is 8 X 4 — 2* + 5^ = 32 — 16 + 25 = 41. 
 
 If a = 16, 6 = 36, c = 9, tZ = 4, e = 1, what are the 
 values of 
 
EXERCISES. 9 
 
 1. 2a-3c + b-4:e-d. 
 
 2. 4 v^ + 2 V~b— A/lel. 
 o ic 8ce , 
 
 ^- T - -ft- + «• 
 
 4. ^7+ Vid + c— -v^T^. 
 
 5. Between the expressions (a + S) — (c — e) and (c? + e), 
 which of the signs =, >, or < is correct ? 
 
 6. Between the expressions -5-^ and ^Z {a + c) d? 
 
 7. A boy expressed his age by saying, that ii a = 6, b — 5, 
 
 and c = 7, he was -'^^ ; years old. How old was 
 
 he? 
 
 8. A laborer's monthly pay was l{a^ + 2ab + ¥) dollars, 
 
 and his annual expenses 54 I ^ J dollars; did he save 
 
 anything if a = 3, 6 = 4, c = 5 ? 
 
 34. Exercises in the Simpliflcatlon of Numbers connected 
 by Parentheses, and the Signs of Operation. 
 
 In reducing such expressions, the operations of multi- 
 plication and division must be performed before those of 
 addition and subtraction. 
 
 Find the reduced values of the following expressions : 
 
 1. 10 + 15-^5 = ? 
 
 2. (10 + 15)^5 = ? 
 
 3. 10 + 15 -f- (5 X 3) = ? 
 
 4. 9 + {(8 - 3) -=- 6} X 2 = ? 
 
 5. 9 + 8-3-f-5 X 2 = ? 
 
 6. 120 -(17 -5) = ? 
 
 7. 4-^-(2x5)-8x2-f-4 + 7=? 
 
 8. v^T+3-\/16-'V^ = ? 
 
10 ALGEBRA. 
 
 35. Exercises in the use of Algebraic Language preparatory 
 to the Solution of Simple Problems. 
 
 Should any difficulty arise in the attempt to answer the 
 following questions, it is recommended that figures be 
 substituted for the letters. 
 
 1. What number is greater than a; by a ? 
 
 2. By how much does x exceed 17 ? 
 
 3. How far can a man walk in a hours at the rate of 5 
 miles an hour ? 
 
 4. If X is one factor of 12, what is the otlier ? 
 
 5. If $ 20 is divided among x persons, how much does each 
 receive ? 
 
 6. What dividend gives a as the quotient when 3 is the 
 divisor ? 
 
 7. By how much does 3 a exceed a ? 
 
 8. If 20 be divided into two parts and one part is as, what 
 is the other ? 
 
 9. The difference between two numbers is 13, and the 
 smaller number is x ; what is the greater ? 
 
 10. If 100 contain x five times, what is the value of a; ? 
 
 11. What is the price, in cents, of 60 oranges, when x 
 oranges cost 10 cents ? 
 
 12. What is the cost of 40 books at x dollars each ? 
 
 13. In x years a man will be 40 years old, what is his 
 present age ? 
 
 14. How old will a man be in a years if his present age is 
 X years ? 
 
 15. If the divisor is x, the quotient y, and the remainder 
 z, what is the dividend ? 
 
 16. If the divisor is m + w, the quotient a; + y, and the 
 remainder s, what is the dividend ? 
 
 17. If a man is a; + y years old now, how old was he 
 X years ago ? How old y years ago ? 
 
AXIOMS. 11 
 
 18. How many hours will it take to walk x miles at 4 
 miles an hour ? 
 
 19. How far can I walk in x hours at the rate of y miles 
 an hour? 
 
 20. A bicyclist rides from Boston to Providence in two 
 days. The first day he rides a hours at as miles an hour, 
 the second day h hours at y miles an hour. Eequired the 
 distance in miles from Boston to Providence. In feet. 
 
 AXIOMS. 
 
 36. The various operations performed upon equations 
 are based upon certain self-evident truths called Axioms, 
 of which the following are the most common: 
 
 1. If equals are added to equals the sums are equal. 
 
 2. If equals are subtracted from equals the remainders 
 are equal. 
 
 3. If equals are multiplied by equals the products are 
 equal. 
 
 4. If equals are divided by equals the quotients are equal. 
 
 5. Like powers and like roots. of equals are equal. 
 
 6. The whole of a number is greater than any of its parts. 
 
 7. The whole of a number is equal to the sum of all its 
 parts. 
 
 8. Numbers respectively equal to the same number, or 
 equal numbers, are equal to each other. 
 
 37. Exercises in the Solution of Simple Problems. 
 
 The Solution of a Problem in Algebra consists, 
 1st. In reducing the statement to the form of an equation ; 
 2d. In reducing the equation so as to find the value of 
 the unknown numbers. 
 
12 ALGEBKA. 
 
 liXAMFLIiS. 
 
 1. The sum of two numbers is 90, and the larger is double 
 that of the smaller. What are the numbers ? 
 
 It is evident that if we knew the smaller number, by doubling it 
 we should obtain the larger number. Suppose we let x equal the 
 smaller number, then 2 x must equal the larger ; and, by the con- 
 ditions of the problem, x the smaller number added to 2x, the larger 
 number, equals 90, or 3 a; = 90. Therefore the smaller number is J 
 of 90, or 30, and 2 x, the larger number, is 60. 
 
 Expressed algebraically the process is as follows : 
 
 Let X = the smaller number. 
 
 Then 2x= " larger 
 By addition x + 2 x = their sum. 
 But 90= " " 
 
 a; + 2 a; = 90 
 3 a; = 90 
 
 X = 30, the smaller number. 
 2 ic = 60, '-' larger 
 
 2. A farmer has a horse, an ox, and a cow ; the horse 
 cost twice as much as the ox, and the ox twice as much as 
 the cow, and all together cost $350 ; how much did each cost ? 
 
 Let X = the numbar of dollars the cow cost. 
 Then 2x= " " « ox " 
 
 and 4 a; = " " " horse " 
 
 By addition a; + 2a; + 4a;=the number of dollars in the 
 
 whole cost. 
 
 But 350 = the number of dollars in the whole cost. 
 
 .-. a; + 2a; + 4 a! = 350 
 
 7 a; = 350 
 
 a;= 50 
 
 2 a; = 100 
 
 4a; = 200 
 
 Therefore the cow cost 150, the ox $100, and the horse $200. 
 
EXAMPLES. 13 
 
 3. Three men, A, B, and C, trade in company and gain 
 $ 300, of which B is to have twice as much as A, and C three 
 times as much as A. Required the share of each. 
 
 Ans. A $50, B$100, C$150. 
 
 4. In a certain garrison of 2700 men there are five times 
 as many infantry and three times as many artillery as cav- 
 alry. How many are there of each ? 
 
 5. A gentleman began trade with a certain sum of money, 
 and continued in trade 8 years. At the end of the first 
 and second years he found he had double what he had at the 
 beginning of those years ; but during the third year he lost 
 as much money as he began business with, when, winding up 
 his affairs, he found he had 11800. How much money did 
 he begin with? Ans. $600. 
 
 6. A man has 3 horses which are together worth $480, 
 and their values are as the numbers 1, 2, and 3 ; what are 
 their respective values ? 
 
 Let X, 2 X, and 3 x represent their respective values. 
 
 7. Divide 200 into three parts, in the proportion of 
 2, 3, and 5. 
 
 8. Two men are 180 miles apart and travel towards each 
 other, one at the rate of 8 miles a day and the other at 10 
 miles a day. In how many days will they meet ? 
 
 9. Four persons, A, B, C, and D, contributed towards a 
 benevolent enterprise $1800. B put in twice as much as A, 
 C put in three times as much as B, and D put in as much as 
 A, B, and C. How much did they each contribute ? 
 
 Ans. A $100, B $200, C $600, D $900. 
 
 10. Required to find such a number that, if it be increased 
 by 8, the result will be equal to 20. 
 
 Let X = the number. 
 
 Then a; + 8 = 20 
 
 But 8 = 8 
 
 By subtraction a; = 20 — 8, or 12 (Ax. 2 ) 
 
14 ALGEBRA. 
 
 11. Required to find such a number that, if it he dimin- 
 ished by 8, the result wiU be equal to 20. 
 
 Let 
 
 X = the number. 
 
 Then x- 
 
 -8 = 20 
 
 But 
 
 8 = 8 
 
 By addition a; = 20 + 8, or 28 (Ax. 1.) 
 
 12. Find two numbers whose sum is 28, and whose differ- 
 ence is 6. 
 
 13. Find a number such that, if we double it and then add 
 20 to it, the result will be 140. Ans. 60. 
 
 14. Two persons agreed to give $60 to a charity, one giving 
 $10 more than the other; what did each give ? 
 
 15. Divide $ 49 between A, B, and C, so that A may have 
 $ 11 more than B, and B $ 7 more than C. 
 
 16. A man walks 10 miles, then travels a certain distance 
 by train, and then twice as far by coach. If the whole jour- 
 ney is 100 miles, how far does he travel by train ? 
 
 Ans. 30 miles. 
 
 17. A and B begin business, each with $4500. B is un- 
 fortunate and loses yearly a certain amount, while A gains 
 yearly the same sum until his money is double that of B's. 
 What does A gain ? 
 
 18. The sum of $5500 was divided among 4 persons; the 
 second received twice as much as the first, the third as much 
 as the first and second, and the fourth as much as the second 
 and third. How much did each receive ? 
 
 19. Three men, A, B, and C, made a joint stock of $3610 ; 
 A put in a certain sum, B put in $ 100 more than A, and C 
 $ 120 less than A. How much did each man put in ? 
 
 Ans. A $1210, B$1310, C$1090. 
 
 20. A person spent $ 410 in buying sheep and cows. If 
 each cow cost $25, and each sheep $5, and if the total number 
 of animals bought was 42, how many of each did he buy ? 
 
POSITIVE AND NEGATIVE NUMBERS. 15 
 
 CHAPTEE II. 
 
 POSITIVE AND NEGATIVE NUMBERS. 
 
 38. We speak of the temperature as iDeing so many- 
 degrees above or below zero ; of the navigator as having 
 sailed so many degrees east or west of a given meridian ; 
 of the merchant as having gained or lost so much money ; 
 of an event as having occurred so many years before or 
 after the Christian era- 
 Such opposition in the direction, character, or quality of 
 numbers is indicated in algebra by the signs + and — . 
 The numbers before which no sign, or the + sign, is placed, 
 are called positive numbers, and those before which the 
 
 — sign is placed, negative numbers. 
 
 Thus it will be seen that the signs + and — , beside 
 their force as signs of operation, which has been explained, 
 have a merdy relative signification. » 
 
 The two meanings assigned to these characters are 
 always in accord, and no confusion can arise from regard- 
 ing these signs in either sense, at pleasure. When, how- 
 ever, it is necessary to denote the character of a number 
 and either addition or subtraction at the same time, we 
 employ two signs, with the parenthesis, thus, + (+ 9), 
 
 — (— 9), the signs within indicating character. 
 
 The sign +, as a sign of character, is frequently omitted ; 
 and when neither the + sign nor the — sign is prefixed 
 to a term, the + sign is to be understood. 
 
 The numbers representing the temperature above zero, 
 north latitude, east longitude, assets, future time, etc., are 
 
16 ALGEBRA. 
 
 usually termed positive, and their opposites negative. 
 But this is purely arbitrary, and there is nothing in the 
 nature of things to prevent a reverse usage. 
 
 39. A clear conception of these numbers in all their 
 relations can best be obtained through the device of a 
 straight line with the zero point at its centre, and positive 
 numbers extending to the right, and negative numbers to 
 the left indefinitely, thus, 
 
 -cc...-l..,-J...O. ..+i...+l... + a), 
 
 a general series, which (imaginaries excepted) embraces 
 all possible numbers; and thus, 
 
 _a> . . . - 4, - 3, -2, -1, 0, + 1, + 2, + 3, + 4, . . . 00, 
 
 a special series, embracing all possible integral numbers, 
 which increase by one indefinitely from left to right, and 
 decrease by one indefinitely from right to left. In these 
 series every negative number is considered to be less 
 than zero, and, in general, every number in these series 
 is considered to be less than any number following it, 
 and greater than any number preceding it, that is, rela- 
 tively so.^ 
 
 In this relative sense, the phrases greater than, less 
 than, less than zero, are to be understood, unless the con- 
 trary is expressed. 
 
 Arithmetic takes into account that part only of the series 
 to the right of the zero, while algebra makes use of the 
 whole series. From this it will be seen at once, that, in 
 algebra, addition, subtraction, multiplication, and division, 
 covering as they do both positive and negative num- 
 bers, must have a more extended signification than in 
 arithmetic. 
 
ADDITION AND SUBTRACTION. 17 
 
 CHAPTEE in. 
 ADDITION AND SUBTRACTION. 
 
 40. Addition in algebra is the process of finding the 
 aggregate, or sum, of two or more algebraic expressions. 
 
 From the series in § 39 it is evident that the numbers 
 added must be all positive, or all negative, or both positive 
 and negative combined. When the numbers are all posi- 
 tive, the algebraic sum will be positive and equal in amount 
 to the number of positive units ; when they are all nega- 
 tive, the algebraic sum will be negative and equal in 
 amount to the number of negative units ; and when they 
 are both positive and negative, the algebraic sum will be 
 positive if the positive units are in the excess, and will 
 be equal in amount to that excess, inegative if the nega- 
 tive units are in the excess, and will be equal in amount 
 to that excess, and zero if the sums of the positive and 
 negative units are equal. Thus, 
 
 (1) + 7 + (+ 5) = -I- 12 ^ 
 
 (2) _ r + (- 5) = - 12 
 
 (3) + 7 + (- 5) = + 2 ^ . . . (A) 
 
 (4) _ r + (+ 5) = - 2 
 
 (5) + 5 + (- 5) = J 
 
 41. In illustration of case (1), group (A), suppose a pencil 
 be moved from the zero point, along the line of numbers, in 
 § 39, second series, in the positive direction (that is, to the 
 right) over seven spaces, then over five spaces, the distance 
 from the starting point would be twelve spaces to the right, 
 and would be represented by + 12 s, the algebraic sum of 
 the distances moved. 
 
 2 
 
18 ALGEBRA. 
 
 In (2) the first movement would be seven spaces, the 
 second five spaces, both in the negative direction (that is, 
 to the left), and the distance from the starting point would 
 be represented by — 12 s, the algebraic sum of the distances 
 moved. 
 
 In (3) the first movement would be seven spaces to the 
 right, the second five spaces to the left, and the distance from 
 the starting point would be two spaces to the right, and would 
 be represented by + 2 s, the algebraic sum of the distances 
 moved. 
 
 In (4) the first movement would be seven spaces to the 
 left, the second five spaces to the right, and the distance from 
 the starting j^oint would be two spaces to the left, and would 
 be represented by — 2 s, the algebraic sum of the distances 
 moved. 
 
 In (6) the first movement would be five spaces to the 
 right, the second five spaces to the left, and the distance from 
 the starting point would be zero, and would be represented 
 by 0, the algebraic sum of the distances moved. 
 
 42. The algebraic sum is not then, as in arithmetic, the 
 entire number of spaces moved, but the distance of the 
 pencil, at the cessation of the movement, from the starting 
 point. 
 
 And, in general, the algebraic sum of several numbers is 
 the deviation of the result from zero, the positive units 
 being counted on, or employed to affect the result, accord- 
 ing to their number, in one way, and the negative units 
 being counted off, or employed to affect the result, accord- 
 ing to their number, in the opposite way. 
 
 43. Illustrative Problems. 
 
 1. Suppose a man to walk along a straight road 100 yards 
 forward and then 70 yards backward, his distance from his 
 starting point is 30 yards. 
 
ADDITION AND SUBTRACTION. 19 
 
 But if he fitst walks 70 yards forward, and then 100 yards 
 backward, his distance from his starting point would be 
 30 yards, but on the opposite aide of his starting point. 
 
 The corresponding algebraic statements would be 
 
 100 yd. + (- 70 yd.) = + 30 yd. 
 70 yd. + (- 100 yd.) = - 30 yd. 
 
 2. Suppose that I have a farm worth 1 6000, and other prop- 
 erty worth 1 3000, and that I owe $ 1000, then the net value 
 of my estate is $6000 + $3000 + (—$1000) = +$8000. 
 Again, suppose my farm is worth $6000, and my other 
 property $4000, while I owe $14000, then my net estate is 
 worth $6000 + $4000 + (- $14000) = - $4000, that is, 
 I am worth — $4000, or, in other words, I owe $4000 more 
 than I can pay. 
 
 3. A boy played two games ; in the first game he won 20 
 points, and in the second he won — 16 points (that is, he lost 
 16 points). How many did he win in all? 
 
 4. A thermometer indicated + 40° (40° above 0) ; it then 
 fell 10°, then rose 30°. What temperature did it then indi- 
 cate ? Had it fallen, instead of risen, these last 30°, what 
 would have been the temperature ? 
 
 5. A ship sailed from the equator due north 40 miles the 
 first day of her voyage, the second day 20 miles due north, 
 the third day 80 miles due south. What was her latitude at 
 the end of the third day ? In this example, which contains 
 the greater number of units, the algebraic or the arithmeti- 
 cal sum? 
 
 SUBTRACTION. 
 
 44. SuiBTE ACTION consists in finding the difference be- 
 tween two numbers. This difference is the number of 
 units which lie between the two numbers, or is what must 
 be added to the subtrahend to produce the minuend. 
 
 It follows, that subtraction is the inverse of addition, and 
 must not be considered a distinct process from addition. 
 
(B) 
 
 20 ALGEBRA. 
 
 45. Prom (3), (4), (1), (2), of group (A), § 40, considering 
 the 7's minuends, and the 5's subtrahends, we produce by 
 addition, that is, by determining what must be added to 
 the subtrahend to produce the minuend, the following : 
 
 (1) + 7 - (- 5) = + 12 
 
 (2) _ 7 - (+ 5) = - 12 
 
 (3) + 7 - (+ 5) = + 2 
 
 (4) -7 -(-5) =- 2J 
 
 46. Wherever the minuend and subtrahend may be situ- 
 ated in the series (§ 39), no spaces, or a certain number of 
 spaces, will lie between them, and this number, with the 
 appropriate sign, will represent their algebraic difference. 
 
 If the movement is to the right, in going from the sub- 
 trahend to the minuend, the + sign would be prefixed ; if 
 to the left, the — sign. 
 
 In the illustration of case (1), group (B), the movement 
 would be from a point five spaces to the left of zero to a 
 point seven spaces to the right ; hence the algebraic differ- 
 ence, 4- 12 s. 
 
 In (2), the movement would be from a point five spaces to 
 the right of zero to a point seven spaces to the left; hence the 
 algebraic difference, — 12 s. 
 
 In (3), the movement would be from a point five spaces to 
 the right of zero to a point seven spaces to the right; hence 
 the algebraic difference, + 2 s. 
 
 In (4), the movement would be from a point five spaces to 
 the left of zero to a point seven spaces to the left ; hence the 
 algebraic difference, — 2 s. 
 
 47. From groups (B) and (A), (2) and (1), it follows 
 that subtracting a positive number is equivalent to adding 
 an equal negative number, and subtracting a negative num- 
 ber is equivalent to adding an equal positive number. 
 
 Therefore, to subtract one number from another, change, 
 the sign of the sultrahend and proceed as in addition. 
 
ADDITION AND SUBTEACTION. 21 
 
 4S. Illustrative Problems. 
 
 1. Suppose I am worth $ 9000 ; it matters not whether a 
 thief steals $4000 from me, or a rogue having the authority 
 involves me in debt $ 4000 for a worthless article ; for in 
 either case I shall he worth only $5000. The thief subtracts 
 a positive quantity ; the rogue adds a negative quantity. 
 
 The corresponding algebraic statements are, 
 
 $9000 - (+ $4000) = + $6000 
 and $9000 + (- $4000) ^ + $5000 
 
 2. Augustus, was born B.C. 63, and died A. D. 14. How'old 
 was he when he died ? 
 
 3. If A has $ 8000 and no debts, and B has no property 
 but owes $4000, how much better off is A than B? 
 
 4. The longitude of Paris is 2° E., that of Boston 71° W. 
 What is their difference of longitude ? 
 
 6. The longitude of San Francisco is 122° W., that of 
 Boston 71* W. What is their difference of longitude ? 
 
 ADDITION AND SUBTRACTION OF ALGEBRAIC 
 LITERAL EXPRESSIONS. 
 
 49. Let a and 6 stand for any positive numbers what- 
 ever, (1) and (3) in group (A), and (3) and (1) in 
 group (B) will become : 
 
 (1) +a + {+b) = a + b\ 
 
 (2) +a + {-b)--^a-b[ ,^. 
 {S) +a-{+b)=:a-bC ' ' ^ ' 
 (4) +a-{-h) = a+b) 
 
 50. To prove the above laws true for all negative values, 
 
 let c = — b, where b is any positive quantity ; then c is any 
 negative quantity, and we have 
 
 -|-e= + (— Si) = — J 
 
 -c = -{-b) = + b 
 
22 
 
 ALGEBRA. 
 
 -b, and 
 
 -cfor +6 in (1), (2), (3), 
 
 Substituting + c for 
 (4), we get 
 
 a + ( — e) = a — e 
 
 a + {+ c) ^=^ a ']- c 
 a — {—c} — a + G 
 a — (+ c) = a — G 
 
 the same laws as before, althougli in a different order. 
 Hence the laws expressed in (C) are true for all values of h. 
 
 51. This proof shows that a letter may stand for any 
 value whatever, and that a letter preceded by the + sign, 
 for example +6, is not necessarily positive. 
 
 Such terms as +5, —a, are called positive and nega- 
 tive terms, because of their outward form, though not so 
 necessarily. The signs before them indicate what is to 
 be done with them when they enter into operations, but 
 nothing as to their reduced, or ultimate value. 
 
 52. The addition of Algebraic Literal Expressions may 
 be conveniently presented under four heads. 
 
 Case I. 
 
 53. To find the Sum of Monomials when they are Similar 
 and have Lilce Signs. 
 
 1. John has 5 apples, Thomas 8 apples, and Frank 3 apples ; 
 how many apples have they all ? 
 
 6 apples, 
 8 apples, 
 3 apples, 
 
 16 apples. 
 
 or, letting a 
 represent 
 one apple. 
 
 8a 
 3a 
 
 16 a 
 
 It is evident that just as 
 5 apples and 8 apples and 
 3 apples added together 
 make 16 apples, so 5 a and 
 8 a and 3 a added together 
 make 16 a. 
 
 In the same way 
 together to — 16 a. 
 
 5 a and — 8 a and — 3 a are equal 
 
ADDITION AND SUBTRACTION. 23 
 
 Therefore, when the monomials are similar, and have like 
 signs, we have the following 
 
 Rule. 
 
 Add the coefficients, and to their sum annex the common 
 letter or letters, and prefix the common sign. 
 
 (2.) (3.) (4.) (5.) (6.) (7.) 
 
 3 ax 
 
 5a^ 
 
 X 
 
 5y 
 
 -5x^ 
 
 — 5by 
 
 9 ax 
 
 3a^ 
 
 5x 
 
 10 2/ 
 
 - 4a;3 
 
 — iby 
 
 7 ax 
 
 8a^ 
 
 8x 
 
 y 
 
 - 8 a;' 
 
 - by 
 
 4: ax 
 
 ia^ 
 
 Ax 
 
 3y 
 
 - 6a;3 
 
 -Sby 
 
 23 ax 18 a! -23a;» 
 
 8. What is the sum of ax^, Aax^, 5 ax^, and 3 aa;'' ? 
 
 Ans. 13 ax': 
 
 9. Wliat is the sum of 4 i a;, 8 6 a;, 7 5 a!, and Zi a; ? 
 
 10. What is the sum oiixy, 5 xy, 10 a; y, and 9 xy? 
 
 11. What is the sum of — 8 a; s, — a; s, — 3 x «, and —xs? 
 
 Ans. —13xz. 
 
 12. What is the sum of - 3 6, — 4 5, —5 b, and — 3 6 ? 
 
 13. What is the sum of —abed, —i abed, -7 abed, 
 and — 5abcd? 
 
 Case II. 
 
 54. To find the Sum of Monomials when they are Similar 
 and have Unlike Signs. 
 
 1. A man earns 8 dollars one week, and the next week 
 earns nothing and spends 5 dollars, and the next week earns 
 
 3 dollars, and the fourth week earns nothing and spends 
 
 4 dollars ; how much money has he left at the end of the 
 fourth week ? 
 
 If what he earns is indicated by +, then what he spends 
 will be indicated by — , and the example will appear as fol- 
 lows : 
 
24 
 
 ALGEBRA. 
 
 + 8 dollars, 
 
 — 5 dollars, 
 + 3 dollars, 
 
 — 4 dollars, 
 
 + 2 dollars, 
 
 or, letting d 
 
 represent 
 
 one dollar, 
 
 r + sd 
 
 -5d 
 + 3d 
 -4.d 
 
 + 2d 
 
 Earning 8 dollars and 
 then spending 5 dollars, 
 the man would Lave 3 
 dollars left ; then earn- 
 ing 3 dollars, he would 
 have 6 dollars ; then 
 spending 4 dollars, he 
 
 would have left 2 dollars ; or he earns in all 8 dollars -\- 3 dollars 
 = 11 dollars; and spends 5 dollars -|- 4 dollars = 9 dollars ; and 
 therefore has left the difference between 11 dollars and 9 dollars 
 = 2 dollars ; hence the sum oi -{- 8d, — 5d, -{- 3d, and — 4d, is 
 + 2d. 
 
 Therefore, when the terms are similar, and have unlike 
 signs, we have the following 
 
 Kule. 
 
 Find the difference between the sum of the coefficients of 
 the positive terms, and the sum of the coefficients of the nega- 
 tive terms, and to this difference annex the common letter or 
 letters, and prefix the sign of the greater sum. 
 
 (2.) 
 
 (3.) 
 
 (4.) 
 
 (5.) 
 
 4a; 2/ 
 
 5y' 
 
 l^ah<? 
 
 
 x'y 
 
 xy 
 
 -2f 
 
 1 ah& 
 
 
 -ISx^'y 
 
 ~&xy 
 
 8 2/' 
 
 — ah<? 
 
 
 11 a.-' y 
 
 ixy 
 
 - 4 2/' 
 
 - 9abo^ 
 
 
 -rsx'y 
 
 ~2xy 
 
 14 y'' 
 
 4 abc^ 
 
 
 25x^y 
 
 5xy 
 
 15 a be' 
 
 
 
 (6.) 
 
 (7.) 
 
 
 
 2Zxyz 
 
 9(x 
 
 + y) 
 
 
 
 — 60 a; y « 
 
 ~5(x 
 
 + y) 
 
 
 
 20 a; 2/ a 
 
 8(x 
 
 + y) 
 
 
 
 — 68 a; 2/ « 
 
 -4 (a; 
 
 + y) 
 
 
 
 9 a; 2/« 
 
 -2 (a; 
 
 + y) 
 
 
 — 73 xy s 
 
 6(x + y) 
 
AUDITION AND SUBTRACTION. 25 
 
 8. Find the sum of 9 x^ y% — 15 x^ y\ 18 x" y\ and — x^ %f. 
 
 9. Find the sum of 9 (« + y), 10 {x + ij), - 2 (a; + y), 
 and {x + y). Ans. 18 (x + y). 
 
 10. Find the sum of — a aj'', + a a;^ + 11 a a;^ + 26 a a;^, 
 and — 14t a x^. 
 
 11. Find the sum of 28 a 6 c, — 34 a 6 c, — 150 a J c. 27 a 6 c, 
 and —13 a bo. 
 
 12. Find the sum of a" x\ - 15 a" x", 18 a« a;», - a' x', 
 23 a' a;', and — a' a;'. 
 
 13. Find the sum of 19 (a + b), — (a + b), (a + b), and 
 - 14 (a + S). Ans. 5 (a + 6). 
 
 Case III. 
 55. To find the Sum of any Monomials. 
 
 From (2) in group (C) it follows that 
 a + (—b) = a~b 
 and that the sum of a, — b, and — c, or 
 
 a + {—b) + (— c) — a — b — e. 
 
 No further reduction is possible, and therefore, to add 
 dissimilar monomials, write them one after the other, each 
 with its proper sign. 
 
 All algebraic expressions can be so written, and the 
 result, without further reduction, is sometimes called an 
 algebraic sum. 
 
 56. It should be remarked that 
 
 a + b + c = a + c + b:^b + a + c = etc. ; 
 
 that is, the sum of any number of algebraic expressions is 
 independent of their order. This is called the Commuta- 
 tive Law of Addition. 
 
 57. Further, 
 
 + {a + b)z= + a + h. 
 
26 ALGEBRA. 
 
 For, to put a and 6 together, and then add the result to 
 what has gone before, is the same as to add both a and h 
 to what has gone before. Similarly, 
 
 a + (6 + c) = (a + Zi) + c 
 
 that is, the sum of any number of algebraic expressions is 
 indepeTident of the mode of grouping them. This is called 
 the Associative Law of Addition. 
 
 58. It follows that 
 
 the sum of 5 a and 6 6 is neither 11a nor 11 h, and can only 
 be expressed in the form of 5 a -\- &h, or6J + 5a; and tlie 
 sum of 5 a and — 4 5 is 5 a — 4 6, or — 4 & + 5 a. In finding 
 the sum of 5 a, 66, 5 a., and — 4 6, the a's can be added to- 
 gether by Case I., and the J's by Case II., and the two results 
 arranged according to §§ 56, 57 ; thus, 5a + 6& + 5a — 46 
 = 66 + 6a-46 + 5a = 66-46 + 5a + .')a = 26 + 10a, 
 or 10 a + 2 6, regardless of the order of the terms. 
 
 1. Find the sum of 7 tZ, — 4 6, x, 3 y, 8 x, — S b, 3 b c, 5 d, 
 5x, lb, 4a;, and — 3bc. 
 
 7 d — 4:b+ x + 3y + 3bo 
 5d — 3b + 8x —3bc 
 
 For convenience, similar 
 terms are written under each 
 other; then by Case I. the 
 + 7 b -\- 5 X first column at the left is 
 
 + 4 a; added ; the second by Case 
 
 12~d +~ 18 x + 3y "•' ^""'^ so on ; + 3 6 c and 
 
 — 3 i c cancel. 
 
 Therefore, to find the sum of any monomials, we have 
 the following 
 
 Knle. 
 
 Add the similar terms according to Cases I. and II., and 
 write after these results, in any order, the dissimilar ones, 
 each v;ith its proper sign. 
 
ADDITION AND SUBTRACTION. 27 
 
 2. Find the sum of 2 a, —3b, + e, + b, — 2c, +3d, —a, 
 + 3 c, and — 2d. 
 
 3. Find the sum of 4 a, —7 a, +3y, — 4 S, + 3 a, + 6 a, 
 — 2/, + 4 i, —2 s, —2 2/, + 4 a, +8b, — «, — 3 a, — 8 6, and 
 -10 c. 
 
 4. 7 + (- 9) + (- 1) + (0) + (+ 5) + (_ 6) =3 ? 
 
 5. 2 a= + (- 3 b^) + (+ 4 b') + (+ 9 a=) + (- b') = ? 
 
 6. a+(-26) + (+3a)-(-&) + (-4a)+(+4:i)=? 
 
 Case IV. 
 
 59. To find the Sum of Polynomials. 
 
 Polynomials are groups of monomials, and hence every- 
 thing necessary to a complete understanding of their addi- 
 tion has been explained in the three foregoing cases. 
 
 60. If in the various operations with polynomials the 
 monomials composing them be first arranged according to 
 the ascending or descending powers of some letter, sym- 
 metry will be secured, and a consequent less liability to 
 error, in the operations to follow. 
 
 The polynomials, 
 
 a;3 + 4 x' + 5 a; + 2 a;°, and 2 x" + 5 x + ix'' + x\ 
 
 are said to be arranged according to the descending and 
 ascending powers of x, respectively. 
 
 We have therefore for the addition of polynomials the 
 following 
 
 Knie. 
 
 Write similar terms under each other, find the sum of 
 each column, and connect the several sums with their proper 
 signs. 
 
28 ALGEBRA. 
 
 1. Findthesumof 3a — 76 + 2c, 6a— 6 + 5 e, — 4a + 
 3 6 - 8 c. 
 
 3a — 7b + 2c 
 
 6 a — b + 5c 
 
 —ia + Sb — So 
 
 5 a — 5 b — c 
 
 (2.) 
 
 -2b^ + 3a¥ + a« 
 
 - ab^+ 5aU-3a» 
 
 Sb' + 8 a» 
 
 + ab^+ 9aH-2a' 
 
 3b^+ Sab-^+UaH + 'ia^ 
 
 (3.) 
 
 Sx" + 5 Va; + T 
 
 -9x+ 2V'S-8 
 — 2a:^ + 4g+ 3a/5 
 
 a;' — 5 a; + 10 V* — 1 
 Add the following : 
 
 4. 4a; — 2y + l, — 3x + 2 — y, x + 3 ij + 3. 
 
 5. x^ — 2 x^ y — 2 X 1/% x^y—3xy^ — if, 3 xy'^ — 
 2y^ — x". 
 
 6. a + b-c+1, -2a-3&-4c + 9, 3a + 2J- 
 16 + 5 c. 
 
 7. 16 x^ — 12-2 X, 11 - 11 a;' - 7 x^ 9 a;' - a; + 
 1 - ai^ 5 + 7 a; + 8 K^, - a;' - 9 a;. 
 
 ^. llx" - 2xy% 3 x^y ~2x'y'' -{■ 1 xf — % y\ 
 8 a;^ — 7 a;^ 1/ + 2/S —12x''+'^3?y\Zx'y'', —Sx'y'' — 
 5xy^ + 7y\ 
 
 9. 3a&2-2J» + < 5aH-ab^-3a', 8a^ + 5b% 
 9a^b — 2a'' + ab\ Ans. 4 a^ + 14 ^2 5 + 3 „ 52 ^ 3 ^s^ 
 
ADDITION AND SUBTRACTION. 29 
 
 10. a» - 2 6» - 11 a'' 5 - 4 a J2 _ J g2^ 4 js _ a 5 c + 
 6o^ + 9an-ac^ + ab\ c^ - 2b' - a^ + ib a" + 3ab\ 
 2an — 3bc^ — 7o^+3abc. Ans. 2abo-ac\ 
 
 11. 3 «'' - 10 / + 5 s'^ - 7 2/ z, -x^ + 4:y^-10z^ + 
 3 xy, z^ + 11 yz + Sxz — 2xy, 4:e^ — Ay« + xz, 
 
 — 2x^ + 6y^ — 9xz — xy. 
 
 12. 4a5+ 12a»-a-10, 6 a^- a«+ 2 a^ - 7, 9 a'^ 
 
 — 3 a= + 4 a, 11 a - 2 a^ - a^ + 9^ 4 aS _ ^4 _ 5^ 
 
 13. 11 a:^ - 9 x^ + 1, 2 a;^ - 3 »* - 2 a;, a;* - a; + 12, 
 a!^ + 4a; — 3, 8a!' + 7ai*. 
 
 14. x^yz — x'z-2xz'-7z*, 3 y* - x* - 2 x^ y^ ~ 
 3a;Sg — 4a;y^ —2a;* — y* — lOxy^z + 4:x''z^ + ixz% 
 3x*+Sxyz^ — 4:x'^z^ + 3y*+7z* — x^y«, 4a:«« + 2y* + 
 2 a;2 2/" + 4 a; 2/' + 10 a; 2/= «, —7y* — 2xz' — Sxyz^ 
 
 15. ^a — ^b, —a + ^b, %a — b. 
 
 16. |a;^ + ix2/ -i2/^ -a;^ - f a; 2/ + 2 2/^ %x^ - 
 
 xy — iy^' 
 
 n. 3a^-iab-lb^, -|a^ + 2«i-f S'', -fa^ 
 -ab + b\ 
 
 18. 4 a' - 2 a= S - I 53, | «^ 6 - J a &= + 2 5», - 1 a« 
 + a 5= + J *'• 
 
 19. 8 2/ Vx — 3x Vy + 5, \/x y + 3x \/y +3, 2 ?/ Vx 
 
 — Va; 2/ ~ 6, 7 2/ Vx — 4 a;A/2/ — 3, 1 + 7 a; \^y —2y \/x. 
 
 20. 4:Vab — SVcd + Qy Vx, —3\fcd — h'\/ab-\- 
 2 y Vx, +7 y Vx — 3 Vab + 8 Ved, 3 ?/ V« + 4 \/a6 
 + 2Vc^- 
 
 SUBTRACTION OF ALGEBRAIC LITERAL 
 EXPRESSIONS. 
 
 61. From group (C) we learn that, in general, the sub- 
 traction of a positive number is equivalent to adding an 
 
30 ALGEBRA. 
 
 equal negative number, and the subtraction of a negative 
 number is equivalent to adding an equal positive number. 
 
 Therefore, for the subtraction of Monomials and Poly- 
 nomials, we have the following 
 
 Change the sign of each term of the subtrahend from + 
 to — , or — to +, or suppose each to be changed, and then 
 proceed as in addition. 
 
 
 (10 
 
 (2.) 
 
 (3.) 
 
 (4.) 
 
 (5.) 
 
 (6.) 
 
 (7.) 
 
 Min. 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 9 
 
 Sub. 
 
 9 
 
 6 
 
 3 
 
 
 
 -3 
 
 -6 
 
 -9 
 
 Eem. 
 
 
 
 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 In examples 1-7, the minuend remaining the same while 
 the subtrahend becomes in each 3 less, the remainder in each 
 is 3 greater than in the preceding. 
 
 (8.) (9.) (10.) (11.) (12.) (13.) (14.) 
 
 Min. 9 6 8 0-3-6-9 
 
 Sub. 9 _9 _9 _9 9 9 9 
 
 Rem. 0-3 ^ ^ ^12 —15 —18 
 
 In examples 8-14, tlie minuend in each becoming 3 less 
 while the subtrahend remains the same, the remainder in 
 each is 3 less than in the preceding. 
 
 (15.) (16.) (17.) (18.) (19.) (20.) (21.) 
 
 Min. 9 6 3 0-3-6-9 
 
 Sub. 9 6 S 0-3-6-9 
 
 Eem. ~^ 
 
 In examples 15-21, both minuend and subtrahend decreas- 
 ing by 3, the remainder remains the same. 
 
ADDITION AND SUBTRACTION. 31 
 
 (1.) (2.) (3.) (4.) (5.) (6.) 
 
 Min. 37a: 28axi/ —Uab —19c iSxyz —GSShcd 
 Sub. 11a; — Taxr/ Bab — 7c —26xyz 29bod 
 
 Rem. 26a: 35axy —19ab — 12c 
 
 (7.) (8.) (9.) (10.) (11.) (12.) 
 
 Min. 20 a;— 11 aic 9xy 25 s— 30a;y 28 c d 
 
 Sub. _46a: 30 a 6 c +15_xj/ 20s 59a;;/ - 560 c <Z 
 — 26 a; —41abc — 6xy 5z 
 
 (13.) (14.) 
 
 Min. 9x-lly-3z 18 a + 22 6 - 33 c 
 
 Sub. 5g!+ 2y + iz - 40 5 + 5 c - 17 t^ 
 
 4a;-13y-7« 18 a + 62 6- 38 c + 17 a: 
 
 15. -7-(-l)=? 
 
 16. -3ab-{+ab)=? 
 
 17. 3a^b-(+aH) + {+5an) = ? 
 
 18. 27 a^ - (+ b^) + (+ 9 b") - (- 2 a") - (+ b'') = ? 
 
 19. From —3a take + 8 a. 
 
 20. From 10 b^ take - 7 b\ 
 
 21. From —12x^y take 16 a;"?/. 
 
 22. From 20 a 6 c take —iabc. 
 
 23. From 25 a — 16 S — 18 c take 4 a — 3 5 + 15 c. 
 
 24. From 7 a^ + 11 a' b + SaH" - 4.ab* - b^ take 
 5aH^ — 'ka^ + 2aH<' + lla*b + &=. 
 
 25. From 3 ab + Bed — 4:ae — 6bd take 3 ab + 
 6cd — 3ac — 5bd. 
 
 26. From 10 a^ & + Sab'^ — 8 aH' - b* take BaH — 
 Gab^-la' b\ 
 
 27. Subtract a;» — a;' + a; + 1 from a;» + a;'= — a: + 1. 
 
32 ALGEBRA. 
 
 28. Subtract J» + c» - 2 a i c from a» + 6» - 3 a & c. 
 
 29. Subtract l — x + x^ — x*—afi from x* — 1 + x — x\ 
 
 30. Subtract a» - 3 a^ 6 + 3 a 5^ + i» from a" + 3 aH + 
 
 31. From 5 Va^ y + 2 a; — 7 a take 3 ^/xy — x + 2 a. 
 
 32. From ^ a; - | y - ^ take - i as + § 2/ - i- 
 
 33. From ^ a» — 2 aa^ - ^ «= a; take ia'x + ^a" - 
 
 34. From 5 x' + 3 a; — 1 take the sum of 2 a; - 5 + 7 a;'' 
 and 3 a;^ + 4 — 2 a;= + a;. 
 
 35. Subtract 5 a;^ + 3 a; — 1 from 2 a:', and add the result 
 to 3 a!^ + 3 a; - 1. 
 
 36. Add the sum of 2y — 3y^ and 1 — Sy' to the 
 remainder left when 1 — 2 y^ + y is subtracted from 5 y\ 
 
 37. What must be added to 5 a;'' — 7 a; + 2 to produce 
 7 a;^ - 1 ? 
 
 38. What must be subtracted from 3a — 5b + c to leave 
 2a-4J + c? 
 
 39. From what must 11 a^ — Sab — 7bc be subtracted 
 to give the remainder 5a^ + 7ab + 7bc? 
 
 40. To what must 7 a;' — 6 a'' — 5 a; be added to make 
 9a;' — 6a; — 7 a;'^? 
 
 41. If 3 a;^ — 7 a; + 5 be subtracted from zero, what will 
 be the result ? 
 
 42. To what must Bab — lib c — 7 c a be added to pro- 
 duce zero ? 
 
 43. Subtract 3 a;= - 7 a; + 1 from 2 a;'^ - 5 a; — 3, then 
 subtract the difference from zero, and add this last result to 
 2 x^ - 2 a;= - 4. 
 
 44. Subtract 3 a;^ — 5 a; + 1 from unity, and add 5 a;*" — 6 a; 
 to the result. 
 
ADDITION AND SUBTRACTION. 33 
 
 45. Take the sum of a;' + 3 a; — 2, 2x^ + x^ — x + 5, 
 and ix^ + 2x^ — 7 X + i, from the sum of 2 a;^ + 9 a; and 
 5 a;» + 3 xK 
 
 46. From the sum of 12x^ + 4:xy* + y^, 2x^ — ix^y — 
 a: y* + 3 y^ and 6x*y + 2x'y^ — 3 a; y*, take the sum of 
 6x^ + 2x^y^ — y\ x^ — 2 x^y + x^ y^ + 2 y\ and 6x^ + 
 4:X*y — 2x^y^ + 3y^. Ans. a;' + x' if. 
 
 Note. — In examples like the last, where several polynomials are to 
 he combined, some by addition and some by subtraction, the result can 
 best be obtained by writing down the polynomials to be subtracted with 
 their signs reversed under those to be added, and then finding the sum of 
 all, thus making use of but a single process. 
 
 Removal and Introduction of Brackets. 
 
 62. The addition or subtraction of a polynomial may be 
 indicated by enclosing the polynomial in a bracket and 
 prefixing the sign + for addition, and the sign — for sub- 
 traction. The polynomial d + e —f added to the poly- 
 nomial a + b + c, that is, a + b + c + (d + e — /), equals 
 by the rules of addition a + b + c + d + e—f; and the 
 polynomial d + e —f subtracted from the polynomial 
 a + b + c, that is, a + b + c — (d + e — /), equals by the 
 rules of subtraction a + b + c — d— e+f. 
 
 Therefore, for the removal of brackets we have the 
 following 
 
 Bule. 
 
 When the bracket with the plus sign before it is removed, 
 the included terms must be rewritten without change of sign ; 
 but when a bracket with the minus sign before it is removed, 
 the included terms must be rewritten with change of sign. 
 
 63. When there are several brackets, they may be re- 
 moved one at a time. 
 
 3 
 
34 ALGEBRA. 
 
 Tims, 
 
 a-2b-[4:a-6b-{3a+{5a- 26 -3a)}] 
 = a-2b-l4:a-Gb- {3a + {5a-2b + 3 a)}] 
 — a-2b-[ia-6b-{3a + 5a-2b + 3a}'] 
 = a-2b-[4:a-6b-3a-5a + 2b — 3a] 
 = a-2b — 'ia + 6b + 3a + 5a — 2b + 3a 
 = 8 a + 2 6 by adding similar terms. 
 
 In the above process the vincnhim was removed first, and then the 
 brackets in succession, beginning with the inner one. 
 
 If we remove the outer bracket first, tlie work will appear as follows : 
 
 a — 26 — [4(1 — 66— j3a+ (5a — 2A — 3a)}] (1) 
 
 = a — 26 — 4a+66+j3a+(5a — 26 — 3a)| (2) 
 
 As the + sign now appears before the next two brackets, these might 
 have been omitted at once from (2) without further change of signs, and 
 at the same time the vinculum over 2 6 — 3 a, and the final expression 
 obtained at once. Thus from (1) we have at once 
 
 a— 26 — 4a + 66 + 3a + 5a— 26 + 3a = 8a+2 6. 
 
 It is recommended that the more advanced students begin with the 
 outermost bracket, as the shorter of the two methods. 
 
 Eemove the brackets, and reduce each of the following 
 expressions to its simplest form. 
 
 1. a + {— b -\- c — d + e). 
 
 2. a — (b — c + d — e). 
 
 3. 3a — 5b + 2c+{2a + 3b-c). 
 4 a - [2 a - {3 5 - (4 c - 2 a)|]. 
 
 5. -{-[-(a-b^^)]l. 
 
 6. _(_(_(_ a;))) -(-(-y)). 
 
 7. 8x- {16y-[3x-(12y-x)-S7/] + x}. 
 
 8. — l_a — \a + (x — a) — (x — a) — a} — 2 a]. 
 
 9. 5 -[4+ {5 -(4 + 6^^)}]. 
 
ADDITION AND SUBTRACTION. 35 
 
 10. 3x— \2y + {5x — 3x+y)}. 
 
 11. {2a:-(5y-3s + 7)}-[4+{a;-(3y+22+5)}]. 
 
 12. 1_[1_|_1_(1_1)_1}_1]. 
 
 ^3. 1 _ (2 - (3 - (4 - (9 - (10 - 11))))). 
 
 14. a - [5 6 - {a - (5 c - 2 c - 6) + 2 a - (a - 2 J - c)} ] . 
 
 15. {3b' + 8cijz)+{(7-b*)-(7-8b')}-i8cyz + 6b'). 
 
 16. {(3 x^ifz^ -2ys')- (xyz-tjz')} - (2 x'^y'^z^ - 
 xyz) — {x^ y'^z'^ — y z"^). 
 
 17. 6 ?)i + ^4 m — [8 re — (2 TO + 4 m) — 22 w] — 7 w} — 
 {7 n + [9 Ml — (3 M + 4 to) + 8 w] + 6 m}. 
 
 18. a - {5 5 + (c - 3 a) + 4 6| + [6 a - (3 J + 2 c)]. 
 
 19. a-26-[4a-6&-{3a-c+(5a-26- 
 3a-c + 26)}]. 
 
 20. -[-2a;- {Sy - (2a; — 3y) + (3a; -2?/)} + 2a;]. 
 
 64. Conversely, from § 62, we have for the introduction 
 of brackets the following 
 
 Knle. 
 
 Wh^n the iraehet introduced is preceded by the plus 
 sign, all the terms enclosed must he written without change 
 of sign ; hut when the bracket ii preceded by a minus sign, 
 all the terms enclosed- must he written with change of sign. 
 
 According to this rule, a polynomial can be written in a 
 variety of ways. 
 
 Thus, x^'-^x'^y + 3xy^-f 
 
 = x^ — (^dx^y — Zxy'^ + y') 
 = x" — Sx'^y — {— 3xy^ + f) 
 = x^ — y^ — {3x^y ^ Bxy") 
 
36 ALGEBRA. 
 
 Place in brackets, with the sign — prefixed, without chan- 
 ging the value of the expression : 
 
 1. The last two terms of 
 
 a + I> — c + d. 
 
 2. The last three terms of 
 
 X — y — z — u. 
 
 3. The first three and the last three terms of 
 
 — 3a-ib + 2c — 3d + e —f. 
 
 4. The last four terms of 
 
 — 2a — 3c — d + 2e + d. 
 
 Bracket, without change of value, the following expressions 
 two together in their order, then three together in their order, 
 with the minus sign before the bracket in each case : 
 
 5. a — b+c — d+f— g. 
 
 6. —2c + 3d-e + 4:f+3a-7b. 
 
 7. ia+6b-5c + 2d — 3e + 3f. 
 
 8. Place, without change of value, a — b + c — d + e—f 
 in the following set of brackets, so that e —f shall stand in 
 the innermost bracket, c — d in the middle, and a — b nearest 
 the left. 
 
 -[ -{ -( )}]• 
 
 Eewrite the following polynomials, so that, without change 
 of value, they shall be composed of binomial instead of mono- 
 mial terms : 
 
 9. a — b — c + d+ e —f. 
 
 10. a' — ab - a c + P — b c + c\ 
 
 11. a'' + b'' + c'' + 2ab-2ac-2bc. 
 
MULTIPLICATION. 37 
 
 CHAPTER IV. 
 MULTIPLICATION. 
 
 65. Multiplication is a short method of finding the 
 sum of the repetitions of a number, or of the repetitions 
 of a certain part of that number. 
 
 The Laws of Multiplication. 
 
 66. In Algebra, as in Arithmetic, ab — ha, and so for 
 any number of factors, as abc = acb = be a, or the 
 product of any number of factors is independent of their 
 order. 
 
 This is called the Commutative Law of Multiplication. 
 
 67. Moreover a (b c) = (a b) e = b (c a), or the product 
 of any number of factors is independent of the order of 
 grouping them. 
 
 This is called the Associative Law of Multiplication. 
 
 68. If <^ is a positive integer, then (a -\-b + c) d = 
 (ffl + 5 + c) + (a + 6 + c) + (a + 6 + c) . . . 
 
 repeated d times, 
 
 = a + ft+c + a + 6 + c + a + & + c... 
 = a + a + a ■\- . . . repeated d times, 
 + 6+6+5 + ... « " 
 
 ■{■ c + c -k- c + . . . « " 
 
 = ad + hd-\-ed'y 
 
 that is, the product of the sum of any number of alge- 
 braic numbers by a third is equal to the sum of the 
 
38 ALGEBRA. 
 
 products obtained by multiplying the numbers separately 
 by the third. 
 
 This is called the Distributive Law of Multiplication. 
 
 69. The multiplier must always be an abstract number, 
 and the product is always of the same nature as the multi- 
 plicand. 
 
 The cost of 5 bushels of potatoes at 75 cents a bushel is 
 75 cents taken, not 5 bushels times, but 5 times ; and the 
 pi'oduct is of the same denomination as the multiplicand 
 75, viz. cents. 
 
 In Algebra the sign of the multiplier shows whether 
 the repetitions are to be added or subtracted. 
 
 1. (+a) X (+4) = + 4a; 
 that is, + a added 4 times is +a + a + a + a:= + 4:a. 
 
 2. {+ a) X (-4) =-4a; 
 that is, + a subtracted 4 times is — a — a — a — o = — 4 a. 
 
 3. (-a) X (+ 4) =-4a; 
 that is, — a added 4 times is — a — a — a ~ a = — 4: a. 
 
 4. (— a) X (— 4) = + 4 a ; 
 that is, — a subtracted 4 times is-|-a-|-a-l-a + ct = -|- 4 a. 
 
 In the first and second examples the nature of the 
 product is + ; in the first, the + sign of 4 shows that 
 the product is to be added, and + 4 a added is + 4 a ; in 
 the second, the — sign of 4 shows that the product is to be 
 subtracted, and + 4 a subtracted is —4 a. In the third 
 and fourth examples the nature of the product is — ; in 
 the third, the + sign of 4 shows that the product is to 
 be added, and — 4 a added is —4a; in the fourth, the 
 — sign of 4 shows that the product is to be subtracted, 
 and —4a subtracted is + 4 a. 
 
MULTIPLICATION. 39 
 
 70. Hence, in multiplication, we have for the sign of the 
 product the following 
 
 Kule. 
 
 Like signs give + ; unlike, — . 
 
 Hence the product of an even number of negative factors 
 is positive ; of an odd number, negative. 
 
 71. Multiplication in Algebra can be presented best 
 under three cases. 
 
 Case I. 
 72. When both Factors are Monomials. 
 
 1. Multiply 3ahy2b. 
 
 3ax25=:3xax2xJ = 3x2xaXZ' = 6a6 
 
 As by the Commutative Law the product is the same in whatever 
 order the factors are arranged, we have simply changed their order 
 and united in one product the numerical coefficients. 
 
 2. Multiply a' by a\ 
 
 As the exponent, if integral and positive, of a number 
 shows how many times it is taken as a factor, 
 
 a' = a X a X a 
 and a^ = a X a 
 
 .•. a'Xa^=(aX»Xa)x(aXa) = aXaXaXaXa: 
 
 ■ a" 
 
 Therefore the product of powers of the same number is that 
 number with an index equal to the sum of the indices of the 
 factors. This is called the Index Law. 
 
 Hence, when both factors are monomials, we have the 
 following 
 
40 ALGEBRA. 
 
 Kule. 
 
 AnTiex the product of the literal factors to the product 
 of their coefficients, rcmevibering that like signs give +, and 
 unlike, — . 
 
 (3.) (4.) 
 
 
 (5.) 
 
 (6.) (7.) 
 
 Ixy Qx^y^ 
 
 
 lab 
 
 -2imn^ - aH* 
 
 2 ah 3xy^ 
 
 — 
 
 9 an 
 
 6 an* -9a:'b 
 
 Uabxy 18 x" y^ 
 
 — 
 
 63 a' b' 
 
 — 144 a m w' 9 a' b^ 
 
 8. a^Xa^=? 
 
 
 15. 
 
 16 x^tfx5x''r/=? 
 
 9. x^Xx*=? 
 
 
 16. 
 
 -Hc'x'xSc^x''^? 
 
 10. c' X (- c") = ? 
 
 
 17. 
 
 -5aHfX (--4ci*y=) = ? 
 
 11. -X* Xx'' = ? 
 
 
 18. 
 
 14a'a;sx5a2a;=0''=? 
 
 12. - a' X (- a') == 
 
 ? 
 
 19. 
 
 -15a;^2/s'x(-4a;22/') = ? 
 
 13. 3ay^x5a'y* = 
 
 :; ? 
 
 20. 
 
 304 a^ i= X (- 8 a= i*) = ? 
 
 14. 613 x^z^x7x^» 
 
 ,4 
 
 ? 21. 
 
 IQ&x^y'^z X 5xifz'^^? 
 
 22. 16b^(^d' X2al 
 
 ,3g5 
 
 X (-4( 
 
 %" d') = ? 
 
 23. Sxy^z* X (- 3 : 
 
 C*?/ 
 
 «0 X (- 
 
 -2x^fz) = f. 
 
 24. 15 Vx y X Bab 
 
 X 2c£Z = ? 
 
 
 26. 3 (a + J) X 5 (a 
 
 + b)^ = 15 
 
 {a + by. 
 
 Note. Any number of terras enclosed in a bracket may be treated as 
 a monomial. 
 
 26. - 10 (a;« - /) X 4 (a;" - y'Y = 7 
 
 27. {a - by X (a- by X (a - by = ? 
 
 28. 3 (a; - «)"■ X 4 (a; - g)» = 12 (x - «)"• + ". 
 
 29. - 8 (c - a;) X (- 3 (c - a;)=) X (- 2 (c-a;)=) = ? 
 
MULTIPLICATION. 41 
 
 Case II. 
 73. When only one Factor is a Monomial. 
 
 1. Multiply X + y + zhy a. 
 
 (x + y + z)a = ax + ay+az, 
 
 oj. These results follow 
 
 a; + 2/ + « from the Distributive 
 
 a Law (§ 68). 
 
 ax + ay + as 
 
 Therefore, for the multiplication of a polynomial by a 
 monomial, we have the following 
 
 Bule. 
 
 Multiply each term of the multiplicand by the multiplier, 
 and connect the several results by their proper signs. 
 
 (2.) 
 2x^ +6x — 12y 
 ixy 
 Sx'y + 24:x^y-4:8xy^ 
 
 3. (ea'-5aH-iab') X (-3ab'>) = ? 
 
 4. (— ab + bc — ca)x(— abe) = ? 
 
 5. (— 5xy^» + 3xys^)xxyz— ? 
 
 6. (abc — a^bc — ab^e)x— abc = ? 
 
 7. (- a^b e + b^ a - c^ ab) x - ab = ? 
 
 8. {9gh — 12ffm-6gn)x(—3ff7i)=:? 
 
 9. (4:a*x-5a^x^-ax* + 2x') X (- 11 asc'^a') = ? 
 
 10. (m^ — 3 m^ n + 3 mn'^ — n^) Xn=? 
 
 11. {-2y+3is-5x''y''z''-7xa^ + 2s^&)X(-3y^x) = ? 
 
 12. (a^ + b^-c^ + d^-e'+f^) X(-a' 6' c*) = ? 
 
42 ALGEBIIA. 
 
 Case III. 
 74, When both Factors are Polynomials. 
 
 1. Multiply X + y + s hy a + b. 
 
 (x + ]/ + z)(a + b) = {a + b)x + {a + b)y+(a + b)z, 
 
 by the Distributive Law, (a + h) being regarded as a single term. 
 The last expression further reduced by the same law becomes 
 ax-\-hx-i-ay-{-by + az-{-bz, and this equals by the Comrau- 
 fcitive Law for Addition (§ 56), ax-i-ay + az-{-bx-\-by-{-bz. 
 
 Hence, for the multiplication of a polynomial by a polyno- 
 mial, we have the following 
 
 Kule. 
 
 Multiply each term of the multiplicand ly each term of 
 the multiplier, and find the sum of the several products. 
 
 2. Multiply 2x' -k-Zxy — y^ by 2,x — 2y. 
 
 2 X + 3 xy y -^g begin at the left, placing 
 
 3 a! — 2y the second result one place to 
 
 6 x' + 9 x' V B xip *^^ "g^tj so that like terms may 
 
 _ 4 a;2 w — 6 X ?/ + 2 «' ^'^^'^ "^ ^^^ ^*™® vertical col- 
 
 umn. 
 
 &x^ + 5x^y — 9xif + 2^f 
 
 3. Multiply 3x-33?-l+ x'hy 3x + x" + 1. 
 
 In Examples 2 and 3 
 
 a;' - 3 a^ + 3 a; 
 
 the multiplicand and mul- 
 
 x^ -\- 3x +1 tiplier were arranged ac- 
 
 3^ — 3 X* + 3x^ — x^ cording to the descending 
 
 + 3 a;^ - 9 x» + 9 a;^ - 3 r« l°Zl"' °l!?5r_!L":lt 
 
 + x' — Bx^+3x — l 
 
 x^ ~ Sx'^ + 5x^ — 1 according to the ascending 
 
 powers as well. 
 
 plying. The polynomials 
 could have been arranged 
 
MULTIPLICATION. 43 
 
 Note. Though the arrangement of the polynomials according to 
 the ascending or descending powers of some letter is not absolutely 
 essential, it should be observed where possible, as symmetry of work 
 tends to minimize errors. 
 
 4. Multiply 2xz — z^-\-2£' — Sjjz + xy by a; — ?/ + 2s. 
 
 2x^+ xy + 2x»—^yii —£' 
 
 X — y + 2« 
 2x*+ a?y + 2x^g—3xyis— x z^ 
 
 — 2x^y —2xyz —xy^+3y''s:+ y z^ 
 
 + 4:x^z+2xyz + 4:xs^ — 6ys^ — 2z° 
 
 2 a;'- x^y + 6x'z-3xys + 3xs^-xrf + 3y'z-5yz'' — 2z^ 
 
 5. {2x' + 3xy-f) x{3x-2y) = ? 
 
 6. (3 x^-x^- 1) X (2 X* - 3 x^ + 7) = ? 
 
 7. (b* - 2 i^ + 1) X -{l>* + 2 &2 + 1) = ? 
 
 8. (a-2 + ax + a') X {x-a) = ? 
 
 9. {x* — ax'+a^x^ — a''x + a*)x{x + a) = ? 
 
 10. (x^-x'y + xf -y>)x(!>i^ + 2xy + 7f) = ? 
 
 11. {a^+2a-b + AaP + Sb') X ia'-4:ab + 4:b) = ? 
 
 12. (a^ + b^ + c' + ab + a c - b c) X (a — b - c) = ? 
 
 13. (4a=+96Hc*--36c-2ac-6afi)x(2a + 36 + e) = ? 
 
 Ans. 8 a" — 18 a J c + 27 6» + c». 
 
 14. (a + 5 — c) (6 + c - ct) (c + a — 6) X (a + 6 + c) = ? 
 16. (a& + c<;?+ac + 6«^) X (ab + c d — a o — b d) = ? 
 
 16. (-3a^i= + 4«6'+15«'i) X {5an^+ab'>-3b*) = ? 
 
 17. (48 a= a; - 64 a' + 27 a;' - 36 a a;2) X (3 a; + 4 a) = ? 
 
 Aus. 81 a;*- 256 a^. 
 
 18. {a? -bab-b^) X (a^ + 5 rt J + 6°) = ? 
 
44 ALGEBRA. 
 
 19. {x" - xy + X + y^ + y + 1) X {x + y -1) = 1 
 
 20. (a^ + J'' + c= - & c - a c - a 6) X (a + 6 + c) = ? 
 
 21. (a;i2 - x' 2/= + «« ?/* - a;'/ + 2/') X (x» + j/^) = ? 
 
 22. (2 o,^ + 2 a + 1 + a* + 3 a^) X (1 - 2 a + a^) =z ? 
 
 23. (3ax2/'^-9ay^-aa:=) X (- aa; - 3 a j/^) = ? 
 
 24. (3a:='2/^— 2a;2/«-2x»2/+a;'+2/*)X(a;H2a!y + ?/2) = ? 
 
 25. (a;= + 4a;*2/'+8a;V + 15a;>«)X(-x'+2a;V— 2/°) = ? 
 
 Ans. - a;8 — 2 a' y2 + 26 a:* ^' - 8 a:» y - 15 x''y^\ 
 
 26. (a;Ha;'y + a!^2/Ha!2/=) X (— a;*?/H a!2/* + a:°2/— 2/0 = ? 
 
 27. (a;" - 2/' + 3 a;2/'' + 2a;'^2') X (2a;2/ - a;^ - ^^) = ? 
 
 28. (21a;^y - Ua;?/' -Itf) X (-3a;^ - a;^?/' + /) = ? 
 
 29. (2 a^ + 3 a^ & + 3 a 5^ + 2 6^ X (4 aH - 6 a= 6^ + 
 6 a^ 6' - 4 a J^) = ? 
 
 Ans. 8 a'5 + 6 0^5' — 6 a.H'^ - 8 a 6'. 
 
 When the coefficients are fractional, the ordinary process is 
 still employed. 
 
 30. Multiply ^a^-^ab + ^b^^3yla + \b. 
 
 \a^ - ^ab +%b^ 
 kcL + ib 
 
 \a' -^an+iaP + ^b^ 
 
 31. (ia'+^a + i)x (Ja-4) = ? 
 
 32. (^x^-2x+^) xiix + i) = ? 
 
 33. (f x^ + o^y + iy')x ax-iy) = ? 
 
 34. (f a;2 - a a; - % a^)X (^ x'' - ^ ax + ^a^) = ? 
 
MULTIPLICATION. 45 
 
 75. A somewhat abridged and simple method of work- 
 ing examples in which the exponents of the letters increase 
 and decrease by a common difference, as in Example 3, 
 is to omit the letters altogether. Thus : 
 
 1-3+3-1 
 1 + 3 + 1 
 
 1-3+3-1 
 +3-9+9-3 
 +1_3+3_1 
 
 1+0-5+5+0-1 
 
 or, x^ — 5 »' + 5 a;^ — 1. 
 
 The insertion of the powers of x depends upon the fact 
 that the highest power in the product is always the product 
 of the highest powers in tlie two factors, and that the rest 
 follow in order. 
 
 2. Multiply a;' + 5a; — 3 by a;^ — 1, or a;' + Oa;^ + 5 a; — 3 
 by a;^ + a; — 1. 
 
 1+0+5-3 
 1 + 0-1 
 
 1 +0+5-3 
 
 _l_0-5+3 
 1+0+4-3-5+3 
 
 Ans. a;6 + a;^ + 4 a;» - 3 a;' - 5 a; + 3, 
 or, x^ + 4 a;» - 3 a;2 - 5 a; + 3. 
 
 The insertion of zero coefficients at the beginning of the 
 above operation, as well as at the close, is necessary iu order 
 to preserve the law of the exponents. 
 
46 ALGEBRA. 
 
 3. Multiply a'-3aH + 3ah^ + b^ hj a^-2ab + b\ 
 
 4. Multiply 5x^ — 3ax^ + 5a^x — a^ by a' + 3 a x + 5 x^. 
 
 5. Multiply 4:x'' — 24:xy + 36]/^ by itself. 
 
 6. Multiply x^-xy^-f by a;' + 3yK 
 
 The process above is called Multiplication by Detached 
 Coefficients. Many of the examples in § 74 will serve as 
 additional exercises under this method. 
 
 76. Exercises in the Omission and Insertion of Bracltets. 
 
 A coefficient placed before a bracket indicates that 
 every term of the expression within the bracket is to 
 be multiplied by that coefficient. 
 
 1. Simplify 94-8 [-lla;-4{-17a; + 3(8-9-5a;)}]. 
 94-8[-lla;-4{-17a; + 3(8-9 + 5a!)}] 
 = 94 - 8 [- 11a; - 4 {- 17 a; + 15a; - 3}] 
 = 94 - 8 [- 11a; - 4 {- 2a; - 3}] 
 = 94-8[-lla; + 8a;+ 12] 
 = 94-8[-3j;+ 12] 
 = 94 + 24 a; -96 
 = 24a;-2 
 
 This example can be done much more briefly as follows : 
 
 94 - 8 [- 11 a; - 4 {- 17 a; + 3 (8 - 9 - 5a:)}] 
 = 94 + 88 a; + 32 {- 17 a: + 24 - 27 + 15 4 
 = 94 + 88 a: + 32 {- 2a; - 3} 
 = 94 + 88 a; -64 a; -96 
 = 24 a; - 2. 
 
 Simplify : 
 
 2. b-(c-a)-lb-a-c-2{c + a-3{a-b)-d}'\. 
 
 3. -20 {a -d) + 3 (b-o)- 2 lb + e + d- 3 {c + d - 
 4:{d— a)}]. Ans. 4a + b + c. 
 
MULTIPLICATION. 47 
 
 4. a - 2 (ft - c) - [- {- (4 a - & - c - 2 {a + S + c} )}]. 
 
 5. - 3 {- 2 [- 4 (- «)]} + 5 {- 2 [- 2 (- a)]}. 
 
 6. x' - [(a; - yy - {{x - y - z^ - (z - x)^}]. 
 
 7. x^-lx''+y'+(x-y) (x + y)- {x^- (^/^ _ .^qil^a)} ] . 
 
 Ans. x'\ 
 
 8. (2 a - J) (2 a + 6) + [a 5 - ft {« - (2 a - 2 a"-^)}]. 
 
 9. (a + ft + c)2 - (a + ft) (a _ c) - (a - ft) (b - c) - 
 
 10. (a + ft + c)2— a(ft+c — a)— ft(a + c — ft)— c(a + ft— c). 
 
 Ans. 2{a^ + h'' + c^). 
 
 In the following expressions bracket together the equal 
 powers of a;, so that the signs before all the brackets shall be 
 positive. 
 
 11. aa^ + bx^ + 2hx — 5x'^ + 2x'^ — 3x. 
 
 ax* + bx'^ + 2bx — 5x'^ + 2x*-3x 
 = ax*+2x* + bx^-Bx^ + 2bx-3x 
 = (a + 2)x*+ (ft - 5) a;^ + (2 ft - 3) x. 
 
 12. 3fta;2-2a; + 6a.a;« + ca;-4a;''-fta;'- 
 
 13. -7x^ + 5ax^ -2cx + 9ax^ + 7x-3x^ 
 
 14. 2 c a;5 — 3 a ft a; + 4 c^a; — 3 ft x'' — a= a;^ + a;^ 
 
 111 the following expressions bracket together the equal 
 powers of x, so that the signs before all the brackets shall be 
 negative. 
 
 15. a a;^ + 5 a;' - a^ a;* - 2 ft a;3 - 3 a;^ - 4a!<. 
 
 16. 7 a;' — 3 c'^ a; — a ft a;« + 5 a a; + 7 a;^ — a ft c a;'. 
 
 17. (7,a:'' + a^a;»- ft a;=- 6 a;2-ca!». 
 
 18. 3 ft^ a;^ — ft a; — a a;^ — c a;* - 5 c^ a; - 7 a;^ 
 
48 ALGEBRA. 
 
 Simplify the following expressions, and in each result group 
 the terms according to the powers of x. 
 
 19. ax'^-^cx — lbx^ -{ex - dx — {hx^-\-Zc x^)} - 
 {r,x^ — Sa;)]. 
 
 20. 5ax=-r(ia;-ca;^)-{65a;^-(3a£B2 + 2fta;)-4ca;'}. 
 
 21. x{x~b — x{a — b x)} + {ax — x {ax — b)}. 
 
 Add together the following expressions in each example, 
 and group the results according to the powers of x. 
 
 22. ax' — 2bx'', bx — ex' — a^, and x' — ax^ + ex. 
 
 23. a" x' — 5 X, 2 a x^ — 5 a x', and 2 x' — bx^ — ax. 
 
 24. ax' + bx — c, q X — r — p x% and a;'' + 2 a; + 3. 
 
 Multiply together the following expressions iu each exam- 
 ple, and group the results according to the powers of x. 
 
 25. a a;^ — 2 6 a; + 3 c and px — q. 
 
 (ax' — 2b X + 3c) (px — q) 
 = apx' — 2bpx'' + 3epx — aqx' + 2bqx — 3cq 
 = (ap)x'— (2bp + aq)x'+ (3op+2bq)x— (3c?) 
 
 26. (x' + ax' — bx — c) (x' — ax' — b x + c). 
 
 27. (x^ — ax^-bx' + ex) (x^ + a x' — b x' - e x + d). 
 
 28. (x'-ax-bx') (x' + p x - q'). 
 
 29. (a;'^ - 3 aa: + 6 a^) (a;^ + 5 5 a; + 8 b'). 
 
 30. (x'-5bx' + 3b) (x' + 3bx'-2bx + b). 
 
 31. (a;' - 3 c^a;^ + d) (x' + 2ex + c). 
 
 32. (a;= + 5ma;'-3TOa;) (a;* — 4ma;2 + 2w). 
 
 33. (x^~Aax* + 2bx'+ e) (x* + 2ax'-b). 
 
 34. (x'' + 3mx*-2nx) (x' - 2 m x + n). 
 
DIVISION. 49 
 
 CHAPTEE V. 
 DIVISION. 
 
 77. Division is finding a quotient which, multiplied by 
 tlie divisor, will produce the dividend. Division is the 
 inverse of multiplication. 
 
 In accordance with this definition and the Kule in § 70, 
 the sign of the quotient must be + when the divisor and 
 the dividend have like signs ; — when the divisor and the 
 dividend have unlike signs ; that is, in division, as in mul- 
 tiplication, we have for the signs the following 
 
 Kale. 
 
 Like signs give + ; unlike, — . 
 
 Case I. 
 
 78. When the Divisor and Dividend are both Monomials. 
 
 1. Divide 9 ax by 3x. 
 
 The coefficient of the quotient must 
 
 9aa:-=-3a; = 3a be a number which, multiplied by 3, 
 
 the coefficient of the divisor, will give 
 
 9, the coefficient of the dividend, that is, 3 ; and the literal part of 
 
 the quotient must be a number which, multipbed by x, will give 
 
 a X, that is, a ; the quotient required, therefore, is 3 a. 
 
 2. Divide a^ by a^. 
 
 W" -i- a' = a°, or ^ aaa = a^ 
 
 aa 
 
 For (§ 72), a^ X a^ = aaa X aa, = a-^- Therefore the 
 
 quotient of two powers of the same number is that number 
 
 tuith an index equal to the index of the dividend minus the 
 
 index of the divisor. 
 
 4 
 
50 ALGEBRA. 
 
 Hence, for the division of monomials we have the following 
 
 Bule. 
 
 Annex the quotient of the letters to the quotient of their 
 coefficients, remembering that like signs give + and unlike —. 
 
 (3.) ' (4.) 
 
 y ax' 1 Irxy 
 
 (6.) (6.) 
 
 -412 tfy^ _ A 3 -54o»a;'y^ _ a 2 g 
 
 7. 2%x^yf'z^2x^f = 'i 
 
 8. 475 a" ¥ c" -^ 2h aV <? =■ 1 
 
 9. -85a;»2/^s-v-5a;2/^ = ? 
 
 10. 68 V" c» d^ -f- (- 17 V c" d') = ? 
 
 11. —135 a^ m* w» -=- (- 15 a to*) = ? Ans. 9 a n\ 
 
 12. 6= «« -^ 52 s = ? 
 
 13. -16 a' -H 4 a= = ? 
 
 14. 128 a^ x" -=- (-16 a' x) = ? 
 
 15. - 310 0^ m^ -h (- 10 c= to) = ? 
 
 16. 238 a' y'^ (-7 a* if) = ? 
 
 17. a'^ ^ a" — a™-". 
 
 18. 27 a;™ 2/" -^ 9 a;" 2/' = ? Ans. 3 a;*"-" 2/"-'. 
 
 19. 342 xfz^~ (- 114 a; j/^ s^) ^ ? 
 
 20. 135 (a + J)5 -J. 9 (a + h^ = ? Ans. 15 (a + J)=. 
 
 21. 27 (a - J)« -=- 3 (a - §)« = ? 
 
 22. -34 (to - ?i)« -;- 17 (to - w)' = ? 
 
 23. 123 (c - rf)5 -^ (_ 41 {c - c?}») = ? 
 
DIVISION. 51 
 
 79. In this took when successive numbers are separated 
 by the sign x or -j-, the operations thus indicated are to 
 be completed in succession from left to right. Thus, 
 
 a-^h X c := 1 xc = -I-, and is not equal to a H- (o X c), or — 
 be 
 
 24- 24 
 
 So 24 -^ 4 X 3 = -r X 3 =: 18, and is not equal to - — -, or 2. 
 .«j< 4 ^4-3 
 
 From thi« ^will be seen that 
 
 a -^ b X c :=»-=- (6 -=-c) 
 
 and a-^(hXc)=a-^b-h-c 
 
 That is, as in addition and subtraction, the removal of a 
 bracket with the sign + before it requires no change of signs, 
 while the removal of a bracket with the sign — before it re- 
 quires that the sign of each term taken out should be changed 
 from + to — , or from — to + (§ 62), so, in multiplication 
 and division, the removal of a bracket with the sign X before 
 it requires no change of signs, while the removal of a bracket 
 with the sign -h before it requires that the signs of the suc- 
 cessive numbers separated by the sign X or -=- should be 
 changed from X to -=-, or from -=- to X. Thus, 
 
 ^ , , ,-, ^ , , 1 abd 
 7x(aXb-^cxd) = 7XaXb-^cXd = 
 
 1c 
 but 1 ^ (a X b -^ c X d) — 1 -^ a -^ b X c -^ d = -—- 
 
 ^ ' abd 
 
 So 5 abe -i- d = 5 X (a X b X c -i- d) = 5 ^ (d -^ a ^ b -i- c) 
 
 1. Divide y^ X y^ -^ y'' X y by y^ -^ y* X y'^ -r- y- 
 
 (fxy*-^fxy)^(s'-^y' x/^y) 
 = y^ xy*-i-y^x y-^/ x y*^f)^y 
 = y'Xy*Xyxy*Xy^y^-^y^^y'' 
 = (lf\Xy*xyxy*xy)^ (y" x t/ x y") 
 = y'' ■^y'^^y' 
 
 The process above la written out merely to illustrate the changes 
 of signs involved in removing and introducing a bracket with the 
 sign -7- before it, and the principle that must be used in applying the 
 Commutative Law in multiplication and division. 
 
52 ALGEBRA. 
 
 2. (2 a -;- a X a) -r- (2 a X a -f- a) = ? Ans. 1. 
 
 3. (24 a;2 H- 2 X H- 3) -=- (2 a: -^ 2) = ? 
 
 4. (5 -=- 2 X 10) H- (10 ^ 5 X 2) = ? Ans. 6^. 
 
 5. (32 -=- 8 X 4 -^ 2) -=- (12 H- 6 X 2) = ? 
 
 Case II. 
 80. When the Divisor only is a Momniial. 
 
 I. Divide ax + ay+azhya. 
 
 From % 73, {x + y + s) a = a X + a y + a s. Conversely, 
 (ax + ay + a»)-^a = x + i/ + s, so that the quotient ob- 
 tained by dividing the sum of two or more monomials by a 
 third is the sum of the quotients obtained by dividing the 
 monomials separately by the third. Hence the following 
 
 Knle. 
 
 Divide each term of the dividend hy the divisor, and 
 connect the several results hy their proper signs. 
 
 (2.) (3.) 
 
 4 g) 28 a JB^ - 12 g x^ -3 a° y°) -21 x" ^f - 18 a;\y' 
 
 7x^ — Zx" 7^ + 6x^ 
 
 4. (a^ - a & - a c) ^ (- a) = ? 
 
 5. (16 x" - 24 x« - 32 cc*) -^ (_8 cc^) = ? 
 
 6. (3a!«-9a;2 2/-12a;2/^^3£B==? 
 
 7. (4 x^ y' - 8 a;3 2/2 + 6 a; 2/^) -^ (-2 a; «/) = ? 
 
 8. (10a*a;2_30aSa,s^40a2£B^y2)H-5a2a! = ? 
 
 9. (-8 a*a:^ + 16 a^ V'x' - 12 J^x«) -=- 4a;2 = ? 
 10. (9a;-12y + 3y)H-(-3) = ? 
 
 II. (36 a= &= - 24 a^ J^ - 20 a^ 5^) ^ 4 „2 j _ 9 
 
 12. (2x^-5xy+|x>^)--(-ix) = ? 
 
 13. (-3 a^— |a6 — 6ac)-^(-Ja) = ? 
 
 14. (;J a^ X — ^5 a 5 c — f a c a;) -^ I a a; = ? 
 
DIVISION. 53 
 
 Case III. 
 81. When the Divisor and Dividend are both Polynomials. 
 
 1. Divide a^ - 3aH + Sab'' - b^ by a'' — 2ab + b\ 
 
 a" - 2 ab + P) a^ - 3 aH + 3 aP - b' {a - b 
 «» - 2 a" & + aP 
 
 — a'''b + 2ab^-b' 
 
 — a^b + 2aP-b^ 
 
 The divisor and dividend are arranged in the order of the powers 
 of a, beginning with the highest power, a', the highest power of a 
 in the dividend, must be the product of the highest power of x in 
 
 the quotient and a^ in the divisor ; therefore, -^ = o must be the 
 
 highest power of a in the quotient. The divisor, a^ — 2 o 6 + 6^, mul- 
 tiplied by a, must give several of the partial products which would 
 be produced were the divisor multiplied by the whole quotient. 
 When (a2 - 2 a 6 + 62) a = aS - 2 a^ 6 + a 62 jg subtracted from 
 the dividend, the remainder must be the product of the divisor and 
 the remaining terms of the quotient; therefore we treat the re- 
 mainder as a new dividend, and so continue until the dividend is 
 exhausted. 
 
 Hence, for the division of polynomials, we have the fol- 
 lowing 
 
 Bule. 
 
 Arrange the divisor and dividend in the order of the 
 powe-rs of one of the letters. 
 
 Divide the f/rst term of the dividend hy the first term of 
 the divisor ; the result will be the first term of the quotient. 
 
 Multiply the whole divisor hy this quotient, and subtract 
 the product from the dividend. 
 
 Consider the remainder as a new dividend, and proceed 
 as before until the dividend is exhaxisted. 
 
54 ALGEBRA. 
 
 2. Divide 8 a" + 8 a- b + i ab^ + b' hj 2 a + b. 
 
 2a + b)8a'+8aH + 4:ab'' + b''{4:a^ + 2ab + b 
 8 a' + 4 «••= 6 
 
 4 
 4 
 
 a''6 + 4 
 aH + 2. 
 
 ab'' 
 
 
 
 2 
 2 
 
 ab' 
 
 
 3. Divide a* — aH + 2 aH'' — ab^ + b* hy a" + b". 
 a' + P) a* — an + 2 aH'' - a b^ + b* {a^ - ab + b^ 
 
 a' + 
 
 an^ 
 
 -aH + 
 
 -an 
 
 -ab^ 
 
 
 a' b^ + 6* 
 a^S^ +b^ 
 
 4. Divide aj^ — 1 by a; — 1. 
 
 £B — 1) a;* + »= + a;2 + X — 1 («« + a;2 + a; + 1 
 
 a;*^ 
 
 x" 
 
 + 
 
 x^ + dx^ 
 a;»- a!^ 
 
 
 + x^ + Qx 
 
 + «'- a; 
 
 or, 
 
 a;-l 
 x-1 
 
 1) a;* — 1 (x' + a;2 + a; + 
 a;* — o;^ 
 
 a;'- 
 
 -1 
 
 
 
 
 a;»- 
 
 -a!» 
 
 
 
 
 
 a;^ 
 
 -1 
 
 
 
 x^ 
 
 — X 
 
 
 
 
 
 X 
 
 — 
 
 1 
 
 
 
 X 
 
 — 
 
 1 
 
DIVISION. 
 
 6. Divide a' + P + c' — 3abc by a + b + e. 
 a^ — 3abc + P + c^fa + b + c 
 
 a^ + a^b+a^o \a!i — ab — ac + b^ — bc + <?. 
 
 — a^b — a? c — 3a6c 
 
 — a'b — ab'^ — abc 
 
 — a^c + ab^ — 2abc 
 
 — a^ c — abc — a c^ 
 
 ab^- 
 ab-" 
 
 abc-\-ac^-{-W 
 
 + 6» + 6^c 
 
 — 
 
 abG-\- ac^—b^c 
 abc — b^G—b<? 
 
 ac^ +bc'' + c^ 
 
 In the examples the dividend, divisor, and successive remainders 
 are arranged in desceiuiing powers of a and x. The ascending powers 
 would have answered as well. The choice of letter and kind of 
 arrangement are immaterial, but it is especially important before 
 beginning the division that some arrangement should be adopted and 
 maintained throughout the operation. 
 
 6. (x' +12x + 20)^(x + 2) = ? 
 
 7. (x^-3x- 70) ^ (a; + 7) = ? 
 
 8. (2 a» - 7 a'' - 3 a + 18) -=- (2 a + 3) = ? 
 
 9. (a;» + a CB^ - 3 a= a; - 6 a») -=- (a; - 2 a) = ? 
 
 10. (x* + 4a!^ + 16) ^ (a;" + 2a; + 4) = ? 
 
 11. (x\- 1) -=- (a; - 1) = ? 
 
 12. {x'' + 2/") ^(x* + y*) = ? 
 
 13. (a;' - 8 2/=) -f- (a: - 2 2/) = ? 
 
 14. (a^ + y')-^ix + y) = ? 
 
 15. (x^ -32i/)-^{x-2y) = ? 
 
 16. (a;* - 8a;» + 21a!2 - 16a; - 7) -^ (a;'^ - 6a; + 7) = ? 
 
 17. (4 a; - 2 a;3 + 2 a;* + 5 - 6 a;'') -=- (2 a; + 1 + a;'') = ? 
 
56 ALGEBRA. 
 
 18. (2x^ + X + 2 + x^) ^ (x + 1 + x^) = ? 
 
 19. {17 x' + 7 X +x^-10x* + 41x^-6)-^(-Sx + x^-6)= ? 
 
 20. (9 a;* - 9 a x» - 13 a^ x=- 16 a» a; - 6 a^ -^ (3 x^ - 
 5ax-3w')::=? 
 
 21. (14a^-45a»5 + 16 aH= - 15aS= - 18 &*) ^ (7a= - 
 5 a 6 + 6 6^) = ? 
 
 22. (5xif- 10 2/^ + 16a;< - 46a:=2/ - 21 a;^?/^) -f- (2 2/== + 
 8x'' — 3xy) — ? Ans. 2 a;^ — Sxy — 5?/^ 
 
 23. (9x2/^ - 2y5 + lla:^^^ + ajf^ - 5xUj - Ux^tj'J -^ 
 (2 2/^ + a;-^-3a;2/) = ? 
 
 24. (a= + 243 J=) -f- (a + 3*) = ? 
 
 25. (a!= - 2/«) -=-(«!« + «'' 2/ + a 2/^^ + 2/») = ? 
 
 26. (x' + Sx'-ix''- 128 a; - 192) -^ (a;^ - 16) = ? 
 
 27. (a^2 + 2 a^S^ + 6^^) -H (a^ + 2 a^ft^ + b') = ? 
 
 28. (x« — a;> + x' 2/^ — a;' - 2/^) ^ (a:' — « — 2/) = ? 
 
 29. (3a;= + 3x2 + a!'-4a;«-3a: + 2)H-(-a; + a;2-2) = ? 
 
 30. (9 a!« + x^ + 2 - a; - 6 a;^ - 5 a;*) -^ (2 + a;2 - 3 a;) = ? 
 
 31. (14a;^+ 78 a;^ y^ + 45 x^ + 45 a; 2/' + 14 ?/*) -H (2 a!^ + 
 5xi/ + Ty'') = ? Ans. 7x^ + 5xy + 2f. 
 
 32. (3 x^ + X + 2 x^ + 1 - a;=) -=- (1 + a;^ — «) = ? 
 
 33. (2 x« + 1 - 3 x^) -f- (1 + 2 X + a;') = ? 
 
 34. (5x'y^ + y^-x^-bxy^) H- (2 x y - x^ — - y'') = ? 
 
 35. (26 x^ 2/'- 8 x' 2/'°-*'- 2 x' 2/^-15 a;'' 2/'^)-=- (2x^2/'- 
 x' — 2/'=) = ? Ans. x^ + Ax*y^+8x^y*+ 15 x^/. 
 
 36. (l-3a6-29a^5^ + 21a*S^)-=-(-l + 5a5-3a2S2) = ? 
 
 37. (14x^2/*+ 28x2y=- 63x^2/ + 42x=2/^— 14xy«-7y') -f- 
 {-Sx^-x'^y^ + 2/^) =? 
 
 38. (x^ - y«) -^ (x= + 2 x" 2/ + 2 X 2/' + s/') = ? 
 
DIVISION. 57 
 
 39. (aj8 + a;*y + /) ^ (aj" -^ a; ?/ + «/=) = ? 
 
 40. (a;«-a:*2/''+3£(;»/-a;''y + ?/«) -^ (a;" _ a; y + 2/=) = ? 
 
 41. (a' — a»2/H 2/^- a''tf)^{2a''y + y» + a^ + 2a'if) = 7 
 
 42. {x^y — xy^) -^ {x^ + x'^y + x^y^ + xy^) — ? 
 
 43. («« + 3 a S c + &» - c=) -^ (ft + 5 - c) = ? 
 
 44. («' + 3 ti'^ 6 + 3 ft 5^ + 6' - 1) ^ (ft + 5 - 1) = ? 
 .45. (ft' + 6 ft 6 c - 8 i= + c*) -f- (a - 2 5 + c) = ? 
 
 46. (ft* + 54 + c*-2ft''S=-2ft^c= — 2 5''c2)-f-(a2 + 2ft5 + 
 ^.2 _ c2) = ? Ans. ft2 - 2 a6 + §2 _ c2. 
 
 47. (a^6''+c=(Z^-ft^c=-62d^)-=-(ft5 + c<i + ac + 6cO = ? 
 
 48. (2 ft"" — 6 a^" i" + 6 ft" &''" — 2 J»») -=- (ft" — &")=? 
 
 49. (6a;"'+*" — 19a;"' + ^» + 20a;"'+" — 7 a;"" — 4a;"'-") -f- 
 (3 a;^" — 5 af + 4) = ? Ans. 2 a;'"+" — 3 a;""— a;'"-". 
 
 50. (ft^" — b*" + ft"c=" + d''"c*" — a"cZ^" — 6^"cZ^") -H 
 (ft" + S^") = ? 
 
 When the coefficients are fractional, the ordinary process 
 may still be employed. 
 
 61. Divide ia;' + -^a ^ y^ + ^y' hy ^x+iy. 
 
 ^x+ iy) ix' + ^^xy^ + ^y' {ix^ - ixy + iy^ 
 |a!»+ T^g x^y 
 
 - tV «' 2/ +1^5 3: 2/' + 1^2/' 
 
 4 «2/' + tV2/' 
 
 52. (^a»-TVa'+TV«-^)-a«-i) = ? 
 
 53. (iaj» + tVs'S'' + 1^2/') -^ (4^3 + i2/) = ? 
 
58 ALGEEKA. 
 
 64. (^% a*-^a^--la'+ ia+ V) -H (f a' - § - a) = ? 
 55. (36x'+ii/' + i-4:Xy-&x+li/)^(^6x-iy-^)^? 
 
 82. When the exponents of the letters in the divisor 
 and dividend increase or decrease by a common differ- 
 ence, the abridged and simple method made use of in > 
 Multiplication, § 75, can be employed with equal facility 
 in Division. 
 
 1. Divide a^ - 5 a* a; + lOa'a;^ - 10 ( 
 hy a'^ — 2ax + x\ 
 
 1-5 + 10-10 + 5-1/1-2 + 1 
 
 (r 
 
 1-2+1 Vl -3 + 3-1 
 
 Ans. a^ — 3 a^ X + 3 a x^ — x^. 
 
 -3 + 
 -3 + 
 
 9-10 Ans. c 
 6- 3 
 
 
 3- 7 + 5 
 3- 6 + 3 
 
 
 - 1 + 2-1 
 
 - 1 + 2-1 
 
 2. Divide a;'' — 1 by a; — 1. 
 1+0+0+0-1 
 
 1 
 
 + 
 
 
 
 1 
 
 -1 
 
 
 
 
 1 + 
 
 
 
 1- 
 
 -1 
 
 
 
 
 1- 
 
 -1 
 
 
 
 1- 
 
 -1 
 
 1+1+1+1 
 
 Ans. a;' + a;^ + a; + 1. 
 
 3. (2y*-162/''+2y='+92y+48)-=-(t/=-52/-12) = ? 
 
 4. (5a;* - 14 a;' y + 31 x^ y''-22xy' + 12 /) ^ (5 a;^ - 
 4a:y + 3y==) = ? 
 
 5. {Qa*b'' + 3an'-4aH* + b'')~{3aH-2ab'+b*)=? 
 
DIVISION. 59 
 
 This process is called Division by Detached Coefficients. 
 Many of the examples in § 81 will serve as additional 
 exercises in this method. 
 
 83. By making use of brackets a neat and concise 
 method is presented for working out certain examples 
 in Multiplication and Division. 
 
 1. Multiply {x + 1)2 + 2 (a; + 1) + 1 by (a; + 1) + 2. 
 
 (CB + 1)^ + 2 (a; + 1) +1 
 
 {x + 1) +2 
 
 (x + 1)' + 2 (a; + ly + (a; + 1) 
 2 (x + 1)" + 4 (a; + 1) + 2 
 
 (a; + 1)» + 4 (a; + 1)2 + 5 (a; + 1) + 2 
 
 2. Divide(a; + l)» + 4(x + l)2+5(a; + l) + 2by(a;+l) + 2. 
 
 (x + Vf + 4(a; + 1)^ + 5{x + 1) + 2 ( {x + 1) + 2 
 
 {x+lf + 2{x + iy \(a; + l)^ + 2(a; + l) + l 
 
 2(a; + l)2 + 5(a: + l) 
 2{x + lf + 4.{x + l) 
 
 (a; + 1) + 2 
 (a; + 1) + 2 
 
 3. Multiply x + a, a; + 5, and x + e, together. 
 
 x 
 
 X 
 
 + a 
 
 
 
 
 
 
 
 x" 
 
 + ax 
 + bx 
 
 + ab 
 
 
 
 
 
 
 x" 
 
 X 
 
 + (a 
 
 + c 
 
 + b)x 
 
 + 
 
 a b 
 
 + b)x 
 
 + 
 
 
 x' 
 
 + (a 
 
 +.b)x^ 
 cx^ 
 
 + 
 
 aba 
 c (a 
 
 abe 
 
 x^ + (a + b + c)x^+{ab + ae+bo)x + abe 
 
60 ALGEBRA. 
 
 4. Divide x^ + (a + b + c) x^ + (ab + ac + be) x + abc 
 
 by X + c. 
 x^+{a + b + c)x^+{ab + ac + bc)x + abc (x+^ 
 
 x^+ cx^ \x^-\-{a + b)x+ab 
 
 (a + b)x^-\- {ab + ac-\-hc)x 
 (a,-\-b)x'^+ {ac + bo)x 
 
 abx + abe 
 abx + abc 
 
 5. Multiply x^ + y^ + ^'^ — ijz — zx — xy hj x + y + s. 
 
 x'^ — (j/ + z)x + {y'^ — y» + z^) 
 a: + (y + g) 
 
 as' - (y + z) 3? + {y^— y« + s'O »; 
 
 + (y + z)x''-(y^-\-2yz + z^)x + (2/° + g°) 
 x^ —'Syzx + y" + z^ 
 
 6. Divide x'^ ~ Zy zx -\- y^ + z* by x -\- y + z. 
 
 a;^ — 3yzx + (?/' + z") (x + (y + z) 
 
 x" + {y + z) x^ V-_(y + z)x+ (i/^-yz + z") 
 
 — (jj-\-z)x'^ — Syzx 
 
 — {y + z)x'^ — {y^ + 2yz + z'^x 
 
 {if — 2? « + «^) a; + (y' + 2°) 
 (/- y»^z^)x^(f\z^) 
 
 7. [jc^ — (5 + c) a: + 6 c] X (aj — a) = ? 
 
 8. \x^ — (a + i?) a;^ + (g' + ai?) sc — a g'] -^ (a: — a) = ? 
 
 9. [3(a;+l)2+4(a;+l)-8]x[-2(a! + l)'+5(a;+l) + l]=:? 
 
 10. [x= + 6a;y-(8?/»-l)] ^ [a; - (2 y - 1)] = ? 
 
 11. (1 — a;= + 8 2/^ + 6 xy) H- (1 - x + 2 ?/) = ? 
 
 12. [(a; + 1)==+ 3 (a; + 1) + 2] X {_{x + 1) + 1] = ? 
 
 13. [3 (a; - 1) - 5] X [(a; - 1) + 1] = ? 
 
 14. (aS _ S3 ^ ^3 ^ 3 ^ J g) ^ („ _ J ^ g) ^ 9 
 
THEOREMS OF DEVELOPMENT. 61 
 
 CHAPTER VI. 
 
 THEOREMS OF DEVELOPMENT. 
 
 84. From the principles already established we are pre- 
 pared to demonstrate the following important theorems. 
 
 THEOREM I. 
 
 85. The square of the sum of tvjo numbers is equal to the 
 square of the first, plus twice the product of the two, plus the 
 square of the second. 
 
 Proof. Let a and b represent any two numbers. Their 
 sum will be a + b; and (a + by = (a + b) (a + b), which 
 expanded becomes a^ + 2 a b + b^, as will appear from the 
 following process : 
 
 a + b 
 a +b 
 
 a^ + ab 
 + ab + b^ 
 
 a^ + 2ab + b'' 
 
 According to this theorem find the square of 
 
 1. 2/ + «. Q. 3x + 5]/. 
 
 2. 2x + J/. ^ns. 9x^+30x]/ + 25 if. 
 
 Ans. ix'' + 4:xi/ + y^- 7. x^ + ■>/. 
 
 3. a + 1. 8. a' + b^. 
 
 4. a + 3 6. 9. a; + 2. 
 
 5. 2 a; + 3. 10. 2 a' + 3 V^. 
 
62 ALGEBRA. 
 
 THEOREM II. 
 
 86. The sqtiare of the difference of two numbers is equal 
 to the square of the first, minus tvAce the produet of the two, 
 plus iJie square of the second. 
 
 Proof. Let a and b represent the two numbers. Their 
 difference will be a — b; and (a — b)^ = (a — b) (a — h), 
 which expanded becomes a^ — 2 a 5 + i^, as will appear from 
 the following process : 
 
 a—b 
 
 a — b 
 
 a b 
 
 ab + b^ 
 
 a^-2ab + P 
 
 According to this theorem find the square of 
 
 1. c — d. 6. a — 3 b. 
 
 2. 2x — 2y. 7. 3x — y. 
 
 Ans. ix^ — 8xy + 4:y''. Ans. 9x^—&xy + y\ 
 
 3. x-3. 8. a^-b\ 
 
 4. 1 — x. 9. x — abc. 
 
 5. x^ — f. 10. 9x-2i/. 
 
 THEOREM m. 
 
 87. The product of the sum and difference of two numbers 
 is equal to the difference of their squares. 
 
 Proof. Let a and b represent any two numbers. Their 
 sum will be a -{■ b, and their difference a — b; and (a + b) 
 {a — b) = a^ — b", as will appear from the following process : 
 
 a 
 
 + 
 
 b 
 
 
 a 
 
 — 
 
 b 
 ab 
 
 
 a^ 
 
 + 
 
 
 
 — 
 
 ab - 
 
 -b^ 
 
 a^ 
 
 
 
 -b^ 
 
THEOREMS OF DEVELOPMENT. G3 
 
 According to this theorem multiply 
 
 1. X -\- a hy X — a. 
 
 2. X + 1 by x — 1. 
 
 3. 1 + 2 a; by 1 — 2 a;. Ans. 1 - 4 a;^. 
 
 4. a^ + V' by a^ - h\ 
 
 5. a + 3bhya — 3b. 
 
 "6. 2ab + 3cd'by 2ab — 3cd. 
 7. bm^n + 2 xy hy 5m^n — 2xy. 
 
 88. This theorem suggests au easy method of squaring 
 iiumbers. 
 
 For, since a^ = (a + h) (a — J) + b^, 
 992 = (99 + 1) (99 - 1) + 1^ :3z 100 X 98 + 1 = 9801 
 
 In accordance with this principle, find the square of 
 1. 97. 
 972 = (97 + 3) (97 - 3) + 3-2 = 100 X 94 + 9 = 9409 
 
 2 95. 3. 498. 4. 45. 5. 995. 
 
 THEOREM IV. 
 
 89. The square of a polyTwmial is equal to the sum of 
 the squares of all its terms, together with tvjice the product of 
 each term into each of the terms that follow it. 
 
 PitoOF. Let a + b + c + etc. be any polynomial. Then 
 (a + b + c + etc.y =z{a + b + c+ etc.) (a + b + c + etc.), 
 
 which expanded becomes 
 
 ^2 + &2 + c2 4. etc. + 2ab + 2ac + etc. + 2 Jc -f etc., 
 
 as will appear from the following process : 
 

 ALGE 
 
 ;bua. 
 
 a + 
 
 b + 
 
 c + etc. 
 
 a + 
 
 b + 
 
 c + etc. 
 
 64 
 
 a^+ ab + ac -\- etc. 
 
 + ab+ ¥ +bc -\- etc. 
 
 + ac + be -{- c^ -\- etc. 
 
 a^ + 6^ + c'* + etc. 
 + 2«6 + 2ac + etc. 
 + 2 6 c + etc. 
 According to this theorem find the square of 
 
 1. a + b + c. 
 
 (a + 5 + c)2 = a= + 4' + c2+2a6 + 2ac + 2Jc 
 
 2. a + 5 — c. 
 
 (a + 6-c)= = a' + b'' + c'^ + 2ab-2 ac-2bc 
 
 3. a + 2 i - 3 c. 
 
 (a + 2 6 - 3 c)2 = a^ + (2 bf + (-3 c)'' + 2 a (2 b) + 
 
 2 a (-3 c) + 4 4 (-3 c) 
 = a2+4 62 + 9c'^ + 4a5-6ac-12ic 
 
 4. a; - 2 y — 3 g. 5. x'' — y"^ — z\ 
 
 6. l + 2a;-3a;=. Ans. 1 + 4x2 + 9a;*+-4a;-6a;'^-12a;«. 
 
 7. a — b — c. 10. m — n — 2? — ?• 
 
 8. a + 2 6 + c. 11. c + rf — a; + y. 
 Q. X — y + a — b. 12. 3 + a; — 2/ + «. 
 
 THEOREM V. 
 
 90. The product of two hinomials of the form of x + a, 
 X + b, is equal to the square of the first term, plus the sum 
 of the second terms into x, plus the product of the second 
 terms. 
 
 Proof. Let x-\-a, x-\-b, represent the binomials. Then 
 {x + a) {x + b) = x'^ -\- {a -\- b) X -\- a b, 
 as will appear from the following process : 
 
THEOREMS OF DEVELOPMENT. 65 
 
 X + a 
 X +b 
 
 x^ + ax 
 + bx+ ab 
 
 x^+ {a + b) x + ab (1) 
 
 91 This includes all possible cases. For, putting in (1), 
 — a, — b, for a and b, 
 then, —a, for a, , 
 
 and finally, — b, for b, 
 
 we get the following three additional cases : 
 
 {x+ (-a)} {x+ (-b)}=x^+(-a-b)x + ab 
 or, (a; — a) (x — b) = x^ — {a + b)x + ab 
 
 {x+ {-a)}{x + b) =x^+(-a + b)x-ab 
 or, {x — a) {x + b) = a;^ — (a — b)x — ab 
 
 (x + a) {x+ (—b)} = x^+ (a-b)x-ab 
 
 or, {x + a) {x — b) = a;^ + {a — b)x — ab 
 
 (2) 
 (3) 
 (4) 
 
 92. The following examples illustrate these four cases. 
 
 (1.) (2.) 
 
 x+ 9 X— 9 
 
 x+ 7 X— 7 
 
 x^+ 9x 
 
 
 + 7x 
 
 + 63 
 
 x^ + lQx 
 
 + 63 
 
 (3.) 
 
 
 a:-9 
 
 
 x + 7 
 
 
 x^ — 9x 
 
 
 + 7x- 
 
 -63 
 
 x^— 9x 
 
 - 7x 
 
 + 63 
 
 a;^ — 16 a; 
 
 (4.) 
 a; + 9 
 
 x-7 
 
 + 63 
 
 a;=' + 9 a; 
 -7a!- 
 
 63 
 
 a;2 _ 2 a; - 63 a;" + 2 a; - 63 
 
66 ALGEBRA. 
 
 According to tnis theorem, multiply 
 
 5. a; + 8 by a; + 5. 11. x — 3 a by x + 2 a. 
 
 6. x-Shy x-5. •^'^^- ^'-(ix-6 a" 
 Ans. x^ — 13a; + 40. 12. a - 1 by a + 1. 
 
 7. a; + 8 by a; — 5. 13. x + 6 by a; — 1. 
 
 8. a; — 8 by a; + 6. 14. a; + 6 a by a; — 5 a. 
 
 9. « — 3 by a + 12. 15. a — 5 6 by a + 10 b. 
 10. X — 4y by X — 10 y. 16. y + 4a; byy — 5a;. 
 
 93. MISCZa^IiANEOUS EXAMPLES. 
 
 1. Find the square of 3 aa; + 2 Jy. 
 
 Ans. 9 a^a;^ + 12 ahxy + 4 J^/. 
 
 2. Find the square of 5 abc — 7 abc. 
 
 3. Expand (5 abc — cf. 
 
 4. Multiply 2 a^ — 3 6 by 2 a^ + 3 4. 
 
 5. Multiply 3 a; + 7 a^i by 3 a; - 7 a'^i. 
 
 6. Expand (5 a^ - i y*f. 
 
 7. Expand (4 a'' A — 5 a by. 
 
 8. Find the square oi xij ■]- yz -\- zx. 
 
 Ans. a;'/ + /z= + a;2z''+2a;/2+2a!2y2 + 2a;y2''. 
 
 9. Find the product of a^" + ¥\ a^ + b\ a^ + i^, a^ ^ j2^ 
 a + 6, and a — 6. Ans. a'-^ — i*''. 
 
 10. Find the product of 1 + a, 1 — a, 1 + a^, 1 _ ««. 
 
 11. Find the product oi (a -\- b + c) {a + b — c). 
 
 12. Find the product of {a — b -{■ c) (b + c — a). 
 
 13. Find the product of (a^ + a + 1) (a'^— a + 1). 
 
 14. Find the product of (a + by (a= — 2ab — b^). 
 
 15. Expand (a + J — c + e)^. 
 
FACTORING. 67 
 
 CHAPTER VII. 
 
 FACTORING. 
 
 94. An algebraic expression which contains no terms 
 in the fractional form is called an integral expression. 
 
 Thus, a? — y^, 2 ax — 3 J, are integral expressions. 
 
 An expression is rational when none of its terms contain 
 square or other, roots. 
 
 95. The Factors of such expressions are the rational and 
 integral expressions whose product will produce these 
 expressions. 
 
 96. A Prime Factor is one that is divisible without a 
 remainder by no rational and integral expression except 
 ± itself and ± 1. 
 
 97. The factors of a purely algebraic monomial are 
 apparent. 
 
 Thus, the factors of a'^hxyz are a, a, b, x, y, and z. 
 
 98. Polynomials are factored in accordance with the 
 principles of division and the theorems of the preceding 
 chapter. 
 
 Case I. 
 
 99. When all the Terms have a Common Factor. 
 
 1. Find the factors of ax — ay + az. 
 
 As a is a factor of 
 
 (ax — ay + az) =z a (x — y + z) each term, it must be 
 
 a factor of the polyno- 
 mial ; and if we divide the polynomial by a, we obtain the other 
 factor. Hence the following 
 
68 ALGEBRA. 
 
 Rule. 
 
 Divide the given polynomial hy the common factor ; take 
 
 the quotient thus obtained for one of the factors, and the 
 
 divisor for the other. 
 
 Note. The greatest monomial factor is usually sought. The two 
 factore may often be still further resolved. 
 
 Find the factors of : 
 
 2. %xy — ZQxy'^ — 24:ax^f. 
 
 Ans. 2, 3, X, y, and 1 — 6y — ^axy^. 
 
 3. 3 a^ — 6 a J. Ans. 3, a, and a — 2 i. 
 
 4. a^ — a X. 
 
 5. ^a''hx^-l'oahx''-20Wx\ 
 
 6. a;' — x^y -\- X y^. 
 
 7. a^h-2a^b^-2 a h\ 
 
 8. 38 a= «= + 57 a''x\ Ans. 19, a\ x}, and 2 a;' + 3 a. 
 
 9. 3 x^y' 2:^ - 6 x'^y^ z' + 12 a'/ 1^ 
 
 10. a" 2/" + x''+'' y^+i. 
 
 11. 7a?/ + 5a;y2 — lOax''. 
 
 12. aa;'» + 2y''+s + Ja;"'+ij;" + 2 + ca;'"y"+'. 
 
 Ans. a;", y".+^, and aa;^^^ ^ j^p^ ^ ^_ 
 
 13. a;^"+»+= + x''+" + <' + 0:''+''+=''. 
 14 12a^x/-18a^a;2 2' + 24aa;=/. 
 
 Case II. 
 
 100. When one Term of a Trinomial is equal to twice the Product 
 of the Square Roots of the other two. 
 
 1. Find tlae factors of x^ ->r 2 xy + y^. 
 
 x"" + 2 xy + y"" = (x + y) {x + y) 
 
 We resolve this into its factors at once by the converse of the 
 principle in Theorem I. § 85. 
 
FACTORING. 69 
 
 2. Find the factors of x^ — 2 x ^ + y^. 
 
 ay' — 2 xy + if = (x — y) {x — y) 
 
 We resolve this into its factors at once by the converse of the 
 principle in Theorem II. § 86. Hence the following 
 
 Bule. 
 
 Omitting the term that is equal to twice the product of the 
 square roots of the other two, take for each factor the square 
 root of each of the other two connected hy the sign of the term 
 omitted. 
 
 Find the factors of : 
 
 3. c^->r2cd+ d\ 10. 4 + 9 x^ - 12 X. 
 
 4. a:^ + 14 a: + 49. 11. 6 a= a; + a= a;^ + 9 a*. 
 
 5. a;= + 6 a; + 9. 12. a;* — 4 a;^ + 4. 
 
 6. a;2 - 22 a: + 121. 13. 25^^ + 1-10 y. 
 
 7. l-6a; + 9a:^ 14. 30 a'^ y + 3 a;^ + 75 /. 
 
 8. x^-6xy+ 9/. 15. 2x"'f- 60 a:^/ + 450. 
 
 9. a;* - 2 a^ x? + a* a;^ 16. (a, + &)H 2 (ac + Jc) + c\ 
 
 17. a;2 + / + «^ + 2a;y + 2xs + 2y«. Ans. (x-\-y+zf. 
 
 Note. This and the following examples can he written, like the 16th, 
 as binomials. 
 
 18. a'' — 2ab + V' + 2bc+(?-2ae. 
 
 19. a;^4-2a;^y + 3a;^/ + 2xy'' + y^. 
 
 20. a;* - 2 a:^ + 3 a;'' - 2 a; + 1- 
 
 Case III. 
 
 101. When a Binomial is the Difference between Two Squares. 
 
 1. Find the factors of x^ — y^. 
 
 x^ —y^={x + y) {x — y) 
 
 We resolve this into its factors at once by the converse of the 
 principle in Theorem III. § 87. Hence the following 
 
70 ALGEBRA. 
 
 Rule. 
 
 Tahe for one of the factors the sum, and for the other 
 the difference, of the sqitare roots of the terms of the bi- 
 nomial. 
 
 Find the factors of : 
 
 2. x^-9. 13. x'^y'-Aa^ 
 
 3. x^-2b.- 14. a'b^c^-x'*. 
 
 4. 4-a;2. 15. 36 «»« - 49 a". 
 Ans. 2 + a; and 2 - X. 16. 1 - 81 a'c'd''. 
 
 5. %l-x\ 17. 16a;i«-4 3^^ 
 
 6. a;^-9al 18. a'' b^ c^ — x" y* z\ 
 
 7. a;^ - 16 a\ 19. 121 a" - 49 i«. 
 
 8. 9x^-25/. 20. 3 -12 ail 
 
 9. 49 a:^ - 4 /. Ans. 3, 1 + 2a;, and l-2a;. 
 
 10. 1-9 a\ 21. 8 - 50 a^ b\ 
 
 11. 121o2-l. 22. 2an'2-8c'. 
 Ans. 11 a + 1, and 11 a — 1. 
 
 12. 49a:*-81al 23. 2a;'^-512. 
 
 102. When one or both of the squares is a polynomial, 
 the same method is employed. 
 
 1. Find the factors of 9 a^ — (6 — c)^- 
 
 Tlie square root of 9 a^ = 3 a. 
 
 The square root of (J — c)'' = J — c. 
 
 Their sum is 3 a + (J — c) = 3 a + 5 — c. 
 
 Their difference is 3 a — (i — c) = 3 a — 5 + c. 
 
 Therefore 9 a^ - (J - c)^ = (3 a + 6 - c) (3 re - J + c). 
 

 
 FACTORING. 71 
 
 ^ini 
 
 d the factors of : 
 
 
 
 2. 
 
 a^-(jb- cy. 
 
 
 10. 4:a^-(b-cy. 
 
 3. 
 
 «= - (6 + cy. 
 
 
 11. 9x^-{3a-2 by. 
 
 4. 
 
 {b - cf - a\ 
 
 
 12. l-{a + by^ 
 
 
 Ans. h — c + ( 
 
 z and b- 
 
 -c — a. 
 
 5. 
 
 (]>\cf-a\ ^ 
 
 
 13. (a! + 3 2,)^-l. 
 
 6. 
 
 {a-Vf-ic- 
 
 dy. 
 
 14. {a + 2by-{3x+ 5yy. 
 
 7. 
 
 (« - Vf - (c + dj. 
 
 15. l-(5a-2i)l 
 
 8. 
 
 (a + by - (c - 
 
 -dy. 
 
 16. (a - 3 a;)^ - 16 /. 
 
 9. 
 
 {x + i/y-i z^. 
 
 
 17. (2a-3i)^-l. 
 
 18. (2 a - 3 5)^ - (c + rf - 2 yy. 
 
 19. (a + i - c)2 - (cB + 2^ - z)2. 
 Resolve into factors and simplify : 
 
 20. {3x+7yy- (2x-3yy. 
 
 {3x + 7yy-{2x-3yy 
 ==(3a;+7y + 2aj-3 2/)(3a; + 72/-2a; + 3y) 
 =^{5x + 4:ij) (x+lOy) 
 
 21. (x-yy-(x + yy. 
 
 22. (a; + 2,)^ -(a: -2,)^. 
 
 23. {5x+2yy-{3x- yy. 
 
 24. 9a;2- {3x-Syy- 
 
 25. 16 a'' - (3 a + l)''- 
 
 26. {3a+ ly- (2 a -1)1 
 
 27. (2 a + 6 — c)2 - (a + 6 + c)", 
 
 28. (2 a: + a - 3)^ - (3 - 2 xy. 
 
 29. (a; + y - 4)^ - (x - 4)'^. 
 
 30. (a + 5 + c)'' - (a - J - cy. 
 
72 ALGEBRA. 
 
 103. Polynomials may often be arranged in two groups 
 with the minus sign between them, and so be factored as 
 above. 
 
 Find the factors of : 
 
 1. a'-2ab + b''-c\ 
 
 a^-2ab + b^-c^ 
 
 = (a _ by - c" 
 
 = (a — S + c) {a — b — c) 
 
 2. 2xy~x''-y' + z\ 
 
 2 xy — a? — y^ -\- z^ 
 
 = 2;^ — (a;^ — 2 xy + y^) 
 
 =zz^ — (x — yf 
 
 = {z-\- x — y){z — x + y) 
 
 3. x2 + a" + 2ax — y\ 
 
 Ans. X -\- a -\- y, and x -\- a — y. 
 
 4. l—x'^ — 2xy — if. 
 
 5. x'^ — &ax + 9 a' — lQb\ 
 
 6. a''-dl>'-2ax + x\ 
 
 7. x" — 4 xy - 9 x'^y^ + 4 yl 
 
 Ans. a; — 2y + 3 xy, and x — 2y — 5 xy. 
 
 8. 2 « i - 1 + a- + i^. 
 
 9. a^ - c^ + S^ - e/^ - 2 a S - 2 erf. 
 
 10. a" + 2 an + n" ~ ¥ — 2 bm — m\ 
 
 Ans. a + w + i + m., and a + n — b — m. 
 
 11. a;'-* + 2 a; + 1 - d^ + 2 ax - x^ 
 
 12. 26 i^ - 1 - 9 6^x^ - 10 a i + a^ + 6 Jx. 
 
 13. 1 - 4 X + 4 x^ - 1 + 6 X - 9 x^. 
 
 14. 4xy — 4x2 + l_y2. 
 
 15. 4 x'-^y^ _ a;^ __y4 _ ^4 _ 2 y.2y2 _|_ 2 x'z^ + 2 y^z^. 
 
FACTORING. 73 
 
 104. Trinomials of the form a;*" + x^'y^'' + y*" can be 
 written as the difference of two squares, and factored by 
 the above method. 
 
 1. Find the factors of x* -{■3?f-\- yK 
 
 x^ + x^y"^ + y = a:* + 2 a;2/ + y* — x^ y^ 
 = {x^^ff-x^f 
 - {x^ + y^ + xy) {a? + y^- xy) 
 
 Find the factors of : 
 
 2. «» + a* 6* + h\ Ans. a^+¥ + a" ¥, and a^ + ¥ -a" b\ 
 
 3. X* — 18 x^ y^ + y*. 6. x* — 3x^y^ + y*. 
 
 4. x' + x^ + 1. 7. x*+ {2 — m^) x'y^ + /. 
 6. x*-5x^ + i. 8. (a + by+{a + by + l. 
 9. x* + 7 x^ + 64. Ans. a;^ + 3 a; + 8, and a;''- 3 a; + 8. 
 
 10. 9x* + 3x''y^ + Ay*. 13. i9 x* — 74: x^ y'' + 25 y*. 
 
 11. a:* - 171 a:^ + 1. 14. a« 4« + aH^ c'^ rf= + c< rf^ 
 
 12. 16 x* + 23x^y^+9 y*. 15. 4 a^ - 21 a* J^ + 9 b\ 
 
 Case IV. 
 
 105. When the Polynomials can be arranged In Groups of two or 
 more Terms, having a Factor common to all the Groups. 
 
 1. Find the factors of a;" — a a; + i a; — a 5. 
 
 a;^ — aa; + Ja: — ai = (a;^— aa') + {bx — ab) 
 — X (x — a) + b (x — a) 
 = (a; — a) (x -\- b) 
 
 Or, 
 
 x^ — ax + bx — ab = {t? + 6 a;) — (a a: + a J) 
 
 = a; (a; + i) — a (a; + J) 
 =: (a; + 6) (a; -~ a) 
 
74 ALGEBRA. 
 
 Hence the following 
 
 Rule. 
 
 Group the terms of the expression so that each group shall 
 have a monomial factor, then factor each group according to 
 Case I., and finally divide by the factor common to all the 
 groups. 
 
 This common factor, and the quotient obtained by the 
 division, will be the factors required. 
 
 Find the factors of : 
 
 2. ac — ad -\- he — hd. Ans. a + h, and c — d. 
 
 3. ax^ + x^ -\- ax ■]- 1. 
 
 4. m,x — my — nx -\- ny. 
 
 5. 5a + ab + -5b + b\ 
 
 6. Sax — bx — 3 ay + by. 
 
 7. 2x^ + iax + 6 bx + 12 ab. 
 
 Ans. 2 X + 6b, and x + 2 a. 
 
 8. 8a^ + 12 ax + 10 ab + 15 bx. 
 
 9. ax^~3bxy — axy + 3by^. 
 
 10. 2x''-x^ + 4:X — 2. 
 
 11. ,/-f + y-l. 
 
 12. 2 ax'' + 3 axy~2bxy — 3by\ 
 
 13. x^ — 2x + x'' — 2. 
 
 14. amx' + bmxy — an xy — h ny'^. 
 16. x'^ — 3 X — xy + 3y. 
 
 16. ax — bx + by + cy ~ ex — ay. 
 
 Ans. X — y, and a — b — c. 
 
 17. a'^x + abx + ac + aby + b^y + be. 
 
 18. a» - a= i + a i^ _ l\ 
 
FACTORING. 
 
 75 
 
 19. 2a+ (a^-4:)x-2a x\ 
 
 20. xy (^a' + lr') — ab (x^ + y^). 
 
 Ans. ax — hy, and ay — hx. 
 
 21. xy (l + z') +z{x' + y% 
 
 22. 2 a;5 + 2 x* + 3 x-8 + 3 a;" + 4 a; + 4. 
 
 23. a^ - a*b + aH"" - a'b^ + ah* - ¥. 
 
 24. a^ + Z a^h + Z a¥ + h\ 
 
 25. a» - 3 a? + 3 a - 1. 
 
 26-. a^b — a'c — ah^+ac^ + Vc — bi^. 
 
 Case V. 
 
 106. A Trinomial in tlie form, a;" + (a + 5) a; + a J, can be 
 separated into two Binomial Factors. 
 
 From the converse of Theorem V. § 90, 
 
 a;^ + (a + i) a; + a 6 = (a; + «) (a; + 6) . 
 x^ — (a -\- b) X -\- ab = {x — a) {x — b) . 
 x^ — {a — b')x — ab^(x — a) (a; + J) . 
 a;^ + (o — 6) X — « i = (a; + a) (x — b) . 
 
 x" + 16a; + 63 = (a; + 9) (a: + 7) . 
 
 a;2 - 16a; + 63 = (a; - 9) (a; - 7) . 
 
 a;2 _ 2a; - 63 = (a; - 9) (a: + 7) . 
 
 a;" + 2 a; — 63 = (a; + 9) (a; — 7) . 
 
 (1) 
 (2) 
 (3) 
 (4) 
 (5) 
 (6) 
 (7) 
 (8) 
 
 By inspecting the above results, we find that, when a 
 trinomial is in this form, 
 
 1. The first term of 'both factors, in the trinomial, is the square 
 root of the first term of the trinomial. 
 
76 ALGEBRA. 
 
 2. The second terms of the factors are such numbers that their 
 product always equals the last term, and their sum the coefficient of 
 the second term. 
 
 Hence, for the factoring of a trinomial of this form, we 
 liave the following 
 
 Bale. 
 
 Find two numbers such that their product shall eqxial the 
 last term of the trinomial, and their sum the coefficient of tlie 
 second term ; join each number, respectively, with its proper 
 sign, to the square root of the first term for the factors 
 required. 
 
 1. Find the factors of a;^ + 6 a; + 8. 
 
 The second terms of the factors must be such that their product 
 is + 8, and their sum + 6. The only pairs of integral numbers 
 that multiplied together make +8 are ±8 and ± 1, ± 4 and ± 2. 
 From these we are to select that pair whose sum is +6. These 
 are 4 and 2. 
 
 The first term of both factors is the square root of x^. 
 
 .-. x^ + 6x+S=^(x + 4:)(x + 2) 
 
 2. Find the factors of a;^ + 4 a; — 12. 
 
 The only pairs of integral numbers that, multiplied together, make 
 — 12 are ± 12 and T 1, ± 6 and T 2, ± 4 and T 3. The pair 
 whose sum is + 4 is + 6 and — 2. The square root of x^ is x. 
 
 .-. x^ + 4.x-12 = (x+6) (x- 2) 
 
 I"ind the factors of : 
 
 3. a:^ + 3 a: + 2. 8. «=" + 17 a; + 72. 
 
 4. a;= + 9 a- + 20. 9. a;^ + 23 a: + 22. 
 
 5. x^ — 9x + 14. 10. x^ — 5xy + 4/. 
 
 6. x^-3x- 10. 11. x^ + 11 xy + 30/. 
 
 7. x^ + 2x- 35. 12. a:^ - 9 a;y + 20/. 
 
FACTORING. 77 
 
 13. x^ — x- 132. 21. x^y^ — llxy-y 110. 
 
 14. x^-x-<a. 22. «'' + 21 a; - 100. 
 
 15. a;2 - a: - 2. 23. x" + 29 a;y - 30 y^. 
 
 16. a;^ + a; - 6. 24. 1 - 7 a: - 30 x\ 
 
 17. a:2 + a;-2. 25. a;« - a;» - 110. 
 
 18. ^y^-hxy-1L 26. 12-7a; + a:''. 
 
 19. a:* —c?x^- 132 a«. 27. 8 a;^ + a;< + 7. 
 
 20. a:^ - 32 a:y - 105 f. 28. a - 20 + a\ 
 
 29. 6 + a; — a;^. Write it, — 1 (a;^ — a; — 6). 
 
 30. 98-7a;-a;«. 37. 120 - 7 a a' - a'^ a;''. 
 
 31. 65 + 8 a;y - x^y\ 38. x^f z^ + 16 xy a - 260. 
 
 32. 10a;2 + 3a;-l. 39. a;» - 3 a;'- - 18 a:. 
 
 33. 1 + 4 a; — 96 x^. 40. ^y — x^y^ — 2xf. 
 
 34. a^ — 18fla;y-243a;=yl 41. 3 a;'' + 51 a:y + 90^/^. 
 
 35. a;* + 13 a" x" — 300 a*. 42. a;« + a;= - 870. 
 
 36. x" -a'x^- 462 a\ 43. 11 a; + 152 - a;^. 
 
 44. x^+ (a + 2i) a: + 2a5. 
 
 45. a;2 + (a + 2 i) a; + (ai + 6^). 
 
 46. a?" -\- (a + b) 3;"^" + a bf- 
 
 47. a:*" + 2 (a= + h'') x + {a' - b^)^. 
 
 48. x^— (3bc + ca + ab)x+3bc(b + c) a. 
 
 49. 6a6a;''3/''z2 — 42aJxj^s— lOSaJ. 
 
 50. 13 m^ n^ — 221 to» w' a:» + 234 a;». 
 
 51. 3 m" a:» + 66 m a:« / - 225 /. 
 
 52. 1b^a? + ^i:bdxy-10bd^f. 
 
78 ALGEBRA. 
 
 107. If the coefficient of the highest power is not unity, 
 and is not a common factor, we can still divide the expres- 
 sion by this coefficient, and then factor the quotient as in 
 the last article. 
 
 1. rind the factors of 3 a;^ + 14 a; + 8. 
 
 Bx^%Ux + 8 = 3 {x'-+ Jg* x + ^) = 3{x''+i^x + Y) 
 
 The first term of each of the two factors oi x" + ^x -{- ^ must 
 be X. Of the second terms + ^ is the product, and + ^ is the 
 sum. 
 
 Hence the second terms must he f , and ^, or 4. 
 
 .-. S{x^+i^x+^)=3(x + i){x + i) = {3x + 2){x + i) 
 
 The denominators of the second terms of the two factors are 
 always the same as the denominator of the sum; in this instance, .3. 
 The numerators are obtained from the numerators of the sum and 
 product, precisely as in the preceding article ; in this instance 12 and 
 2, from 14 and 24, regarded respectively as the sum and product. 
 
 Further, it is evident that the denominator of the fraction which 
 is the product of the second terms must be the square of the denomi- 
 nator of the fraction which is the sum of these terms. Thus above 
 we must use *^ instead of f. 
 
 The process, without explanation, will appear as follows : 
 
 3x'+Ux+8-^3(x'+jX + ^^) 
 
 = 3(x+^)(x + 'L) 
 ^(3x + 2)(x + i) 
 
 2. Find the factors of 3x'^ — 10 x+jS. 
 
 10 . 24\ 
 
 3x^-10x1^8 = 3 (x^-—x + 
 
 (3fJ 
 = 3 (x + 5) (x - 4) 
 :='{3x + 2) {x — 4) 
 
FACTORING. 79 
 
 Find the factors of 7x^-19x- 6. 
 
 
 / 10 
 
 42N 
 (If) 
 
 = 7(x + f)(z- 
 
 3) 
 
 = (7a;.+ 2)(a:- 
 
 ■3) 
 
 Find the factors of 10 a;^ — 13 a; — 3. 
 
 
 / IS 
 10a:''-13a;-3 = 10 (^'-r^^ 
 
 30 > 
 (10)^> 
 
 = 5(x + ^)2{x-H) 
 = (5 a; + 1) (2 a; - 3) 
 
 5. Find the factors of 2 a;= — 5 aiy — 3 jf". 
 
 2ar - 5a;y - 3/ = 2 (a;= - ^a. - ^J) 
 
 = 2(x + fj(x-3y) 
 ^(2x + y){x-3ij) 
 
 Find the factors of : 
 
 6. 3a:= + 5a; + 2. 15. 12a;2 - 23 a;y + 10/. 
 
 Ans. a; +1, and 3a; + 2. 16. 24a;= - 29 a;^^ - 4j^=. 
 
 7. 2ar^ + lla: + 5. Ans. 3 a: — 43/, and 8 a; + y. 
 
 8. 4a;»+lla;-3. 17. 6 a:^ + 31 a; + 35. 
 
 9. 3a;^+14a;-5. 18. 20 - 9 a: - 20 a;". 
 
 10. 6 x'- 31 a; +35. 19. 21z= + 26x^-15/. 
 
 11. 3+ 11a; -4a;''. Ans. (7a;-32/)(3a; + 5^/). 
 
 Ans. 1 + 4a;, and 3-a;. 20. aa;^ + (a + i) a; + b. 
 
 12. 2 — 3a: — 2 x\ 21. ax^ + (a — b)x — b. 
 
 13. 3a;=' + 19a;-14. 22. ahx''- (a'^-P) x- ab. 
 
 14. 2 a;'' + 15 a; — 8. Ans. (ax + b) (bx — a). 
 
80 * ALGEBRA. 
 
 Case VI. 
 
 108. When the Expression is a Binomial of the Form, a" ± If, 
 n being a Positive Integer. 
 
 (1) a -f ft is a factor of a" + 6" when n is odd, but not when n is 
 
 even. 
 
 (2) a + 6 is a factor of a" — 6" when n is even, but not when n is 
 
 odd. 
 
 (3) a — 6 is a factor of a" — ft" always. 
 
 (4) a — ft is a factor of a" + ft" never. (See Preface.) 
 
 I. To prove (1) : 
 
 It is evident that at each successive step of the division of a" + 6" 
 by a + 6, the exponent of a in the successive remainders will dimin- 
 ish by one, and hence eventually become zero. 
 
 At this stage of the process, let Q represent the quotient and R the 
 remainder, if any. K will not involve a, as a" = 1. The product of 
 the divisor by the quotient plus the remainder equals the dividend. 
 
 .-. e(o+ft)+iJ = a» + 6» 
 
 Now the equation must be true whatever value we assign to a, 
 and B will remain unchanged, since it does not involve a. 
 
 Let a= —b 
 
 then Q(— ft-|-ft) + i? = (— 6)»+ft" 
 
 but (3(— 6 + ft) = 
 
 .-. R = {—by+b« 
 
 But ( — ft)" + ft" = 0, when n is odd, and 2 ft", when n is even ; 
 showing that a + ft is a factor of a" + ft" when n is odd, but not 
 when n is even. 
 
 II. To prove (3) : 
 
 As before, when the exponent of a becomes zero in the division, 
 let Q represent the quotient and B the remainder. 
 
 Then Q (a — b) + B = a" — b' 
 
 Substituting a for ft in the equation, and recollecting that, since B 
 does not involve a, it will remain unchanged, we have 
 
 QCa — ay + B^a' — a' 
 
FACTORING. 81 
 
 From which it is at once seen that E = 0, whether n is odd or even, 
 and that hence a — b must he a factor of a" — 6", whether n is odd 
 or even. 
 
 The proofs of statements (2) and (4) are reserved as exercises for 
 the student. 
 
 Tlie law of the formation of the quotients, or second fac- 
 tors, is simple, and may be determined by actual division, 
 thus: 
 
 a + 6) a" + 6" (a"-^ — a^'H ... . + b"-^ 
 a" + a"-»& 
 
 — «"-'& + b" 
 
 + a &»-' + b" 
 + ab"-^ + b" 
 
 The following are the general expressions for the factors of 
 a" + b", and a" — b", when n is odd : 
 
 (1) a- + S» = (a + b) (a-i- a-'' b + a'-n\...-a b"-^ + 6"-^). 
 
 (2) a" - i" = (a - b) (a"-i + a"-^ 5 + a-» 5^ . . . + a S""' + b"-^). 
 
 1. Factor a^ + b'. 
 Substituting 5 for n in (1), we have 
 
 a^ + b^ = (a + b) (a* - a'b + aH^ - a&» + 6*). 
 
 2. Factor a^ — bK 
 Substituting 5 for n in (2), we have 
 
 a^-¥ = {a- b) (a* + c^b + a^ft" + a J» + 6«). 
 
 3. Factor 8 + c». 
 
 8 + c» = 2' + c» = (2 + c) (2= - 2 c + c»)" 
 Find the factors of : 
 
 4. a« + V. 6. a;^ - /. 8. 343 - 2». 
 
 5. o» - 6». 7. c^ + 32. 9. a" + 243. 
 
 6 
 
82 ALGEBRA. 
 
 109. The factors of such examples can be written out by 
 inspection. Attention to the following laws will enable 
 cue readily to do this : 
 
 1. The terms of the quotient, or second factor, are all positive 
 when the divisor is a — h, and alternately positive and negative when 
 the divisor is a + 6. 
 
 2. The number of terms always corresponds to the degree of the 
 binomial. 
 
 3. u, appears in the first term, b in the last term, and a and 6 in 
 all the intermediate terms. 
 
 4. The exponent of a in the first term is one less than the degree 
 of the binomial, and decreases regularly by unity in each successive 
 term ; the exponent of 6 in the second term is 1, and increases regu- 
 larly by one in each successive term, till in the last term it becomes 
 the same as the exponent of a in the firet term. 
 
 5. The sum of the exponents of a and b in any intermediate term 
 is always the same, and is equal to the exponent of a in the first 
 term, a and b stand for any letters or expressions. 
 
 , Factor by inspection the following expressions : 
 
 1. c^ + d\ 4. x^ + 1. 7. 3 a;* + 24/. 
 
 2. c» - d'. 5. x^ — 1. 8. 27 a:' + 1. 
 
 3. 125 + a«. 6. 8a;»-y». 9. 1-8/. 
 
 10. 27 a;' -8/. 
 
 27a;'-8/=(3a:)»-(2 2/)3 
 
 = {3x — 2y) {9x'+ Qxy + 4:f) 
 
 11. 16a'&' + 250x». 12. a» + 343 5». 
 
 13. 216 - a». Ans. 6 - a, and 36 + 6 a + a\ 
 
 14. a«6» + 512. 18. 343- 8 a». 
 
 15. 216 if - z\ 19. 343 x" + 1000 z\ 
 
 16. l-343x=. 20. a;»-27/. 
 
 17. 40 a;' -135/. 21. a«-729 6«. 
 
FACTORING. 83 
 
 22. 8a»a;»-27i»/. 30. 27a»-64 2/' 
 
 23. 27 m'-UnK 31. (a;-2y)'- (y-2ic)'. 
 
 24. 125 a;« + 64 /. 32. (x + 2yf + (y + 2 a;)'. 
 
 25. (x + yf + {x- y)\ 33. 729 a» - 64 h\ 
 
 26. (4a:'^-l)»-(4*'^ + l)». 34. 270 - lOpOO a;«. 
 
 27. 10 «» - 640 y\ 35. a» 4' - J c». 
 
 28. a;'2/»-216z». 36. 27 x' y' - ^ »'. 
 
 29. 4 a^ 6' c' - 4. 37. {a + by — c'. 
 
 110. When n is ewri aud greater than 2, there will be 
 three or more factors in each case, and they can be more 
 expeditiously determined by Case III., with other princi- 
 ples already explained. 
 
 1. Find the factors of a;* — y^. 
 
 x'-7/=(x' + y')(x'-i/) 
 
 - («" + y') (a; + y) (x - y) 
 
 2. Find the factors of as' — y\ 
 a;» — 2/' = (3? + 2/') (x^ — y') 
 
 = (as + y) («'' — a= ?/ + y*^ (« — «/) {x^^-xy-\- y^ 
 
 3. Find the factors of x^ — 1. 
 
 a;= - 1 = (a:* + 1) {x* - 1) 
 
 = (x* + 1) (a;^ + 1) (x^ - 1) 
 
 = {x^ + 1) (a;» + 1) (a; + 1) {x - 1) 
 Find the factors of : 
 
 4. x^ — y\ 10. 1-a:*. 16. a}^ - aK 
 
 5. a^-Jo. 11. a;" -2/'". 17. «>2 _ ji2_ 
 
 6. a;*-l. 12. a;« - 64. 18. 64 a;' -a;. 
 
 7. a;' — 1. 13. a;" — 2/". 19. si" — a»i». 
 
 8. 1 - a;». 14. a« i» - e^cf. 20. a' - 729 a'. 
 
 9. l-a;". 15. a>2 - a*. 21. x' - ^y. 
 
84 ALGEBRA. 
 
 111. Though a" + 6" is not divisible by a + 6 when n is 
 even, it is possible to find a binomial factor in every case 
 except when w is a power of 2, such as 2, 4, 8, 16, etc. 
 
 1. Find the factors of a« + h\ 
 
 a« + J^ = {ay + (py 
 
 = (a= + b') {(ay -aH^+ (h")"^ 
 = (a'' + ¥) (a^-aH^ + b*) 
 
 2. Find the factors of a'^ + ¥\ 
 
 al2 + J12 ^ („4)3 + (^4)3 
 
 = (a* + b^) {(ay -aH* + (by} 
 = (a* + b*)'(a^ - aH* + b') 
 
 Find the factors of : 
 
 3. a'" + b^\ 5. x"^ + 1. 7. 64 «» + 1. 
 
 4. a« + 1. 6. a;!" + 1024 2/"'. 8. a" + &"• 
 
 112. The examples in all the cases, thus far, save those 
 in Case I, can be factored by Case IV. 
 
 1. Factor x'^ — 2xtj + %f (Case II.). 
 
 x'^ — 2xy-\-y'^=^x'^ — xy — xy-\-'i^ 
 
 = x(x — y) — y(x — y) 
 
 = {x-y) (« - y) 
 
 = (x-yY 
 
 2. Factor x^ - if (Case III). 
 
 a;2 — y^ 1^ x^ -\- xy — xy — y^ 
 
 = x(x-\- y) — y (x-\- y) 
 ^(x + y) (x- y) 
 
 3. Factor a;^ + 6 a; + 8 (Case V.). 
 
 a:^ + 6a; + 8=a;2 + 2a; + 4a; + 8 
 
 = a; (.r + 2) + 4 (a: + 2) 
 = (a; + 2) (a; + 4) 
 
FACTORING. 85 
 
 4. Factor a» + b' (Case VI.). 
 
 a» + 68 = a8+ aH-aH-ab^+ab^ + b^ 
 
 = »='(«+&) — a&(a+&) + S^(a + 5) 
 = (a + b){a'-ab + b^) 
 
 By this method find the factors of the following examples : 
 6. x^ + 2xy + y\ 10. 4 a;^ + 4 a;y + «/l 
 
 6. a;^ - 6 X + 8. 11. 8 - a;». 
 
 7. a!=-2/=. 12. a^-a;-6. 
 
 8. x^ — 4. 13. a;' + a; — 10. 
 
 9. x^-1. 14. a!^-l. 
 
 113. To be expert in factoring, it is necessary to become 
 familiar with the following algebraic expressions : 
 
 a^ + b^ is prime. 
 
 a^-p=(a+ b) (a-b). 
 
 a" + b» = (a + b) {a^ - ab + b% 
 
 a?-b^={a-b) {a^ + ab+ b^. 
 
 a^ + 6* is prime. 
 
 a< - 5« = («2 + 52) (a + 5) (a - b). 
 
 a= + JB = (o + 5) (a^ - a»S + a^ i^ _ a J8 + ^4)^ 
 
 a5- J6 = (a - 5) (a< + a»6 + a^i^ + „ J' + 6')- 
 
 a6 + i6 ^ (^2 ^ J2) (^4 _ ^252 + J4') 
 
 «=- J« = (a + 5) (a^ _ aJ + W) (a - b) («*+ ab + J'). 
 «'+ JT^ (a + b){a^-an + a*6^- a»6» + a'6'- «&« + &')- 
 a'- 6'= (a-5)(a''+ «'& + a'i' + a'S" + a'5* + «&' + 5*)- 
 a^ + b' is prime. 
 
 ffiS _ 58 ^ (ft4 ^ J4) (^2 ^ J2) (^ + J) (a _ J). 
 
80 ALGEBRA. 
 
 a'+2ab + ¥ = {a + bf 
 
 a^—2ab + h^ = {a - by. 
 
 a*+2aH'' + b*=(a^ + ¥)\ 
 
 a*-2aH^ + ¥= {a" - ¥f = (a + If {a - by. 
 
 a« + 2 a«6= + 6° = («" + 5")' = (a + if (a''-ab+ ¥)\ 
 
 a^_2a?¥ -\-W= {a? — ¥Y = (a — bf (a^ + afi + by. 
 
 1. What does a? + ab ■{- V^ suggest ? 
 
 2. Wliat does a^ + 2ab + V suggest ? 
 
 3. What does a^ — ab + b'^ suggest ? 
 
 4. Wliat does a? — 2 ab + b^ suggest ? 
 
 5. What does {a* + b") (a^ + b^) (a + b) {a — b) suggest ? 
 
 114. Division by Factors. 
 
 Divide : 
 
 1. a'-x^ hy a — x. 13. 9a;2 - 1 by 3a; + 1. 
 
 2. a=-a;» by a - a;. 14. 25 0;^- 9 by 5a;2/-3. 
 
 3. a« - a;= by a - a;. 15. 8 a;' - 1 by 2 a; - 1. 
 
 4. a;' + 2/» by a; + 2/. 16. a;^ - 9 by a; + 3. 
 
 '5. x^ + f hj x + y. 17. 1 + 27 a;' by 1 + 3 a;. 
 
 6. x^ + y' by a; + y. 18. x« + 27 by a; + 3. 
 
 1. x — yhy y~x. 19. x'' — %f h^ x — 2 y. 
 
 8. a;'' - y2 by 2/ - a;. 20. a;^ - 4 by x + 2 
 
 9. 7?-'f by a;-^ + a;2/ + y2. 21. a;'' + 6 a; + 5 by a; + 5. 
 
 10. a;' - 2/= by 2/ - a:. 22. a;^ - 8a; + 12 by a; - 6. 
 
 11. a;= - 2/= by y - x. 23. a;2-6aa; + 9a2bya;_3a. 
 
 12. 4x^-1 by 2 a:- 1. 24. a;^ - 2a: + 1 by a; - 1. 
 
FACTORING. 87 
 
 25. ix^ + 12xi/ + 9f by 2a! + 3y. 
 
 26. x'-Sx^a + Sxa^-a^ hy a?-2ax + a\ 
 
 27. x» + 3ji:''+3x+ Ihy x + 1. 
 
 28. {x — ly (SB + 4) by a; - 1. 
 
 29. {x + 1) {x^ - 4) by a; + 2. 
 
 30. (a;= - a^") (a; + a) by a;^ + 2 aa; + a». 
 
 31. (a; + 2/) 2 — s^ by x + y — s. 
 
 32. x^ — 2xy + y' — z^ by a; — ?/ + s. 
 
 33. a!* + 64 by a;2 + 4a;+8. 
 
 34. a!< + £C= + 1 by a;= — a; + 1. 
 
 35. a;* + 9 a;2 + 81 by a;" - 3a; + 9. 
 
 36. a:2 + 3^" + «^ + 2ai2/ + 2a;s + 2yiS!bya! + y + 2. 
 
 37. aa; + ay + 6a; + 6y by a; + y. 
 
 38. ac — bc — ad + hd \)j a — b. 
 
 39. a;" + y^'' by a;* + yK 
 
 40. a;' + J/* + «' — 3xys by a; + y + ». 
 
 115. MISCELLANEOUS EXAMPLES. 
 
 Factor the following examples : 
 
 1. a;2+73a; + 72. 8. Sa;^^^ 26aa!y + 35al 
 
 2. a;2-14ax + 45o^ 9. x^ - aK 
 
 3. 12 a;^ - 60 a;^ - 288. 10. a;»= - a;». 
 
 4. aa;^ — 4a2a; + 4 a'. 11. x^^ - x''. 
 
 Ans. a, a: — 2 a, a; — 2 a. 12. 2 — 50 a^. 
 
 5. 2a:«y -3a;y^- 4 a; y. 13. 18x^-S2y\ 
 
 6. a;''— (a — S)a; — aS. 14. (a; + l)^ _ a;^. 
 
 7. (a-5)'-ll(a-J) + 18. 15. 4 (a; - y)» - (a; - y). 
 
88 ALGEBRA. 
 
 16. 2ah-2cd-&-\-a?-d}^V. 
 
 Ans. a -\- h -\- c -\- d, and a ■\- h — c — d. 
 
 17. a» + i^ + a + 6. 2\. x^ - i.y' ■\- x -2y. 
 
 18. a' — W^a — h. 22. 5 a* i^ — 5 a 6. 
 
 19. a^-y'-'iyz- z\ 23. a^ - 9 6^ + a + 3 5. 
 
 20. 1 — (a; — 2/)'. 24. 1 — (w= + W^) + 2 m re. 
 
 25. x'^ y — 3? y^ — 3? y^ -\- x y'^. 
 
 Ans. a; 2/, a; + 3/, and (a; — y)\ 
 
 26. <x^ + 6* - c* - cZ^ + 2 a^S^ - 2 c^ (^. 
 
 27. a^ + a;^ — {%/ + s^) — 2 (ys — ax). 
 
 28. i + ix + 2 ay + x^ - a^ — yK 
 
 29. 21 a;2 + 82 a; — 39. Ans. 3 a; + 13, and 7 a; — 3. 
 
 30. a^-8a^6». 38. x^-{a^ + b'')^^ + aH\ 
 
 31. (a + 5)^-1. 39. a;" + a;' + l. 
 
 32. 250 (a - by + 2. 40. x* + ix^ + 16. 
 
 33. 6 a:= - a; - 77. 41. x* + 25 a:'' + 625. 
 
 34. c' d^-e'- a" c' <F + a^. 42. a:^" + 16 a;" + 48. 
 
 35. a;^-4096. 43. .0001 a:* -1. 
 
 36. 75a*-48i^ 44. 3a:'' + a;-2. 
 
 37. 3 a!= + 96. 45. 4 a:^ - a;^ - 2 a; - 1. 
 
 46. a;^a2-8/a'2-4a;'a2 + 322/'a2_ 
 
 Ans. 3, a'', 2?/ — a;, and 4 2/^^ + 2 a; 2/ + a'''- 
 
 47. 1 _ a; - a;' + x^. 51. «» - 64 a' - a» + 64. 
 
 48. x^ -^ [a -\-^\x + 1. 52. 216 a» - ^ . 
 
 49. a;* — 4 2/^ + a;'^ + 2 y'^. 53. a;* - 15 x'' 2/" + 9 /■ 
 
 50. 3 a;*" - 3 2/2 + 6 a; + 6 2/. 54. J a» + 5)7 h\ 
 
FACTORING. 89 
 
 55. a;' — 9 a;^ + 9 a; — 1. Ans. x — 1, and a;" — 8 a; + 1. 
 
 56. a^x^ + 2a^bx + a^b''-a*b\ 
 
 57. x^ — x*+l- x\ Ans. x^ + 1, (x+ 1)% and (a: — 1)^ 
 
 58. a* + aH^- ¥ <? — c\ Ans. a= - c\ and a'' + J^ + c\ 
 
 59. (a;^ + a; - 20) (as* - a; - 30). 
 
 60. a^ — a^h^a^h^-d^U^-^aV-W. 
 
 61. a;2/ (1 + s^) + s («^ + y). Ans. x-\-yz, and 2/ + a:». 
 
 62. 32 a^ - 243 6". 65. 3 a;^ + 3 a: - 18. 
 
 63. 12a:=' + 14a: + 4. 66. 6a!= + 7a; - 6. 
 
 64. 8/+18y + 7. 67. 2a:2-3a;+ 1. 
 
 68. 2a;»— a;= + 8a; — 4. 
 
 69. 5 a;' + 15 a;« - 5a;» - 15 x\ 
 
 70. a? -W - e^ ^ d^ -2{ad-h c). 
 
 71. (a:2 + 3»2-2T-^^'/- 
 
 72. a?' — 2a?hc-V' — V^ c" — c*. 
 
 Ans. a?-hc-\-}?-\-(?, &udi a' — lc — h''- c\ 
 
 73. (a;»-3a;=')^-(3a;-6)2. 
 
 74. (a + J)2 + (c + <Z)^ + 2 (ac + a<? + Sc + St?). 
 
 75. a^ + 5" + c= — 2 a c - 2 a J + 2 5 c. 
 
 76. a* + 54 + c*-2aH2-2aV-262c^. 
 
 77. a' + ft' + c'-3a6c. 
 
 78. a» + J»+ l-3a6. 
 
 79. 4 (a 5 + cd)'= - {a? ^^V'-c^- d?f. 
 
 80. a;»-3a!2y + 3a;2/2-y». 
 
 81. a;' + 3 a;'^ + 3 a; + 1. 
 
90 ALGEBRA. 
 
 CHAPTEE VIIT. 
 GREATEST COMMON DIVISOR. 
 
 116. A Common Divisor of two or more algebraic expres- 
 sions is an expression which will divide each of them 
 without remainder. 
 
 117. The Greatest Common Divisor of two or more alge- 
 braic expressions is the expression of the highest degree 
 which will divide each of them without remainder. 
 
 118. From this definition it is evident that tlie greatest 
 common divisor of two or more algebraic expressions must 
 contain all the factors common to the expressions, and no 
 others. 
 
 Note. The names greaiest common measure, highest common measure, 
 highest common divisor, highest common factor, are used by different 
 autliors to mean the same thing as greatest common divisor. 
 
 Case I. 
 
 119. To find the Greatest Common Divisor of Monomials and 
 Polynomials which can be resolved into Factors by Inspection. 
 
 1. Find the greatest common divisor of 12 a' l>*c\ 36a^i', 
 and 8a^b c\ 
 
 12an*c*^2^-ZaH*c* 
 36 aH^ =2^-32aH» 
 
 G. C. D. =2^ •a'b 
 
 It is evident that the highest power of 2 which will divide all 
 three expressions is 2^ ; of a, a^; of 6, b ; and that c will not divide 
 them all, therefore the greatest common divisor is 2^0' 6. 
 
GllEATEST COMMON DIVISOR. 91 
 
 2. Find the greatest common divisor of y? — y^, x^ + 2xi/ 
 + if, and 7? -y xy. 
 
 x" — y = (£B + y) (a; — y) 
 x^ -\- 2 xy + y^ = {x + yY 
 x^ + xy = x {x +'y) 
 
 .-. G. C. D. =x + y 
 From these examples we derive the following 
 
 Kule. 
 
 Separate each expression into its prime factors ; then take 
 every factor common to the given expressions tJie least numler 
 of times it occurs in any one of them for the greatest common 
 divisor required. 
 
 Find the greatest common divisor of : 
 
 3. dx^y^z", 12xifz. 
 
 4. 17 a" bH^, 34 a J c», 51 a V c. 
 
 5. 24 a» ¥ cS 16 a« h* c», 40 a^ b* ^. 
 
 6. 2hx'yz\ 100 a;» ?/«««, \2^xf. 
 
 7. a? -\- X y, 3? — y^. 
 
 8. {x + y)\ x' - y\ 
 
 9. 2^ — 2xy, 3? — x^ y. 
 
 10. 6a^-9a;y, i:X- — 9y'-. 
 
 11. a' — c^x, a' — a a?, a* — a x'. 
 
 12. a^ — x^, a^ — ax, a^x — a x^, 
 
 13. x^ + x, (x + ly, a? + 1. 
 
 14. 6 (a + b)\ 15 (a + by. 
 
 15. 24 (a^'- 9), 16 (a -37. 
 
 16. x^-1, a;»-l, (x-l)2. 
 
 17. a^ - i^ (a - i)=, (a + 5)^ a^ - b\ 
 
92 ALGEBRA. 
 
 18. 30 a; - 6, 100 x" - 4. 
 
 19. 3?-2xy + if, (x-y)\ 
 
 20. ix'-l, (2x + ly. 
 
 21. x'-5x + 'i, x^ + x-2. 
 
 22. x^-lSx + 45, 2 (x" - 9). 
 
 23. a;'' - a; - 20, a!^ - 9 a: + 20. 
 
 24. 2a;2-a;-l, 3a;2-a;-2. 
 
 25. x'+ Sy% x- + xi/-2f. 
 
 26. c^a^ — cP, aica:^-ica; + a(Za: — 5rf. 
 
 27. a^ + (a + 6) a; + a 5, x^ + (a + c) a; + a c. 
 
 28. a;'' — (a — c) — a c, x^ — (a + c) X + a c. 
 
 29. a^ + (3 + 2/) a; + 3 2/, K^ + 6 a; + 6. 
 
 30. ab (x + a),' a (a^ + a X — b X — a b), b (a? + ax). 
 
 31. 15 (a;^ - %f), 6 (x + y) (x= - j,'). 
 
 32. 2 x' + 9 X + 4, 2 x^ + 11 a; + 5, 2a;' - 3 a; - 2. 
 
 33. ft x' + 2 a' X + a», 2 a x^- 4 a' X — 6 a', 3 (a a; + a^'. 
 
 34. 3a2+ 9aJ, ft'-9ftJ^ «'+ 6a=6 + 9a6l 
 
 35. x« - 125, x' - 25, x' - 10 x + 25. 
 
 Case II.* 
 
 120. To find the Greatest Common Divisor of Polynomials 
 which cannot be factored by Inspection. 
 
 To deduce a rule for finding the greatest common divisor 
 of two or more numbers, we demonstrate the two follow- 
 ing theorems : 
 
 THEOREM I. 
 
 121. A common divisor of two polynomials is also a divi- 
 sor of the sum or the difference of any multiples of each. 
 
 * See Preface. 
 
GREATEST COMMON DIVISOR. 93 
 
 Let A and B be two polynomials, and let d be their common divi- 
 sor ; d is also a divisor oi m A ± n B. 
 
 Suppose A -i- d = 'p ; i. e. ^= dp, and mA = dmp, 
 
 and B -i- d=: q; i.e. B = dq, and nB ^dnq; 
 
 then mA ± nB = dmp ±dnq = d(mp ± nq). 
 
 That is, d is contained in m A ■{■ nB, mp •}■ nq times, and in 
 mA — nB, mp — nq times ; that is, d is a divisor of the sum or 
 the difference of any multiples of A and B. 
 
 THEOREM II. 
 
 122. The greatest common divisor of two polynomials is 
 also the greatest common divisor of the less and the remain- 
 der after dividing the greater hy the less. 
 
 Let A and B be two expressions, and A not lower 
 in degree than B, and let the process of dividing be E) A (5 
 
 as appears in the margin. Then, as the dividend is „ 5 
 
 equal to the product of the divisor by the quotient 
 
 plus the remainder, 
 
 A=r + <lB. (1) 
 
 And, as the remainder is equal to the dividend minus the product of 
 the divisor by the quotient, 
 
 r=A — qB. (2) 
 
 Therefore, according to the preceding theorem, from (1) any divisor 
 of r and B must he a divisor of A ; and from (2) any divisor of A 
 and B, a divisor of r ; that is, the divisors of A and B, and B and r, 
 are identical, and therefore the greatest common divisor of A and 
 B must also be the greatest common divisor of B and r. 
 
 In the same way, the greatest common divisor of B and r is the 
 greatest common divisor of r and the remainder after dividing B 
 by r. 
 
 123. Hence, to find the greatest common divisor of any two 
 polynomials, we have the following 
 
 Kule. 
 
 After removing every monomial factor possible from each 
 expression, arrange the restdting expressions, according to the 
 
94 
 
 ALGEBKA. 
 
 descending powers of some common letter, and divide the ex- 
 pression which is of the higher degree bg the other. Continue 
 the division until the remainder is of a lower degree than the 
 divisor. Then make the remainder a new divisor and the 
 divisor a new dividend ; and continue the process until there 
 is no remainder. The last divisor, together with the common 
 monomial factors, removed at the beginning of the operation, 
 will he the greatest common divisor. 
 
 Note. Monomial factors should be rejected when possible, and intro- 
 duced only to avoid fractious. If, after the removal of such factors at any 
 point of the process, polynomials of the same degree appear, either may be 
 used as the divisor, though it is better to take as the divisor the one whose 
 first term has the smaller coefficient. 
 
 1. Find the greatest common divisor of 4 a;* — 3 a;^ — 24 a; 
 — 9 and 8 a;' - 2 a;" - 53 a; - 39. 
 
 4xs- 
 
 -3j;=- 
 
 -24x-9)8a;»- 
 8a;3- 
 
 .-. G. C. D 
 
 -6a:=- 
 ix"- 
 
 = X 
 
 53.r- 
 -48a- 
 
 -5x- 
 -3. 
 
 -39(2 
 -18 
 
 ix'—hx"—2\x 
 
 -9(x 
 
 -ix- 
 -6x- 
 
 -21(2 
 -18 
 
 -3x- 
 -6x 
 
 
 
 
 Zx"— 8x- 
 
 -9) ix^- 
 4a;2- 
 
 
 
 
 X- 
 
 - 3)2,;! 
 2x'- 
 
 -9(2x-|-3 
 
 
 
 Sx- 
 3x- 
 
 -9 
 -9 
 
 The following arrangement saves rewriting the divisor when it 
 becomes the dividend : 
 
 X 
 
 ix'- 
 
 - 3 a;^ - 24 a; - 9 
 
 
 4a;»- 
 
 - 5 a;'' - 21 a: 
 
 2x 
 
 
 2x'- 3 a; -9 
 20?- 6x 
 
 3 
 
 3a;-9 
 
 
 
 3x-9 
 
 8 
 
 8 
 
 x^ 
 a;" 
 
 -2x'- 
 -6x'- 
 
 53 X 
 
 48 X 
 
 — 
 
 39 
 
 18 
 
 
 
 4x2- 
 4x^- 
 
 5x 
 6 X 
 
 — 
 
 21 
 18 
 
 
 X 
 
 — 
 
 3 
 
 G. C. D. == X - 3. 
 
GREATEST COMMON DIVISOR. 
 
 95 
 
 2. Find the greatest common divisor of 10 x' + 3oa;^ + 30 a: 
 and 4a;* + 26 a:» + 58 x^ + 42 x. 
 
 2 a;|4 a;* + 26 ar" + 58 a;= + 42 a; 
 
 X 
 
 10a;»+35a;^ + 30 
 
 X 
 
 2a!^+ 7a; + 6 
 2a;^+ 3a; 
 
 2 
 
 4a; + 6 
 4 a; + 6 
 
 2a;» 
 2^ 
 
 + 13a:2-f29a: 
 + 7a;^+ 6a; 
 
 + 21 
 
 
 6a;=+23a; 
 6 a;^ + 21 a; 
 
 + 21 
 + 18 
 
 G. C. D. = a; (2 a: + 3). 
 
 2a; + 3 
 
 3. Find the greatest common divisor of 3 a;" — 2 a; — 1 and 
 3a:» — 9x+ 6. 
 
 3a;''- 
 3a;^- 
 
 2a;- 1 
 i2a;+ 9 
 
 
 3 a;' - 9 a; +6 
 3 a;» _ 2 a;2 - a; 
 
 10 
 
 lOaj-lO 
 
 2 
 
 2 a:= - 8 a; +6 
 
 
 X- 1 
 
 a;'' - 4 a; +3 
 a?- X 
 
 G. C. D = a; - 1. 
 
 -3a; + 3 
 -3a; + 3 
 
 -3 
 
 4. Find the greatest common divisor of 20 x'^ + 12 a; — 11 
 and6a:' + a;^-l. 
 
 ]0a: 
 
 n 
 
 20a;2+12a:-ll 
 20 a:^- 10 a; 
 
 22a;-ll 
 22X-11 
 
 10 
 
 a;''-l 
 
 60 a;' 
 60 a:' 
 
 + 
 + 
 
 Wx'- 
 36 x^- 
 
 10 
 33 a; 
 
 
 
 — 
 
 2Ga?-{- 
 10 
 
 33 a; 
 
 - 10 
 
 
 — 
 
 260 a;^^ 330^ 
 260 a;'' -156 a; 
 
 -100 
 + 143 
 
 243 
 
 G. C. D. = 2a; 
 
 486 a; -243 
 2a;-l 
 
 3 a; 
 
 13 
 
96 ALGEBRA. 
 
 Find the greatest common divisor of : 
 
 5. 6a;2 + 7a;-3, 12x2+ 16a;-3. 
 
 6. a^ + 8 cc + 15, a;2 + 9 cc + 20. 
 
 7. a:^ + 12 a; + 35, ce^ + 13a; + 42. 
 
 8. a;» + 3 x^ + 4a: + 4, a;3 + 4 x^ + 5 a; + 6. 
 
 9. 2a? + 18x + 36, 2x^ + 20x2 + 48 x. 
 
 10. 3x*-x-3x'- 2x^-1, -3x'-x + 6x*-x2-l. 
 
 Ans. 3 x^ + 1. 
 
 11. 2x«-2x^ + x»+3x=-6x, 8x»-4x* + 6x=-18x. 
 
 12. 2a''x — 2tt» + 3x'-3aa!^ 12 ax^ + Sa'' + 3x^ + 2a^x. 
 
 13. 2x= - 14x + 20, 20a; - 25x^ + 4x' + 25. 
 
 14. 12ax-9a+a' — 4ax^ 2a2-4ax2 + a' + 8ax -3a. 
 
 15. 2x^ + 9x' + 14x + 3, 3x* + 14x'+ 9x + 2. 
 
 Ans. x^ + 5 X + 1. 
 
 16. x= + 4x'' + 16x, 2x' - 8x - x^ + lex^. 
 
 17. 5x=+2x- 7x^3x^ + 2, 5 + 12x + 2xH 3x»-2xl 
 
 18. y'-2y' + 37/-6y, ,/-f-t/-2f. 
 
 19. 24 x^y + 72 x'^ - 6 x^/ - 90 xy^ 6 xUf + 13 x",/ 
 -4x^2/* — 15x^/^ Ans. xy (2x2 + xjf-32^''). 
 
 20. 54 x^ a^ + 4 a;« a^ + 10 x^ a* - 60 x' a\ 24 x^ a' + 30 «= a= 
 -126x=a«. 
 
 21. 4x«+ 14x^4- 20a;=+ 70x2, 56x» + 8 x' + 28 x«- Sx^ 
 -12x*. 
 
 22. 72a=x-420a» + 72x'-12ax2, 42ax2+ 18x'-282a2a; 
 + 270 a\ Ans. 6 (3 x - 6 a). 
 
GREATEST COMMON DIVISOR. 97 
 
 23. 7x*-10ax'' + 3a'x^-ia^x + 4:a*, 8a;<-13aa:'+6aV 
 
 — 3a'x + 3 a*. 
 
 24. Sufi- BSx' + 3, 3a;5 - 55a:2 + g. 
 
 25. 4 a;S + 14 aa:* - 18 a»a;^, 24 a j!» + 30 a' a; + 126 a*. 
 
 26. 2cB='(a:*^2a:«-7a^+16a: + 7), 6a;(a:»-6a;='-26a;-9). 
 
 Ans. 2x{x'^ + 3x + 1). 
 
 27. 2 a;« — 2 asc^ — 16 a*x\ 3 ax^ — 3 a" x!* + 6 a*a? 
 
 — ^a*x^ — 3Qa^x. 
 
 28. 2\y> - 14 ay" + 21 a^'y - 14 a», 54 / 2 - 16 a' z. 
 
 29. 6a:« - 13a;» + Sa:" + 2a;, 6a;* - 9a;» + 15a;'' - 27a: 
 
 — 9. P\ 
 
 30. 4 ax* + 2 aa;» - 16 aa:* - 2 aa: + 12 a, 4 a;* + 12a;« 
 
 — a;" - 27 a; - 18. 
 
 124. To find the Greatest Common Divisor of tliree or more 
 Expressions. 
 
 Find the greatest common divisor of two of them, then of 
 this result and a third, and so on ; the lust divisor will be 
 the greatest common divisor sought. 
 
 Find the greatest common divisor of : 
 
 1. a' + 3ab + 2b% a^ + ab — 2b% a^+2aH-ab''-2b'. 
 
 2. 39a:=-178a; + 39, 39 3:^-184 a; + 65, 27a;2-114a;-18. 
 
 3. a;» - 6 a;" + 11 a; - 6, a;' - 9 a;^ + 26 a; - 24, a;» - 8 a;" 
 + 19 a: -12. Ans. (a: - 3). 
 
 4. a;''-9a*+26a:-24, a;»-10a;2+31a;-30, a;»-lla;= 
 + 38 a; — 40. 
 
 5. a:' — 3 a:'' + 3 a; — 1, a:' - a;' - a; + 1, a;* - 2 a;' + 2 a; 
 
 — 1, a;* — 2a;' + 2a;'' — 2a;+ 1. 
 
 7 
 
98 ALGEBKA. 
 
 CHAPTER IX. 
 LEAST COMMON MULTIPLE. 
 
 125. A Common Multiple of two or more algebraic ex- 
 pressions is an expression that can be divided by each of 
 them without remainder. 
 
 126. The Least Common Multiple of two or more algebraic 
 expressions is the expression of the lowest degree that can 
 be divided by each of them without remainder. 
 
 127. It is evident from the above definitions that a com- 
 mon multiple of two or more expressions must contain 
 the factors of these expressions ; and the least common 
 multiple of two or more expressions must contain only the 
 factors of these expressions. 
 
 Case I. 
 
 128. To find the Least Common Multiple of Monomials, and Poly- 
 nomials which can be resolved into Factors by Inspection. 
 
 1. Find the least common multiple of 8 aH'^, 24 a^&V, and 
 4, a be". 
 
 4a b 0^ = 2^ aba" 
 .-. L. C. M. =:^ 2'> ■ 3 a*b^c" = 24a'b^c" 
 
 It is evident that no number which contains a power of 2 less than 
 2», of a less than a*, of b less than b% of c less than c», and which does 
 not contain 3, can be divided by each of these numbers ; therefore 
 the least common multiple is 2' • 3 a* 6" c. 
 
LEAST COMMON MULTIILE. 99 
 
 2. Find the least common multiple of 4 (a:^ — 1), 6 (a; — 1'), 
 8 (a^" + 2 a; + 1). 
 
 4.(3^-1)^2^ (a;+l)(a;-l) 
 6(x-iy = 3-2(x- ly 
 8 (a^ + 2 a: + 1) = 2» (x + ly 
 
 77 1^ C. M. = 2»- 3 (a; + ly (« - 1)" 
 From these examples we derive the following 
 
 Knle. 
 
 Separate each expression into its prime factors, and then 
 take every factor the greatest number of times it occurs in any 
 one of the expressions for the least common multiple required. 
 
 Find the least common multiple of : 
 
 3. 8aH% 24:a*Pc% 18 a be'. 
 
 4. 15 an*, 20 a^Pc\ SOac'. 
 
 5. 35a;V«> 4:2x't/s^, dOxfe\ 
 
 6. 21 a!», 7 a;" (x + 1). 11. x — ij, x + y, x^ — y\ 
 1. x^ - 1, x" ■\- X. 12. l-Zx,l^Zx,l-Zx\ 
 
 8. 6 a;'' — 2 a;, 9 a;2 — 3 a;. 13. a; — 2, a; + 2, a;'' + 4. 
 
 9. 4 a^y - y, 2 a;" + a;. 14. 6 (a; + yY, 9 (a; + yf. 
 10. a; - 2, (x- 2)1 15. 4 (a; + 1)^ 6 {3? - 1). 
 
 16. 3 (« - 1), 2 (a - 1)^ (a - Vf. 
 
 17. c (a^ - c2), (a + c) c, c. 
 
 18. 8 (1 - x), 8 (1 + x), 4 (1 + a:''). 
 
 19. a + b, a — b, a^ + 6^ a* + b\ 
 
 20." 4 a' (a + «), 4 a' (a - a;), 2 «» (a'' - x^). 
 
 21. 3 a + 1, 9 a^ - 1, 27 a» + 1. 
 
 22. a» + a" J, a (c - 6), a'' - h^. 
 
 23. 2axy{x-y), Sax'^x^-y^ iy^(x + yY. 
 
100 ALGEBItA. 
 
 24. 6 (x^ - 9), 9 (a; + 3), 15 (a; - 4), 10 {x^ - x - 12). 
 
 25. 4:a + ib, 6a^-24:b% a^-3ab + 2 b\ 
 
 26. 2 a - 3, 6 a + 9, 3 (4 a^ - 9). 
 
 27. 9 — a% a + 3, 3 — a. 
 
 28. a!» + f, 7?y — y\ x^ — if. 
 
 29. x"^ — y\ x'^ — xy + if, x" + xy + f. 
 
 30. i.{aH-a P), 8 (a» - a b''), 12 (a b^ + ¥). 
 
 31. «» — 2/', a:' + 2/'. « / — /> a:» — a; «/*. 
 
 32. a;'^ - 5 a; + 6, a;'' - 6 a; + 9. 
 
 33. a;' + a; - 2, a;'' - 2 a; + 1. 
 
 34. 2a!'' -7a; + 3, 2x^ + Bx-3. 
 
 35. 3 a;2 + 7 a; + 2, a;^ - a; - 6. 
 
 36. a;2 — (a + 6) a; + aS) a;" — (a — 6) a; — ab. 
 
 37. (a; - yf -{a- bf, {x - af - (y - b)'. 
 88. (a; + 2)2 - (a; - 2)^ x*. 
 
 39. (a + by -c\ (a + 6 + c)^. 
 
 40. 3 (a; - 2), 7 (a; - 3) (a; - 2). 
 
 41. (3 a; -2) (2 a; -5), (2a;-5) (a; + 7), (a; + 7) (a; - 1). 
 
 42. (a+6)', (a + S) (a^-afi + i''), a a; («■' - a i + J^). 
 
 43. (a — h) {a — c), (J — c) (ft — a), (c — a) (c — b). 
 
 44. a (a^-S'^) (a'^-tr'), b (b^-c^) {b'-a^), c {o^-a") (c^—b^). 
 
 45. 1 - a;, (1 - x^f, (1 + a;)l 
 
 46. (6c=-aJc)^ b^iac'-a"), a" (? -{■ 2a<? ^ c*. 
 
 47. 1 + a; + a:^ 1 + a;'' + a;*, 1 — a; + a;''. 
 
 48. a;^" — y^", (a;» — i/")''- 
 
 49. a;^" + 2 a;"?/" + /», a;''" — i/'", a;=» — 2 ai'-'j^"" + y^"- 
 
 50. 21x{xy-yy, Zb {x* y^ - x"" y% 15y (a:^ + a;y)=. 
 
LEAST COMMON MULTIPLE. 
 
 101 
 
 Case II. 
 
 129. To find the Least Common Multiple of Polynomials which 
 cannot be readily factored by Inspection. 
 
 Let A. and B stand for any two algebraic expressions, and let G 
 stand for their greatest common divisor, and L for their least common 
 multiple. 
 
 Let the quotients of A and B by G be represented, respectively, 
 by g and 5'. 
 
 Since G is the greatest common divisor of A and B, g and g' can 
 have no common factor; therefore the least common multiple of 
 A and B must be G • g ■ g' ; that is, £ = G ■ g ■ g', or 
 
 L-o-r<ii. 
 
 •^■l-^o" 
 
 130. Hence, to find the least common multiple of two 
 polynomials which cannot be readily factored by inspec- 
 tion, or, indeed, of any two algebraic expressions, we have 
 the following 
 
 Bule. 
 
 Divide one, of tlie eospressions hy their greatest common 
 divisor, and multiply this quotient iy the other expression for 
 the least common multiple required. 
 
 1. rind the least common multiple of 6 k'' + 7 x + 2, 
 2Ax^ + Ux + 21. 
 
 3x 
 2 
 
 6x^ 
 6x^ 
 
 + 7x 
 + 3x 
 
 + 2 
 
 
 4:X 
 
 ix 
 
 + 2 
 + 2 
 
 2ix^ + 54:x + 21 
 24 a;-^ + 28 a; + 8 
 
 13 
 
 26 a; + 13 
 
 L. C. M. 
 
 G.C.D.= 2a;+ 1 
 
 ^ (6a!^+7a; + 2)(24a!°+54a:+21) 
 
 "~ 2a; + 1 
 
 = 3 (2a! + 1) (3a; + 2) (4a; + 7) 
 
102 ALGEBEA. 
 
 Find the least common multiple of : 
 
 2. 2 a;'' + 3 X - 20, 6 a:' - 25 a;'' + 21 a; + 10. 
 
 3. a;2 - 15 a; + 36, a;' - 3 a;^ - 2 a; + 6. 
 
 4. a:' — a;= — 7 a; + 15, a:» + a;'' — 3 a; + 9. 
 
 5. a" — 6 a* + 9 a" — 4, a" + a^ - 2 a'' + 3 a^ — a — 2. 
 
 Ans. (a^ - ly (a^ - 4) (a« - a + 2). 
 
 6. a:' - a;^ + a; + 3, a;* + a;= - 3 a;= - a; + 2. 
 
 7. a;* - 5 a!» + 20 a; - 16, X* - 2 a!» - 3 a;2 + 8 a; - 4. 
 
 8. a' + 6 a^ + 11 a + 6, a» + 10 a^ + 29 a + 20. 
 
 Ans. (a + 1) (a + 2) (a + 3) (a + 4) (tt + 5). 
 
 131. To find the Least Common Multiple of three or more 
 Expressions. 
 
 Find tJie least common multiple of tvjo of them, then of 
 this result and a third, and so on; the last common multiple 
 will he the least common multiple sought. 
 
 Find the least common multiple of : 
 
 1. Sx^ — lx^y+Bxif — if, x'-y + dxi/ — Bx^ — y', 
 3 x^ + 5 x^ y + X y^ — y\ 
 
 2. x^—3ax + 2a% 3a;''-19aa; + 28 a^ x'^ — Sax + ia^ 
 
 Ans. 3 a:^ — 28 a a;8 + 91 a^ a.^ — 122 a' a; + 56 a*. 
 
 3. x' + 3x''-6x~8, x^-2x^~x + 2, a;^ + a; - 0. 
 
 4. 2a:*+a;'-8a;2— a; + 6, 4a;^+ 12 a;»- a;''— 27a;-18, 
 4^5 + 8a;< — 13a.= -26a;2 + 9a; + 18. 
 
 Ans. 4 a;' + 8 x^ - 13 x' - 26 a.-^ + 9 a; + 18. 
 
 5. a:»-3a:2 + 33;-l, a;' - a;^ _ j. + 1, a;*-2a;' + 2a: 
 — 1, a;* - 2 a;= + 2 a;2 — 2 a; + 1. 
 
FRACTIONS. 103 
 
 CHAPTEE X. 
 FRACTIONS. 
 
 132. When division is expressed by writing the dividend 
 over the divisor with a line between, the expression is 
 called a Fraction. As a fraction, the dividend is called the 
 numerator, and the divisor the denominator. 
 
 Thus T means a -=- 6. Hence it follows that t X 5 = a. That is, 
 a " 
 
 the algebraic iiaction ^, wlwre a and b are supposed to have any values 
 
 whatever, is that expression which, when multiplied by b, becomes a. 
 
 133. The words fractional and integral, as applied to 
 algebraic expressions, always apply to the algebraic form 
 of the expression, and not to its numerical value for par- 
 ticular numerical values of the letters involved. 
 
 134. The Terms of a fraction are its numerator and de- 
 nominator. Tliese Terms may be polynomials, and hence 
 each be composed of several tervis (§ 14). Their dividing 
 line also. performs the office of a vinculum, and any removal 
 of the line and sign preceding it calls for a corresponding 
 change of the signs of the terms of the numerator, in ac- 
 cordance with the laws whicla govern the omission of 
 brackets. 
 
 135. A Simple Fraction is one whose terms are integral 
 
 . - a a + c 
 
 in form: as, -, ^ =. 
 
 o — a 
 
 136. A mixed Expression, or Mixed Number, is an ex- 
 pression composed of both integral and fractional forms ; 
 as, _ , J _ I c — tZ 
 
 a + -, a — b — 
 
 c e + / 
 
104 ALGEBRA. 
 
 137. From the principles of division, and § 134, it fol- 
 lows that 
 
 ab _ — ab _ _ — ab _ _ ah _ _|_ j 
 a —a a — u 
 
 andthat ^^^ = ^ =-^ =-b; that is, 
 
 (a) The value of a fraction is not changed, 
 
 (1) If the sign of every term of the numerator and de- 
 nominator is changed. 
 
 (2) If the sign of every term of the numerator and the 
 sign before the fraction are changed. 
 
 (3) If the sign of every term of the denominator and the 
 . sign before the fraction are chan/jed. 
 
 (b) But the value of a fraction is changed, 
 
 (4) If every sign of the numerator, or denominator, or 
 the sign before the fraction, is changed. 
 
 138. From the rules of multiplication it follows that 
 
 (a)(6)(c) = + ubc 
 (— a) (b) (c) = — aba 
 (- a) (—b)(G) = + abc 
 (— a) (— b) (— c) = — ab e 
 (a — b) {c — d) =^ ac ~ a d — b c ^b d 
 (b — «)(c — d) =^ — ac + ad + be — bd 
 and (b — a) (d — c)= ac — ad — b c + b d 
 
 Hence, 
 
 abc _ (-a) i-b) (c) ^ _ (-a)ib)(c) ^ _ (-a) (-5) j-c) 
 
 IT 1 1 • 1 
 
 Also 
 
 {a-b)(o-d) (b-a)(d-c) (b-a){c-d) 
 
FRACTIONS. 105 
 
 That is, the value of a fraction is not changkd, if the signs 
 of two of its factors, or of an even number of its factors, in 
 the numerator or denominator, or in both, are changed ; but 
 the value of a fraction is changed, if the sign of one of 
 its factors, or of an odd number of its factors, in either the 
 numerator or denominator is changed. 
 
 139. EXERCISES. 
 
 Express the following fractions in four different ways : 
 
 1. 
 
 a — b b — a b — a a — &_ 
 
 o—d d—c c — d d — c 
 
 X t, X — V — s o a — b 
 
 2. ^ - 5. ^ — y — '^ . 8 
 
 4. 
 
 y—z e—f (b — c)(c — d) 
 
 a — b ^ „ a — b + d ^ q (a — b){c — d) 
 
 c ' ' ^Tf^ ' ' (a-b)(c-d)(e-f) ' 
 
 a — b n a, ^f^ a (e — d) 
 
 c-d—e' ' (a-b){b-c)' ' (e -/) (6 - c) ' 
 
 Write the following fractions so that a and b shall become 
 positive : 
 
 X —a 
 
 11. -Ilii. = ^l_. 14. 
 
 
 - a 
 
 _ a 
 
 X 
 
 -b 
 
 b-x 
 
 
 X - 
 
 - a 
 
 
 -y 
 
 -b 
 
 — 
 
 a — 
 
 ^ + y . 
 
 12. - ''-"' ■ 15. 
 
 13. ZLl^z£±y.. 16. 
 
 y — b 
 
 x + y — a 
 X — y — b 
 
 — a (b — c) 
 
 z — a d — b 
 
 140. We are now to prove that -j = t — , and that 
 "^ b b m 
 
 '^ = - , for all values of a, b, and m. 
 b m b 
 
106 ALGEBHA. 
 
 -= a-7- b, by definition, 
 b 
 = a-^b-i-mXin, oy axiom, 
 
 =: a X m ^ b -i- m, by law of commutation, 
 
 = (a m) -=- (b m), by law of association, 
 
 am 
 
 b m 
 
 n T am a 
 
 Conversely, = - • 
 
 b m b 
 
 That is, multiplying or dividing both numerator and 
 denominator hy the same number does not change the value 
 of the fraction. 
 
 141. REDUCTION OF FRACTIONS. 
 
 The reduction of fractions, that is, the changing of their 
 form without changing their value, depends upon the 
 preceding principles, and may be conveniently presented 
 under the following cases. 
 
 Case I. 
 142. To reduce a Fraction to its Lowest Terms. 
 
 A fraction is in its lowest terms when its numerator and 
 denominator have no common factor. 
 
 Since the cancellation of the same factors from eacli 
 term, or the dividing each term by the same factor, as, for 
 instance, the greatest common factor, does not change the 
 value of the fraction (§ 140), we have for the reduction of 
 a fraction to its lowest terms the foUowinsr 
 
 Rule. 
 
 Besolve the numerator and denominator of the fraction 
 into their prime factors, and cancel all the common factors. 
 Or, 
 Divide both terms by their greatest common divisor. 
 

 FRACTIONS. 
 
 
 1. 
 
 ^ , 36 a*Pc ^ .^ . 
 Keduce ^-. ., „ to its lowest terms. 
 90 a^b e" 
 
 
 
 SQa^b^c 2^-3^a^b'-e 
 
 2b 
 
 
 90a'bc'~2-3^-5a'bc^~ 
 
 Sac 
 
 2. 
 
 a;4 _ JJ.2 
 
 Reduce — r 5- to its lowest terms. 
 
 x*—l 
 
 
 
 x^-x^ _ x" (a;2 - 1) 
 
 x" 
 
 107 
 
 X-- 1 {x^ + 1) {x^ - 1) ~ a;^ + 1 
 Reduce the following fractions to tlieir lowest terms : 
 
 3. 
 
 14 a b"- c' 
 
 21 «« 6^ c 
 
 ^ 125a:^j/^g"' 
 IbOx^ifz^ ' 
 
 2ab 
 
 a" + ab 
 a'-b^' 
 
 8 ("'-I)' 
 ■ x'-l' 
 
 g X^ — X^f 
 
 10. 
 
 11. 
 
 12. 
 13. 
 
 (a= 
 
 + xyy 
 
 a;* 
 
 + x^ 
 
 x^ 
 
 -\ 
 
 a 
 
 -b 
 
 P 
 
 -a?' 
 
 a 
 
 -3 
 
 9 -a'' 
 
 b-'j-a^ 
 
 (a -by' ' (a» - b') (a* - b') 
 
 Id 
 
 flj2 a;2 y2 _ a;« 2/* 
 
 
 a;* y* — a* 
 
 
 Ans. -=^'2/' 
 
 15 
 
 a* + 2 a'' 6^ + J* 
 
 
 a* -6* 
 
 16. 
 
 a;«-^a;^ 
 x'-x^' 
 
 17 
 
 1 - 5 a + 6 a^ 
 
 
 l-7a + 12 a''' 
 
 
 A 1 - 2 a 
 
 Ans. _ 
 
 1 — 4a 
 
 18 
 
 a^y + 2x^y + ixy 
 
 
 a;»-8 
 
 IP 
 
 (a'-J»)(-a5 + aHi') 
 
 
 («»+&')(«'+ "6 + 6^) 
 
 />0 
 
 x*-Ux^-5i 
 
 
 a;4_2a!^-15 
 
 
 A a:'' - 17 
 
 Ans. -^ g- 
 
 x^ — 5 
 
 ^i 
 
 (a' -b'^{a- b) 
 
108 ALGEBRA. 
 
 22 x(b — c) 25 (tt + by — «" 
 
 ■ • - ■ ' {a + b + cy 
 
 Qo a- — a o -t- ax — ox nc (^ + vY — (» + bY 
 
 • -^ ^ ^- • (x-aY^{y-by 
 
 24. 
 
 28. 
 29. 
 30. 
 31. 
 
 (a 
 
 — 
 
 b){c- 
 
 -b)(a- c) 
 
 a" 
 
 — 
 
 ab + 
 
 ax — bx 
 
 a^ 
 
 + 
 
 ab + 
 
 ax + bx 
 
 
 X* 
 
 ' + a^: 
 
 x^ + a^ 
 
 27 {c,+b-^oy-i<^-b-cf 
 
 x'^ + ax^—a^x — a"' ' 'la {b^ + 2bc + c") 
 
 2 
 
 5 + c 
 
 Ans. "- -"'•^ T • Ans. 
 
 x^ — a^ 
 
 a b x^ + a^ x y -\- ab y^ + b^ X y 
 ab x^ -\- a'xy — ab y^ — b^ xy 
 
 a^ + b^ + c'' + 2ab + 2ac + 2bo 
 d'- — y — G'-2bG 
 
 { a + b) {{a + bf - c'} , g+S 
 
 \b''e^-{a^-¥-cy ' {c-\-b-a)(c-b-\-a)' 
 
 a;' - 23 a; + 10 
 
 5 £B» - 23 a;^ + 4 
 By § 123 the G. C. D. is found to be x^ — 5 x + 2, and 
 
 (a!° - 23 a; + 10) ^ (a;'' - 5a; + 2) _ a; + 5 
 (5a;»- 23x^+ 4) -- (a;^- 5a: + 2) ~ 5a; + 2' 
 
 32. 
 33. 
 
 a;' — 3 a; + 2 
 
 2: 
 
 B» - 3 a;'' + 1 " 
 
 
 3a;2-8a; + 6 
 
 a;» 
 
 -4x^ + 5a:-2' 
 
 
 a;' + 5a;2+7a; + 3 
 
 x" 
 
 + 3a!a + 4a;2 + 3a;+l' 
 
 
 a;* — a;' — a; 4- 1 
 
 X* 
 
 -2a:«-a;^-2a; + l 
 
 2 
 
 a;» + 9a;'' + 7a! — 3 
 
 3^_ a;^ + 5x^+7a; + 3 . x + ^ 
 
 35. 
 
 3g Zg;'' + 9a;' + 7a!-3 . 2 a; -I- 3 
 
 3a;» + 5a;^-15a: + 4' 3ir^4 ' 
 
37. 
 
 FRACTIONS. 109 
 
 5a^+10a*x + 5a^x' 
 
 a^x + 2a^x^ + 2aa:^ + x* 
 
 38 Sa^x* — 2ax^ — l ^^^ 3aa;^4-l 
 
 39. 
 
 4:a^x''—2a^x*-3ax^+l' Aa'x*+2ax'-l 
 
 a;^ + 2 a;' -3 a;^ — 7a; — 2 
 
 41, 
 42. 
 
 2a;^ + a!»-6a:''-5a;-l 
 
 4Q aa!° + 40a^ — 5 a'a;' — 99 a" a; y. a (a; + 8 a) . 
 
 ■ ««-6aa;»-86a^x'' + 35a=a;' ^" a; ^a; + 7 a) " 
 
 - a:' - 48 a; + 12 a,' - 35 
 19 a;^ - 32 a; + 28 a!» - 15 " 
 
 7 a:° - 2 a;' y - 63 a: y'' + 18 y' " 
 
 5 a;* — 3 a;' y - 43 a;^ 2/'' + 27 a; 2/« - 18 ?/* ' 
 
 Case II. 
 
 143. To reduce a Fraction to an Integral or Mixed Number. 
 
 2 x^ 7 a; 1 
 
 1. Keduce ^ to an integral or mixed number. 
 
 X — tj 
 
 a;-3)2a;''-7a;-l(2a;-l+ ~* 
 
 2a;'-6a; or, '^ ~ ^ 
 
 - a;-l 2a;-l ~ (§134) 
 
 1 o X — 6 
 
 — x + 3 
 
 Since a fraction represents the quotient of the numerator divided 
 by the denominator, we perform the indicated division, adding to 
 the quotient the fraction formed by placing the remainder over the 
 divisor. Hence, 
 
 To reduce a fraction to an integral or mijced number, we 
 have the following 
 
 Kule. 
 
 Divide the nwrmrator ly the denominator, and if there is 
 a remainder place it over the divisor, and add the fraction 
 so formed to the quotient. 
 
110 ALGEBRA. 
 
 Reduce the following to integral or mixed numbers : 
 
 2. 
 
 4. 
 
 14. 
 15. 
 16. 
 
 17. 
 
 a^ — 2ah + c g tc° — y° 
 
 g Zx^+^x + 2 
 
 3a; 
 
 2x^+ax — Za^ 
 X + a 
 
 Ans.2x-a- ''^' 
 
 X + a 
 
 6 '"' + 16 
 
 x + 2 
 
 g a;^ - 9 y" + 13 s' 
 K — 3y 
 
 ^_ x'-if 
 
 a; + y 1 — a; 
 
 
 x — y 
 
 
 9. 
 
 (a - bf 
 
 
 10 
 
 a;2 - 3 a; + 1 
 
 
 
 a;-2 
 
 
 Ans. X — 1 
 
 X — 
 
 2 
 
 11 
 
 a^ + a^ + 1 
 
 
 
 a + 1 
 
 
 12. 
 
 ^= + 1 
 a;-l ■ 
 
 
 13. 
 
 l + a;= 
 
 
 x^ — 3x^a — Sxa'' + a° 
 a; — a 
 
 2 x^ - 6 a:^ + 13 a;^ - 15 a; + 8 
 a;2 - a; + 3 
 
 a;5 — a;' - 2 a!= — 2 X — 1 
 
 as'^ — a; — 1 
 
 Ans. x^ + x^' + x- '*'"'■■'" 
 
 a;^ — a; — 1 
 
 x* + x' + x^ + x + 1 
 x^ — x -\- 1 
 
 Case III. 
 144. To reduce a Mixed Number to a Fractional Form. 
 
 It will be observed by referring to Case II., of which this is the 
 converse, that the integral part of the mixed expression always 
 stands for the quotient, the denominator of the fractional part for the 
 divisor, and the numerator for the remainder ; and that the dividing 
 line also performs the office of a vinculum for the numerator. Hence, 
 
FRACTIONS. Ill 
 
 To reduce a mixed number to a fractional form, we have 
 the following 
 
 Kule. 
 
 Multiply the integral part by the denominator of the 
 fraction ; to the product add the numerator if the dgn of 
 the fraction is plus, and subtract it if tlie sign is minus, 
 and under the result write the denominator. 
 
 2 52 g2 
 
 1. Reduce a — b -\ 5— to a fractional form. 
 
 a -\- 
 
 ^ 26= -a^ a'-b^ + 2b''-a^ b^ 
 
 (1) «-* + ^T^= ^-fTb ^^Tfb 
 
 ^ a'-2P a''-b^-a'' + 2b^ P 
 
 (/) a — 
 
 a + b a + b a + b 
 
 ,„. 2b^-a^ 2b^-a^ + a^-b^ P 
 
 a+b a+b a+b 
 
 ,.. 2P-a'' , , 2P-a^-P + a^ P 
 
 W „ I z. b + a = 
 
 a + b a + b a + b 
 
 2P-a^ 2P-a^-(-a?+l-^ _ P 
 
 ^ a + b ^ ' a + b a + h 
 
 (2), (3), (4), and (5) differ from (1) in form only. They are 
 derived from (1) through §§ 137, 57, and 64. 
 
 a* + 6* 
 
 2. Reduce a + b A r to a fractional form. 
 
 a — b 
 
 , a' + P a'-P + ia^+b'^ 
 
 a + b ■] r = ^ 
 
 a — b a — b 
 
 «"-&= + «' + &' 
 a — b 
 _ 2a^ 
 a — b 
 
112 ALGEBRA. 
 
 a' + h^ 
 
 3. Reduce a + h r to a fractional form. 
 
 a — b 
 
 a2 + j2 a^-h^- (a^ + J^) 
 
 a + fi r- = 7 
 
 a—b a—b 
 
 a'^-b^-a^- b^ 
 
 a — b 
 — 2 b'' 2b^ 
 
 a — b b — a 
 
 Eeduce to fractional form the following examples : 
 
 4 « + -. 10. x'-xy^f- ^ 
 
 a ' " x + y 
 
 ^ , , 2xv ^. , , a?—ay+y'- 
 
 5- 1+^^T7- 11. x-a+y+ ^^^ . 
 
 6 l_-l^. Ans. ^1+^i^+i^. 
 
 a;^ + w^ a; + a 
 
 r. a-25-^. 12.2.-3- ^^-1 
 
 a — i ■ Sx-' — a; + 2 
 
 „ a(b — c) a^ — ¥. . 
 
 8. a ^-^. 13. ^^-(«-J). 
 
 9. -^^ + (x-l). 14x^-2-^. 
 
 a; + 1 ^ a; — 1 
 
 8 aH + 11 a^ i^ - 12 aW +46* 
 
 15. 5 a^ - 4 a i + 2 6'' + 
 
 Ans. 
 
 2 a^ - 2 6^ 
 10 a* + 5a''62-4a&' 
 
 .- a;*+2a;^ + l . 2 .^ . 2a:* + 3a;^+2 
 
 JO. — J- — 5 (a; — a!'' — 1). Ans. — ., • . ^ • 
 
 x^ + x + 1 ^ ' x' + x + l 
 
 17. a — b — ^ =- . 19. a^ — b" -\ 5^^ — 75— • 
 
 a — b cr — b^ 
 
 18 i_«' + 5'-c^ '"'' 
 
 - «--f-^i^T- 
 
 2ab 
 
 21. 1 + a; + a;^ + a;' + :;-^— • Ans. _1- 
 
 1 — a; 1 — 
 
FRACTIONS. 113 
 
 Case IV. 
 145. To reduce Fractions to their Least Common Denominator. 
 
 1. Eeduce , , and — to equivalent fractions 
 
 mx my mz 
 
 having the least common denominator. 
 
 The required fractions must have lor their denominators mxyz, 
 the least common multiple of the given denominators. 
 
 If we multiply the numerators, a, b, c, by the quotients of mxy» 
 divided hy mx, my, m z, respectively, we shall have, by § 140, the 
 equivalent fractions required. 
 
 That is, 
 
 a _ a X (yz) ays 
 
 mx mx X (y«) mxyz 
 b _ b X {xz) _ hxz 
 
 and 
 
 my my X (xz) mxyz 
 
 c_cx(xy)_cxy 
 mz mzx{xy) mxyz 
 
 Hence the following 
 
 Kule. 
 
 FiTid the least common multiple of the denominators for 
 the least common denominator. For new numerators, mul- 
 tiply each numerator hy the quotient arising from dividing 
 this multiple by its denominator. 
 
 Note 1. Fractions should always "first be reduced to their lowest 
 terms. 
 
 Note 2. The familiar method of multiplying together the denomina- 
 tors for a new denominator, and each numerator by all the denominators 
 except its own, for a new numerator, is sometimes useful. lu cases where 
 the denominators are mutually prime, it is identical with the process 
 above. 
 
 Note 3. Every integral form may be considered as a fraction with unity 
 for its denominator ; that is, a = ~ . 
 
 8 
 
114 AT.GEBRA. 
 
 2. Eeduce ^ , -^ , and ^^—^ to equivalent frac- 
 
 6xy 5x^y 15 a;/ 
 
 tions having the least common denominator. , » 
 
 The L. C. D. is 15 x^ -f. 
 
 «" 6 , ft' 
 
 . — ;"• and ; 
 
 Zxy bx^y Ibxy* 
 
 are equal, respectively, to ^^^, -^,-^, and ^^^^^^ 
 
 3. Eeduce —, ^, and —. — ■ — r to a common denom- 
 
 x{x — y) yi^ + y) 
 
 inator. 
 
 TheL. C. D. is xyix'^ — y^. 
 
 35(1 — S>' y(x + y) 
 
 .■ 1 . y(x + y) ^ (^ — y) 
 
 are equal, respectively, to ;^y^^i_y-i^ , ^ j, (^.2 _ ^,2^ 
 
 Eeduce to equivalent fractions having their least common 
 denominator : 
 
 a + 3 a + 7 „ 1 a; + y 
 
 *■ ~5~' 10 ■ 2a; -Sj^' ix'-Qf 
 
 ^ a-h b-c „ 4a;2 a; y 
 
 5. ^-, — ; . <• 
 
 «6 ' 6c • 3(a + i)' 6(a^-6-0' 
 
 8a;'(a-&) a;?/ 
 
 a — a;a. + a; ^ ' ^ ' 
 
 1 (a + 2a;)' - 
 
 a -2a;' a" - 8 a;" " 
 
 10. It 7,2 ' „2 ■ 
 
 11. 
 
 Ans 
 
 1 1 
 
 4a?(a + a:)' 4a,= (a-a;)' 2 a^ (a'' - a;'') ' 
 
 a — X a + X 2 a 
 
 4:a\a^-x'')' 4 a= («'' - a;') ■ 4 a» (a" - a;'') ' 
 
12. 
 
 FRACTIONS. 115 
 
 a b 
 
 (a-b) (b-c)' (a-b)(c-b)' 
 
 3x 2 2 _x_ _a^ 
 
 1 — a-, a; — 11 + a; a + x 04— cc 
 
 15. 
 
 d'-U" (a + bf {a -by 
 
 1 
 
 16, 
 17. 
 
 a'(a'-&^) b^{a-bf ab(a + hf 
 
 (a'-h^f (a^-by (a" -by ■ 
 2 . 1 
 
 a:* + 5aa;+6a=' x' + 4:ax + 3a'" a^ + 3ax + 2a^ 
 Ilia* 
 
 ia\a + x)' 4a»(a!-a)' 2a'(a^ + x'')' a' — x' ' 
 
 18. i, 1 ^ 
 
 x' X — ?/' a;^ + a;2/ + ^' 
 
 .. 6x 2x 3x^ . 
 
 n^^' r+2^' 4 a;'' -1' *" 
 
 146. ADDITION AND SUBTEACTION OF FRACTIONS. 
 
 1. Eind the sum of — and — , and also their difference. 
 a a 
 
 From the distributive law of division it follows that 
 
 (ab + ac)-7-a = ab-^a + ac-^a = b + a 
 
 Or, expressed in the fractional form, 
 
 ah 4- nc ah , ae ,, 
 
 = \- =b + a 
 
 a a a 
 
 ., ah — ac ah ac , 
 
 Also, = = 6 — a 
 
 a a a 
 
 _ , ah , ae ah -\- ae 
 
 Conversely = 
 
 a a a 
 
 . ah ac ah — ae 
 
 a a a 
 
116 ALGEBRA. 
 
 Hence, for adding or subtracting fractions, we have the 
 following 
 
 Bule. 
 
 Reduce the fractwns, if necessary, to equivalent fractions 
 having their least common denominator ; then add or sub- 
 tract their numerator's, as the sign before the fraction directs, 
 and write the result over the least common denominator. 
 
 ^ -^. ■• ■. , ,3b + a7b — 4:a 
 
 2. Eind the value of — = rr: • 
 
 5 « 10 a 
 
 TheL. C. D.islOo. 
 
 &h + a 7 6 — 4a 
 
 5 a 10 a 
 
 2 (3 6 + g) + (7 6 — 4 g) 
 
 " lOo 
 
 66 + 2n + 76 — 4g 136 — 2a 
 
 10 g 10 a 
 
 3. Pind the value of -i + -^ • 
 
 xp ay ax 
 
 TheL. C. D. is axy. 
 
 X — 2 V , 8w — a 3x — 4a 
 
 .-. ■ + — 
 
 XT/ ay ax 
 
 a(x—2y) ■\- X (Zy — g) — y {^x — 4 a) 
 
 axy 
 ax — 2ay ■}■ 3 xy — ax — 3xy -{- iay 
 
 axy 
 _2ay _2 
 axy X 
 
 Simplify : 
 
 x — 1 X + 3 X + 7 
 
 a-2b a-5b a + 7b 3(a + 3 b) 
 
 2a 4a 8a 8a 
 
 a — X a + X a^ — x^ 
 b. 1 
 
 2ax 
 
FRACTIONS. 117 
 
 2 By' — a;" xy + if a;° + y" 
 
 rf. 5 + 5 — 5 — ■ ■ aIXS. s ^ ' 
 
 - a — hb — cc — a 
 
 o. 7 — I 7 — -1 . 
 
 ao ca 
 
 9. 
 
 be ca ab 
 
 ^^ a , b , c . a + bx + cay' 
 10. -„ + ^ + ^- Ans. 
 
 a' + y (a -by (a + by 2a-2b' 
 c c c 2e 
 
 a + 5b — 7c3a — 7b + 5c 2a—b + 5c 
 15 "*■ 20 30 
 
 3(2a-36) 2(3a-5J) , 5(a-b) 
 ^'^- 8~ ~ ~~3 + 6~ ■ 
 
 ^ , 3a — 46 2a — S — c 15 a — 4c 
 
 14. ■ • 
 
 2 3 ^ 12 
 
 25 a -20 b 
 
 Ans. 
 
 15. Simplify 
 
 12 
 2a: — 3a 2a; — a 
 
 X — 2 a X — a 
 
 The L. C. D. is (a; — 2 a) (a; — a). 
 2x — 3a 2a: — a 
 
 X — 2 a X — a 
 (2x — Za){x — a)—{2x — a){x—2a) 
 
 (x — 2 a) (a; ^ a) 
 
 2a:'' — 5aa: + 3a''^(2a:'— 5aa:+2o'^ 
 
 (a: — :2 a) (x ^ a) 
 
 2x^—5ax+ Sa^ — 2x^+ 5a: — 2a'' 
 (x — 2 a) (x — a) 
 a" 
 
 (a: — 2 a) (x — a) 
 
118 ALGEBKA. 
 
 16. Simplify — ^ + "^ ^ 
 
 x + y 3? — xy -^ 'f x" + y^ 
 The L. C. D. is a;» + y". 
 
 1 X — y x^ — xy 
 
 X -\- y x'^ — xy -\- y'^ a* + ^' 
 
 _ a' — xy + ;/' + {x-\-y) (x — y) — (x^ — x y) 
 x^ + ^' 
 x^ — X y -\- y^ -\- x^ — y'^ — x^ -\- xy x^ 
 
 xS -\- y^ i' + y^ 
 
 Simplify : 
 
 17. — —- + 
 
 X + y x — y 
 
 a b 
 
 a — b a + b 
 
 3 1 
 
 a:— 6 X + 2' 
 
 20. 1±-%J-^^ 
 
 1 — X 1 + X 
 
 a + X a — X 
 
 18. 
 19. 
 
 21. 
 
 a — X a -{- X 
 
 a a „„ y 
 
 24. 
 
 a; a; 
 
 1 - a;-^ 1 + a:" ■ 
 
 25. 
 
 1 1 1 
 
 26. 
 
 1 1 
 
 (a; -2,)^ (a: + #- 
 
 27. 
 
 1 (« + 2 a:)2 
 a-2x a»-8a;S 
 
 
 . 2 aa; 
 
 Ans. -—5 -.■ 
 
 8 a;' — a' 
 
 22. — — - -^— , • 28. -7-/— 5, + 
 
 a; — a x — a 
 
 xix^ — y^) y(x^ + y^) 
 
 ^"'- 2a; -3y ix'-9y^' ^•'- x+y ^ x-y ' 
 
 Ans '" + ^y 00 1 3__ 
 
 ■4 a;'' -9 J/'' "^ a;^-7a;+12 x^-5a;+6' 
 
 a; + 7 a; + 2 
 
 31. 
 
 a;2 — 3a; — 10 x^ + 2 x — 35 
 
 10 a; + 45 
 
 Ans. 
 
 1 
 32, 
 
 (a; + 2) (a; + 7) (x - 5) 
 
 1 3 
 
 2 a:'' — a; — 1 6a;'*-a;-2' 
 
33. 
 
 FRACTIONS. 119 
 
 1 
 
 2 3^ — x-l 2x^+x — 3 
 
 34.4-.+ ' '" 
 
 X 
 
 a + b a — b d^—U^ 
 
 OK ^ , 1 3 a; . 
 
 2x + 1/ 2x — y 4 a;'' — y* 4 a;'— y* 
 
 36. -^ + ^ 
 
 37. 
 
 a + X ' a — X a' — x^ 
 
 5 3 a; 4 — 13 a; 
 
 l + 2a; l-2a;~l-4a:''' 
 
 38 ^ + y _ ^ — y I ^.y' 
 
 a; — y a; + y a;** — y" 
 
 39. -^ - 1 - , ' ,, • Ans. 1. 
 a — 1 a (a — 1) a 
 
 40. 1-2 ^ 1 
 
 41. 
 
 a a + 1 a + 2 
 
 1 1 a; + 3 
 
 a; - 1 2 (x + 1) 2 (x" + 1) 
 
 42. -^ + -^ + 1. 
 a + X a — X 
 
 43. T + X T • Ans. 5" ; 
 
 X — 1 X — 1 X — 1 
 
 ..3 5 2x-7 
 
 X 2x-l 4x^-1 
 
 2- x-3 a!» 
 
 X + 4 x'' - 4x + 16 "^ x' + 64 ■ 
 
 , „ x' + a x'' X (x — a) 2 a x 
 46, - 
 
 2 
 
 ax" — a? (x + a) a x'' — a' 
 
 6 ab a}? , J» 
 
 47. — —^ — 7 — r-jTi — 7 — nrs • Ans. 
 
 a + b {a+by (a + bf (a + by 
 
120 
 
 ALGEBRA. 
 
 48. 
 
 x-y 1 xy 
 
 3?-xy^f x + y ar' + y" 
 
 49. 
 
 ^1^1^ 
 
 4 a^ (a' + a;^) ' 4 a« (a? - x") ' 2 a* (a* + x*) 
 
 50. 
 
 1 1 2x 4.« ,„, 8a.' 
 
 147. Sometimes a modification of the foregoing general 
 method will considerably shorten the work. 
 
 1. Simplify ^ + ^ + ^ + ^,. 
 
 
 1 ,1=2 
 1-a; 'l + a; l-x^ 
 
 
 then 
 
 2 2 4 
 
 1 — a;2 ' 1 + z2 1 _ a;i 
 
 
 and finally 
 
 4 4 8 
 
 
 1-1* ' 1 + s* l-a;8 
 
 
 2-S-piify,i3 ,!i + 4i 
 
 1 
 
 a; + 3 
 
 
 1 1 6 
 
 
 
 a;— 3 a; + 3 a;2_9 
 
 
 and 
 
 3 3 -6 
 
 a: + l x — \. x2_l 
 
 
 then 
 
 6 —6 6x2-6 — 6a 
 
 :=+54 
 
 
 a;'-' — 9 ' x^ — l (x2 — 9) (x^ 
 
 -1) 
 
 
 48 
 
 
 
 (X'' — 9) (x2 - 
 
 -1) 
 
 „ „. ,., a 2x a(3x — a) 
 
 3. Simplify — - — H 1 ^ =-^ ■ 
 
 X + a X — a a' — x^ 
 
FRACTIONS. 121 
 
 Note 1. In working examples like this and like Ex. 4, the symmetric 
 arrangement of the denominators should first be secured by changing the 
 form of any fraction or fractions necessary, in accordance with the prin- 
 ciples of § 137 and §138. 
 
 a 2x a(Zx — a) 
 
 x-\- a X — a a^ — x^ 
 
 a 2x _,a (a — 3 a;) 
 
 x-\- a X — a X* — a^ 
 
 a(x — a) -\-2x(x -{- a) -\- a(a — Sx) 
 
 a;2 — a2 
 
 ax — a'' + 2x'' + 2ax+a^ — Sax 
 ~ x^ — a^ 
 
 2a;2 
 
 4. Simplify ^p— — — — — — + 
 
 (x-2) (x-3) (2-x) (x-l) ^ (x-3) (1-x) 
 1 . 2 3 
 
 (a; — 2) (x — 3) (2 -x)(x — l)^{x- 3) (1 - x) 
 12 3 
 
 (1-2) (x — 3) ' (a:-l)(a;-2) (ar_3)(a;-l) 
 
 (a;— 1) + 2 (a; — 3) — 3 (a: — 2) 
 {x — 1) (a; — 2) (x — 3) 
 
 z — l+2a; — 6 — 3a: + 6 — 1 
 
 (a; - 1) (a; — 2) (a; — 3) ~ (a; — 1) (a; — 2) (a;— 8) 
 1 
 
 (1 — x) (a; — 2) (a; — 3) 
 
 6. Simplify 777 r + 77 777 r + 
 
 (a—b) (a—c) (b—c) (b—a) {p—a) (e—b) 
 
 Note 2. Where the differences of three letters are involved, as in this 
 example, the work is made the shortest and easiest possible by choosing 
 out of the different methods of symmetric arrangement for the denomina- 
 tors what is known as the cyclic order ; that is, an an-angement where J 
 follows a, c follows b, and a follows c ; thus, abc,bca, cab, also a — J, 
 b — c, c — a. 
 
122 ALGEBRA. 
 
 -' + .. ' „. + 
 
 (a — J) (a — c) ^ (6 — c) (6 — a) ' (c — a) (c — 6) 
 1 1 1 
 
 (a — i) (c — a) (6 — c) (a — i) (c — a) (6 — c) 
 6 — c — (c — a) — (a — i) 
 
 ft — c — c -^ a — a -\- b 
 ' (a — 6) (6 — c) (c — a) 
 
 2 (6 - c) 2 
 
 (a — 6) (i - c) (c — a) (a - 6) (c — a) 
 
 Simplify : 
 
 1 1 2 a; 4x« 
 
 "• ; 5—; — 7, 7—, — 7 ■ Ans. 
 
 a—x a+x a' + x' a* + a;* 
 
 x — 1 x + 1 x — 2 x + 2 
 
 3 2 5« . 1 
 
 Ans. 
 
 9. 
 10. 
 
 1 + a l — a a?~l 1 — a^ 
 
 c d 
 
 {a — b)(c + d) {b -a){d + c) 
 a" V~2ab 
 
 (a — b) {r. — a) {b — a) {b — c) 
 
 , 3a^h-a'c + b''c-ab''-2abc 
 
 Ans. — 
 
 (a — b) {b — c) (c — a) 
 
 11. .-^-^4 1 I 4.x 
 
 t. 9 ^ 1 ~ „ _ 1 -I T^ 
 
 a;-2 x-l^x a; + l"^x + 2' 
 
 gj^_2a 2(rt"-4ax) 3a 
 
 a: + a. a^ — x^ a; — a ' """' x~+~a^ 
 
 13. ^~^« _|. 2(rt"-4ax) 3a^ x + 3a 
 
 14. 
 
 + 
 
 1 
 
 (x-3)(a;-4) (x-2)(x-4) "^ (x-2)(x-3) 
 
FRACTIONS. 123 
 
 IK 1 _L 
 
 ■ (a - i) (a — c) (i - c) (A - a) "^ (c - a) (t - h) ' 
 6 33; , 4-13x , l-6a:2 
 
 16. :; fi q S h -; — r, ^ • Ans. 
 
 l + 2x 1 — 2a: ' 4a!--l " l-4z-'' 
 
 y x^ y + a; ry' 
 
 ,6 ,fi 
 
 17. — = ^ — + 
 
 a? + y° a;' — y^ x" — y' 
 
 24 a; 3 + 2 a; 3-2a ; 
 
 9^12 a; + 4a;2 3-2a;'^2a;+3" 
 
 2 a 5 4 (3 a + 2) 12a'-4:a + 7 
 
 2a-3 6a+9 3(4a^-9;' 3 (4 a^ - 9) " 
 
 20 5a; 1 1 
 
 6 (a;-^ - 1) 2 (a; - 1) ^ 3 (a; + 1) 
 
 21 3 ^1 a-x 
 
 8{a-x)^ 8(a + x) 4 (a^ + x^) ' 
 
 a; + a x^ — a'' a — x x'' -\- a' 
 
 1 1 1 . 18 
 
 6a -18 6a + 18 a'' + 9 ^ «^ + 81 
 
 36 o* 
 
 Ans. 
 
 a' - 6561 
 
 8 -8a; 8 + 8x^4 + 4a;^ 2 + 2a;* 
 
 ax'' + b 2 (/)a! + ax'^) _ ax^ — b 
 2V^^'^ 1-4 a;^ 2a; + l" 
 
 2b — a _ 3x{a — b) b — 2a 
 X — b b^ — x'' X + b 
 
 a + c , b + c . X + c 
 
 27. 7 7-^ c + 77 TT R' -'^r- 
 
 (a — b) (x — a) (b—a)(x — b) ' {x—a)(x—b) 
 
124 
 28. 
 29. 
 30. 
 31. 
 
 32. 
 33. 
 34. 
 35. 
 
 36. 
 37. 
 38. 
 39. 
 
 ALGKBRA. 
 b-e 
 
 a — b) (x — a) (b — a) (i — x) 
 
 2a + y a + b + y x + y — a 
 
 — a) (a — b) {x — b) (b — a) (as — a) (x — b) 
 111 
 
 a2_62) (c2+i'> {b^-a,^){c'+aF) {c^+a^){c^+b^) 
 
 £8-3 
 
 x-1 2 (x - 2) 
 
 x-2)(x-3) ' (3-a;)(a;-l) (x-l)(2-a;) 
 
 2 
 
 Ans. 
 
 + 
 
 {x — l)(x — 2) (x — 3) 
 
 + 
 
 « — i) (a — c) (6 — a) (6 — c) (c — a) (c — J) 
 
 + 
 
 + 
 
 — i) (a — c) ^ (6 — c) {b — a){e — a) (c — i) 
 
 + 
 
 c a 
 
 + 
 
 aS 
 
 a — b){a, — c) (6 — c) (J — a) (c — a) (c — 6) 
 a , b c 
 
 + 77 
 
 a — b) (a — c) {b — c) (b — a) {a — c) (c — b) 
 Ans. , ,. ,. ^ ^ , : = 0. 
 
 1 
 
 (a — S) (6 — c) (c — a) 
 1 1. 
 
 + 
 
 aXa-b)(n-c) b\c-b){b~a) c\o-a){c-b) 
 
 + 
 
 + 
 
 o» + h* 
 
 a {a" — ¥)' b {a^ + b^) ' ab {¥ - a') ¥ - w 
 X +1 x — 1 2 
 
 x'^ + X + 1 X^~X + 1 X* + x' +l' 
 
 + 
 
 x^-5xt/ + 6f x''—4:X7/+3f "^ £8^— 3xy + 2/' 
 
 
 
 Ans. 
 
 (x-y) (x-2y) {x-3y) 
 
 0. 
 
FRACTIONS. 125 
 
 148. MULTIPLICATION AND DIVISION OF FRACTIONS. 
 
 Multiplication and division of fractions are simply par- 
 ticular cases of the laws of association and commutation 
 for multiplication and division, as will appear from the 
 following examples. 
 
 1. Multiply I ^y I • 
 
 |x^=(a-^ft) x(c-f-rf) 
 
 = a -e- 6 X c -f- d (§79) 
 
 ^aXc-i-b-v-d 
 = (a c) -H (6 d) 
 ac 
 
 From this it follows that 
 
 a c e ac e ace 
 b^ d'^f^ Fd^f^ bdf 
 and so for any number of fractions. 
 
 •^^ n a a a a' 
 b^bb^¥ 
 <a\" a" 
 
 faV a 
 Further, (h)^b^ 
 
 and in general M = -^ 
 
 Hence, for the multiplication of fractions, we have the 
 
 following 
 
 Bule. 
 
 Multiply the numerators together for a new numerator, 
 and the denmninators for a new denominator. 
 
 €t C 
 
 , 2. Divide -; hy -,• 
 b d 
 
 J -H ^ = (« ^ *) ^ (c -^ rf) 
 
 = a -f- 6 -^ c X rf (§79) 
 
 = aXd-i-b-i-c 
 = (ad)-i- (b c) 
 
 ad 
 ^b~c 
 
126 ALGEBRA. 
 
 Hence, for the division of fractions, we have the follov^ing 
 
 Rule. 
 
 Invert the divisor, and then proceed as in multiplication 
 of a fraction by a fraction. 
 
 Note 1. As a mixed number can be reduced to a fractional form, 
 g 144, and an integral expression can be expressed as a fraction by writitig 
 under it 1 as its denominator, this rule covers all possible cases in multi- 
 plication and division of fractions. 
 
 Note 2. Always caned, if passible. 
 
 3. Multiply together ^^, y^^ , and -^^,. 
 
 Sa-^b 6acd^ 14crf° 
 4:c^d ^ 7 62c2 ^ 9aH^ 
 3 • 6 • 14 • flSfic^rfs ad* 
 
 4 -7 • 9 • a^b*c*d b^c'^ 
 
 J Ti- 1 ii 1 n ^^ — y^ x(x + 2y) 
 4. Find the value of -^ — K- X -^ — - — f^ X 
 
 2 a:?/' xy + 1/^ x' — xy 
 
 a^ - y' ^ xjx + 'iy) ^ 3?/' 
 2x y xy -\- y^ x^ — xy 
 
 ^ (^ + .V) (^ - y) ^^ a:(r + 2,v) ^ _ 3 y' 
 _Z(x+2y) 
 
 5. Simplify ^— X 
 
 9x^-Aa^ • 3ax + 2a? 
 
 6x2 — a 1 — 2 n" j. _ „ 2a; + (i 
 
 X • 
 
 ax — a^ 9a;2_4(i2 ■ 3 a i + g a^ 
 
 _ 6 a:' — ax — 2 a^ x — a 3ox + 2a2 
 
 ~ ax — a' '**'9x2 — 4a2^ 2x + a 
 
 ^ (3x — 2o)(2x + a) x — a a(3x + 2a) 
 
 a{x — a) (8x + 2a)(8x — 2a) ^ 2x+a 
 
 = 1 
 
FRACTIONS. 127 
 
 (• + ^Jx;(-?)-('+9' ■ 
 
 a + X a^ — x^ (r -f af 
 
 == X -i — ■ — 5 — 
 
 o — X- ax x^ 
 
 a -\- X (a-\-x')(a — r) x^ ' x 
 ^— X ^^ — - — — ^ X 
 
 a — X ax {x-\- (if a 
 
 Simplify the following : ^ 
 
 6bd* ^ 28 a* ■ 
 
 „ 12xyz ISa^x 10?/' 
 ° X ^„ ' X •' 
 
 11. 
 
 14. 
 15. 
 
 25 a« 16/z 9aa;2 
 
 l^^_3xV_ 2^_ 
 
 14a^i^ ■ 2an2 7a&-^' 
 
 /„ 4a;2 12a;« 8 a;» 
 10. ,^„ X -— .^ • 
 
 3/ 7^'' ■ 35yz^ 
 
 2x\ ( 7p^ \ (_3p£ 
 3p) \nqV \ Zbxy. 
 
 a? — y xy 
 
 12. ^^-^Xy:r^,.- 
 
 ^„x^ — 4a^ 2a ao 
 
 13. —-—J X s- • Ans. 2. 
 
 14a:2-7a; 2a;-^l 
 
 12a;»+24a;^ " x"" -\- '1 x 
 
 a" - 121 . a + 11 
 a2 _ 4 • a + 2 ■ 
 
 25 a" r- &' a: (3 g + 2) 
 
 9a^!B'?-4a;^ 5a + * 
 
 a;^ — a' _ {x — aY 
 
 x^~+~a^ ~ x^ — a^ ' a;^ — a; a + a" 
 
 a;^ — a' (x — a)'' . a:' + a: a + a' 
 
 17. - -; — 5 -• s— — 5 • -^iis. • ■ 
 
128 ALGEBRA. 
 
 ^ - ab + b^ a^ c — b^ c 
 
 a^— ab a^x + b^x 
 
 x^-1 a;2 _ 12 a: + 35 
 
 a;^ — 3a;-10 ■ x^ -\-Zx + i 
 
 2 a;2 + 13 a; + 15 . 2 a;^ + 11 a; + 5 
 
 4 a:^ — 9 ' 4 sb^ 
 
 {a -by 
 
 af^-b^ (a-5)° («'' + 5')(a + i)' 
 
 19. 
 
 20. 
 
 21. 
 
 22. 
 
 23. 
 
 24. ,-" ' -;, X "„ . ", X 7 . "",, • Ans. 1. 
 
 25. 
 
 a* 
 
 + a''J^+6^ ■ 
 
 a« 
 
 _ 6« • 
 
 a» 
 
 -8 4= 
 
 
 a ■ 
 
 4-25 
 
 a^ 
 
 -4 5'^ 
 
 ''^ «^ + 2 
 
 a6 + 44''' 
 
 x^ 
 
 -6a:2 
 
 + 36 a; 
 
 
 a;" + 216 x 
 
 
 a;^- 
 
 49 
 
 
 a;2_a;_42 
 
 a 
 
 (IT 
 
 + a; 
 
 
 x^ 
 
 a^-a:* 
 
 ^ (« + xy • 
 
 
 5m^n 
 
 -5n' 
 
 
 w'' — m '« + n^ 
 
 w? n -\- 2 mn^ ■\- It? ' m -\- n 
 
 - X a fx a\ 
 
 ■^7. — ; — X I !• Ans. X — a. 
 
 x+ a \a XI 
 
 29. t+^^(_^ — _y_\ 
 
 3? + y^ \x—y x + y) 
 30. 7x{a-b) -4- ^^ 
 
 w'-2ab+b^ 
 
 ^8 „,3 
 
 31 
 
 ^'-f ^ a:^-2a!^/ + y* . x^ - y'^ 
 
 2y(7? + y^) x* + 7?y + a;V ' ^xy 
 
 x (a;2 + y^ 
 
FRACTIONS. 129 
 
 32. 
 
 fa + b a — b\ _ (a + b a — b\ 
 \a — b a + b/ ' \a — b a + b/ 
 
 a:" + a; - 2 x^ + Sx + i . ( x'' + 3x + 2 x+3 \ 
 
 Ans. X. 
 
 12 a; (a:» + 1) 3 a:^ + 6 a:' + 3 . 24 (a;" + xf 
 a^ + x^ ^ (x^-x + iy ' x* + l 
 
 3(x^+l)(x^ + l) 
 ®- 2a!»(a;'' + l) 
 
 36. ^' + ^; + 3.M. + 5) ^ /_ 5JN ^ / 5y. 
 2 a (a — by \ a I \ a) 
 
 3^_ (g + by - e^ ^ a ^ {a - by - c^ 
 
 a^ + ab — ac {a + ey ~ b^ ab — b^ — be 
 
 a^ + 2ab + ¥-c^ c^-b^ + 0^-2 ac 
 ^*'- ftS _ 62 _ c2 _ 2 Ac ^ c» - a" + 62 _ 2 J c ■ 
 
 39 o^+^b+^l+ab'^ ¥^ ± 
 
 *^^- an + aH''+ al^+b'^a' ■ ■ a'. 
 
 ,^ 4a'' + ¥-c^ + 4:ab 2a + b + c 
 ia^-p-c^-2bG ■ 2a-b — c 
 
 b_ 
 a^x' 
 
 41 
 42. 
 
 \be ac ab aJ \ a + b + c/ 
 , x*-a* ^ ( x* - 2 a'a:" + a* x' - a* \ 
 
 . x^ — ax + a" 
 
 r- — ax -\- a- 
 (x — a) (a;' — a') ' 
 
130 ALGEBRA. 
 
 149. COMPLEX FRACTIONS. 
 
 The division of fractions is sometimes expressed by 
 
 a 
 
 writing tlie divisor under the dividend; thus, — . Such 
 
 y 
 an expression is called a Complex Fraction. A complex 
 
 fraction can be reduced to a simple one by performing the 
 division indicated, or by multiplying its numerator and 
 denominator by the least common multiple of the denomi- 
 nators of the fractional parts. 
 
 X 
 
 1. Reduce ^ to a simple fraction. 
 
 or. 
 
 
 
 y 
 
 9 
 
 X 
 
 ~y 
 
 9 X 
 • l-y 
 
 '5_5x 
 
 
 
 5 
 
 
 
 
 
 
 X 
 
 y 
 
 9 
 5 
 
 X 
 
 y 
 
 '9 
 5 
 
 X 5y 
 X5i/ 
 
 5x 
 
 Oy 
 
 U( 
 
 a 
 
 =eL 
 a 
 
 c 
 
 toi 
 
 a simple 
 
 fraction. 
 
 
 b 
 
 d 
 
 
 
 
 a 
 T. 
 
 + ~ 
 
 
 
 
 
 b d ^ (a , c\ _^ (a _ c\ 
 a_c^ [b '^ d) • \b d) 
 
 g rf -|- & e . ad — he 
 
 ~ Td ' Vd 
 _ ad + lc hd _ ad+ he 
 
 bd ad — he~ad — be 
 
FliACTIONS. 131 
 
 a , c fa c\ 
 a c (a c\ 
 
 ad + 6c 
 
 
 
 
 ad — b( 
 
 s 
 
 3. 
 
 Eeduce 
 
 as — 
 
 a" 
 — ^ to its simplest 
 
 form. 
 
 
 *• + 
 
 a2 
 a; 
 
 = (-?)*('- 
 
 -S) 
 
 
 X — 
 
 a* 
 
 J.8 
 
 
 
 
 _ a;2 + a= . x* — a* 
 
 I 
 
 
 X • x» 
 
 
 
 
 = 
 
 X2 + a2 3.8 
 
 a;" 
 
 
 a: ^^ a;* - a' 
 
 ' x-' — a^ 
 
 
 x + 
 
 X — 
 
 a-' 
 
 X 
 
 = 
 
 ■ a;2 (a;2 + a^) 
 
 x*-{- a2a;2 
 a;* — a* 
 
 x^ 
 
 
 (j:2 + a2) (a;-^ — a' 
 
 !) a;2 _ a2 
 
 A 
 
 Simplify 
 
 X 
 
 1 
 
 
 
 f 1 1 
 
 1 
 
 
 • X — 1 + - 
 X 
 
 1 
 
 
 + 1+ \ 
 
 In simplifying this form of a complex fraction, called a continued 
 fraction., we begin at the bottom. The fractions to bp simplified 
 alternate from a mijced number to a complex iraction, with an inte- 
 gral _expression foj ita JUJDieK(|or and a sinjpje' fraction for its denom- 
 inator. The parte beloy the oblique lines, which are in many ways 
 helpful to the beginner, show this. 
 
132 
 
 ALGEBRA. 
 
 
 
 x^ 
 
 + 2^-1 
 
 Simplify 
 
 the 
 
 following : 
 
 
 1 + 
 
 c 
 
 
 5. 
 
 h 
 
 X 
 
 
 
 
 1 
 
 
 
 X 
 
 
 
 
 1 
 
 1 
 
 
 
 - + 
 
 
 
 6 
 
 X 
 
 2/ 
 
 
 
 1 
 
 1 
 
 
 
 X 
 
 y 
 
 
 
 H 
 
 
 
 
 1.^ ■ 
 
 -?, 
 
 
 
 6 
 
 
 
 V. 
 
 2 
 
 3 
 
 
 
 X — 
 
 _ 
 
 
 8. 
 
 
 V 
 
 X 
 
 2 
 
 
 2h 
 
 
 4 
 
 -r + 
 
 x-1 
 
 9 
 
 X — 
 
 i 
 
 X 
 
 
 
 1 
 
 1 
 
 10. 
 
 X* + x^ — 1 
 
 a a' 
 
 11. 
 
 l--„ 
 
 X + 5+ - 
 
 X 
 
 a; + 4 
 
 12. 
 
 13. 
 
 X x^ x" 
 
 a + X- 
 
 a^ — ax + x'^ 
 
 -1 
 
 a + X — 
 
 a^—ax + x^ 
 
FRACTIONS. 133 
 
 a + X a — X 21 
 
 ^. « — X a + X -, , 
 
 a + X a — X f , ^ 
 
 T i — '' 1 _ 
 
 15. 
 
 a — x a 
 a X 
 
 + x 
 
 «•' ax 
 
 1 ■ 
 
 Ans. a -\- X. 
 
 22. 
 
 ab ac he 
 a'-jh- cf ' 
 a 
 
 Ans. 
 
 bc(b — a — c) 
 
 16. 1 + - 23 0'i«'-1>)-b(a + h) 
 
 2x' ' ■ „, h 
 
 l + a; + 
 
 1- 
 
 X 
 
 a -\- b a — b 
 
 ab \a bl 
 
 2 3a; + 2 24. ^^^'5 + 
 
 17. 1 ■ {a+b) 
 
 25. 
 
 18, 
 
 a + b a — b 
 a — b a + b 
 
 a' + P" 
 (a + bf 
 
 a-\-h 
 
 Ans. ^(^). 
 a — 
 
 19. 
 
 20. 
 
 1 1 1 
 
 -+T + - 
 
 a b c 
 aba 
 
 1 + - + - 
 b c a 
 
 9 a;" - 64 
 
 26. 
 
 27. 
 
 « + J + 
 
 1 
 
 
 
 1 
 
 1+ b 
 
 
 a-b + - 
 c 
 
 Ans 
 
 d'-b^ + l 
 '• a" _ §2 + 2 ■ 
 
 a;= + y 
 
 a;" — 2/= 
 
 
 x^ - ?/ 
 
 x^ + y^ 
 
 
 « + 2/ 
 
 x — y 
 
 
 x—y 
 
 X + y 
 
 
 1 + x 
 
 l + x'' 
 
 
 l + a;" 
 
 1 + a;» . 
 
 
 1 + x^ 
 
 l+a!» 
 
 
 -1- 
 
 1 + x' 1 + x* 
 
 1 + x* 
 
 Ans. 
 
 4 + a; ■ a; (1 + a;^) 
 
134 ALGEBRA. 
 
 150. MISCELLAimonS EXAMPLES. 
 
 Simplify the following fractions : 
 
 1. if-1 L\ ^ 
 
 b\a — b a-+2b/ a^+ab — l 
 
 26" 
 Aus. 
 
 a'+ ab — 2b'' 
 a {a?-V)x a {c? - W) x" 
 ' b V" '^ ¥(})-\-ax) ' 
 
 „ ^a{a? — 03°). _ \a? — ax a^ + 2 ax + x'l - , ; 
 
 ■ 3 S (c^ - x^) "^ [bc + bx ^ c'-2cx + x^y 
 
 . j X* — a* x^ + ax\ x^ — a^x^ /x a\ 
 
 Ix^ — 2ax + a^' x — a ) x" + a^ ' \q, xj ' 
 
 a' — a;' 
 x'^ + ax + a^ ■ a^ — x^ 
 
 Ans. 
 
 x+3 11 1 
 
 (a — x) (a + xy 
 
 2xV+9a; + 9^2 2a;-3 9 
 
 4a; 
 
 (1 4- aa!)2 - (a + a;)^ • 2 Vl -a; "^ 1+ x/' 
 
 g 2a;° + 5a;^y + a;.y''-3y° 
 
 3 a;* + 3 x'y - ^x'^if -xy» + y*' 
 
 9 1 ("t+J^ _ 1 « + «' f « \ ' 
 ■ 2\a^-xy 2' a-x \a + x) ' 
 
 . a(a^ + x^ 
 
 Ans.' ^^^- — - ■ 
 
 {x — a) (x -\- ay 
 
FRACTIONS. 135 
 
 10. ^(^. l_U_flziZ 
 
 2\x — y X + yj x^y ■\- x 
 
 y' x + y 
 
 11. -J-- r^f ^ I Mx "''-^^ 1- 
 
 a; + y L2\a! + 2/ x — y) x'y + xy^l 
 
 12 I? — -^— -4- -J_l -^ f?-i_f _ ?JZ^\ 
 ■ (a; a+x a — x) ' \a — x a + xj' 
 
 a — b b — c 
 
 1 + ab 1 + be a — c 
 
 ■*■"• 7 7\~7I r~ ■ Ans. 3 
 
 {a — b){b — c) l + «c 
 
 (l + a«.) (1 + ic) 
 
 , a — 6 a — b 
 
 o + :; — : 7 a — ; 
 
 J l + gj l-a6 f /g V\ 
 
 / 1 _ ('^ - ^) ^ 1 _ «(«-.&) f ■ U ~ J 
 
 1 + ai 1 — ab 
 
 15. — <— r— X 
 
 r_i »:' + 
 
 2/ a; 
 
 16- — 7 — ; r — T^ X 
 
 X 
 
 abc 
 
 
 a' + J» + c' - 3 a 5 c 
 
 
 a2 + 6^ + c^"— a6 — Jc — 
 
 ca 
 
 a^ — 6^+ c^ — 2ca 
 
 
 a Jc 
 
 
 (ft + 6 — c) (J c + c a + ft J) 
 
 1+ " 
 
 1 + a; . (a: - 1)' — a;' 
 
 ^^- , 1 ■ x^-\-x^\ ' 
 X + 1 I - 
 1 + a; 
 
3a; + 2 
 
 136 ALGEBRA. 
 
 18. 3 J Ans, 
 
 5x 
 1 1 
 
 19. 
 
 1 1 
 
 X q- a; + 
 
 1 *"^ 1 
 a; + - a: 
 
 X X 
 
 f a + b y ( a — b y 
 
 \a _ J \a + b) _ ^^^ iab(a' + b^ 
 
 \a — b) "^ W + 5/ 
 2^_ ^(2x + H)-i(2-3a:) ^^^_ 6a, 
 
 l|-i(2a; + 4i) l-2a 
 
 22 f ^^ I ^ I ^M 
 V3a; + 4^3a;-4^16-9a;V 
 
 «'- (5 - c)^ &^ - (c - a)^ c^- (a- &)' 
 ''■ (a + c)^ -6^ (a + by -c'^ {b + cf-a^' 
 
 24 /'1±^ J. 4a! _ 8a; _ l-x \ 
 \1 - a; ■•■ 1 + a;= a;* - 1 x+ll 
 
 ■ U - a;2 + 1 + a;* a;^ + 1/" 
 
 Ans. 2J1±^, 
 
 X 
 
 25. „ 1 ~ „„ , -I + q -; + 
 
 a;" — 1 x" + 1 l — x" x" + l' 
 
FRACTIONS. 137 
 
 26. (2-3_%«-|!p/--^)^/i- 
 \ m, VI'' + 2 mnj m 
 
 An' 
 m — Zn — ■ 
 
 27. 2 + L^ . 
 
 ^ 1 Ans. o ^1 • 
 
 9j -*- 2a; — 1 
 
 a; — 1 
 
 28. Find the value of — -, when x = —^ — ^7, and 
 
 X — y + 1 ao + 1 
 
 y = — , I ., • Ans. a. 
 
 ao + 1 
 
 !-»' l + a' 
 
 29. Find the value of ^ — , — ^ „ , when o = - . 
 
 1 — a° 1 + a^ 4 
 
 1 — a 1 + a 
 
 . 25 
 
 Ans. — . 
 
 12 
 
 30. If o = J, b = 0, X = — I, y = — 1, find the value of 
 xy — ab a — x 
 
 (f a 
 a? X 
 
 a^ 
 
 + X' 
 
 
 
 
 Simplify : 
 
 
 
 
 
 
 m — 
 31. 
 
 n ■ 
 
 -2n 
 
 (m 
 
 m 
 
 ± 
 
 n 
 
 m — n _ m -\- n 
 m 
 
 32 1 I 1 I ^ . 
 
 ''^* a(a-b)(a-o) ^ b{b-c)(b-a) ^ c(c-a) (e-b) 
 
 Ans. —r- • 
 abc 
 
133 ALGEBRA. 
 
 33. ^^+A__ + __£+^__ + '^+^ 
 
 a{a — b){a — o) b{b — c) (6 — a) c (c — ct) (c — ii) 
 
 (I — h — c 6 — c — « c — a — b 
 
 34. 7 TT-7 ^ + Ti W7 ^ + 
 
 (a — b) (a— c) (b — c) {h — a) (c — a) {c — b) 
 {a — b) (b — c) (c — a) 
 
 c + a - a + b b + 
 
 i55- 7 7T~7 ^ + 77 — ^ n~77 \ + 
 
 36. 
 
 {a — b) (a — b) (6 — c) (b — a) (c — a) (c— b) 
 
 3 5 7 
 
 (a!-2/)(y-«) («-«)(y — a;) (a; — 2) (« — y)' 
 
 12?/- 2s- 10a; 
 
 Ans. 
 
 {x — y) {» — X) (y — s) 
 
 bc(a + d) ca (b + d) ah {c + d) 
 
 {a — b){a — e) "^ (& - c) (i - a) ^ (c _ «) (c - b) 
 
 Qg 2^ ? ^ J^! ^ 1 
 
 y X \y XI 
 b 
 
 a 
 
 1-7 
 
 39. ((j« - 5') ~ ■ 
 
 b 
 
 a + 
 
 ' + I 
 
 a;2+/(a5 — 2/ x' — y^) ' \x — y «? — y^) ' 
 
 Ans. 1. 
 
^1- W^-^ + xvUv + W V- 
 
 43. 
 
 FRACTIONS. 139 
 
 X 
 
 a. - 15 
 
 Ans. 
 
 44. 
 
 2 (3 a - 2) 
 
 / a^ a'' 2 ax \ [x -^ a X — a\ 
 
 \{x — of ~ (x + af "^ a;' — aV ~ U — a a; + J 
 
 a X 
 Ans. 
 
 ofi- 
 
 1 1-i 
 
 X X 
 
 x + ' ' 
 
 X X 
 
 ^ 1-^ 
 
 X a; 
 
 1 + i ^ 
 
 a; a; 
 
 45. 4-^ ^X -^-^A- An§. 1. 
 
 1 X + a 1 a; — a 
 ,^ i"~^T^ , i ~ a;^ + a^ 
 
 46. 3 ; 1- ^ 
 
 1 a + X 1 a — X 
 
 a a^ + x^ a a^ + x^ 
 
 / Sx + x" \2_ 9^ _ 33 - a;' 
 
 Vl + 3a;V _ x^ 3x^+1 ^ a; (a; + 1) 
 
 47. -s— 5 5 '- o ' o /-2 I ox ■ Ans. 
 
 3a;''-l . ■ 3 2 (x^ + 3) °- x'+ix + l 
 
 ?^ 3 a: "^ a;-'' (a;» - xf 
 
140 ALGEBRA. 
 
 48. ~--l--3lib-2{la-b-3(ia-i 6)}]. 
 
 Ans. ^a — 2b. 
 a^ + 24 fi c — 16 i^ — 9 c^ . a + ib — So 
 
 49. 
 
 16b' — 9c^ — 6ac — a^ ' a + ib + 3o 
 
 1 x-a-x) 10x^-^-3 
 
 _1_ +a; + 3-(l-3a!)'^4a;'' + 4a; + l' 
 ■^l + a; 
 
 16 a:" + 23 jg - 12 
 '^"®' 2 (a; + 2) (2 a; + 1) 
 
 8 [ G(a + c) j l4(5 + e) . g^-c^M 
 
 6-c ■ b {b^-c^)^\9(a-b) ■ a^-b^ii' 
 
 52. 
 
 4 y (a;" — a; y + 2/^) 2 a;' — ai^'y — Sa;?/'^ 4^^ 
 a;' + 2/' x' + 2/* a; + y 
 
 2a;^ — 3x1/ 
 
 + 
 
 x^ — xy ■\- y^ 
 
 n' + &' _ a + 5 _ 1 / «-5 1_\ 
 
 • a* - S* a^ - 6^ 2 l^a'^ + 6'' a - ftj 
 
 54. When a; = 1, ?/ = — ^, z = 0, fiud the value of 
 ^-[y-^- {2 a^-2y-H3^ -?/)}] 
 
 ^ i 
 
 « — 
 
 a; 
 
SIMPLE EQUATIONS. 141 
 
 CHAPTER XL 
 SIMPLE EQUATIONS. 
 
 15L An Equation is a statement of the equality of two 
 expressions. 
 
 The parts of an equation to the left and right of the sign 
 of equality are called members, or sides, and are distin- 
 guished as the first member and second member, or the left 
 side and right side. 
 
 152. An Identical Equation, or simply an Identity, is a 
 statement which is true for all values of the letters in- 
 volved ; as, a" + b'' = (a + b) (a^ - ab + b^). 
 
 153. An Equation of Condition is a statement which is 
 only true for particular values of the letters involved ; as, 
 £c — 3 = 5 is true only when x = 8. 
 
 154. A Literal Equation is one in which some or all of 
 the known numbers are represented by letters ; as, 
 
 ax — 2b = 5bx — 3a. 
 
 155. The Degree of an equation, when in its simplest 
 form, is shown by the sum of the indices of the unknown 
 factors in that term in which this sum is the greatest. 
 
 156. A Simple Equation is an equation of the first degree; 
 as, 5 X — a a; = 12. 
 
 157. A ftuadratic Equation is an equation of the second 
 degree ; as, a;" + 2 a: = 8, or Zx^ — '2xy = 1. 
 
 158. A Cubic Equation is an equation of the third degree; 
 as, x' + 2 x"^ y = 5. 
 
142 ALGEBRA. 
 
 159. The letter whose value it is required to find is 
 called the unknown number. The process of finding its 
 value is called reducing the equation. The value so found 
 is called the root of the equation. 
 
 160. If, on the substitution of the root for its unknown 
 symbol, the equation becomes an Identity, the root is said 
 to be verified. 
 
 161. In the reduction of an equation the processes in- 
 volved depend upon the Axioms given in § 36. These 
 processes can be best understood by considering an equa- 
 tion as a pair of scales which balance as long as an equal 
 weight remains in both sides ; whenever on one side any 
 additional weight is put in or taken out, an equal weight 
 must be put in or taken out on the other side, in order that 
 the equilibrium may remain. So, in an equation, whatever 
 is done to one side must he done to the other, in order that 
 the equality may remain. That is, 
 
 1. If anything is added to one member, an equal amount 
 must be added to the other. 
 
 2. If anything is subtracted from one member, an equal 
 amount must be subtracted from the other. 
 
 3. If one member is multiplied by any number, the other 
 member must be multiplied by an equal number. 
 
 4. If one member is divided by any number, the other 
 member must be divided by an equal number. 
 
 6. If one member is involved or evolved, the other must 
 be involved or evolved to the same degree. 
 
 TRANSPOSITION. 
 
 162. Transposition is the changing of terms from one 
 member of an equation to the other, without destroying 
 the equality. . 
 
SIMPLE EQUATIONS. 143 
 
 This is effected through the use of Axioms 1 and 2, as 
 will appear from the following processes : 
 
 Let X + a ^b 
 Now a = a 
 
 (1-) 
 
 (2.) 
 
 By subtraction x = b — a (Ax. 2.) 
 
 Let X — a — 
 Now a = a 
 
 By addition as = c rl- a (Ax. 1.) 
 
 It appears from these examples that any term, as a, 
 which disappears from one member of an equation, re- 
 appears in the other with the opposite sign. Hence, any 
 tertn may be transposed from one member of an equation to 
 the other, provided its sign is changed. 
 
 It follows from this that the signs of all the terms of an 
 equation may be changed without destroying the equality; 
 for this is equivalent to transposing all the terms, and then 
 making the right and left hand members change places. 
 For example : 
 
 Let 5x— 1 = 7 X — 15 
 
 Transposing, — 7a; + 16 = — 5a;-t- 7 
 
 or, — 5a;+7 = — 7a;+15 
 
 The same result could be obtained by either multiplying 
 or dividing the equation throughout by —1. (Ax. 3 or 4.) 
 
 REDUCTION OF SIMPLE EQUATIONS CONTAINING 
 BUT ONE UNKNOWN NUMBER. 
 
 163. When the Equations are Integral. 
 
 1. Reduce 8x + 7 = 4a; + 39. 
 
 Transposing, 8a;^4a; = 39 — 7 
 
 Combining like term?, 4ar = 32 
 
 Dividing both sides by 4, .r = 8 (Ax. 4.) 
 
144: ALGEBRA. 
 
 Hence, for the reduction of simple equations containing 
 but one unknown number, we have the following 
 
 Transpose the known terms to one ^member and the unknown 
 to the other. Combine like terms, and divide both members 
 by the coefficient of the unknovm number. 
 
 2. Eeduce 5 (8 - a;) - 3 (60 - 6 a;) = 2 (4 - a;) - 40. 
 
 Removing brackets, 
 
 40 — 5 a; -180 + 15 a; = 8 — 2. T- 40 
 Transposing, — bx -[■ I5x + 'ix = —40 + 180 + 8-40 
 Uniting terms, 12 a; = 108 
 
 Dividing by 12, a; = g 
 
 3. Reduce 5a;-(4a;-7)(3a;-5) = 6-3(4a;-9)(a;-l). 
 Removing bracket^!, 
 
 5 a; — 12 a;2 + 41 a; - 35 = 6 - 12a;2 + 39 a; - 27 
 o'"' 5 a; + 41 a; — 35 = 6 + 89 a; — 27 
 
 Transposing, 5 .-c + 41 a; — 39 a; = 35 + 6 — 27 
 
 Uniting, 7 a: = 14 
 
 a; = 2 
 
 4. Eeduce 7 x ~ &{x ~ {! ~ Q {x ~ 3)}] = 3a; + 1. 
 Removing brackets, 
 
 7a; — 5a; + 35 — 30a; + 90 = 3a;+l 
 Transposing, 7a;-5a; — 30 a; — 3a; = — 35-90 + 1 
 
 ^''"'"'g. - 31 a; = - 124 
 
 a; = 4 
 
 The student should verify his results, especially when the answers 
 
 ri" •^"^'"' "' f^P^'''"^'^ '" § 160. For example, if we substitute 
 8 tor X in Example 1, we have 
 
 .,, . 8X8+7 = 4X8 + 39 
 
 or, 
 
 7i = 71, an identity. 
 
SIMPLE EQUATIONS. 145 
 
 Again, if we substitute 2 for x in Example 3, we have 
 
 10 — (8 — 7) (6 — 5) = 6 — 3 (8 — 9) (2 — 1) 
 that is, 10 — 1 = 6 — 3 (— 1) 
 
 or, 9 = 9, an identity. 
 
 Beduce the following equations : 
 
 5. 12 X + 7 = 9 a; + 13. 
 
 6. 17 a; -50 = 2 a; -5. 
 
 7. 12 a; + 4 + 15 a; = 8 + 11 a; + 28. 
 
 8. 21 - 7 a; = 2 a; + 57. 
 
 9. 29a;-ll = 19-lla;. 
 
 10. 15 -67 -18 a; + 12 -21 = 2. 
 
 11. 42a! - lla; + 100 + 13a; - 121 - 23 = 0. 
 
 12. 8 (a; - 3) - (6 - 2 a;) = 2 (a: + 2) - 5 (5 - x). 
 
 13. 157 - 21 (a; + 3) = 163 - 15 (2 a; - 5). 
 
 14. 179 - 18 (a; - 10) = 158 - 3 (x - 17). Ans. 10. 
 
 15. 2 (a; - 1) - 3 (a; - 2) + 4 (a; - 3) + 2 = 0. 
 
 16. 5a; + 6(x + l)-7(a; + 2)-8(a; + 3) = 0. 
 
 17. 3(2a;-l)-4(6a;-5)=12(4a;-5)-22. 
 
 18. a; - [3 + {a: - (3 + a;)}] = 5. Ans. 5. 
 
 19. 25a;-19-[3- {4a;-5}] = 3a;-(6a!-5). 
 
 20. (a; + l)2-(a:^-l) =a;(2a; + l)-2(a; + 2)(a; + l) + 20. 
 
 21. 10(2a;-9)-7(4a;-19) + 5 = 4a!-3(2a!-3). 
 
 22. 20(2-a:) + 3(a!-7)-2[a; + 9-3{9-4(2-7)}]=22. 
 
 23. 3(5-6a!)-5[a!-5{l-3(a;-6)}]=23. Ans. 4. 
 
 24. 3 (a: - 1)" - 3 (a;'' - 1) = a; - 15. 
 
 25. 2x''=(x+iy+ (x + sy. 
 
 10 
 
146 ALGEBRA. 
 
 26. 3a;2 = (a; + 1)^ + {x + 2f + (a; + 3)=. 
 
 27. (a; -2) (a; -5) + (a; -3) (a; -4) =:. 2 (a; - 4) (a; -5). 
 
 28. {x - 1)'' + 4 (a; - 3)^ = 5 (a; + 5)1 
 
 29. 5 (a; + 1)2 + 7 (a; + 3)^ = 12 (a: + 2)^ 
 
 30. (a; - 1) (x - 4) = 2 a; + (a; - 2) (a; - 3). 
 
 31. (a!-l)=+(a:-2)»+(a;-3)= = 3(a;-l)(a;-2)(a;-3). 
 
 32. 4 (a; - 6) - 2 {3a; - (a; - 8)} = 5 (13 - 3a;). 
 
 33. 2 (3a; + 4) = 5 {2a; - 3 (a; + 4) + 9}- (1 - 3a;). 
 
 Alls. —3. 
 
 34. 8{2(a; - 1) - (a; + 3)} = 5[a; + 7 {x - 2 (4 - a;)}]. 
 
 164. Equations with Fractional Terms. 
 1. Keduce? + ^-4 = |f-| + 2i. 
 
 Multiplying every term by 20, the L. C. D., 
 
 4a:+ 10a; — 80 = 6a; — 5a; + 50 (Ax. 3.) 
 4a;+10a; — 6a; + 5a; = 80 + 50 
 13 a; = 130 
 a; = 10 
 
 Hence, for the solution of fractional equations, we have 
 the following 
 
 Kule. 
 
 Multiply each term of the equation iy the least common 
 multiple of tlie denominators, and then proceed as in the 
 preceding article. 
 
 NoTF, 1. In multiplying a fractional term, divide tlie multiplier by the 
 denominator of the fraction and multiply the numerator by the quotient. 
 
 Note 2. Before clearing of fractions it is better to nnite terms which 
 can readily be united. 
 
 Note 3. When the sign — is before a fraction and the denominator 
 is removed, as the dividing line has the same effect as a bracket, the sign 
 of each term that was in the numerator must be changed. 
 
SIMPLE EQUATIONS. 147 
 
 2. Keduce ^-1 + 20 = 30-^^. 
 
 O O 
 
 _ . 2x X ,^ X — S 
 
 Transposing 20, -= ^ = 10 r: — ■ 
 
 Multiplying by 30, 12 a; — 10 x = 300 — 5 a; + 1 5 
 12a; — 10a; + 5a; = 300+ 15 
 7a; = 315 
 a; = 45 
 
 Note 4. The sign of the numerator of — ? is +, and must be changed 
 
 to — when the denominator is removed ; for — (+ 10 a;) = — 10 a; ; 
 
 X — 3 
 and so the sign of each term of the niunerator of the fraction — — - — 
 
 must be changed when the denominator 6 is removed ; for — (+ 5 a; — 15) 
 
 = — 5a;+ 15. 
 
 3. Eeduce -^ ^ f- — ^ — = H -i jg 
 
 Multiplying by 12, ^:^:?^±^ - 8a; + 72 + ea; + 9 = 64 + 3a; + 4 
 
 12 a; + 72 
 Transposing and uniting, — 5 x = — 13 
 
 Multiplying by 1 1 , 12 a; + 72 - 53 a; = — 143 
 
 43 a; = 213 
 a; ^ 3 
 
 Note 5. It will be seen from the above, that it is sometimes advanta- 
 geous to clear of fractions at first partially, then transpose and unite like 
 terms before the remaining fractions are removed. 
 
 Eeduce the following equations : 
 
 
 4. I + --f-^ = 10. 
 
 X — 5 x + 5 „ 
 10 + 5 =^ 
 
 . x+5 x+1 x+3 
 ^- 6 9 = 4- 
 
 
 i-5x l-2a; 13 
 ^'6 3 ~42 
 
 Ans. ^- 
 
148 ALGEBRA. 
 
 4(x + 2) 6(a;-7) _ 
 
 11- -\^ + H + ^-^^i-^-n- Ans. 0. 
 
 12. f(4a!-l)-^(3a; + 2) = 6 + i(5a;-2). 
 
 13. i (4x - 11) + 4 (3a; - 4) = 3i - T^ (3a; + 13). 
 
 14- -5- - ^(a; - 11) = |(a; - 25) + 34. 
 
 15.3 + | = |(4-^)-J + |(ll_|). Ans. 3,. 
 
 ifl -'^/'^ Q^i^* ^* a; — 12 a; + 3 
 
 ^^- 3U~'^/' + T-T=-^ 3~- 
 
 IT /^o 2 a; — 5\ 1 .„ __. 5 
 
 17. x-(3a;-^^j=g(2a;-57)-3. 
 
 a; a; + 10 85-2 
 
 18. J ^+4j = a;-l 3-. 
 
 5a;-8 10 a; - 7 , ^ 5a; 
 
 -ly- ^2 3 + ^ - X - !*• 
 
 20. 4a;-3{5a;-8(a; + ^)} =12a; + 10. Ans. -2. 
 
 21. 12(x + J)-(2. + 5) = 15a;-^^±^. 
 
 22. 3 (2 a; -3)-^^+—=--=^^. 
 
 23. 2(a;-l) + i^-l^ + ^^ = 0. 
 
 ^ ' 4 9 2 
 
 24. J (a; - 1) + ^ (2 a; - 3) - 3 = 2 (a; + ,5). 
 
SIMPLE EQUATIONS. 149 
 
 25. l(5x + l)-lcix-2)-^-^ = l(x-l). 
 
 Ans. — 1. 
 
 26. 2(3a; + l)-f(a! + 4) + 20+ i(* + 7)=0. 
 
 27. 5x + ^^^ - I (6a! - 1) - 2 (2 + a:) + 5 = 0. 
 
 28. 4(a; + 6)-?-g(a:+10) = ^(a; + 6)-li. 
 
 29. 5\2x + l-3(x + l)}-'^ + ji3x + 5l) + 2i^0. 
 
 Ans. — i. 
 
 30. 0.5 a; + 2 - 0.75 x = OAx- 11. 
 
 Writing the decimals as common fractions, 
 
 ix+2-fa; = f2;-ll 
 Multiplying by 20, the L. C. D., 
 
 10 a; + 40 — 15 a; = 8 a; — 220 
 13 a; = 260 
 
 a; = 20 
 Or, 
 Transposing, 0.5 x — 0.75 a; — 0.4 a; = — 2 — 11 
 
 Combining like terms, 0.65 a; = 13 
 
 Dividing by 0.65, a; = 20 
 
 31. 0.5 a; -0.25 a; = 1.5. 
 
 32. 2.25 X - 0.125 = 3 a; + 3.75. 
 
 34. 0.6 a; + 6.3 - 3.5 x = 0.25 x. 
 
 35. 0.6 a; - 0.7 a; + 0.75 x - 0.875 a; + 15 = 0. 
 
 36. 0.125 a; - 0.0625 a; = 0.375. Ans. 6. 
 
 X + 0.75 a;-0.25 _,^ 
 ^'- 0.125 0.25 ~ 
 
)U 
 
 
 
 
 ALGEBRA. 
 
 38. 
 
 0.5 a; 
 
 0.45 
 
 X — 
 0.6 
 
 0.75 
 
 1.2 
 
 ~0.2' 
 
 39. 
 
 1.5 = 
 
 0.36 
 0.2 
 
 0.09 a; — 
 0.9 
 
 0.18 
 
 3 X — 0.6 
 0.9 
 
 Ans. 5. 
 
 165. When the Equations are Literal. 
 
 1. Reduce 2x (a + J) - 3 a& = 2 a (a; - i) - i=. 
 
 Removing the brackets, 
 
 2aa; + 2Jx — 3a& = 2aa; — 2a5 — 6= 
 Dropping lax from both members, transposing, and uniting, 
 2hx = ah — lfl 
 
 li (a — h) a — 6 
 ^= 20 =n^ 
 
 -_. ab + X b^ — X X — b ab — x 
 Z. ICeduce — -, j-.- = — ., j-i — * 
 
 Multiplying by a^V^, the L. C. D., 
 
 a^h -\- oP' X — I' + bx = b^x — 6' — a^h -\- a'x 
 Dropping a^x and — i' from both members, and transposing, 
 bx — b^x = — a^ b — a'i 
 (J — 62) X = — 2 a' b 
 
 — 2 qS ft — 2 flS 
 
 an" 
 
 6 — 62 i_i, j_i 
 Reduce the following equations : 
 
 3. 2 (x - a) + 3 (a; - 2 a) = 2 a. 
 
 4. i{x + a + b) + l(x + a — b) = b. 
 
 5. (a + b)x + (a — b)x — a\ Ans. ^ . 
 
 6. (a. + 6) a; + (5 — a) a; = 6. 
 
 7. ^ (a + a;) + ^ (2 a + a:) + J (3 a + a;) = 3 a. 
 
 8. ^ + ^ = a2 + Z,2. Ans „j_ 
 
 o a 
 
SIMPLE EQUATIONS. 151 
 
 9. (a + bx) {b + ax) = ab (x^ — 1). 
 
 10. a (x + a) + b (b — x) = 2 ab. 
 
 11. (x + a + by+(x+a-by=:2 x\ 
 
 12. (a; — a){x — b) + {a + bf = (x -\- a) {x -{■ b). 
 
 Ans. ^ (a + b). 
 
 13. (x+a-\-b+c){x+a—b—c) — (x—a—b+c)(x—a+b—c), 
 
 14. a a; (a; + a) + 6 x (sB + 6) = (a + &) (a; + a) (x + J). 
 
 15. (x - aY-\-{x-bf + {x-cY = Z(x—a) (x—b) (x-c). 
 
 Ans. ^ (a + b + c). , 
 
 ^^ 9a 3a; 4S 2a; 
 
 16. -r — = 
 
 b a a 
 
 18.J«(.-.)-(i±--y=^-(.-|). 
 
 19. b(a-x)--^(b + xy + ab (^^ + l)'== 0. Ans. a. 
 
 20. a;''+a(2a-a;)-^= ^a:-|) +al Ans. a + i. 
 
 166. Since, in the solution of problems, the relations of 
 the numbers involved are often best expressed in the form 
 of a proportion, we introduce here the necessary definitions. 
 
 167. Ratio is the relation of one number to another of 
 the same kind ; or, it is the quotient which arises from 
 dividing one number by another of the same kind. 
 
 Ratio is indicated by writing the two numbers one af- 
 ter the other with two dots between, or by expressing the 
 
152 ALGEBRA. 
 
 division in the form of a fraction. Thus, the ratio of a to 
 h is written, a:h,ov-.; read, a is to 6, or a divided by h. 
 
 
 
 168. The Terms of a ratio are the quantities compared, 
 whether simple or compound. 
 
 The first term of a ratio is called the antecedent, and the 
 other the consequent; and the two terms together are called 
 a couplet. 
 
 169. Proportion is an equality of ratios. Four numbers 
 are proportional when the ratio of the first to the second is 
 equal to the ratio of the third to the fourth. 
 
 The equality of two ratios is indicated by the sign of 
 equality (=) or by four dots (: :). Thus, a: b = c : d, or 
 a :b :: c : d, ov ^ = ~; read, a to & equals c to d, or a is 
 to & as c is to d, or a divided by b equals c divided by d. 
 
 In a proportion the antecedents and consequents of the 
 two ratios are respectively the antecedents and consequents 
 of the proportion. The first and fourth terms are called 
 the extremes, and the second and third the means. 
 
 170. If four numbers are in proportion, the product of 
 the means is equal to the product of the extremes. 
 
 that 
 
 Let a : 6 == c : (Z 
 
 a c 
 
 Clearing of fractions, a d =^bc 
 
 Hence, if any three terms of a proportion are given, the 
 fourth may be found. Thus, ii a, c, d are given, then 
 ■,_ad 
 c 
 
 A proportion is an equation ; and making the product of 
 the means equal to the product of the extremes is merely 
 clearing the equation of fractions. 
 
SIMPLE EQUATIONS. 153 
 
 171. QUESTIONS LEADING UP TO PROBLEMS. 
 
 1. Form a proportion out of the following numbers : 3, 9, 
 16,5. 
 
 2. Arrange the ratios 5:6, 7:8, 41 : 48, 31 : 36, in the 
 order of their magnitude. 
 
 3. For what value of x will the ratio 3 + a; ; 4 + a; be equal 
 to 8 : 9 ? 
 
 4. What number must be added to each term of the ratio 
 11 : 17 to make it equal to the ratio of 2 : 3 ? 
 
 5. Write four consecutive numbers, of which x is the least. 
 
 6. Write three consecutive numbers, of which y is the 
 greatest. 
 
 7. Write five consecutive numbers, of which x is the middle 
 one. 
 
 8. What is the next even number after 2n? 
 
 9. What is the odd number next before 2 a; + 1 ? 
 
 10. Find the sum of three consecutive odd numbers, of 
 which the middle one is 2 n + 1. 
 
 11. Al horse eats a bushels and a donkey b bushels of corn 
 in a week. How many bushels will they together consume in 
 n weeks ? 
 
 12. If a man was x years old five years ago, how old will 
 he be y years hence ? 
 
 13. A boy is x years old, and five years hence his age will 
 be half that of his father. How old is the father now ? 
 
 14. What is the age of a man who y years ago was m times 
 as old as a child then x years old ? 
 
 15. A's age is double B's,.B's is three times C's, and C is 
 X years old. Find A's age. 
 
 16. Find a number exceeding the sum of x and y by their 
 difference. 
 
154 ALGEBRA. 
 
 17. A post 6 X inches in length is half in the mud, and one 
 sixth in the water. How many inches are above the water ? 
 
 18. There are two casks containing 20 gallons each. If x 
 gallons are poured from the first to the second, how many 
 gallons will each cask contain ? 
 
 19. I pour X gallons of milk and y of water into a bucket. 
 What fraction of the whole is mill?:, and what water ? 
 
 20. If I rowed x miles in 10 hours, at what rate did I row ? 
 
 21. A and B, starting from the same point, walk in oppo- 
 site directions at the rate of x and y miles an hour respect- 
 ively. How far apart will thej' be in 20 minutes ? 
 
 22. A and B, starting from the same point, walk in the 
 same direction at the rate of x and y miles an hour respect- 
 ively. How far apart will they be at the end of one hour ? 
 In three hours '/ 
 
 23. If A gives B a start of one mile, and then overtakes 
 him in x hours, how manj- miles an hour does A walk faster 
 than B? 
 
 24. How many miles can a person walk in 46 minutes, if 
 he walks a miles in x hours ? 
 
 25. How long will it take a person to walk h miles, if he 
 walks 20 miles in c hours ? 
 
 26. If a train goes a miles in h hours, how many feet does 
 it move through in one second ? 
 
 27. A train is running with a velocity of x feet per second. 
 How many miles will it run in y hours ? 
 
 28. How long will x men take to mow y acres of corn, if 
 each man mows z acres a day ? 
 
 29. A beats B by a; yards in a mile race. At the same rate 
 of running, how far ahead will A be when B comes to the end 
 of the mile ? 
 
 30. A room \a X -\- y feet long, and x~y feet broad. What 
 is its area ? 
 
SIMPLE EQUATIONS. 155 
 
 31. What is the area of a square court, each side of which 
 is 2 a; yards long ? 
 
 32. A Mock of buildings 3 x yards square is placed in the 
 centre of a court 4x yards square. Find the area of the 
 unoccupied space. 
 
 33. Tf a room is a feet long, b feet broad, and c feet high, 
 how much carpet would be required for the floor ? 
 
 34. If a room is p feet long and x yards in width, how 
 many yards of carpet two feet wide will be required for the 
 floor? 
 
 35. What is the cost in dollars of carpeting a room a yards 
 long, and b feet broad, with carpet costing c cents a square 
 yard ? 
 
 36. If A can do a piece of work in x days, what fraction of 
 the work can he do in one day ? 
 
 37. If A can do a piece of work in - days, what fraction of 
 the work can he do in one day ? 
 
 38. A can do a piece of work in x days, B in y days. What 
 fraction of the work can they do together in one day ? 
 
 39. If a pipe discharges x gallons of water in y hours, how 
 many does it discharge in one hour ? 
 
 40. A pipe is letting off water from a full cistern contain- 
 ing z gallons. If it could empty the whole in y hours, how 
 ipuch water would there be in the cistern at the end of one 
 hour ? How much at the end of a hours ? 
 
 41. If a pipe can fill a cistern in y hours, what fraction of 
 the cistern can it fill in three hours ? 
 
 42. There are x men in each outer face of a hollow square, 
 and y men in each inner face. How many men are there in 
 the square ? 
 
 43. The digit in the units' place of a number is n, and in 
 the tens' place m. What is the number ? 
 
156 ALGEBRA. 
 
 44. The digits of a number beginning from the left are 
 a, h, 0. What is the number ? If the order of the digits of 
 the number is reversed, what is the number thus formed ? 
 
 45. A number consists of 4 digits, the first of which is 
 a, the second 0, the third b, and the fourth c. What is the 
 number ? 
 
 46. Find the interest on $ 1 for n years at r per cent. 
 
 47. Find the amount of $ 1 for n years at r per cent. 
 
 48. If there are 4 stones in a row, what number from the 
 end is the second from the beginning ? 
 
 49. If there are 20 stones in a row, what numbers from the 
 beginning are the seventh and nth from the end, respectively? 
 
 PROBLEMS 
 
 PRODUCING EQUATIONS OF THE FIRST DEGREE CONTAININa 
 BUT OSE UNKNOWN NUMBER. 
 
 172. The problems given in this chapter must either 
 contain but one unknown number, or the unknown num- 
 bers must be so related to one another that if one becomes 
 known the others also become known. 
 
 173. With beginners the chief difficulty in solving a 
 problem is in translating the statements or conditions of 
 the problem from common to algebraic language ; that is, 
 in preparing the data and forming an equation in accord- 
 ance with the given conditions. 
 
 It is impossible to give a general rule for the solution of 
 problems, applicable to all cases. Each problem must be 
 considered, and its meaning thoroughly understood, before 
 an equation can be formed. 
 
 The following suggestions may be of service : 
 
 1. Let X {or some one of the latter letters of the alphabet) 
 represent the unknown number ; or, if there is mare than 
 
SIMPLE EQUATIONS. 
 
 157 
 
 one unknown number, let x represent one, and find the 
 others by expressing in algebraic form their given relations 
 to the one represented by x. 
 
 2. With the data thus prepared, form an equation in 
 accordance with the conditions given in the problem. 
 
 3. Reduce the equation. 
 
 The three steps may be briefly expressed thus : 
 
 1st. Preparing the data ; 
 2d. Forming the equation ; 
 3d. Eeducing the equation. 
 
 1. Find two numbers whose sum is 28, and whose differ- 
 ence is 4. 
 
 Let 
 
 X = the smaller number ; 
 
 then 
 
 a; + 4 = the greater number. 
 
 By addition, 
 
 a: + a; + 4 = their sum ; 
 
 but 
 
 28 = their sum ; 
 
 .•. 
 
 a; + a; + 4 = 28 
 
 
 2 a: = 24 
 
 
 X = 12, the smaller number, 
 
 and 
 
 a; + 4 = 16, the greater number. 
 
 Verification, 
 
 16 + 12 = 28 
 
 2. Divide $ 59 among A, B, and C, so that A may have $ 9 
 more than B, and B $ 7 more than C. 
 
 Let 
 
 X = the number of dollars in C's part ; 
 
 then 
 
 ~a; + 7== " « " B's " 
 
 and 
 
 a; + 7 + 9 = " " " A's " 
 
 By addition. 
 
 3 X + 23 = the number of dollars to be divided ; 
 
 but 
 
 59 = " " " " " 
 
 
 .-. 3 a; + 23 = 59 
 
 
 3 a: = 36 
 
 
 a: =12 
 
 
 C has $12, B$19, A $28. 
 
 Verification, 
 
 12 + 19 + 28 = 59 
 
158 ALGEBUA. 
 
 3. Charles is now twice as old as Henry, and eight years 
 ago he was six times as old. What are their present ages ? 
 
 Let a; = the number of years in Henry's age now ; 
 
 then 2x= « " " Charles's " 
 
 X — 8 = the number of years in Henry's age eight years af;o, 
 2 1 — 8= " " " Charles's " " 
 
 6 (a; — 8) = " " " • " " " " 
 
 .-. 2 a; — 8 = 6 fa; — 8) 
 2x— 8 = 6a: — 48 
 — 4x = — 40 
 x = \0 
 
 Therefore Henry and Charles are 10 and 20 years old, respectively. 
 
 4. One number exceeds another by 5, and the sum of the 
 two is 29. What are the numbers ? 
 
 5. Pind three consecutive numbers whose sum is 96. 
 
 6. The difference between two numbers is 8, and if 2 be 
 added to the greater the result will be three times the smaller. 
 Find the numbers. Ans. 13, 5. 
 
 7. A man walks 10 miles, then travels a certain distance by 
 train, and then twice as far by coach. If the whole journey 
 is 70 miles, how far does he travel by train ? 
 
 8. If 288 be added to a certain number, the result will be 
 equal to three times the excess of the number over 12. Knd 
 the number. 
 
 9. Find a number such that, if 5, 15, and 35 are added to 
 it, the product of the iirst and third results may be equal to 
 the square of the second. Ans. 5. 
 
 10. If the difference between the squares of two consecutive 
 numbers is 121, what are the numbers ? 
 
 11. A father is four times as old as his son, and in 24 years 
 he will be twice as old. Find their ages. 
 
SIMPLE EQUATIONS. 159 
 
 12. A is 25 years older than B, and A's age is as much 
 above 20 as B's is below 85. Find their ages. 
 
 Ans. A, 65; B, 40. 
 
 13. The length of a room exceeds the breadth 3 feet. If 
 the length is increased 3 feet, and the breadth diminished 2 
 feet, the area will remain the same. Find the dimensions. 
 
 14. The length of a room exceeds the breadth 8 feet. If 
 each dimension is increased 2 feet, the area will be increased 
 60 square feet. Find the dimensions of the room. 
 
 15. Find two numbers which differ by 4, and such that one 
 half the greater exceeds one sixth of the less by 8. 
 
 Let X = the smaller number ; 
 
 then a; + 4 =: the greater number, 
 
 X 
 
 o r= one sixth of the less, 
 ,-j + 2 = one half of the greater, 
 
 X X 
 
 ij + 2 — ^ = the excess of ^ the greater over J the less. 
 But 8 = " " " 
 
 •■•i+^-i-» 
 
 X = 18, the less number, 
 E + 4 = 22, the greater. 
 
 16. A has $180, and B $84. How much must A give to B 
 in order that A may have five sixths as much as B ? 
 
 Let X = the number of dollars A must give to B ; 
 
 then 84 + a; = " " " B will then have, 
 
 and 180 — a; = " " " A " 
 
 But I (84 + a;) = " « " A " " 
 
 .-. 180 — a; = I (84 + a:) = 70 + |a; 
 — ajLa; = _llo 
 a;= 60 
 That is, A must give $ 60 to B. 
 
160 ALGEBRA. 
 
 17. Find the number whose fifth, fifteenth, and twenty-fifth 
 together are 23. Ans. 75. 
 
 18. Four times the difference between the fourth and fifth 
 parts of a certain number exceeds by "4 the difference between 
 the third and seventh parts. What is the number ? 
 
 19. Find three consecutive numbers such that, if they are 
 divided by 10, 17, and 26 respectively, the sum of the quo- 
 tients will be 10. 
 
 20. From a certain number 3 is taken, and the remainder is 
 divided by 4 ; the quotient is then increased by 4 and divided 
 by 5 and the result is 2. Find the number. Ans. 27. 
 
 21. A sum of money is divided among three persons. A, B, 
 and C, in such a way that A and B have together $ 60, A and 
 C $ 65, and B and C $ 75. How much has each ? 
 
 22. Four persons, A, B, C, T>, have certain sums of money, 
 such that A and B together have $ 49, A and C $ 51, B and 
 C $53, and A and U $47. How much has each ? 
 
 23. Divide 15 dollars among 3 men, 5 women, and 20 chil- 
 dren, giving to each man one dollar more than to each woman, 
 and to each child half as much as to each woman. 
 
 24. A man leaves one half of his property to his wife, one 
 third to his son, and the remainder (which is $2000) to his 
 daughter. How much did he leave in all ? Ans. $ 12000. 
 
 25. A man left his property to be divided among his three 
 children in such a way that the share of the eldest was to be 
 twice that of the second, and the share of the second twice that 
 of the youngest. It was found that the eldest received $ 750 
 more than the youngest. How much did each receive ? 
 
 26. An estate of 8000 acres is to be divided among three 
 persons, A, B, and C, so that B has 276 acres less than A, and 
 G 1112 acres more than B. How many acres does each get ? 
 
 Ans. A, 2480 ; B, 2204 ; C, 3316. 
 
SIMPLE EQUATIONS. 161 
 
 27. A person has a flock of sheep and goats, together num- 
 bering 75. He has two goats to every three sheep. How 
 many are there of each ? 
 
 28. If $ 1200 is divided between A and B in the proportion 
 of 2 to 7, how much does A get less than B ? 
 
 29. Find a number which, increased by its half, its third, 
 and its fourth, will amount to 50. 
 
 30. The width of a room is two thirds of its length. If the 
 width had been 3 feet more, and the length 3 feet less, the 
 room would have been square. Find its dimensions. 
 
 31. A spends a sum of money ; B spends half as much as 
 A ; C spends three fourths as much as A and B together. If 
 A, B, and C together spend $1060, what does each spend ? 
 
 32. 2850 acres of land are divided among three persons, 
 A, B, and C, so that the shares of A and B are in the ratio of 
 6 to 11, and C has 300 acres more than A and B together. 
 What does each get ? 
 
 33. $7500 is divided among a mother, two sons, and three 
 daughters, so that each son has twice as much as each daugh- 
 ter, and the mother $ 500 more than all the children together. 
 How much does each get ? Ans. $500, $1000, $4000. 
 
 34. A post is a fourth of its length in the mud, a third of 
 its length in the water, and 10 feet above the water. What is 
 its length ? 
 
 35. A marketwoman, being asked how many eggs she had, 
 replied, " If I had as many more, and one less than half as 
 many more, I should have 104 eggs." How many had she ? 
 
 36. Find two numbers, whose difference is 25, such that 
 the second divided by the first gives 4 as quotient and 4 as 
 remainder. Ans. 7, 32. 
 
 11 
 
162 ALGEBRA. 
 
 37. Find a fraction whose denominator exceeds its nu- 
 merator by 4, while, if the numerator is diminished 1 and 
 the denominator increased 1, the fraction becomes equal to ^. 
 
 38. A person paid $ 800 for 4 horses. For the second he 
 gave half as much again as for the first ; for the third, half as 
 much again as for the second ; and for the fourth, as much as 
 for the first and third together. What was the cost of the 
 fourth horse? Ans. $325. 
 
 174, The following fractional equations are more difficult 
 to reduce than those thus far given, since their denomina- 
 tors are generally polynomials rather than monomials, and 
 generally involve the unknown number. 
 
 The method already given still applies, but it is often 
 unnecessarily prolix, and hence for the ready solution of 
 such examples special artifices must be frequently resorted 
 to. 
 
 1. Reduce 5 -; + — 
 
 3x-6 6-2a; 8 (42 - 35 a; 4- 7 a;") 
 
 Factoring the denominators and writing them in symmetrical form, 
 
 4 1 a: 4-3 
 
 3 (a; — 2) 2 (a — 3) ~ 21 (a; — 2) (x — 3) 
 
 Multiplying by 42 (x — 2) (x — 3), the L. C. D., 
 
 56 (s — 3) — 21 (x — 2) = 2 (x + 3) 
 Freeing from brackets, 
 
 56 X— 168 — 21 X -I- 42= 2x -I- 6 
 33 X = 132 
 
 x = 4 
 
 Verification, 
 
 ^ I 1 _ (4) + 3 
 
 3(4) -6^ 6-2(4) 3 142 - 35 (4) 4- 7 (4)2j 
 
 That is, ^ = ^ 
 
 6 6 
 
SIMPLE EQUATIONS. 163 
 
 „ _ , 8a; + 23 5a; + 2 2a; + 3 , 
 2. Eeduce -^^ 3^-^ = -^ - 1. 
 
 Multiplying by 20, 
 
 20C5a;+ 2) 
 
 Transposing and uniting, 
 
 20 (5 a; + 2) 
 
 = 31 
 
 3a; + 4 
 
 Multiplying by 3 a; + 4, 100 a; + 40 = 93 a; + 124 
 
 7 a; =84 
 a; = 12 
 Verification, 
 
 8 (12) + 23 5 (12) + 2 _ 2(12) + 3 
 
 
 20 
 
 
 3 (12) + 4 
 
 
 5 
 
 
 That is. 
 
 
 
 
 22 
 
 5 '' 
 
 22 
 ~ 5 
 
 
 
 3. Eeduce 
 
 X 
 
 X — 
 
 -2 + 
 
 a!-9 
 x-7 
 
 x 
 
 X 
 
 + 1 
 -1 
 
 n^ 
 
 -8 
 -6 
 
 Transposing, 
 
 
 X 
 
 x+l 
 x—1 
 
 X 
 
 x 
 
 — 8 
 
 X — 
 
 9 
 
 X ■ 
 
 _ 2 
 
 — 6 
 
 X — 
 
 7 
 
 Combining, 
 
 a:(a;-l) — (a:-2)(a:+l) ^ (3:-7) (a; — 8) — (a; - 6) (a:-9) 
 
 (x - 2) (a; - 1) (a; _ 6) (a; - 7) 
 
 2 2 
 
 Or == • 
 
 (x — 2) (a: - 1) (x — 6) (a; - 7) 
 
 Dividing by 2 and clearing of fractions, 
 
 (x - 2) (a: - 1) = (a: - 6) (a; - 7) 
 
 a;2 — 3 ar + 2 = a;2 — 13 a; + 42 
 
 10 a; = 4^ 
 
 a; = 4 
 
 „.„,. 4,4-9 4 + 14-8 
 
 Verification. ^—^ + ^_^ = -—^ + __ 
 
 That is, 2 + f = 2 + I 
 
164 ALGEBRA. 
 
 4. Keduce 
 
 X + 2 x + 6 
 Reducing the frantions to mixed numbers, 
 
 11 
 
 x+2 x+6 
 
 5 _ 11 
 
 ■'■ X + 2~ X + 6 
 
 Clearing of fractions, 
 
 5a; + 30=llz + 22. 
 
 6a; = 8 
 
 „ _ , a'' + 7a;^ + 24x + 30 2a!' + lla;H S6a: + 45 
 ^- ^'^'''' a=^ + 5:.+ 13 = 2^^ + 7:^ + 20 
 
 Reducing the fractions to mixed numbers, 
 
 a;2 + 5 x + 13 ^ ^ 2x^+7 x + 20 
 
 Removing a; + 2 from both members, and inverting the fractions, 
 a:'+5z+13 2a:^ + 7a: + 20 
 X + 4 ~ 2a; + 5 
 
 Reducing to mixed numbers, 
 
 q 15 
 
 X + 4 ^ ^2x + 5 
 
 9 15 
 
 ■ X + 4~ 2x + ?> 
 
 18a;+45 = 15s + 60 
 Sx= 15 
 
 Ex. 3 may be written in the following form, and then solved like 
 Exs. 4 and 5 : 
 
SIMPLK EQUATIONS. 165 
 
 a — 2 + 2 a:-7 — 2 _ a: — 1 + 2 a; — 6 — 2 ,. 
 
 x — 2 ■*" x — ^ x — 1 + a; — 6 ^' 
 
 Reducing to mixed numbers, 
 
 l + x-^ + l-^ = l + .4i + l-^ (2) 
 That is, 2 2 2 2 
 
 x—2 x—7 x—1 x—6 
 -10 -10 
 
 •• {x-2){x-7) (a;-l)(a;-6) 
 a; = 4 
 Note. Eq^uation (2) can also be found directly, without writing (1). 
 
 Reduce the following equations : 
 2 3 11 
 
 4a;2 — 1 l-2a; 2a; + l 
 
 2a; + 1 8 _ 2a!-l 
 
 2a;-l'*'l-4a:2~2a; + l" 
 
 2 1 7 
 
 8. 
 
 a; -4 2 a: -9 2x^-173; + 36 
 
 9 _^ 2 I 1 
 
 2(3a; + 7) 3x^ + 22a; + 35 ^ 2a: + 10 ~ 
 
 Ans. —2. 
 2a: — 5 , a;-3 _ 4a:-3 
 ^"- ~5~ +2a:-15~7lO ^' 
 
 4 (x + 3) _ 8 a; + 37 7a: - 29 
 9 ~ 18 5a; -12' 
 
 (2a;-l)(3x + 8) 
 ^''- 6x(a; + 4) -^ - "• 
 
 1o _J 2^ 5 2i 
 
 ;a; + 3 a; + 1 2a: +6 2 a; + 2 ' 
 
166 ALGEBRA. 
 
 U JL ^^.^2i ^. Ans.-10. 
 
 ^*- a;-4 5x-30 32:-]2 x-6 
 
 15. 25-1 I 16. + 4t_p , 23 
 
 a; + l 3x + 2 X + 1 
 
 ,„ 30 + 6 a: ,60 + 8 a: .. , 48 
 16. ij 1 —5- = 14 + 
 
 17. 
 
 18. 
 
 X + 1 X + 3 X + 1 
 
 1111 
 
 a; — 1 a; — 2 a; — 3 x — 4 
 
 a:— 7 a; — 9_a: — 13 a; — 15 
 a; — 9~a; — 11~ a: — 15 ~ a; — 17 ' 
 
 2x + l . 2a; + 9 2a! + 3^2a; + 7 . _ 
 
 a; + l a; + 5 a: + 2 a; + 4 
 
 4 a; -17 10 a: — 13 _ 8 a; — 30 5 a; — 4 
 a;-4 "^ 2a:-3 ~ 2a; - 7 "^ a; - T ' 
 
 Sx — 8 6x-44 _ 10a; — 8 _ a; — 8 
 X — 2 X — 7 a; — 1 x — 6 
 
 a;" + 2 a; — 2 a:^ — 2a; — 2 _ 2a:^ — 6a' + 2 
 X — 1 a; + l X — 3 
 
 .„ 4.r'' + 4,7;^ + 8x4-1 2a.^ + 2a; + l . 
 
 2x^ -\- 2x + 3 X + 1 
 
 24 a^ + 4q + & 4x + ffi + 2i ^g 
 a; + a + 6 a;+a — 6 
 
 25. 
 
 26. 
 
 2a; + 3a_2(3.r+2a) 
 X -\- a 3 X -\- a 
 
 a b b"^ — a^ 
 
 X — b W —hx 
 
SIMPLE EQUATIONS. 167 
 
 27. (2x-a)fx+^fj=ix(l-x\-^(a-4x)(2a + Sx). 
 
 Ans. 2^- . 
 
 3abc a'b^ (2a + b)b^x _ bx 
 
 ■ a + fi"*" (a + 6)»+ a (a + 6)^ -dca; + — . 
 
 2g 2a;-3 0.4 a: -6 
 
 Cr.3 X — 0.4 0.06 X — 0.07 
 
 30 l-l-4a! ^ 0-r(a;-l) 
 ■ 0.2 + a; 0.1 -0.5 a;' 
 
 „, (0.3 x-2) (0.3 a; - 1) 
 
 31. ^ q:2x-\ - * (^-^ X - 2) = 0.4 a: - 2. 
 
 Ans. 20. 
 32 --+Ag-H3.-4) + (^-^H2-3) ^..-A. 
 
 a; 4- w a; — m w* 
 
 33. 
 
 31. 
 
 x'' ■\- m X ■\- rri^ x' — mx + m? x{x*+m^x^+ m*) ' 
 
 _x—a x~a—l_ x—b x—b—1 
 
 x — a — 1 X — a — 2 x — b — 1 x — b — 2' 
 
 Ans. |(fl + J + 3). 
 
 35 ^~^ I ' ^ ~ ¥ _ a^ — I , a: — V 
 X - g- a; - V ~ « - ^ a; - -V- ' 
 
 36. ^(^-2) + 3(6^ ^ -7) _¥~^_i^. 
 
 37 »-, A(2a:-3)-H3a:-l) _3 x'-^^ + 2 
 ■ ^ U«-l) 2 3a;-2 ■ 
 
 Ans. 0. 
 
168 ALGEBRA. 
 
 175. ADDITIONAL PKOBLEMS. 
 
 1. A man's age was to that of his wife at the time of their 
 marriage as 3:2, and their ages 9 years after were as 4:3. 
 What was the age of each at the time of their marriage ? 
 
 Let Zx = No. of yrs. in the man's age at marriage ; 
 
 then 2x= " " wife's " " 
 
 3 a; + 9 = No. of yrs. in the man's age 9 yrs. after marriage, 
 2a; + 9= " " wife's " " " 
 
 By the conditions of the question, 
 3a;+9:2a;+ 9 = 4:3 
 .-. 9 a; + 27 = 8k + 36 
 x = 9 
 3 2 ^ 27, man's age, 
 2 X =: 18, wife's age. 
 Verify the example. 
 
 2. A can do a piece of work in 3 days, and B can do it in 
 5 days. In what time can they do it together ? 
 
 Let 
 
 
 X = the number of days it will take A and B together. 
 
 then 
 
 
 - = the part A and B can do in one day. 
 
 Now 
 
 
 - = the part A can do in one day, 
 3 
 
 and 
 
 
 5 
 
 .-. 
 
 1 
 3 
 
 + - = the part A and B can do in one day, 
 
 or, 
 
 1 
 3 
 
 + 1-1 
 
 8 1 
 15 ~" a; 
 
 . = L5 = iZ 
 8 ^8 
 
 3. Two trains are running on parallel tracks, in the same 
 direction, one at the rate of 30 miles an hour, the other at the 
 
SIMPLE EQUATIONS. 169 
 
 rate of 28 miles an hour. The slower of the two trains is 12 
 miles in advance of the other. 
 
 (1) How long before the faster train will be within 4 miles 
 of the slower train ? 
 
 (2) How long before it will be up with it ? 
 
 (3) How long before it will be 6 miles in advance of it ? 
 
 Let X = the number of hours in case (1). 
 
 30 — 28 ^ the number of miles gained in one hour. 
 Then x (30 — 28) = « " « « x hours. 
 
 But 8 = " " " " « 
 
 .-. a;(30 — 28) = S 
 2a; = 8 
 X = i, the number of hours in case (1). 
 
 Let a;' =; the number of hours in case (2) ; 
 
 then x' (30 — 28) = 12 
 2a;' = 12 
 a:' = 6, the number of hours in case (2). 
 
 Let x" = the number of hours in case (3) ; 
 
 then a;" (30 — 28) = 18 
 2 x" = 18 
 x" = 9, the number of hours in case (3). 
 
 It is left to the pupil to determine the number of miles the trains 
 run in each case. 
 
 4. Suppose the trains in Ex. 3 were 174 miles apart, and 
 running towards each other, how many hours before they 
 would meet ? How many miles would each train run ? 
 
 Let X = the number of hours. 
 
 30 + 28 = the rate of approach in miles an hour. 
 Then (30 + 28) x = the number of miles passed over in x hours. 
 But 174 = " " " " " 
 
 .-. (30 + 28) a; = 174 
 58 a; = 174 
 
 a; ^= 3, the number of hours. 
 The number of miles would be 90 and 84. 
 
170 ALGEBRA. 
 
 6. Suppose in Ex. 3 the tracks were circular, and 20 miles 
 long. If the trains were side by side at 12 o'clock m., when 
 would they next be in the same position ? 
 
 Let X = the number of hours, 
 
 and 30 — 28 = the number of miles gained in one hour; 
 
 then (30 — 28)x= ■' " " " a; hours. 
 
 But 20= " " 
 
 .-. (30 — 28) x = 20 
 2 a; = 20 
 X = 10, the number of hours. 
 The time would be 10 o'clock, p. m. 
 
 6. Find the time between 4 and 5 o'clock, when the hands 
 of a clock are, 
 
 (1) Together. 
 
 (2) Opposite each other. 
 
 (3) At right angles to 6ach other. 
 
 (4) Two minute-spaces from each other. 
 
 In all these cases, let the position of the hands at 4 o'clock be their 
 starting position. Then, in case (1), the minute-spaces to be gained 
 are 20; in case (2), 50; in case (3), 5, or 35; in case (4), 18, or 22. 
 
 The rates an hour of the minute-hand and hour-hand are 60 
 minnte-spaces, and 5 minute-spaces, respectively. 
 
 Let X = the number of minutes past 4 in case (1). 
 
 Now 60 — 5 = the number of spaces gained in one hour, 
 
 and — = " " " " a minute. 
 
 60 
 
 Then — — — x = j^ = the number of spaces gained in x minutes. 
 
 But 20 = « « " « « 
 
 ■• ^' = - 
 
 X = 21^, the number of minutes. 
 Hence, the time is 21 ^x minutes past 4 o'clock. 
 
 Let a/ = the number of minutes past 4 in case (2) ; 
 
 11a;' 
 then — =- = the number of spaces gained in x' minutes. 
 
SIMPLE EQUATIONS. 171 
 
 But 50 = the number of spaces gained in x' minutes ; 
 
 ••■ ¥=» 
 
 a;' = 54^\, the number of minutes. 
 Hence, the time is 54j*j minutes past 4 o'clock, or 5^\ minutes of 5. 
 Let k" = the number of minutes past 4 in case (3). 
 
 Then lA^ = 5, or 35 
 
 x'' = 5^\, or 38^j, the number of mmutes. 
 Hence, the time is either 5^^ minutes or 38^ minutes past 4 o'clock. 
 
 Let a"' = the number of minutes past 4 in case (4). 
 
 11 t'" 
 Then tlJL. = 18, or 22 
 
 .12 
 x'" = 19^\, or 24, the number of minutes. 
 Hence, the time is either 19^'^ minutes or 24 minutes past 4 o'clock. 
 
 7. A dog pursues a hare. The hare gets a start of 50 of 
 her own leaps. The hare makes 6 leaps while the dog makes 
 5, and 7 of the dog's leaps are equal to 9 of the hare's. How 
 many leaps will the hare take before she is caught ? How 
 many leaps will the dog take ? 
 
 Let \ of the hare's leap be the unit of measure. 
 Then 7 = the length, in these units, of the hare's leap, 
 
 and 9= " " " " dog's " 
 
 Since the hare takes 6 leaps to the dog's 5, 
 therefore 42 = the distance the hare goes in a certain unit of time , 
 
 and 45 = " " dog goes in the same " " 
 
 .*. 45 — 42 = the gain of the dog in a imit of time. 
 Let X = the units of time before the hare is caught ; 
 
 then (45 — 42) x = the distance the dog gains. 
 But 50 X 7 = " " " 
 
 3 a; = 350 
 X = 116f , the units of time. 
 
 116?^ X 42 
 .•. . S = TOO, the number of leaps the hare takes, 
 
 and llS^iiS = 583J, " " " dog " 
 
172 ALGEBRA. 
 
 8. A and B are talking of their ages ; A says to B that if 
 three fourths, three tenths, and four fifths of his age are added 
 to his age, the sum will be 3 less than three times his age. 
 What was A's age ? 
 
 9. A grocer has some tea worth 40 cents a pound, and some 
 worth 70 cents a pound. How many pounds must he take of 
 each sort to produce 100 pounds of a mixture worth 50 cents 
 a pound ? 
 
 10. A person had $1000, of which he lent part at 4 per 
 cent, and the rest at 5 per cent. The whole annual interest 
 received was $ 44. How much was lent at 4 per cent ? 
 
 Ans. $600. 
 
 11. An army in a defeat loses one sixth of its number in 
 killed and wounded, and 4000 taken prisoners. It is then 
 reinforced with 3000 men, but retreats, losing one fourth of 
 its number in doing so. There remain 18000 men. What 
 was the original number ? 
 
 12. In a certain weight of gunpowder the saltpetre was 6 
 pounds more than half of the weight, the sulphur 5 pounds 
 less than a third, and the charcoal 3 pounds less than a 
 fourth. How many pounds were there of each of the three 
 ingredients ? 
 
 13. A general, having lost a battle, found that he had left 
 fit for action 3600 men more than half of his army. 600 more 
 than one eighth of his army were wounded, and the remainder, 
 forming one fifth of the army, were slain, taken prisoners, or 
 missing. What was the number of the army at first ? 
 
 Ans. 24000. 
 
 14. A colonel, on attempting to draw up his regiment in 
 the form of a solid square, finds that he has 31 men over, and 
 that he should require 24 men more in his regiment in order 
 to increase the side of the square by one man. How many 
 men were there in the regiment ? 
 
SIMPLE EQUATIONS. 173 
 
 15. Divide the number 90 into four parts such that the 
 first increased by 2, the second diminished by 2, the third 
 multiplied by 2, and the fourth divided by 2 will all be equal. 
 
 16. A can do half as much work as B, B can do half as 
 much as C, and together they can complete a piece of work in 
 24 days. In what time can each alone complete the work ? 
 
 Ans. 168, 84, 42. 
 
 17. A regiment was drawn up in a solid square. After 
 295 men had been removed from the field, it was again drawn 
 up in a solid square, and it was found that there were 5 men 
 less in a side. What was the original number of men in the 
 regiment ? Ans. 1024. 
 
 18. A and B shoot at a target. A puts 7 bullets out of 12 
 into the bull's-eye, B 9 out of 12, and both of them put in 32 
 bullets. How many shots did each fire ? 
 
 19. A mother is 70 years old, her daughter is exactly half 
 that age. How many years have passed since the mother was 
 3^ times the age of the daughter ? 
 
 20. A basket of . oranges is emptied by one person taking 
 half of them and one more, a second person taking half of 
 the remainder and one more, and a third taking half of the 
 second remainder and six more. How many did the basket 
 contain at first ? Ans. 54. 
 
 21. A man leaves two fifths of his property to his eldest 
 son, one half of the remainder to be divided equally between 
 two younger sons, and the other half equally among three 
 daughters. What is the value of the property, if the younger 
 sons each get $ 1250 more than each of their sisters ? 
 
 Ans. 125000. 
 
 22. A trader allows $100 for expenses, and increases that 
 part of his capital which is not expended by ^ of it. At the 
 end of 3 years his stock is doubled. What had he at first ? 
 
174 ALGEBRA. 
 
 23. A man meets 3 beggars. To the first he gives half of 
 the coppers in his pocket and one more ; to the second, half of 
 the remainder and one more; and to the third half of what 
 he has left, and one more. After this he finds he has but 
 
 3 coppers. How many had he at first? Ans. 38. 
 
 24. The length of a floor exceeds the breadth by 4 feet. If 
 each dimension is increased a foot, the area of the room will 
 be increased 27 square feet. Find the dimensions. 
 
 25. A and B have the same income. A saves a fifth of his, 
 but B, by spending annually $80 more than A, at the end of 
 
 4 years finds himself 1220 in debt. What was their income ? 
 
 26. A can do half as much as B, B can do one and a half 
 times as much as C, and together they can complete a piece of 
 work in 10 days. How long would it take each separately to 
 do the work ? 
 
 27. A and B can dig a trench in 7 days, A and C in 8 days, 
 and A alone in 15 days. How many days would it take A, B, 
 and C together to dig it ? 
 
 28. A can do a piece of work in 5 days, B can work twice 
 as fast as A, and C one half as fast. Hov/^ long would it take 
 A, B, and C together to do the work ? 
 
 29. A and B can do a piece of work together in 4 days. 
 A working with a different companion, C, can do it in 3| 
 days, while B and C working together can do it in 5 f days. 
 How many days will A, B, and C each take to do it alone ? 
 
 Ans. A 6 days, B 12, C 9. 
 
 30. A tank is filled through a pipe in 30 minutes. It is 
 emptied through a waste-pipe in 50 minutes. In what time 
 will the tank be filled if both pipes are opened at once ? 
 
 31. A cistern can be filled by means of two pipes in 20 
 minutes and 30 minutes, respectively, and emptied by means 
 of a third in 40 minutes. In what time would it be filled if 
 all three were running together ? 
 
SIMPLE EQUATIONS. 175 
 
 32. A tauk contaiiiing 800 gallons has three pipes. The 
 first lets in 8 gallons in 2^ minutes, the second 10 gallons in 
 3J^ minutes, and the third 12 gallons in 5 minutes. In what 
 time will the tank he filled by the three pipes all running 
 together ? 
 
 33. A can do a piece of work in 6 ilays, B in 7 days, and C 
 in 10 days. How long wiU it take them all working together 
 to do it ? 
 
 34. A can build a boat in 18 days, but with the assistance 
 of B he can do it in 12 days. How long would it take B 
 woxking alone to do it ? 
 
 35. A can do a piece of work in 5 days, B in 3J days, and 
 C in 5^ days. How long would it require them working to- 
 gether to do it ? 
 
 36. A performs ^ of a piece of work in 15 days ; he then 
 calls in B to help him, and the two together finish the work 
 in 8 days. In how many days can each alone do the work ? 
 
 37. A can do in 20 days a piece of work which B can do in 
 30 day&. A begins the work, but after a time B takes his 
 place and finishes it. B worked 10 days longer than A. 
 How long did A work ? Ans. 8 days. 
 
 38. A deer running at the rate of "40 rods a minute was 
 first seen by a huntsman when 30 rods in advance of a hound 
 pursuing at the rate of 50 rods a minute. The huntsman 
 waited until 25 rods intervened between the hound and the 
 deer, when he shot the deer. How long did he wait ? 
 
 39. There are two places 154 miles apart, from which two 
 persons start at the same time toward each other. One travels 
 at the rate of 3 miles in 2 hours, and the other at the rate of 
 5 miles in 4 hours. Where will they meet ? 
 
 40. A privateer sailing at the rate of 10 miles an hour dis- 
 covers a ship 18 miles off sailing at the rate of 8 miles an 
 hour. How many miles can the ship sail before it is over- 
 taken ? 
 
176 ALGEBRA. 
 
 41. A freight train passes through a station at 20 miles an 
 hour, and is followed at a distance of 2 miles by an express 
 going 60 miles an hour. Where will the collision occur ? 
 
 42. A courier sets out from a certain place and travels 17 
 miles in 4 hours, and 2^ hours afterward a second courier, 
 travelling 13 miles in 3 hours, is sent after him. How long 
 and how far will the first go before being overtaken ? 
 
 43. A person walked to the top of a mountain at the rate 
 of 2 J miles an hour, and down the 'same way at the rate of 3^ 
 miles an hour, and was out 5 hours. How far did he walk ? 
 
 44. A har^ takes 4 leaps to a greyhound's 3, but 2 of the 
 greyhound's leaps are equivalent to 3 of the hare's. The hare 
 has a start of 50 leaps. How many leaps must the greyhound 
 take to catch the hare ? Ans. 300. 
 
 45. A merchant bought two pieces of cloth, the first at the 
 rate of $4 for 9 yards, and the second at that of $3 for 4 
 yards ; the second piece contained as many times 3 yards as 
 the first contained times 4 yards. He sold each piece at the 
 rate of $5 for 9 yards, and lost 15 by the bargain. How 
 many yards were there in each piece ? 
 
 Ans. First, 144 ; second, 108. 
 
 46. At what time between 5 and 6 o'clock will the hands of 
 a watch be together ? 
 
 47. At what time between 7 and 8 o'clock will the hands of 
 a watch be opposite one another ? 
 
 48. At what times between 6 and 7 o'clock will the hands 
 of a clock be at right angles ? 
 
 49. A certain fraction is equal to |, and if its numerator is 
 increased by 5 and its denominator by 9 it becomes f . Knd 
 the fraction. 
 
 50. Two laborers are employed at $3 and $5 a day each. 
 The sum of the days they worked was 40. They each received 
 the same sum. How many days was each employed ? 
 
EQUATIONS OF THE FIRST DEGREE. 177 
 
 CHAPTER XII. 
 
 EQUATIONS 
 
 OF THE FIRST DEGREE CONTAINING TWO OR MORE 
 UNKNOWN NUMBERS. 
 
 176. Independent Equations are such as cannot be derived 
 from one another, or reduced to the same form. 
 
 Thus, X + y = lO, 5 + 1 = 5, and 4x + 3y = 40 — y, are not 
 
 independent equations, since any one of the three can be derived from 
 any other one ; or they can all be reduced to the form x -\- y ^ 10. 
 But X -{- y =10 and 4x = y are independent equations. 
 
 177. Simnltaneons Equations are those which are satisfied 
 by the same values of the unlcnown numbers. 
 
 Thus, X -\- y ^ 10, and x -{- y ^7, though independent, are not 
 simultaneous equations, since no values of x and y will satisfy both 
 equations. But a; + y = 10 and ix = y are simultaneous, as well 
 as independent equations. 
 
 178. Two independent simultaneous equations are neces- 
 sary to determine the values of two unknown numbers. 
 
 For from the equation i + y = 10 we cannot determine the value 
 of either a; or y in known terms. If y is transposed, we have 
 1=10 — y; but since y is unknown, we have not determined the 
 value of X. We may suppose y equal to any number whatever, and 
 then X would equal the remainder obtained by subtracting y from 10. 
 It is only required by the equation that the sum of two numbers shall 
 equal 10 ; but there is an infinite number of pairs of numbers whose 
 sum is equal to 10. But if we have also the equation 4 a; = y, we 
 may put this value of y in the first equation, x -\- y = 10, and obtain 
 a; + 4 a; = 10, or x = 2 ; then 4 a; = 8 = y, and we have the value 
 of each of the unknown numbers. But from the two equations 
 X ■\- y = 10 and x + y = 7 we can find no values of x and y that 
 will satisfy both equation^. 
 
178 ALGEBRA. 
 
 ELIMINATION. 
 
 179. Elimination is the method of deriving from the given 
 equations a new equation, or equations, containing one 
 (or more) less unknown number. The unknown number 
 thus excluded is said to be eliminated. 
 There are three methods of elimination: 
 I. By substitution. 
 II. By comparison. 
 III. By combination. 
 
 Case I. 
 
 180. Elimination by Substitution. 
 
 . „-.. f3x + 72/ = 27. (1) 
 
 ^- ^°^^^ l5x + 22/ = 16. (2) 
 
 ,^^J^ (3) 5.+ 2(?^^) = 16 (4) 
 
 ^. „ 3.5a;+54-6a;=112 (5) 
 
 y = -J-'=^ (7) ^ = 2 (6) 
 
 Transposing Zx in (1) and dividing by 7, we have (3), which 
 
 gives an expression for the value of y. Substituting this value of 
 
 y in (2), we have (4), which contains but one unknown number ; 
 
 that is, y has been eliminated. Reducing (4) we obtain (6), or a; = 2. 
 
 Substituting this value of x in (3), we obtain (7), or 3/ = 3. Hence, 
 
 the following 
 
 Knle. 
 
 Find an expression for the value of one of the unknown 
 nmnbers in one of the equations, and substitute this value 
 for the same unknown number in the other equation. 
 
 Note. After eliminatiDg, the resulting equation is reduced by the rule 
 in Art. 163. The value of the unknown number thus found must be sub- 
 stituted in one of the equations containing the two unknown numbers, and 
 this reduced by the rule in Art. 163. 
 
EQUATIONS OF THE FIRST DEGREE. 179 
 
 Solve the following equations by substitution : ' 
 
 j Sx:^7y. S2x-3y = -U. 
 
 \l2y = 5x-l. l5y-4a:= 26. 
 
 (3a; + 42/ = 10. j 7t/-21=5a;. 
 
 I4.X+ y= 9. l21a;-9y = 75, 
 
 - y = 34. 
 + 82, = 53. 
 
 Case II. 
 181. Elimination by Comparison. 
 
 (8a; 
 ( X 
 
 i3x + 1y = 27. 
 \5x + 2y=16. 
 
 (1) 
 
 (2) 
 
 --" (3, „»7^' 
 
 (4) 
 
 27-72^ . 16-22, 
 3 5 
 
 (5) 
 
 135 — 35 !, = 48 — 6 y 
 
 (8) 
 
 y = S 
 
 (7) 
 
 . --« 2 
 
 (8) 
 
 Finding an expiession for the value of x from both (I) and (2), we 
 have (3) and (4). Placing these two values of a; equal to each other 
 (Art. 36, Ax. 8), we form (5), which contains but one unknown 
 number. Keducing (5) we obtain (7), or y = 3. Substituting this 
 value of y in (4), we have (8), or a; = 2. Hence the following 
 
 Bule« 
 
 Find an expression for the value of the same unJcTWwn 
 number from each equation, and put these expressions equal 
 to each other. 
 
 Solve the following equations by comparison : 
 
 <2x + 3y= 7. (19a! + 172,= 0. 
 
 \5x + 7y = 19. I 2a!- y = 53. 
 
 ( x + 8y = 17. (14a;- 3y = 39. 
 
 l7x-3y= 1. I 6a; + 17y = 35. 
 
180 ALGEBRA. 
 
 Case III. 
 
 1. Solve 
 
 182. Elimination by Combination. 
 
 + 72/ = 27. (1) 
 
 (3a; 
 
 15 a; + 62,= 48 (4) 
 
 + 22/ = 16. (2) 
 
 15 a; + 35 y = 135 (3) 
 
 29y= 87 (5) 5x + 6 = 16 (7) 
 
 2,= 3 (6) a:= 2 (8) 
 
 If we multiply (1) by 5 and (2) by 3, we have (3) and (4), in 
 which the coefficients of x are equal ; subtracting (4) from (3), we 
 have (5), which contains but one unknown number. Eeducing (5) 
 we have (6), or 3/ = 3 ; substituting this value of y in (2), we obtain 
 (7), which reduced gives (8), or a; = 2. Hence the following 
 
 Bule. 
 
 Multiply or divide the equations so that the coefficients of 
 the unknmon number to he eliminated shall become equal; 
 then, if the signs of this number are alike in both, subtract 
 one equation from the other ; if unlike, add the two equon 
 tions together. 
 
 Note. The least multiplier for each equation will be that which will 
 make the coefficient of the unknown number to be eliminated the least 
 common multiple of the two coefficients of this number in the given equa- 
 tions. It is always best to eliminate that unknown number whose coeffi- 
 cients can most easily be made equal. 
 
 Solve the following equations by comhination : 
 
 (5 a; + 6 2/ =17. ( 21 a; - 50 7/ = 60. 
 
 ' X&x-\-5y = lQ. ■ (-272/ + 28a; = 199. 
 
 ( 15 a; + 77 2/ = 92. ( 28 a; - 23 2/ = 33. 
 
 ' X 5x- 3y= 2. ■ ( 63 a; - 25 2/ = 101. 
 
 ( 62/ -5a; = 18, f 5 a; + 3 2/ = 68. 
 
 ■ (12 a; -9 2/= 0. ' (2 a; + Sj/ = 69. 
 
EQUATIONS OF THE FIRST DEGREE. 181 
 
 183. Solve the following equations : 
 
 Note. Which of the three methods of elimination should he used de- 
 pends upon the relations of the coefBoients to each other. That one which 
 will eliminate the number desired with the least work is the best. 
 
 
 2y)-{3x + lly) = U. (1) 
 
 9y -3(a;- 42/) =38. (2) 
 
 5a; + 103/— 3a;— 112^ = 14 7a; — 9;/ — 3a: + I23; = 38 
 
 2x— 2, = 14 (8) 4a;+3y = 38(4) 
 
 From (1) and (2) we derive (.3) and (4); reducing (3) and (4) by 
 any of the methods of elimination, we have a; = 8, and y = 2. 
 
 2. < 
 
 3.-?^-^=i^2^. (1) 
 
 ^y^-i(2.-5) = y. (2) 
 
 42a; — 2y+ 10 = 28a; — 21 9y + 12 — 10a; + 25 = loy 
 
 14a; — 23f = -3l' (3) ]0a; + 63, = 37 (4) 
 
 From (1) and (2) we obtain (3) and (1) ; reducing (3) and (4), 
 we find a; = — ^|, and y = ^^. 
 
 4:X-i(y-3) =bx-3. 
 
 4. ^|-Hy-2)-i(a;-3)=0. 
 «'-i(2/-l)-i(^-2) = 0. 
 
 ■'■{ 
 
 (x + l)(y + 5) = (x + 5) (y + 1). 
 xy + x + y=(x + 2) (y + 2). 
 
 Ans. a; = v = — 2. 
 
 6. | + | = 3a;-72/-37 = 0. 
 
 „ x + 3_8-y_3Jx±yl 
 ~T~~ 4 ■" 8 ■ 
 
182 ALGEBRA. 
 
 8 ' ^ + ^ = 3y — 5 _ x—y ^ ( a; = 19. 
 
 10 2 - 8 ■ ' \y= Z. 
 
 10. |*(^ + 2') + *(*-2') = 3^- 
 
 i (a; + y) - i (a; - 2/) = 2^. 
 
 11. J0.7a;-0.02^, = 2. (,. = 3. 
 
 <-0.7a; + 0.02y = 2.2. ty = 5. 
 
 J 2 \ 1.75 a; = 20- 0.625 y. 
 
 l2a; — 1.75 y = 0.6 a; + 7. 
 
 13. \ ^ ^ 
 
 . 8 — x „,, 2m + 1 
 
 14. ^ 
 
 .+!-. 
 
 + 7 = 0. Ans, 
 
 
 = 4. 
 
 ^-^ • "°^-l2, = 12. 
 
 3y-10(a;-l) a; - y , ^ _ ^ 
 6 + 4 +^-"- 
 
 f 2w-a; „^ 59 -2 a; 
 « - „;r = 20 ■ 
 
 15. { 
 
 16. <^ 
 
 23 - a; 2 
 
 «-j/^ 1 
 a! + y 5' 
 
 2^ 5 2/ 3a; 2/ . ^ ( a; = 18. 
 
 Ans. i^ = 
 
 3 12 2 3^2. ■ (y = 12. 
 
EQUATIONS OF THE FIRST DEGREE. 
 
 183 
 
 j,^ < ax + bi/ —0. 
 \a' X + b' y = c'. 
 
 Multiplying (1) by b'. 
 Multiplying (2) by b, 
 
 Subtracting (4) from (3), 
 
 Multiplying (1) by a', 
 Multiplying (2) by a, 
 
 Subtracting (6) from (5), 
 
 18. |«a' = *y- 
 
 Xbx + ay = 
 
 20. 
 
 21. 
 
 23. 
 
 24. 
 
 aV X -\-bb' y = b' c 
 a' bx-{-hV y =^bc' 
 
 (ah' — a'b)x = b'c — bc' 
 _ b'c — bc' 
 •■• '^~a¥ — a'b 
 a a' X + a'by = a! c 
 aa' X -\- ab'y =■ ac' 
 
 (a'b — a¥) y ^ a' c — ac' 
 
 a'c — ac' 
 
 .: y 
 
 (1) 
 (2) 
 
 (3) 
 (4) 
 
 (5) 
 (6) 
 
 19. 
 
 :b — aV 
 
 \ ax + by = a", 
 [bx + ay = b\ 
 
 a^ b~ 
 
 1 
 
 ab 
 
 
 X 
 
 y 
 
 ~b'~ 
 
 1 
 a'V 
 
 
 3a; 
 a 
 
 A^ 
 
 = 3. 
 
 
 9a; 
 a 
 
 6y 
 b 
 
 = 3. 
 
 
 qx 
 
 -rb = 
 
 --p{a- 
 
 y)- 
 
 qx 
 a 
 
 + r = 
 
 --p(l + 
 
 y\ 
 b) 
 
 Ans. 
 
 X = 
 
 y = 
 
 a + 
 
 a'b-\- 
 ¥ - 
 
 aib + 
 
 a' 
 
 ~abi' 
 b 
 aV' 
 
 22. 
 
 Ua — b)x=(a + b)y. 
 
 X + y = c. 
 
 Ans. 
 
 X = 
 
 y- 
 
 ap 
 
 br 
 P 
 
 25. { 
 
 {a — b)x+ {a + b)yz=2a^ — 2b\ 
 {a + b)x — (a — b) y = iab. 
 
 Ans. r = '* + 
 
 X 
 
 a + b. 
 b. 
 
 a b 
 
 3a'^ 6b 3" 
 
 26. 
 
 
 b a b 
 
184 
 
 27. 
 
 
 ALGEBRA. 
 
 x—a y-b_ 
 e — a c — b 
 
 = 1. 
 
 x+a y—a 
 ' a-b' 
 
 a 
 c 
 
 Ans. {-=;• 
 (y = b. 
 
 28. 
 
 29. 
 
 bx -\- cy =■ a -\- b. 
 
 {a-b)x+{a + h)y = 2{a^- 
 ax — by = a^ -\- b^. 
 
 8 9 
 
 b + a) 
 
 a+b 
 
 30. . 
 
 X y 
 
 L ^ y 
 
 Multiplying (1) by 2, 
 
 Multiplying (2) by 3, 
 
 Adding, 
 
 Whence, 
 
 Substituting in (1), 
 
 31. ^ 
 
 [5 + 5 = 3. 
 
 X y 
 
 32. ' 
 
 ■^+^ = 79. 
 X y 
 
 L6_U44. 
 a: y 
 
 An 
 
 X = a + b. 
 a — b. 
 
 (2/ = 
 
 (1) 
 
 (2) 
 
 16_18_ 
 X y ~ 
 
 = 2 
 
 a; y 
 
 15 = 23 
 
 a; 
 
 2 = 2 
 
 2/ = 3 
 
 33. < 
 
 f-^-^.2. 
 X y 
 
 -^? = 30. 
 
 a; y 
 
 34. < 
 
 '?-S^^ = i 
 
 a; 2/ 
 20f? + 3^ 
 Va; yJ 
 
EQUATIONS OF THE FIRST DEGREE. 
 
 185 
 
 36. 
 
 36. 
 
 37. 
 
 a , b 
 
 - + - = TO. 
 
 X y 
 
 
 a b 
 ix~y~'^' 
 
 
 ax by 
 
 
 ' b='- 
 ax by 
 
 
 f2 5 4 
 
 
 X 3y~2j' 
 11 11 
 
 38. < 
 
 .4a! "^y 72' 
 
 
 Ans. 
 
 Ans. 
 
 2a 
 
 m -\- n 
 26 
 
 1 
 x = - ■ 
 a 
 
 1 
 
 b' 
 
 m n 
 
 — -\ = m + w. 
 
 wa! my 
 
 n , m 2,2 
 
 X y 
 
 184. In order to solve equations which contain two un- 
 known numbers we have seen (§ 178) that we must have 
 two independent simultaneous equations. In like manner, 
 to solve equations which contain more than two unknown 
 numbers we must have as many independent simultaneous 
 equations as there are unknown iiumbers. 
 
 1. Solve 
 
 -6a: + 2y 
 8x + 3y- 
 J X + by- 
 
 5s = 13. 
 2 s = 13. 
 3s = 26. 
 
 6a; + 2y — 6z = 13 (1) 
 6x + 6j^ — 4z = 26 (4) 
 
 iy-\- 2 = 13 (5) 
 
 12+ z = 13 (12) 
 2=1 .(13) 
 
 (2) 
 (6) 
 
 Sx+ Zy— 22 = 13 
 21a; + 21j;— 14a = 91 
 21a; + 15g^— 9z = 78 (7) 
 
 (8) 
 
 Ta + Sj/ — 32 = 26 (3) 
 
 6y-^ 52 = 13 
 20 y + 5 2 = 65 
 
 (9) 
 
 262/ 
 V 
 
 = 78 (10) 
 = 3 (11) 
 
 7a; + 16- 
 
 3 = 26 a*) 
 = 2 (15) 
 
 Multiplying (2) by 2, we obtain (4), from which we subtract (1) and 
 obtain (5) ; multiplying (2) by 7 and (3) by 3, we obtain (6) and (7) ; 
 subtracting (7) from (6), we obtain (8). We have now two equations, 
 (5) and (8), containing but two unknown numbers. Multiplying (5) 
 by 5, we obtain (9), which added to (8) gives (10)^ which reduced 
 
186 ALGEBRA. 
 
 gives y = 3. Substituting this value of y in (5) and reducing, 
 we obtain z = 1. Substituting these values of y and z in (3), and 
 reducing, we obtain x = 2. 
 
 Hence, for solviug equations containing any number of 
 unknown numbeis, we have the following 
 
 Kule. 
 
 From the given equations deduce equations one less in 
 number, containing one less unknown number; and continue 
 thus to eliminate one unknown number after another, until 
 one equation is obtained containing but one unknown number. 
 Reduce this last equation so as to find the value of this un- 
 known number; then substitute this valvs in an equation 
 containing this and but one other tmknown number, and, 
 reducing the resulting equation, find the value of this second 
 unknown number ; siibstitute again these values in an equa- 
 tion containing no more than these two and one other un- 
 known number, and reduce as before ; and so continue, till 
 the value of each unknown number is found. 
 
 Note. The process can often be very much abridged by the exercise of 
 judgment in selecting the unknown number to be eliminated, the equa- 
 tions from which the other equations are to be deduced, the method of 
 elimination which shall be used, and the simplest equations in which to 
 substitute the values of the numbers which have been found. 
 
 Find the values of the unknown numbers in the following 
 equations : 
 
 ' X + y -\- s + w = 19. 
 p+«-\-w+u = 22. 
 X -{■ z + w -\- u — 21. Ans. { z =b. 
 
 M = 6. 
 
 w = 7. 
 
 a; =3. 
 
 2/ = 4. 
 
 X + y +w-\- u = 20. 
 X + y + z -\- u — lB. 
 
 Note. If these equations are added together and the sum divided by 
 4, we shall have a; + y + s + to + m = 25 ; and if from this the given 
 equations are successively subtracted, the values of the unknown numbers 
 become known. 
 
12. 
 
 EQUATIONS OF THE FIRST DEGREE. 187 
 
 2 a; + 3 y + 4 » z= 20. 
 3a; + 4?/ + 5« = 26. 
 
 3 a; + 5 y + 6 s = 31. 
 
 ix — y — s = b. (a: = 2. 
 
 4. •]3a!-4y+ 16 = 62. Ans. \y^-2. 
 3y + 2(s-l) =a;. (» = 5. 
 
 ^7a; + 3y-2s = 16. 
 
 5. ■]2a; + 52/ + 3s = 39. 
 '5a;— y + 5« = 31. 
 
 j5a; — 6?/ + 4z = 15. / a; = 3. 
 
 6- ■]7a; + 4y — 32 = 19. Ans. ■] y = 4. 
 
 2a; + y + 62 = 46. (» = 6. 
 
 2a! + 4y + 5«=:19. 
 
 7. ■|_3a; + 5y + 7«= 8. 
 
 8x-3y+5s = 23. 
 
 5a; + 6y — 12z = 5. r x = 2. 
 
 2a; — 2. y— 6z = — 1. Ans. ] ?/ = f 
 
 4x-5y+ 32 = 7^. (-- = §■ 
 
 ;*a;+ Jy + is = 23. ra; = 30. 
 
 \h<« + i!/+h^ = 27. Ans. -^y= 6. 
 
 i* + iy+43 = 17. ls = 30, 
 
 y + s — 86 = 7s — 5a;. 
 10. ^ 93-is-^2/=i2/-2g. 
 
 ■ix + iy = 12-is. , 
 11- -|i2/ + is= 8 + ^ a;. 
 .ia:+,is = 10. 
 
 a(a;-l) -1(2,-2) = Vtt(« + 3). 
 + iz==ix + 5. 
 
 I a;-J(22,-5) = lJ-Ti^2. 
 
188 
 
 ALGEBRA. 
 
 13. 
 
 X y 
 
 y z 
 i + Ur. 
 
 X z 
 
 Note. Subtract from half the 
 sum of the three equations each 
 equation successively. 
 
 14. < 
 
 ri i_ 1 
 
 a; 2/ 4 
 
 1 + 1==!. 
 
 y^ z 36 
 1 1_ 5 
 
 La: 
 
 18 
 
 Ans. 
 
 ^i-? + 4 = 0. fi+^ + ^=36. 
 
 X y X y z 
 
 15. <!?:_?: + 1 = 0. 16. <; 1 + ? _ - = 28. 
 
 2 X y z 
 
 ? + ? = 14. - + 3-+^=20. 
 
 g a; La; oy Zz 
 
 2 _A+l_3l 
 a: ^y z 27 
 
 17. ^J.+ l+? = 6l^ 
 
 A_l +l = 12l. 
 .6 a; 2/ =^ 36 
 
 18. ^ 
 
 ~ + - = a. 
 X y 
 
 X z 
 
 1 1 
 
 - + - = c. 
 
 y 2 
 
 Ans. ' 
 
 a 
 
 + b- 
 2 
 
 c 
 
 a 
 
 -6 + 
 2 
 
 c 
 
 J + c — a 
 
EQUATIONS OF THE FIRST DEGREE. 
 
 189 
 
 19. 
 
 21. 
 
 22. 
 
 23. 
 
 ri 
 
 X 
 
 1 
 
 V 
 1 
 
 - H !-- = «• 
 
 a; y 3 
 
 a; 2/ 2 
 1 _ 1 ^ 1 ^ ^_ 
 .a; y « ■ L« 
 
 Z y + m a; = w. 
 w a; + Z 3 = m. 
 ^mz + ny =: I. 
 
 2x + 2y-4:z = 2c — a—b. 
 .ax + by + cz — bo + ca + ab. 
 
 a; + 2/ = 22. 
 y + « = 24. 
 2 + w = 38. 
 w + M = 50. 
 M + a; = 26. 
 
 1 
 
 2 
 1 
 X 
 
 1 
 
 z 
 
 1 
 
 X 
 
 1 
 
 = H5 + «). 
 
 Ans. -j y = |(c + a), 
 s = i(a+b). 
 
 PROBLEMS 
 
 PKODDCING EQUATIONS OF THE FIRST DBGKEB CONTAINING TWO 
 OR MORE UNKNOWN NUMBBRS. 
 
 185. Many of the problems given in Chapter XI. contain 
 two or more unknown numbers ; but in every case these 
 are so related to one another that, if one becomes known, 
 the others become known also; and therefore the problems 
 can be solved by the use of a single letter. But many 
 problems, on account of the complicated conditions, can- 
 not be performed by the use of a single letter. No 
 problem can be solved unless the conditions given are 
 sufficient to form as many independent equations as there 
 are unknown numbers. 
 
190 ALGEBRA. 
 
 1. Find two numbers such that the greater exceeds twice 
 the less by 3, and twice the greater exceeds the less by 27. 
 
 Let X = the greater number, 
 
 and y = the less ; 
 
 then, by the conditions, x — 2 ;/ = 3 
 and 2 a; — 2^ = 27 
 
 Solving these equations by either of the preceding methods, we 
 have X = VI and y = 1. 
 
 2. If the numerator of a fraction is increased by 2 and the 
 denominator by 1, it becomes equal to f ; and if the numera- 
 tor and denominator are each diminished by 1, it becomes 
 equal to \. Find the fraction. 
 
 X 
 
 Let - = the fraction : 
 
 y 
 
 X JL 2 5 X 1 1 
 
 then, by the conditions, —^ = - and ——j = - . 
 
 Solving these equations, we have a = 8 and y = 15, and the 
 fraction is ^. 
 
 3. A number consists of three digits whose sum is 10; the 
 middle digit is equal to the sum of the other two, and, if 99 
 is added to the number, the order of the digits is reversed. 
 Find the number. 
 
 Let X = the digit in the units' place, 
 
 y = " " tens' place, 
 
 z = " " hundreds' phice ; 
 
 then 100 2 -f 10^ + x = the number. 
 
 By the conditions, x -\- y -\- z= \Q 
 x + z = y 
 and 100z+ 10?/ + a; + 99= 100a: + 10^/ + z 
 
 Solving these equations, we have x ^ S, y == 5, z = 2. 
 .•. The number is 253. 
 
 4. A and B walk for a wager on a course of one mile in 
 length. At the first heat, A gives B a start of 45 seconds, 
 
EQUATIONS OF THE FIRST DEGREE. 191 
 
 and beats him by 110 feet. At the second heat, A gives B a 
 start of 484 feet, and is beaten by 6 seconds. Required the 
 rates at which A and B walk. 
 
 Let X = the number of feet A walks a second. 
 
 y— il >> B " 
 
 By the conditions of the problem, 
 
 5280 5170 
 
 __ + 4^ = __ (1) 
 
 5280 4796 
 
 6 = (2) 
 
 X y ^ ^ 
 
 374 
 Subtracting (2) from (1), 51 = — 
 
 Substituting this value of y in (2), we find 
 
 x==S 
 .•. A and B walk 8 and 7J feet per second, respectively. 
 
 6. I row 8 miles with the stream in 1 hour 4 minutes, and 
 return against the stream in 2^ hours. At what rate would I 
 row in still water, and at what rate does the stream flow ? 
 
 Let x ^ number miles an hour in still water, 
 
 y = number miles the stream flows an hour ; 
 then, by the conditions of the problem, 
 
 8 15 
 
 x + y = 
 
 ^H 
 
 2 
 
 
 8 
 
 7 
 
 x — y = 
 
 -2f- 
 
 2 
 
 Add 2 a: =11 
 
 a; = 5^ 
 Substituting this value of x in first equation, we find y = -• 
 
 6. The denominator of a fraction exceeds the numerator by 
 4 ; and if 6 is taken from each, the sura of the reciprocal of 
 the new fraction and 4 times the original fraction is 5. Find 
 the original fraction. 
 
192 ALGEBRA. 
 
 7. A, B, C, and D, together, bave $270. A has three times 
 as much as C, and B five times as much as D ; and A and B 
 together have |50 less than eight times what C has. Find 
 how much each has. 
 
 8. One eleventh of A's age is greater by two years than 
 one seventh of B's, and twice B's age is equal to what A's age 
 was thirteen years ago. Find their ages. 
 
 9. Find two numbers in the ratio of 3 to 4, such that, if the 
 first is increased by 7 and the second doubled, these numbers 
 will be in the ratio of 2 to 3. 
 
 10. A certain fraction becomes ^ when its numerator is 
 doubled and its denominator increased by 1 ; but if its nu- 
 merator is increased by 1 and its denominator diminished by 
 1, it becomes ^. Find, the fraction. 
 
 11. A certain fraction is doubled by adding 14 to its nu- 
 merator and 6 to its denominator, and it is trebled by adding 
 7 to its numerator and taking 4 from its denominator. Find 
 the fraction. Ans. -/j. 
 
 12. If § is added to the numerator of a certain fraction, the 
 fraction will be increased by ^ ; and if ^ is taken from its 
 denominator, the fraction becomes f . Find the fraction. 
 
 13. The middle digit of a number between 100 and 1000 is 
 zero, and the sum of the other digits is 11. If the order of 
 the digits is reversed, the number so formed exceeds the 
 original by 495. Find the number. Ans. 308. 
 
 14. A number of 3 digits has the right-hand digit zero. 
 If the left-hand and middle digits are interchanged, the num- 
 ber is diminished by 180 ; if the left-hand digit is halved and 
 the middle and right-hand digits are interchanged, the num- 
 ber is diminished by 454. Find the number. 
 
EQUATIONS OF THE FIRST DEGKEE. 193 
 
 15. In a number of 4 digits the sum of the first and third 
 is equal to the sum of the second and fourth ; and, if the order 
 of the digits is reversed, the number is increased by 1089. 
 The difference between the third and fourth digits is three 
 times the difference between the first and fourth, and the sum 
 of the digits is 20. Find the number. 
 
 16. If I divide a certain number by the sum of its two 
 digits, the quotient is 6 and the remainder 3. If I reverse 
 the order of the digits and divide tlie resulting number by 
 the sum of the digits, the quotient is 4 and the remainder 9. 
 Find the number. 
 
 17. If I divide a certain number by the sum of its two 
 digits diminished by 2, the quotient is 5 and the remainder 1. 
 If I reverse the order of the digits and divide the resulting 
 number by the sum of the digits increased by 2, the quotient 
 is 5 and the remainder 8. Find the number. Ans. 36. 
 
 18. Having 1 45 to distribute among a number of persons, 
 I find that for a distribution of $3 to each man and $1 to 
 each woman I shall lack $1, but I can give $2J to every man 
 and $ 1 J to every woman, and have nothing left. How many 
 men and women were there ? 
 
 19. A man put $12000 at interest in three sums, the first 
 at 5 per cent, the second at 4 per cent, and the third at three 
 per cent, receiving for the whole $ 490 a year. The sum at 
 5 per cent is half as much as the other two sums. What are 
 the three sums ? Ans. 14000, $5000, $3000. 
 
 20. As John and James were talking of their money, John 
 said to James, " Give me 16 cents, and I shall have four times 
 as much as you will have left." James said to John, "Give 
 me 8 cents, and I shall have as much as you will have left." 
 How many cents did each have ? 
 
 Ans. John, 48 cents ; James, 32 cents. 
 13 
 
194 ALGEBKA. 
 
 21. An income of $120 a year is derived from a sum of 
 money invested partly in 3J per cent stock, and partly in 4 
 per cent stock. If the stock should be sold out when the 3J 
 per cents are at 108 and the 4 per cents at 120, the capital 
 realized would be $3672. How much stock of each kind is 
 there ? Ans. $2400 3^ per cent, $900 4 per cent. 
 
 22. A market-man bought eggs, some at 3 for 5 cents and 
 some at 4 for 5 cents, and paid for the whole 1 4.60 ; he after- 
 ward sold them at 24 cents a dozen, clearing $0.80. How 
 many of each kind did he buy ? 
 
 23. A fishing-rod consists of two parts ; the length of the 
 upper part is to the length of the lower as 5 to 7 ; and 9 times 
 the upper part together with 13 times the lower part exceeds 
 11 times the whole rod by 36 inches. Find the lengths of the 
 two parts. 
 
 24. There is a rectangular floor, such that, if it were two 
 feet broader and three feet longer, it would be sixty-four 
 square feet larger ; but if it were three feet broader and two 
 feet longer, it would be sixty-eight square feet larger. Find 
 the length and breadth of the floor. Ans. 14, 10. 
 
 25. A rectangle is of the same area as another which is 6 
 meters longer and 4 meters narrower; it is also of the same 
 area as a third, which is 8 meters longer and 5 meters nar- 
 rower. What is its area ? 
 
 26. Find the length and breadth of a rectangle, such that, 
 if 3 feet were taken from the length and added to the breadth, 
 its area would be increased by 6 square feet ; but if 3 feet 
 were taken from its breadth and 4 feet added to its length, its 
 area would be diminished by 20 square feet. 
 
 27. The fore-wheel of a carriage makes 6 revolutions more 
 than the hind-wheel in going 120 yards, but if the circumfer- 
 ence of j;he fore-wheel were increased by a fourth of its present 
 length, and that of the hind-wheel by a fifth of its present 
 
EQUATIONS OF THE, FIRST DEGEEE. 195 
 
 length, the fore-wheel would make 4 revolutions more than 
 the hind-wheel in going 120 yards. Find the circumferences 
 of the two wheels. Ans. 4 yards and 5 yards. 
 
 28. A crew which can row 12 miles an hour down a river, 
 finds that it takes twice as long to row up the river as to row 
 down. At what rate does the water flow ? 
 
 29. A man sculls for 1 hour and 40 minutes down a stream 
 which runs at a rate of 4 miles an hour. In returning, it 
 takes him 4 hours and 15 minutes to arrive at a point 3 miles 
 short of his starting-place. Find the distance he scuUed down 
 the stream, and the rate of his sculling. 
 
 Ans. 20 miles ; 8 miles an hour. 
 
 30. A person swimming in a stream which runs 1^ miles 
 an hour finds that it takes him 4 times as long to swim a mile 
 up the stream as it does to swim the same distance down. At 
 what rate does he swim ? 
 
 31. A boat's crew row 9 miles with the tide in f of an 
 hour, and when the tide is flowing at half its former rate the 
 same crew row 9 miles against the tide in 1 J hours. Required 
 the rate of the stronger tide, and the rate at which the crew 
 can row in still water. 
 
 32. A and B can perform a piece of work together in 4 
 days, A and C in 3| days, and B and C in 5| days. Find 
 the time in which each can perform the work alone. 
 
 33. A and B can together perform a certain work in 30 
 days ; at the end of 18 days, however, B is called off, and A 
 finishes it alone in 20 more days. Find the time in which 
 each can perform the work alone. 
 
 34. A and B working together can do a piece of work in 
 2| days. It can also be done if A works 3 days and B 2 days. 
 In what time can each of them do the work alone ? 
 
 Ans. 6 days ; 4 days. 
 
196 ALGEBRA. 
 
 35. A cistern has 3 pipes opening into it. If the first 
 should be closed, the cistern would be filled in 10 minutes ; if 
 the second, in 15 minutes ; and if the third, in 20 minutes. 
 How long would it take each pipe alone to fill the cistern, and 
 how long would it take the three together ? 
 
 36. A and B together completed a piece of work in 2^ 
 days; but if A had worked one half as fast, and B twice as 
 fast, they would have finished it in 3}^ days. In how many 
 days could each alone perform the work ? 
 
 37. 24 Ovids and 12 Caesars will just fill a certain shelf. 
 6 Ovids and 10 Caesars will fill half of it. How many of each 
 alone will fill it ? 
 
 38. A and B run a mile. At the first heat A gives B a 
 start of 20 yards, and beats him by 30 seconds. At the 
 second heat A gives B a start of 32 seconds, and beats him by 
 9/j- yards. Knd the rate at which A runs. 
 
 39. A fox is pursued by a greyhound, and is 60 of her own 
 leaps before him. The fox takes 3 leaps in the time that the 
 gi'eyhound takes 2 ; but the greyhound goes as far in 3 leaps 
 as the fox does in 7. In how many leaps will the greyhound 
 catch the fox? 
 
 40. Three men. A, B, and C, had together a certain sum of 
 money. If ow, if A gives to B and C as much as they already 
 have, and then B gives to A and C as much as they have after 
 the first distribution, and again C gives to A and B as much 
 as they have after the second distribution, they will each have 
 $ 6. How much did each have at first ? 
 
 Ans. A $9|, B $5^, and C $3. 
 
 41. A and B run a mile. First, A gives B a start of 44 
 yards, and beats him by 51 seconds ; at the second heat, A 
 gives B a start of 1 minute 15 seconds, and is beaten by 88 
 yards. Find the time in which A and B each can run a mile. 
 
EQUATIONS OF THE FIRST DEGREE. 197 
 
 42. Two baskets contain mixtures of wheat and barley. 
 In the first there is 3 times as much wheat as barley, and 
 in the second 6 times as much barley as wheat. Find how 
 much must be taken from each to fill a third basket which 
 holds 7 liters, in ofder that its contents may be half wheat 
 and haK barley. 
 
 Ans. 4 liters from the first, and 3 from the second. 
 
 43. There are twcf alloys of silver and copper, of which one 
 contains twice as much copper as silver, and the other three 
 times as much silver as copper. How much must be taken 
 from each to weigh a kilogram, of which the silver and the 
 copper shall be equal in quantity ? 
 
 44. A and B are travelling on roads which cross each other. 
 When B is at the point of crossing, A has 675 meters to go 
 before he arrives at this point, and in 5 minutes they are 
 equally distant from this point ; and in 40 minutes more, they 
 are again equally distant frojn it. What is the rate of each ? 
 
 Ans. A's, 75 ; B's, 60 meters a minute. 
 
 45. A sets out from C to D. Three hours afterward B sets 
 out from D to C, travelling 2 miles an hour more than A. 
 When they meet, their distances are as 13 : 15. Now, had A 
 travelled 5 hours less, and B 2 miles an hour faster, their dis- 
 tances would have been as 2 : 5. How many miles did each 
 go an hour, and how many hours did each travel before 
 meeting ? 
 
 46. A and B, whose times to reap an acre are as 2:3, 
 engaged to reap a field in 12 days. Finding themselves un- 
 able to finish it, they take in C the last few days. C's rate of 
 working was such that, if he had worked with them from the 
 beginning, they could have finished it in 9 days. And the 
 time in which C could reap it with A and B severally is as 
 7:8. When was C called in ? 
 
 Ans. C was called in after A and'B had worked 6. days. 
 
198 ALGEBRA. 
 
 CHAPTER XIIT. 
 
 GENERALIZATION. 
 NEGATIVE ANSWERS. DISCUSSION OF PROBLEMS. 
 
 186. Since letters stand for any numbers whatever, the 
 answer to a problem in which the given numbers are 
 represented by letters is a general expression including all 
 cases of the same kind. Moreover, the operations per- 
 formed with, letters are not, as with figures, lost in the 
 combinations of the process, but all appear in the resulting 
 expression. Such an expression is called a formv.la, and 
 when expressed in words a, rule. 
 
 187. To illustrate the making of formulas, and their use, 
 the following problems are solved. 
 
 1. If A can do a piece of work in a hours and B can do it 
 in h hours, how long will it take A and B together to do the 
 work ? 
 
 Let X = the required number of hours ; 
 
 *i, 1,11 
 
 then - -|- 
 
 a h X 
 hx -{- ax ^ ah 
 a h 
 
 the required number of hours. 
 
 a + 6' 
 
 That is, — -j-r is the formula that expresses, in terms of the time 
 
 in. which each can do it alone, the time it will take two men together 
 to do a given piece of work. Stated in words, it will be as follows : 
 Given the time it takes each of two men to do a piece of work, to find 
 the time it will take the two to do it together. 
 
GENERALIZATION. 199 
 
 Rule. 
 
 Divide the product iy the sum of the numbers expressing 
 the time it will take each to do it alone. 
 
 2. If A can do a piece of work in 5 days and B can do it 
 in 7 days, how long will it take A and B together to do tl^e 
 work ? A 5 • 7 oi 1 1 
 
 An infinite number of examples can be made by changing the 
 numbers to represent a and b. 
 
 3. If A can do a piece of work in a days, B in J days, and 
 C in c days, how long will it take A, B, and C together to do 
 the work ? 
 
 Let X = the required number of days ; 
 
 then i + l + -^ = ^ 
 
 a c X 
 
 hex -\- acx -\- ahx = abc ' 
 
 ahc 
 
 ^ — , , , , , the required number of days. 
 ab -{- be -\- ac 
 
 That is, ■ , , — I is the formula that expresses, in terms of 
 
 ' ab + bc -\- ac '■ ' 
 
 the time it takes each to do the work, the time it will take three men 
 together to do a piece of work. Stated in words, it will be as fol- 
 lows : Given the time it takes each of three men to do a piece of 
 work, to find the time it will take the three to do it together, 
 
 Kale. 
 
 Divide the product hy the sum of as many products as can 
 he made hy taking the three given numbers in pairs. 
 
 4. If A can do a piece of work in 3 hours, B in 4 hours, 
 and C in 5 hours, how long will it take A, B, and C together 
 to do the work ? 
 
 Ans. , or , or lii hours. 
 
 3.4 + 4.5 + 3.5' 12 + 20 + 15' '^ 
 
 Apply this Rule to E.ics. 33, 35, p. 175. 
 
, the greater part ; 
 , the less part. 
 
 200 ALGEBKA. 
 
 5. Find the formula and state the rule when 4 men are 
 employed. 
 
 6. Find the formula and state the rule when 5 men are 
 employed. 
 
 7. Divide a into two parts such that the greater exceeds 
 the less hy b. 
 
 Let a; = the greater part ; 
 
 then X — 6 = the less part. 
 
 .-. 2a; — 6 = a 
 
 22 = 0+6 
 
 a + b 
 
 ^ = ^- 
 
 , a — b 
 
 In this example a represents the sum and b the difference of two 
 
 numbers, and , — - — , are respectively the formulas for the 
 
 two numbers. Stated in words,.it will be as follows : Given the sum 
 and difference of two numbers, to find the numbers, 
 
 Rale. 
 
 (1) Divide the sum plus the difference of the two numbers 
 hy two, and it will give the greater. 
 
 (2) Divide the sum minus the difference of the two num- 
 bers hy two, and it will give the less. 
 
 8. By provision of the father's will, the sum of $5500 is to 
 be divided between his two sons so that the elder shall receive 
 $500 more than the younger. Find the share of each. 
 
 9. A and B together have 730 sheep, and A has 100 more 
 than B. How many has each ? 
 
 10. The sum of two numbers is 13.5, and their difference 
 is 1. Find the numbers. 
 
 11. The sum of two numbers is 2 a, and their difference is 
 2 h. Find the numbers. 
 
GENERALIZATION. 201 
 
 12. Three men,- A, B, and C, form a partnership, and ad- 
 vance money in sums represented by ^, p\ p", respectively. 
 They gain a certain number of dollars, which we will repre- 
 sent by g. How ought the gain to be divided ? 
 
 Let px = the number of dollars in A's share, 
 
 then p'x= " " « B's " 
 
 and p"x= " " « Cs « 
 
 .-. px + p'x -\-p" X = g 
 
 ^ = 9. 
 
 P + P' + P" 
 gp 
 Whence, — ; — rn — r, = A's share, 
 
 P+ P +P 
 
 ^^ ^^ = B'3 « 
 
 p + p'+ p" 
 
 ^P - Cs 
 
 P+P'+ P" 
 
 13. If each of the sums named in Example 12 is invested a 
 certain time, say t, t', t", respectively, how ought the gain to 
 be divided ? 
 
 In this case ptx, p' I' x, pf' t" x, would represent respectively the 
 number of dollars in the share of each. That is, 
 pt X ■]- p' t' X -\- p" t" X = g 
 
 9 
 pt + p't' + p"t" ■ 
 
 •"• pt + p't' +p"f''^-^'^''^^''^' 
 ffP't' 
 p i + p' t' + p" t" 
 gp"t 
 
 pt+p't'+ p" t" 
 
 = B's 
 = C's 
 
 14. A, B, and C form a partnership. A puts in $4000, 
 B $ 5000, C 1 6000. They gain $ 3000. How ought the gain 
 to be divided ? 
 
 15. Divide $200 between John, Charles, and William, so 
 that John shall have 2 dollars as often as Charles has 3 and 
 William 5. 
 
202 ALGEBRA. 
 
 16. A, B, and C form a partnership. A furnishes $700 for 
 11 months, B $ 1100 for 8 months, and C $ 900 for 12 months. 
 They gain $ 1365. What is each man's share of the gain ? 
 
 17. A, B, and C hought a horse for $ 200, and sold him for 
 $250, by which A gained $25 and B $10. How much had 
 A, B, and C each paid for the horse ? 
 
 18. What is the interest of f dollars for t years at r per 
 cent ? What the amount ? 
 
 Let i represent the interest, and a the amount. 
 
 The interest = principal, X time, X rate. 
 .•. i^ptr, and a=^ p -\- ptr 
 
 These formulas contain four different things, any one of which 
 may be determined when the others are known. Deducing, for ex- 
 ample, the value of t, we have 
 
 i a — p 
 
 t: 
 
 pr p r 
 
 19. At what rate must $300 be put on interest to gain $18 
 in 2 years ? 
 
 20. What principal at 6% will amount to $130.39 in 8 
 months ? 
 
 21. How long must $254 be on interest at 5% to gain 
 $44.45? 
 
 22. How long will it take any sum of mone}' to double 
 itself on interest at 5% ? 
 
 23. How long will it take any sum to quadruple itself on 
 interest at 10% ? 
 
 24. What is the amount of p dollars, at compound interest, 
 for n years, at r per cent ? 
 
 The amount of one dollar, at compound interest, for one year, will 
 be represented by 1 -|- r ; that oi p dollars will be therefore p (1 + r). 
 For the second year^ (1 -J- r) will be the principal, and its amount 
 will he p {1 + r) (1 + r), or p (l + ry-, for the third year jo (1 + r)\ 
 and so on. Therefore, putting a for the amount for n years, we have, 
 a =;) (1 + )■)". 
 
GENERALIZATION. 203 
 
 25. What is the amount of $200 for 4 years, at compound 
 interest, at 5% ? 
 
 Ans. a =^ (1 + »•)" = $200 (1.05)* = $243.10+. 
 
 26. What principal, at compound interest at 6%, will 
 amount to $357.3048 in 3 years? 
 
 27. Find the amount of $300, at compound interest, for 
 2 years 6 months, at 8%, the interest being compounded 
 semiannually. 
 
 188. MISCELLANEOUS EXAMPLES. 
 
 1. Two persons have together $a. One has m times as 
 much as the other. How much has each ? 
 
 2. If $ a are divided among three persons, A, B, and C, so 
 that B has $p more than A, and G $q more than B, how 
 much has each ? 
 
 3. If the mth and «th parts of a sum of money amount to 
 $«., what is the sum ? 
 
 4. A certain number, a, is to he divided into two parts so 
 that the mth part of the first, together with the wth part of 
 the second, is equal to h. What are the parts ? 
 
 Ans. (nb — a), (mi — a). 
 
 n — m m — n 
 
 5. A man spends the mth part of his income on board and 
 lodging, the Mth part on clothes, the ^th part on amusements, 
 the 2'th part in charities, and at the end of the year finds that 
 he has $ a left. What was his income ? 
 
 6. The number a is divided into two parts, so that the ^ 
 of the first exceeds ^ of the second by b. Find the parts. 
 
 7. If $s are divided among A, B, and C so that B has $a 
 more than A, and C $& more than B, how much has each ? 
 
 Ans. A$i(s-2a-b),B$i(s + a-b),C$i{s + a + 2b). 
 
204 ALGEBKA. 
 
 8. If A can do a piece of work in a hours, B in J hours, and 
 C in c hours, liow long will tliey take working all together ? 
 
 9. I row a miles down stream in h minutes, and return in 
 c minutes. Find the rate at which I row in still water, and 
 the rate at which the stream flows. 
 
 10. A has to write m lines, which he can do in jp minutes. 
 If B helps him they can do the work together va. p — q min- 
 utes. How long would it take B to do the work? 
 
 Ans. {p ^ q) min. 
 
 11. How many pounds of tea at m cents a pound must be 
 mixed with p pounds at n cents a pound, that the mixture 
 may be sold at s cents a pound ? 
 
 12. A courier, A, travelling p miles in q hours, is followed 
 at an interval of m hours by another, B, travelling r miles in 
 s hours. How many hours will elapse before B overtakes A ? 
 
 Ans. JItPl_. 
 q r — ps 
 
 13. A and B can do a piece of work in a hours, B and C 
 in c hours, A and C in 6 hours. How long will each take 
 separately ? 
 
 14. Two men set out at the same time to walk, one from A 
 to B, and the other from B to A, a distance of a miles. The 
 former walks at the rate of p miles, and the latter at the rate 
 of q miles an hour. At what distance from A will they meet? 
 
 15. A man spends c dollars in buying two kinds of silk, 
 at a dollars and h dollars a yard, respectively. He could buy 
 three times as much of the first and half as much of the 
 second for the same money. How many yards of each does 
 
 J^eW? Ans. ^, l£. yards. 
 
 6 a 56 
 
 16. A man rides one third of the distance from A to B at 
 the rate of a miles an hour, and the remainder at the rate of 
 2 h miles an hour. If he had travelled at a uniform rate of 
 
GENERALIZATION. 205 
 
 3 c miles an hour, he coulil have ridden from A to B and hack 
 
 2 11 
 
 again in the same time. Prove that - ^ - -) — . 
 
 cab 
 
 17. Divide the numher a into three parts, so that the 
 first is to the second as m : n, and the second to the third 
 as p : q. 
 
 18. A, B, and C start at the same time for a town p miles 
 distant. A walks at a uniform rate of m miles an hour, 
 and B and C drive at a uniform rate of n miles an hour. 
 After a time B gets down and walks forward at the same 
 rate as A, while C returns to meet A. A mounts with C, 
 and they enter the town at the same time as B, C driving 
 uniformly throughout. Show that the time of the journey is 
 p _ 3n + m ^^^^^_ 
 
 n 3 m + ra 
 
 INTERPRETATION OF NEGATIVE RESULTS AND OF 
 
 THE FORMS ^, ^, % 
 A 
 
 A 
 189, To explain negative results and the forms -, , — , 
 
 and -, we work the following problems. 
 
 1. A man and his son are employed to build a wall. At 
 one time the father worked 7 days, and his son was with him 
 5 days, and for this time the payment was $ 16.75. At another 
 time the father worked 5 days, and his son was with him 3 
 days, and for this time the payment was $ 12.25. What were 
 the daily wages of the father and of the son ? 
 
 Ans. Father's, $2.75; son's, —$ 0.50. 
 
 The negative answer for the son's wages shows that the son, instead 
 of assisting the father, was a hindrance to him, and caused the father 
 a loss of $ 0.50 every day the son was with him. Is it right to say 
 that the son is " employed," and to speak of the son's " wages "I If 
 the example stated that the son did no work, hut every day he was 
 with the father he had to pay |0.50 a day for his hoard, the answers 
 would hoth he positive. 
 
206 ALGEBRA. 
 
 2. A is 40 years old, and twice as old as B. How many- 
 years hence will A be three times as old as B ? 
 
 Ans. — 10 years. 
 
 — 10 years "hence" means 10 years ago (§ 38). Change "hence 
 will A be " to ago was A, and the answer will be positive. The 
 negative answer shows that one cannot be three times as old as an- 
 other at a point of time after he is twice as old. Arithmetically the 
 problem is impossible (§ 39, last paragraph). 
 
 Negative answers to problems can usually be interpreted, and the 
 problem can generally be worded so as to change the negative to 
 positive answers. 
 
 3. M and N are points on a straight road, m miles apart. 
 A starts from M toward N at a miles an hour, and, A hours 
 after A starts, B starts from N at J miles an hour. How far 
 from N will A and B be together ? 
 
 A B 
 
 M N R 
 
 Let M R represent the road ; M, A's starting point ; N, B's start- 
 ing point ; and R, the point where A and B are together. 
 
 Let X = the number of miles from N to R. 
 
 Then — — — == the number of hours A travels, 
 
 a 
 
 and - ^= the number of hours B travels. 
 
 
 
 Then ^ = T + * 
 
 a 
 
 bx -\- bm = ax -\- abh 
 
 ax — bx = bm — abh 
 
 ^^ ^^'"-f^ .NtoR. 
 a — 
 
 (1) Let a > 6, and m > ah. 
 
 Then x, or N R, is positive, and R, the point where A and B are 
 together (in the positive direction, § 39) is at the right of N. This 
 is as it should be; for since m> ah, B is at the right of A when 
 B starts, and, as a > 6, A travels faster than B, and wiU overtake 
 B somewhere at the right of N. 
 
GENERALIZATION. 207 
 
 (2) Let a <.b, and m < ah. 
 
 Then x, or N R, iis positive, and E, the point where A arid B are 
 together, is at the right of N. This is as it should be ; for since m<a}i, 
 A has already passed N when B starts, and A is at the right, or, ahead 
 of B, when B starts, and, as a < 6, B travels faster than A, and will 
 overtake A somewhere at the right of N. 
 
 (3) Let o < 6, and m> ah. 
 
 Then x, or N R, is negative, and R, the point of meeting, if it take 
 place, must be (in the negative direction) at the left of N. This is 
 as it should be; for since m > aA, B is at the right of A when B 
 starts, and, as a < 6, A travels slower than B, and can never overtake 
 B. To make the problem possible under this hypothesis, we must 
 suppose that, at the moment of B's starting, A turns round and walks 
 ill the opposite direction, and B follows him toward M. Then R, 
 where they are together, will be at the left of N. 
 
 (4) Let a > 6, and m <, ah. 
 
 Then X, or N R, is negative, and R, the point of meeting, if it take 
 place, is at the left of N. This is as it should be ; for, as in (2), 
 A has passed N when B starts,- andj as A travels faster than.B, B can 
 never overtake A. To make the problem possible, when B starts, 
 both A and B, as in (3), must walk toward M. 
 
 (5) Let a > 6, or a < 6, and m = ah. 
 
 Then x, or N R, = — ^ = 0. 
 
 a — 
 
 This is as it should be ; for since m = ah, A. and B are together at 
 
 N, and, since they travel at different rates, they will never be together 
 
 again ; therefore R, the place of meeting, is at N, and nowhere else. 
 
 (6) Let a = 6, and m> ah, ov m <ah. 
 
 Then :r, orNR, = ^("'~"^). 
 
 Such a result may be regarded as a sign of impossibility; for whatever 
 time we allow the two travellers, they can never come together, since, 
 being once separated a given distance, and travelling equally fast, 
 that distance will always be preserved. Nevertheless, such expres- 
 
 A 
 
 sions as -jr are considered by mathematicians as a value, to which they 
 
 give the name of infnity. For if we suppose a — 5 to be very small. 
 
208 ALGEBKA. 
 
 say one inch an hour, in the course of many miles A and B would 
 come together, and if this small gain were divided by a number as 
 large as we can imagine, R would be at an infinite distance from M. 
 
 A 
 Hence we say — ^ oo, which means that, by taking the denomina- 
 tor as small as we like, we can make the result as great as we can 
 conceive. 
 
 (7) Let a = 6, and m = ah. 
 
 Then x = 1- 
 
 
 
 Since m^ ah, A and B are together at N, and since a = b, they are 
 always together; that is, R is at any point on the line N R, and there 
 are as many solutions as one pleases. Therefore, the expression 
 
 - is called a symbol of indetermination. 
 
 EXAMPLES FOR PRACTICE. 
 
 190. Solve the following problems, and write each 
 problem in a form to make it a possible arithmetical 
 problem. 
 
 1. A, who is 30 years old, is three times as old as B. How 
 many years hence will he be four times as old ? 
 
 2. What two numbers are those whose difference is 48, and 
 sum 22 ? 
 
 3. A cistern is being filled by three pipes. One pipe alone 
 would fill it in 20 minutes, another in 30 minutes, and the 
 cistern is filled in 40 minutes. In how many minutes would 
 the third pipe fill it ? 
 
 4. A starts upon a walk at the rate of 4 miles an hour, and 
 after 15 minutes B starts from the same place at the rate of 
 3| miles an hour. Where will B overtake A ? 
 
 5. A and B have $8, A and C $10, and B and C have $35. 
 How many has each ? 
 
GENERALIZATION. 209 
 
 6. A garrison of 1000 men was victualled for 30 days. 
 After 10 days it was reinforced, and then the provisions 
 lasted 25 days more. How many men were there in the 
 reinforcement ? 
 
 7. A man invested $ 3000 in 3 per cents at 98, and on their 
 rising sold out, and reinvested the proceeds in 4 per cents at 
 120, but did not change his income. Find the amount of the 
 rise. 
 
 8. I began business with $ a, but having lost a certain sum 
 I find that b times what I have left is equal to c times what I 
 had at first. How much did I lose ? 
 
 If 6 = 3 and c = 4, what modification must be made in the 
 question to make it possible ? 
 
 9. T began business with f a, but having lost a certain sum 
 I find that b times what I have left is the difference between 
 c times what I had at first and d times what I lost. How 
 many dollars did I lose? 
 
 Examine and interpret the results : 
 
 (1) When rf = 4, 6 = 7, c = 8. 
 
 (2) " rf = 4, 6 = 7, c = 6. 
 
 (3) " rf = 4, 6 = 4, c = 6. 
 
 (4) " d = 6 = c = 4. 
 
 10. A cistern whose capacity is d gallons is filled by two 
 pipes, A and B, and emptied by a waste-pipe, C, through 
 which there flow a, b, and c gallons a minute, respectively. 
 If A is opened for p minutes, and then closed, and then B 
 and C are opened simultaneously, in how many minutes after 
 B and C are opened will the cistern be full ? 
 
 Let 
 
 
 X = the number of minutes. 
 
 Then 
 
 (6- 
 
 - c) X = d — ap 
 
 - 
 
 
 X = -^ the number of minutes. 
 
 h — c 
 
 14 
 
210 ALGEBRA. 
 
 Interpret the results : 
 
 (1) If 6 = 5, c = 6. 
 
 (2) " 6 = 4, c = 3, d = 40, a = 5, jo = 8. 
 
 (3) " 6 = 6, c = 6, d = 40, a = 5, J9 = 7. 
 
 (4) " 6 = 6, c = 6, rf = 40, a = 5, ;^ = 8. 
 
 (1) X = -= — , a negative result. 
 
 ' 5 — 6 
 
 That is, the discharge pipe c being the greater, more runs out than in. 
 The question should then read. In how many minutes will the cis- 
 tern be emptied? 
 
 40 — 5 . 8 „ 
 
 That is, the cistern is full at the moment A is closed. 
 
 40 — 5 ■ 7 _ 5 
 (3) ^— 6 — 6 "0' 
 
 That is, when A is closed, the pistern lacks 5 gallons of being full, 
 and the number of gallons in the cistern will neither be increased nor 
 diminished after A is closed. But if c is very little less than b, say 
 one drop, then in the course of millions of minutes the cistern would 
 be full. By decreasing the size of the drop, we could increase the 
 number of minutes to as many as we please ; we therefore consider 
 the answer infinitely great, or infinity. 
 
 40 — 5 . 8 
 
 (4) - = -«z:6- = o- 
 
 That is, when A is closed, the cistern is full, and will always remain 
 full, since just as much flows in as flows out. The value of x is 
 therefore indeterniinate. 
 
 11. A hound pursues a hare, and makes b leaps while the 
 hare makes c leaps ; but d hound-leaps equal e hare-leaps. The 
 hare has a start of a leaps ; when will the hound overtake 
 the hare ? 
 
 Interpret the results : 
 
 (1) When a = 50, ft = 7, c — d, d == 5, e = 6. 
 
 (2) " a = 50, 6 = 9, c =- 7, rf = 6, c = 5. 
 (5) " a = 0, i = 7, c = 9, rf = 5, e = 6. 
 (t) " a == 50, ft = 6, c = 8, d = 3, e = 4. 
 (.j) " „ = 0, ft = 6, c = 8, d = 3, e = 4. 
 
INDETERMINATE EQUATIONS. 211 
 
 CHAPTEK XIV. 
 
 INDETERMINATE EQUATIONS. 
 INEQUALITIES.* 
 
 191. In order to solve equations containing, unknown 
 numbers, there must be as many equations as there are 
 unknown numbers (§§ 178, 184). But with a less number 
 of equations than there are unknown numbers, there may 
 be conditions, or limitations, that determine what the solu- 
 tion or solutions must be. 
 
 Thus, suppose it is required to find the positive integral values 
 of X and y in the equation i x -\- S i/ = 10. If a; = 0, y is not 
 integral; if x = 1, y = 2 ; if a: = 2, y is not integral ; if a; > 2, 
 y is negative ; therefore, the only solution with the conditions given 
 is, a: =: 1, ^ = 2. 
 
 192. Equations containing more unknown numbers than 
 there are equations are called Indeterminate Equations. 
 
 Solve in positive integers : 
 1. 3a; + 5?/ = 20. 
 
 If 
 
 x = 0, 
 
 y = 4 
 
 0) 
 
 if 
 
 x^l. 
 
 y = 
 
 (2) 
 
 if 
 
 x = 2, 
 
 y^ 
 
 (3) 
 
 if 
 
 x = S, 
 
 y = 
 
 (4) 
 
 if 
 
 x=^i, 
 
 y = 
 
 (5) 
 
 if 
 
 x = 5, 
 
 2^ = 1 
 
 (6) 
 
 if 
 
 x = 6. 
 
 y = 
 
 (7) 
 
 In equations (2), (3), (4), (5), (7), as it can be seen at once that 
 the value of y is a fraction, its value is not found. So it is clear that, 
 if a; > 6, y must he negative. Hence, the only possible answers are, 
 a; = 0, y = i; or a;.= 5, y = 1. 
 
 * See Preface. 
 
212 ALGEBRA. 
 
 2, 7a; + 2y = 18. 4. 3a; + 11 2/ = 42. 
 
 3. 5a; + 42/ = 38. 5. 6a: + 5?/ = 48. 
 6. 3 a: + 5 y = 16. 
 
 3a; = 16 — 5?/ 
 
 
 - = 5-, + l-2. 
 
 Now as X, 
 
 5, and y are integers, ~ ^ is an integer. 
 
 Let 
 
 1 — 2a 
 
 5 — - = »» 
 
 o 
 
 Then 
 
 l — 2y = Sm 
 
 and 
 
 2y=-l — 3m 
 
 
 1 — Ml 
 
 V — — m 
 
 
 ^2 
 
 As before, 
 
 — — - must be an integer. 
 
 Let 
 
 1-m 
 2 " 
 
 Then 
 
 1 — m = 2n 
 
 and 
 
 m = 1 — 2 n 
 
 
 1 — (1 — 2 n) „ „ . 
 
 y = '^^ - (1 - 2 «) 
 
 
 = 3n — 1 
 
 
 .-5 (Zn n,l-2(3»-l) 
 
 = 7— 5n 
 
 From y = 3 n — 1 , it is evident that, in relation to y, n must be 
 positive, and not ; while from a; = 7 — .5 n, it is evident that, in 
 relation to x, n cannot have a positive value greater than 1. Hence 
 u must equal 1. 
 
 Ans. 
 
 (x — 2. 
 
 Note 1. It is better to find, in terms of the other, the value of the 
 unknown number that has the smaller coefficient. 
 
INDETERMINATE EQUATIONS. 213 
 
 7. 5 a; - 8 2/ = 7. 
 
 Let '-^ = - 
 
 5 
 
 Then 3 y + 2 = 5 m 
 
 anil « = m -I- 
 
 2 (m — 1) 
 
 Then — - — must be an integer. 
 
 o 
 
 T . m — 1 
 
 Let . — - — =: n 
 
 o 
 
 Then m — 1 = 3 » 
 
 m = 3n + 1 
 
 „ ,,,2(3n + l — 1) r J_^ 
 y = 3n + 1+ *■ g ^ = 5n+ 1 
 
 K _L 1 r 1 , 3 (« w + 1) + 2 a , „ 
 
 a; = 5ra + l + l + — i "^ — = 8 n + 3 
 
 5 
 
 If n = 0, a;= 3, y= 1 
 
 if n = 1, a; = 11, y = 6 
 
 if n = 2, a; = 19, 2^ = 11 
 
 &c. &c. &c. 
 
 Note 2. From Exs. 6 and 7 it will be seen that, if x and y are in the 
 same member of the equation and have like signs, the number of positive 
 integral solutions is limited ; hut if unlike signs, the number of solutions 
 is infinite. 
 
 8. 4x + 9i/ = 75. 9. 7a; — 4y = 53. 
 
 Q <3x + 5y+ z = 25. 
 l5a; + 4y + 3s = 34. 
 
 Ans. X = 2, y — 5, s = i. 
 
 Note 3. Eliminate one of the unknown numbers ; then proceed as 
 before. 
 
 11. In how many ways can $7 be paid in two-dollar bills 
 and fifty-cent pieces ? 
 
 Let X = number of two-dollar bills ; 
 
 y = number of fifty-cent pieces. 
 
214 ALGEBRA. 
 
 Then 2x + ^- = 7 
 
 4x + 2/ = 14 
 
 If x = 0, y = 14 
 
 if x = l, 2/ = 10 
 
 if a; = 2, y = 16 
 
 if a; = 3, j^ = 2 
 
 Ans. 3 ways. 
 
 12. In how many ways can $0.50 be paid in three-cent and 
 five-cent pieces ? 
 
 13. In how many ways can $27 be paid in five-dollar bills 
 and two-dollar bills ? 
 
 14. A owes B $8.25. If A has only fifty-cent pieces and B 
 only three-cent pieces, what is the simplest way for them to 
 square accounts ? 
 
 15. Divide 55 into two parts, so that one shall be divisible 
 by 2 and the other by 3. 
 
 16. A drover buys sheep, turkeys, and hens. The whole 
 number is 100, and the whole price $100. For the sheep he 
 pays $3.60, for the turkeys, $1.33J, and for the hens $0.50 
 each. How many of each does he buy ? 
 
 INEQUALITIES. 
 
 193. A statement that one number is greater or less than 
 another (§ 26) is an inequality. 
 
 Thus, 7 > 5 and a < J are inequalities. 
 
 194. Inequalities of the same direction are those in which 
 the sign > (or <) points the same way in both ; otherwise, 
 the inequality is reverse. 
 
 Thus, 7 > 5 and a > 6 are of the same direction; but 7 > 5 and 
 a <.b are reverse. 
 
INEQUALITIES. 215 
 
 195. If equals are added to, or subtracted from, the mevi- 
 ters of an inequality, the inequality remains the same. 
 
 Thus, if c is added to both members of a > 6, it is evident that 
 a + c is as much greater than 6 + c as a is greater than b. So 
 a — c is as much greater than b — c as a is greater than b. 
 
 From this it follows that, as in equations, a term can be 
 transposed from one member of an inequality to the other, 
 provided its sign is changed, without changing the inequality. 
 It also follows that, if tfoe signs of the terms of an inequality 
 are changed, the inequality is reversed. 
 
 Thus, if a > 6, then —a< — b (§39). 
 
 196. If each member of an inequality is midtiplied or 
 divided by the same positive number, the inequality will be 
 in the same direction as before. 
 
 Thus, if a > 6, then 10 a > 106. 
 
 But if the multiplier is negative, the inequality is reversed. 
 Thus, if a > 6, then — 10 a < — 10 6. 
 
 197. Like powers of both members of an inequality, with 
 both members positive, are ineqtialiiies of the sa?ne direction; 
 but if both m^embers are negative, the odd powers of the in- 
 equality are of the same, and the even powers of the reverse 
 direction. 
 
 Thus, if a > 6, then a" > 6"; but if — a > — 6, then (— o)" >(— t)" 
 if n is odd, but ( — a)" < ( — J)" if n is even. 
 
 198. Like roots of both memhers of an inequality are in- 
 equalities of the same direction. 
 
 1 ^ 
 Thus, if a > 6, then a" > b". 
 
 In case n is even, the inequality of the possible negative roots is 
 reversed. 
 
216 ALGEBRA. 
 
 199. The sum of the corresponding members of several 
 inequalities of the same direction is an inequality of the 
 same direction. 
 
 Thus, 7 > 5 
 
 3 > 2 
 6 > 5 
 
 16 > 12 
 
 200. If one inequality is subtracted from another of the 
 same direction the remainder is not necessarily an inequality 
 of the Same direction. 
 
 Thus, 8>4 8>4 8>4 
 
 5>3 6>2 7>1 
 
 3>1 2 = 2 1<3 
 
 201. These principles enable us to reduce an inequality 
 so that the unknown number may stand alone as one 
 member of the inequality. 
 
 Reduce : 
 
 - 7a; 1 „ 5 
 
 X 1 
 Transposing and uniting, q > 7 
 
 2. 5a; — 8<3a: — 5. Ans. a; < |. 
 
 3. 42 a; - 11 a; + 100 > 121 - 13 ar. 
 
 4. 2 a;*" - (a; + 1)^ > (a; + 3)^. 
 
 5. 4{(2 X - 1) - 2 (a; + 1)} < 3 (a; + 5). 
 
 6. 5 (a; + 1)= > 12 (a; + 2)^ - 7 (a; + 3)^. 
 
 7 riven 53(a;-5)-8>2(a;- 4) ) to find the limits 
 l5(a; + 3) + 2<3(a; + 17)i of x. 
 
 Ans. j^>15. 
 ( a; < 17. 
 
INEQUALITIES. 217 
 
 8. I . paid for a horse an even number of dollars. If three 
 times the price of the horse plus f 20 is more than twice the 
 price plus 1169, and four times the price of the horse minus 
 $67 is less than three times the price plus $84, what did I 
 pay for the horse ? Ans. $150. 
 
 9. What number is that whose third minus its fifth is 
 greater than 2, while its half plus its sixth is less than 12 ? 
 
 202. The exercises given below depend upon the follow- 
 ing proposition : 
 
 If . a — h=±d 
 
 then a2 — 2a6 + J2 = d2 
 
 and a" + 62 = 2 a 5 + d2 
 
 If ±d = Q, that is, if a = J, then a" + 6" = 2 «6 ; but if ± d 
 does not ec[ual 0, that is, if a does not equal 6, then a^ + Jfi > 2 ah. 
 
 If the letters used below are positive and unequal, prove 
 that 
 
 1. a' — 5'>3a''5 — 3a6l (Divideby a — 6.) 
 
 3. (a + J -of + {a-b + ef + {-a + b Jr cf > ab 
 + ao + bo. 
 
 4. 4= + -4- > V^ + V6. 
 yb ya 
 
 5. a^ + b'^ + c' > ab + ac + b c. 
 
 6. a -\ — > 2, whenever a does not equal 1. 
 
 a 
 
 7. (a' + ¥) (a« + 6=) > (a' + b^f. 
 
 8. ah{a-\-b) + ac(a + c) + bc{b + e)>&abc. 
 
 9. ab(a+b)-\-ac{a + c) + bc{b+c)< 2(a»+6° + c')- 
 
 10. (a + 5) (a + c) (i + c) > 8 a6c. 
 
 11. a' + J'+c» >3a6c. 
 
218 ALGEBRA. 
 
 CHAPTEE XV. 
 INVOLUTION AND EVOLUTION. 
 
 203. Involution is the process of raisiug a number to a 
 power. 
 
 204. A number is involved by taking it as a factor as 
 many times as there are units in the index of -the required 
 power. 
 
 205. According to Art. 70, 
 
 (+ a ) X •(+ a) = + a^ 
 (+ a) X (+ a) X (+ a) = (+ a^) x (+ a) = + a", 
 and so on ; 
 
 and (-a) X(-a)= + a^ 
 
 (- «) X (- a) X (- a) = (+ a^) X (- a) = - a», 
 (-a) X (- a) X (- a) X (-a) = (-a^) X (-a) = +,a\ 
 and so on. 
 
 Hence, for the signs we have the foUowinof 
 
 Rule. 
 
 Of a positive number all the powers are positive. 
 Of a negative number the even powers are positive, and 
 the odd powers negative. 
 
 INVOLUTION OF MONOMIALS, 
 
 206. To raise a monomial to any required power. 
 
 1. Find the third power of 3 a^ b'. 
 
 (3a2J8)s = 3o2j8x 3a2J3 X 3a2js (1) 
 
 = 3 . 3 . 3 . a2 a" a2 ja J8 j8 ^g) 
 
 = 27a«io (3-) 
 
INVOLUTION. 219 
 
 According to Art. 204, to raise 3 a^ 6' to the third power, we take it 
 as a factor three times (1) ; and as it makes no diflference in the pro- 
 duct in what order the factors are taken, we arrange them as in (2) ; 
 performing the multiplication (§ 72) expressed in (2), we have (3). 
 Hence, 
 
 Rule. 
 
 Multiply the exponent of each letter hy the index of the 
 required power, and prefix the required power of the nu- 
 merical coefficient, remembering that the odd powers of a 
 negative number are negative, while all other powers are 
 positive. 
 
 Note 1. It follows that the power of the product is equal to the product 
 of the powers. 
 
 Note 2. It follows from Art 148 that a fraction is raised to a power 
 by involving hoth numerator and denominator to the required power. 
 
 Find the indicated power in the following : 
 
 2. (2xyy. 
 
 3. (anr- 13. (-|J5)^ 
 
 4. (5x^y^e*y. 
 
 5. (-2a'cy. 14. (-^T 
 
 6. {—Sb'c^'xy. 
 
 7. (-3a?b<ff. 15. (^^^]\ 
 
 8. {2a^H'"'c')K 
 
 9. {4.x^fx2xyY. 16. (-|^„) 
 
 10. (3 a: 2/2)". 
 
 11. (-4.5^0)". 17. (1^)" 
 
220 ALGEBRA. 
 
 INVOLUTION OP BINOMIALS. 
 
 207. A tinoraial can be raised to any power by suc- 
 cessive multiplications. But when a high power is re^ 
 quired, the operation is long and tedious. The Binomial 
 Theorem, first developed by Sir Isaac Newton, enables. us 
 to expand a binomial to any power by a short and speedy 
 process. 
 
 208. In order to investigate the law which governs the 
 expansion of a binomial we will expand a + b and a — h 
 to the fifth power by multiplication. 
 
 a+b 
 a+b 
 
 a^ + ab 
 
 ab +t^. 
 
 a^ + 2ab + b^ 2d power. 
 
 a +h 
 
 a^ + 2aH+ a¥ 
 
 a^b+ laV + ¥ 
 
 a» + 3a=J + 3a6^ + J' 3d power. 
 
 a+b 
 
 a^ + 3a^b+ Sa^fi"-!- aP 
 
 a'^b+ Za^b'^+ Sab' + b* 
 
 a* + 4taH+ 6 a^^ + 4 a J» + 4* 4th power. 
 
 a+b 
 
 a^ + 4ca*b+ 6a''b^+ 4.aH^ + ab* 
 
 a*b+ ian^+ 6a'b^ + 4ab* + b^ 
 
 a^ + Sa^b + lOaH^ + lOa^^+Bab^ + b^ . . 5th power. 
 
INVOLUTION. 221 
 
 a — b 
 a — b 
 
 a' — ab 
 
 - ab +6* 
 
 a^ — 2 a 5+6^ 2d power. 
 
 a — b - 
 
 a^ — 2a^b+' ab^ 
 
 - a^fi+ ^ab"" -b^ 
 
 a' — Sa^b+Sab^-b'' 3d power. 
 
 a — b 
 
 a* — 3aH+ 3aH^- a¥ 
 
 - a^b+ 3aH^— 3a6»+6* 
 
 a*-4ci»6+ 6«^6^— 4ai.«+6*. .... 4th power. 
 a — b 
 
 a? — 4:aH+ 6an^- 4a^6»+ ab* 
 ■ — an+ Aa*W— 6aH^ + 4:ab*—l^ 
 
 a^ -5a*b + 10 an^ - 10 aH^ + 5 ab* -b^ . . 5th power. 
 
 By examining the different , powers of a+b and a — b 
 in these- examples, we shall find the following invariable 
 laws governing the expansion : 
 
 1st. The leading number (that is, the first number of 
 the binomial) begins in the first term of the power with an 
 exponent equal to the index of the power, and its exponent 
 decreases regularly by one in each successive term till it dis- 
 appears; the follovAng number {that is, the second number 
 -q/' the binomial) begins in the second term of the power with 
 the exponent one, and its exponent increases regularly by one 
 till in the last term' it becomes the same as the index of the 
 power. 
 
222 ALGEBRA. 
 
 Thus, in the fifth power the 
 
 Exponents of a are 5, 4, 3, 2, 1. 
 Exponents of h are 1, 2, 3, 4, 5. 
 
 It will he noticed that the sum of the exponents of the 
 letters in any term is equal to the index of the power. 
 
 2d. The coefficient of the first term, is mie; of the second, 
 the same as the index of the power; and universally, the co- 
 efficient of any term multiplied hy the exponent of the lead- 
 ing quantity, and this product divided by the eocponent of the 
 following quantity increased hy one, will give the coefficient 
 of the succeeding term. 
 
 Thus, in the fifth power, 5, the coefficient of the second 
 term, multiplied by 4, a's exponent, and divided by 1 plus 1, 
 
 6's exponent plus 1, = = 10, the coefficient of the 
 
 third term. 
 
 The coefficients are repeated in the inverse order after 
 passing the middle term or terms, so that more than half 
 of the coefficients can be written without calculation. 
 The number of terms is always one more than the index 
 of the power ; that is, the second power has three terms ; 
 the third power, four terms; and so on. When the 
 number of terms is even, that is, when the index of 
 the power is odd, the two central terms have the same 
 coefficient. 
 
 2id. When both terms of the binomial are positive, all the 
 terms of the power are positive ; but when the second term is 
 negative, those terms which contain odd povjers of the follow- 
 ing quantity are negative, and all the others positive; or 
 every alternate term, beginning with the second, is negative, 
 and the others are positive. 
 
INVOLUTION. 223 
 
 1. Expand (x + yY. 
 
 According to the law, the first term will be i? 
 
 and the second term + 8 a;' y 
 
 4 
 The coefficient of the third term will be •* y, '^ 
 
 and the third term + 28 x'^y^ 
 
 2 
 
 The coefficient of the fourth term will be ^— ^ 
 
 and the fourth term -f- 56 x^ y" 
 
 14 
 
 The coefficient of the fifth term will be ^^^ ^ . 
 
 and the fifth term 70 x* y* 
 
 Having found the preceding coefficients and the -coefficient of the 
 middle term, we can write the others at once. Hence, 
 
 Expand by the binomial theorem the following : 
 
 2. {x + yy. 
 
 Ans. x'' + 7xi'y + 21x''y' + a5x''y<^-[-S5x'y* + 21x^y^ + 7xy' + y''. 
 
 3. (a -by. 5.- {a-by\ 
 
 4. (e + dy. 6. {x+yy\ 
 
 7. (a + iy. 
 
 Ans. a».+ 6 a? + 15 a* + 20 a» + 15 a2 + 6 a + 1. 
 
 Note. Since all the powers of 1 are 1, 1 is not written when it appears 
 as a factor ; but its exponent must be used in obtaining the coefficients. 
 
 8. (1-yy. 
 
 Ans. 1 — 8y + 28j''— 56 3(» + 702;*— 66y6+28/— 82/T + /. 
 
 9. (c + l)«. 10. (1-z)". 
 
 209. When the terms of the binomial have coefficients or 
 exponents other than 1, the theorem can be made to apply 
 by treating each term as a single literal quantity. In the 
 
224 ALGEBRA. 
 
 expansion, each factor should be enclosed in a parenthesis, 
 and after the expansion of the binomial by the binomial 
 theorem, the work should be completed by the expansion 
 of the enclosed factors, according to the rule for the expan- 
 sion of monomials. 
 
 11. (3a^-b^y. 
 
 (3 a»)» — 5 (3 a')* (fts) + 10 (3 a=)s (6S)' — 10 (3 0")= (I^)' + 6 (3 a") ( J*)* — (6')' 
 Expanding each factor as indicated, we have 
 
 243 o" -, 405 a« fr' + 270 a« W — 90 a* 69 + 15 o^ 6" — b"> 
 
 12. {2x- 5)*. 14. Expand (2 a + 3 by. 
 
 13. Expand (3 a; — 2 2^)*. 
 
 15. Expand (2 x" - ly. 
 
 Ans. 64 a;" — 192 si" + 240 x' — 160 a;' + 60 x* — 12 a:' + 1. 
 
 16. Expand (2 a; — 3 y)^ 
 
 Ans. 32 3fi — 240 x^y + 720 a' y^ — 1080 x^y^+SlOxy* — 243 y^. 
 
 17. Expand (2- iJ- 
 
 18. Expand (a -^5)*. Ans. a^-2o»J+|a''J•■'-^aJ»-|-Jg• 
 19. Expand ^2 a - 5) • 
 
 20. Expand (^ - J)'- Ans. ^ - A + _^ _ _L + 1^. 
 
 21. Find the middle term of (1 + «)«. 
 
 22. Find the two middle terms of (a — 6)". 
 
 Ans. 1716 a' i^ and -1716 a' 6'. 
 
 (a a;^■''' 
 - + - • 
 X al 
 
 24. Find the term independent of a; in [^o(? — —-\ . 
 
 \Z 3 xl 
 
EVOLUTION. 225 
 
 25. Eind the two middle terms of (3a — ^J • 
 
 . 189 a" , 21 „ 
 Ans. — - — . , aad — r^ or- 
 
 210. The Binomial Theorem can be applied to the ex- 
 pansion of a polynomial. Thus, in a + 6 — c, a + 6 can be 
 treated as a single term, and the quantity can be wiitten 
 (a + 6) — c. In like manner, a -{■ h -\- x — y can be written 
 {a-{-h) + (x-y). 
 
 26. (x — y — 2f. 
 
 (a;-jf-2)'={(a;.-2^)-2is 
 = (a; - 2,)s - 3 (r - 2,)2 (2) + 3 (cc - J,) (2)2 - (2)8 
 = x'^ — Zx^y+Sxy^ — f — Qx^-\-l2xy — Qy^-\-Vix — \2y — % 
 
 Note. A single letter, as a and h, might be substituted for a; — y and 2, 
 and, after expanding (a — hf, the values of a and 6 substituted. 
 
 27. (x — y + 0)». 29. {2x — ^y-m- If. 
 
 28. (a-2b-c + lY. 30. (| - | + 2)'. 
 
 EVOLUTION. 
 
 211. Evolution is the process of extracting a root of a 
 number. It is the reverse of Involution. 
 
 212. A Root is one of the equal factors into which a 
 number may be resolved (§ 12). 
 
 A root is indicated by the radical. sign ^. Thus, 
 
 '\/~x indicates the square root of x. 
 ^/x " " cube " « 
 " " with « " 
 
 213. Since Evolution is the reverse of Involution, the 
 rules for Evolution are derived at once from those of In- 
 volution. And therefore, as according to Art. 205 an odd 
 
 15 
 
226 ALGEBRA. 
 
 power of any number has the same sign as the number 
 itself, and an even power is always positive, we have for 
 the signs in Evolution the following 
 
 Kule. 
 
 An odd root of a number has the same sign as the number 
 itself. 
 
 An even root of a positive number is either, positive or 
 negative. 
 
 An even root of a negative number is imaginary. 
 
 SQUARE ROOT OF ARITHMETICAL NUMBERS. 
 
 214. To find the square root of a number is to resolve it 
 into two equal factors, that is,. to find a number which, mul- 
 tiplied into itself, vnll produce the given number. 
 
 Numbers, 1, 10, 100, 1000. 
 
 Squares, 1, 100, 10000, 1000000. 
 
 215. Comparing the numbers above with their squares, 
 we see that the square of any arithmetical integral number 
 less than 10 has either one or two figures ; the square of 
 any arithmetical integral number less than 100 and over 
 9 has either three or four figures ; and. so on. That is, 
 the square of an arithmetical number consists of twice 
 as many figures as the root, or of one less than twice as 
 many. Hence, to find the number of figures in the square 
 root of an arithmetical number. 
 
 Begin at units and mark off the number into periods of 
 two figures each, and there will be one figure' in the root for 
 each period of two figures in the square, and another figure 
 in the root if a figure remains at the left of tlie full periods 
 of the square. 
 
EVOLUTION. 227 
 
 216. To extract the squai'e root of an arithmetical 
 number. 
 
 1. Find the square root of 6889. 
 
 From the preceding explanation, it is evident that the square root 
 
 of 6889 is. a number of two figures, and that the tens figure of the 
 
 ■root is the square root of the greatest perfect sqiiare in 68; that is, 
 
 ■v64, or 8. Now, if we represent the tens of the root by a and the 
 
 units by h, a -\- b will represent the root ; and the number will be 
 
 (a + 6)2 = a'' + 2a6 + 6« 
 
 Now a^ = 802 = 6400 
 
 therefore 2ab + b^ = 6889 — 6400 = 489 
 
 But 2 a 6 + 62 = (2 a + S) 6 
 
 If therefore 489 is divided by 2 a + 6, it will give b, the units of 
 the root. But b is unknown, and is small compared with 2 a ; 
 we can therefore use 2 a =: 160 as a trial divisor. 489 -f- 160, or 
 48 -^ 16 = 3, a number that cannot be too smallj but may be too 
 great, because wo have divided by 2 a instead of 2 a + b. Then 
 6 = 3, and 2 a + 6 = 160 + 3 = 163, the true divisor; and 
 (2 a + 6) 6 = 163 X 3 = 489 ; and therefore 3 is the unit figure 
 of the root, and 83 is the required root. The work will appear as 
 follows : 
 
 6^ Tg ( 8 3 a = 80 
 
 6 4 6= 3 
 
 2 a + A = 163)489 
 (2 a + 6) 6 = 489 
 
 Hence, to extract the square root of an arithmetical 
 number, 
 
 Kule. 
 
 Beginning at units, separate the number into periods of 
 two figures each. Find the greatest square in the left-hand 
 period, and place its root ,0't iheright. Subtract the square 
 of this root figure from the left-hand period, and to the re- 
 mainder annex the next period for a dividend. DoiMe the 
 
228 ALGEBRA. 
 
 root already found for a trial divisoe, and, omitting the 
 right-hand figure of the dividend, divide, and place the quo- 
 tient as the next figure of the root, and also at the right of 
 the trial divisor for the TRUE DIVISOR. Midtiply the true 
 divisor hy this neiv root figure, subtract the product from the 
 dividend, and to the remainder annex the next period, for a 
 new dividend. Double the part of the root already found 
 for a trial divisor, and proceed as before, until all the 
 periods Imve been employed. 
 
 Note 1. When a root figure is 0, annex also to the trial divisor, and 
 bring down the next period to complete the new dividend. 
 
 Note 2. If there is a remainder, after using all the periods in the given 
 example, the operation may be continued at pleasure by annexing succes- 
 sive periods of ciphers as decimals. 
 
 2. rind the square root of 119025. 
 
 We suppose at iirst that a repre- 
 
 4-> ,Ck ,c> , c, . ,- sents the hundreds of the root, and 
 
 11902o(345 14.1.4. J- -t:^, 
 
 . ^ o the tens ; proceeding as m Ex. 1, 
 
 we have 34 in the root. Then let- 
 
 6 4)290 ting a represent the hundreds and 
 
 ^ 5 Q tens together, that te, 34 tens, and 
 
 685)3425 * the units, we have 2a, the second 
 
 3 4 2 5 trial divisor, = 64 tens; and there- 
 
 fore 6^5; and 2 o + 6 = 685 ; and 
 
 345 is the required root. 
 
 3. Find the square root of 7527.5. 
 
 75 27.50 ( 8 6.7 6 + 
 64 
 
 166)1 127 
 996 
 
 1 7 2.7 ) 1 3 1.5 
 1 2 0.8 9 
 
 1 7 3.4 6 ) I 0.6 I 
 10.407 6 
 
47)3.5 
 3.2 9 
 
 EVOLUTION. 229 
 
 Find the square root of : 
 
 4. 29929. 9. 42849. 14. 1677.7216. 
 
 5. 67081. 10. 927369. 15. 360840.49. 
 
 6. 37636. 11. 290521. 16. 0.2116. 
 
 7. 762129. 12. 9703225. 17. 2330.24. 
 
 8. 401956. 13. 21418384. 18. 171819.6. 
 
 19. What is the square root of 7.5 ? 
 
 7.50 ( 2.7 3 111 ^^^ example we can only ap- 
 
 proximate to the root. By annexing 
 successive periods of ciphers we can 
 approximate nearer and nearer to the 
 root. 2 is the square root to the near- 
 5 4 3)2100 ggt unit; 2.7, to the nearest tenth; and 
 
 16^29 2.74 (as the thousandths ' figure will be 
 
 5 4 6)471 more than 5), to the nearest hundredth. 
 
 Note 3. When once the decimal point has heen placed in the root, no 
 further attention need be paid to it in the remainders or in the divisors. 
 
 Find the square root of : 
 
 20. 78 to the nearest thousandth. 
 
 21. 523 to the nearest tenth. 
 
 22. 52.3 to the nearest hundredth. 
 
 Note 4. As a fraction is involved by involving both numerator and 
 denominator (§'206, Note 2), the square root of a fraction is the square root 
 of the numerator divided by the square root of the denomiiiator. 
 
 23. ^. 24. M. 25. \%. 26. iff. 
 
 Note 5. If both terms of the fraction are not perfect squares, and can- 
 not be made so, reduce the fraction to a decimal, and then find the square 
 root of the decimal. A mixed number must be reduced to an improper 
 fraction, or the fractional part to a decimal, before its root can be found, 
 
 27. ?. 28. -^^-g. 29. T^j. 30. ^. 
 
230 ALGEBRA. 
 
 CUBE ROOT OF ARITHMETICAL NUMBERS. 
 
 217. To find the cube root of a number is to resolve it into 
 three equal factors; that is, to find a number which, taken 
 three times as a factor, will produce the given number. 
 
 Numbers, 1, 10, 100, 1000. 
 
 Cubes, 1, 1000, 1000000, 1000000000. 
 
 218. Comparing the numbers above with their cubes, we 
 see that the cube of any arithmetical integral number less 
 than 10 has less than three figures ; the cube of any arith- 
 metical integral number less than 100 and more than 9 
 has less than seven and more than three figures; and so on. 
 That is, the cube of an arithmetical number consists oi three 
 times as many figures as the root, or of one or two less than 
 three times as many. Hence, to find the number of figures 
 in the cube root of an arithmetical number, 
 
 Begin at the right, and mark off the number into periods 
 of three figures each, and there ivill be one figure in the root 
 for each period of three figures in the cube, and if there are 
 one or two figures besides full periods in the cube, th^re ipill 
 he a figure in the root for this part of a period. 
 
 219. To extract the cube root of an arithmetical number. 
 
 1. Find the cube, root of 79507. 
 
 From the preceding explanation, it is evident that the cube root of 
 79507 is a number of two figures, and that the tens figure of the rfjot 
 is the cube root of the greatest perfect cube in 79 ; that is, t 64, or 4. 
 Kow, if we represent the tens of the root by a and the units by b, 
 a -\- b will represent the root, and the number will be 
 
 (a + 6)8 = aS + 3 a26 + 3 a fi2 + b» 
 Now a8 = 408 = 64000 
 
 therefore, 3a^b + 3ab^ + b^ = 79507 — 64000 = 15507 
 But 3a'b+3ab'^+b^=^(^Sa^+3ab + b')b 
 
EVOLUTION. 231 
 
 If therefore, 15507 is divided by 3 a^ + 3 ai + 6^ it will give b, 
 the units of the root. But 6, and therefore Sab -{- h% a part of 
 the divisor, is unknown, and we must use 3 a^ = 4800 as a trial 
 divisor. 15507 -v- 4800, or 155 -f- 48 = 3, a number that cannot 
 be too small, but may be too great, because we have divided by 3 a^ 
 instead of the true divisor, 3 a" + 3 a 6 + 6^. Then 6 =: 3, and 
 Sa^ + Zab + b^ = 4800 + 360 + 9 = 5169, the true divisor ; and 
 (3 a2 + 3 a 6 + 62) 6 = 5109 x 3 = 15507 ; and therefore 3 is .the 
 units figure of the root, and 43 is the required root. The work 
 will appear as follows : 
 
 79507(43 a = 40 
 6 4 6=3 
 
 Trial divisor, 3 a= = 4 8 1 
 
 3ai= 360 
 6^= 9 
 
 True divisor, S a'' -\- Sab + b^ = 51Q9 
 
 15507 
 
 15507 
 
 Hence, to extract the cube root of au arithmetical number, 
 
 Bule. 
 
 Beginning at units, separate the number into periods of 
 three figures ea,ch. Find the greatest cube in the left-hand 
 period, and place its root at the right. Subtract this cube 
 from the left-hand period, and to the remainder annex the 
 next period for a dividend. Square the root figure, annex 
 two ciphers, and multiply this result by three for a TEIAL 
 DIVISOR ; divide the dividend Inj the trial divisoi', and plack 
 tJie quotient as the next figure .of the root. Multiply tMs 
 root figure by the part of the root previously obtained, annex 
 one cipher, and multiply this result by three; add the last 
 product and the square of tlie last root figure to the trial 
 divisor, and the sum will be the true divisor. Multiply 
 the true divisor by the last root figure, subtract the product 
 from the dividend, and to the remainder annex the next pe- 
 riod for a dividend. Find a new trial divisor, and proceed 
 as before, until all the periods have been employed. 
 
232 
 
 ALGEBRA. 
 
 Note 1. The notes under the rule in square root (§ 216) apply also to 
 the extraction of the cube root, except that 00 must be annexed to the trial 
 divisor when the root figure is 0. 
 
 Note 2. As the trial divisor may be much less than the true divisor, 
 the quotient is frequently too great, and a less number must be placed in 
 the root. 
 
 2. rind the cube root of 303464448. 
 
 ^^64 178(672 
 216 
 
 1st trial divisor, 
 
 3a2=10800 
 
 Sah= 1260 
 
 i- = 4 9 
 
 87464 
 
 1st true divisor, 3a= + 3a6+6==1210 9j 
 
 2d trial divisor, 3 a" =1346700 
 
 Sab= 4020 
 
 62= 4 
 
 84763 
 
 2701448 
 
 2d true divisor, 3 a2 + 3a6 + 62 = 135072 4j 
 
 2701448 
 
 We suppose at first that a represents the hundreds of the root, and 
 h the tens: proceeding as in Ex. 1, we have 67 in the root. Then, 
 letting a represent the hundreds and tens together, that is, 67 tens, 
 and 6 the units, we have 3 a^, the 2d trial divisor, = 1346700 ; 
 and therefore 6 = 2; and 3 a' + 3 a 6 + 6^, the 2d true divisor, 
 = 13.50724; and 672 is the required root. 
 
 Though the 1st trial divisor is contained more than 8 times in the 
 dividend, yet the root figure is 7. 
 
 3. rind the cube root of 129554.6. 
 
 129 554.6 00(50.6 + 
 125 
 
 750000 
 
 9000 
 
 36 
 
 759036J 
 
 4 5 5 4.6 
 
 455421 6 
 
 (See Note 3, §216.) 
 
 S S i 
 
EVOLUTION. 
 
 233 
 
 Find the cube root of : 
 
 4. 4330747. 
 
 6. 2924207. 
 
 6. 12326391. 
 
 7. 34786542. 
 
 8. 8120601. 
 
 9. 4789.65. 
 
 10. 0.07348. 
 
 11. 0.8754321. 
 
 12. 46.78134. 
 
 13. 5187.6423. 
 
 14. 10073.2456. 
 
 15. 0.9073468. 
 
 16. What is the cube root of 7854 ? 
 
 7 854(19.8, Ans. 
 1 
 
 300 
 
 270 
 
 81 
 
 6854 
 
 651 
 
 108300 
 
 4560 
 
 64 
 
 58 59 
 
 112924 
 
 9 9 5.0 
 
 9 3.392 
 
 In this example we can only ap- 
 proximate to the root. By annex- 
 ing successive periods of ciphers 
 we can approximate nearer and 
 nearer to the root. 20 is the cube 
 root to the nearest unit; 19.8, to 
 the nearest tenth. 
 
 9 1.6 8 
 rind the cube root of : 
 
 17. 10 to the nearest hundredth. 
 
 18. 560 to the nearest tenth. 
 
 19. 0.08 to the nearest thousandth. 
 
 Note 3. As a fraction is involved by involving both mimeratorand 
 denominator (§ 206, Note 2), the cube root of a fraction is the cube root 
 of the numerator divided by the cube root of the denominator. 
 
 20. §J. 
 
 21. m- 
 
 22. ftf. 
 
 23. 
 
 Note 4. If both terms of the fraction are not perfect cubes, and cannot 
 be made so, reduce the fraction to a decimal, and then find the cube root of 
 the decimal. A mixed number must be reduced to an improper fraction, or 
 the fractional part to a decimal, before its root can be found. 
 
 24. f. 
 
 25. A- 
 
 26. 5f. 
 
 27. 17§. 
 
234 ALGEBRA. 
 
 EVOLUTION OF MONOMIALS. 
 
 220. As Evolution is the reverse of Involution, and since 
 to involve a monomial (§ 206) we multiply the exponent of 
 each letter by the index of the required power, and prefix 
 the required power of the numerical coefficient, therefore, 
 to find the root of a monomial, 
 
 Bule. 
 
 Divide the exponent of each letter by the index of the re- 
 quired root, and prefix the required root of the numerical 
 coefficient. 
 
 Note 1. The rule for the signs is given in Art. 213. As an even root 
 of a positive number may be either positive or negative, we prefix to such 
 a root the sign ± ; read, plus or minus. 
 
 Note 2. It follows from this rule that the root of the product o/ several 
 factors is eqital tothe product of theroots. Thus, V^ = Vi V9 = 6. 
 
 Perform the operation indicated in the following examples : 
 
 1. ^/21x''y\ 6. VSla'fi'. 
 
 2. vT^t^. 7. v^-243a"i«. 
 
 3. v^l6m^w>i=. 8. -v^125kV- 
 
 4. ^-8a'R 9. v^ 626 a^i'c^^. 
 
 5. 'V^32a;^V- 10. '^-729a;V. 
 
 Note 3. As a fraction is involved by involving both numerator and 
 denominator ( § 206, Note 2), a fraction must be evolved by evolving both 
 numerator and denominator. '. 
 
 ^1 ./9x^ . .3 a; 
 
 12. 47IIZl\ 13. 4/I6ZZ. 
 
EVOLUTION. 235 
 
 14. ^JK. 16. 4)^. 
 
 15. y^z. 17. iXf!^. 
 
 SQUARE EOOT OF POLYNOMIALS. 
 
 221. la order to discover a method for extracting the 
 square root of a polynomial, we will consider the relation 
 of a + & to its square, a? + 2ab + V^. The first term of 
 the square contains the square of the first term of the 
 root; therefore the square root of the first term of the 
 square will be the first term of the root. The second 
 term of the square contains twice the product of the two 
 terms of the root; therefore, if the second term of the 
 square, 2 ab, is divided by twice the first term of the 
 root, 2 a, we shall have the second term of the root b. 
 Now, 2 ab + bf^= (2 a + b)bi therefore, if to the trial 
 divisor 2 a we add b, when it has been found, and then 
 multiply the corrected divisor by 6, the product wUl be 
 equal to the remaining terms of the power after a? has 
 been subtracted. 
 
 The process will appear as follows : 
 
 o I o L ■ !■>/- 11 Having written a, the square 
 
 ., root 01 a\ in the root, we subtract 
 
 its square (o^ from the given poly- 
 
 2a + b)2ab+b^ iiomial, and have 2ab + b^ left. 
 
 2ab -j- Dividing the first term of this re- 
 
 mainder, 2 ab, by 2 a, which is 
 double the term of the root already found, we obtain b, the second 
 term of the root, which we add both to the root and to the divisor. 
 If the product of this corrected divisor and the last term of the root 
 is subtracted from 2ab -{-b^, nothing remains. 
 
236 ALGEBRA. 
 
 222. Since a polynomial can always be written and in- 
 volved like a binomial, as shown in Art. 210, we can apply 
 the process explained in the preceding article to finding 
 the root, when this root consists of any number of terras. 
 
 1. Find the square root oi a' + 2ab-\- b'^ — 2ac — 2bc-[- c\ 
 a^ + 2 ab + b'' — 2 ac — 2b c + c^(a + b — c 
 
 2a + b)2ab + b^ 
 2a6 + 6a 
 
 2a + 2b — c) — 2ac — 2bc + c^ 
 — 2ac — 2bc + c^ 
 
 Proceeding as before, we find the first two terms of the root a + 6. 
 Considering a + 6 as a single number, we divide the remainder 
 — 2ac — 2bc + c^ by twice this root, and obtain — c, which we 
 write both in the root and in the divisor. If this corrected divisor 
 is multiplied by — c, and the product subtracted from the dividend, 
 nothing remains. 
 
 Hence, to extract the square root of a polynomial, 
 
 Bule. 
 
 Arrange the terms according to the powers of some letter. 
 
 Find the square root of the first term,, and write it as the 
 first term of the root, and subtract its square from the given 
 polynomial. 
 
 Divide the remainder by double the root already found, 
 and annex the result both to the root and to the divisor. 
 
 Multiply the corrected divisor by this last term of the root, 
 and subtract the product from the last remainder. Proceed 
 as before with the remainder, if there is any. 
 
 Find the square root of : 
 
 3. a«-2a» + 3 a= - 2 a + 1. 
 
EVOLUTION. 237 
 
 4. 4 a^ + 4 a» - 7 a^ — 4 a + 4. Ans. 2a^+a — 2. 
 
 5. 4:x^ + 9y^ + 25z'' + 12xy-30yz — 20xs. 
 
 6. 16 a:= + 16 «' - 4 a;8 - 4 a;» + a;". 
 
 Ans. 4 a;» + 2 a:* — a;^ 
 
 7. a;« - 22 a;* + 34 x» + 121 a;" - 374 a; + 289. 
 
 8. 25 a;^- 30 a a;' + 49 a" a;'' -24 a* a; + 16 a*. 
 
 9. 4a:* + 4a;=2/2 — 12a;''»2 + /-6/«''+ 9 z*. 
 
 10. 6a6«c-4a26c + a2i2^.4^2g2^9jag2_jL2a6cl 
 
 11. -6b^c^+ 9c* + A^- 12 c^a^ + 4 a* + iaH^ 
 
 Ans. 2a'+b^-3c^. 
 
 12. 4a!* + 9y* + 13a;'y^— 6a!2/» — 4a;»3^. 
 
 13. 67a;= + 49+9a;*-70a!-30a:». Ans. 3 a;» - 5 a; + 7. 
 
 14. 1 - 4a; + 10a;2 - 20a;« + 25x« - 243:" + 16a;«. 
 
 15. 6aca;' + 4 62a:* + a'^a;'" + gc" - 12&ca;= - 4a6a;'. 
 
 Ans. a a:' — 2 Sx^ + 3 c. 
 
 16. i?^_?^+24-^%^:. 
 
 x' X y y 
 
 16y' 32y ^^ 9.x x^Uy ^ x 
 
 x^ x y y^\ "! y 
 
 8.V A 32, ^^ 
 
 X ) X 
 
 32 « 
 / + 1« 
 
 ^'-8 + ^)8- 
 8- 
 
 8x a;'' 
 
 ~ y^y^ 
 
 9,x x^ 
 
 ~ y ^y'' 
 
 . ^:+"%25. 
 
 
238 ALGEBRA. 
 
 ^^- 64+8-'^ + ^- 
 
 19. x*-6x' + .%9- a;= — 2 a; + ^. 
 
 Note ] . According to the principles of Art. 200, the signs of the an- 
 swers given ahove may all be changed, and still be correct. 
 
 Note 2. No binomial can be a perfect square. For the square of a 
 monomial is a monomial, and the square of the polynomial with the least 
 number of terms, that is, of a binomial, is a trinomial. 
 
 Note 3. By the rule for extracting the square root, any root whose 
 index is any power of 2 can be obtained by successive extractions of the 
 square root. Thus, the fourth root is the square root of the square root ; 
 the eighth root is the square root of the square root of the square root j 
 and so on. 
 
 Find the fourth root of : 
 
 21. 16x* — 32a^y^ + 24:x^y* — 8xy'' + f. Ans. 2x—f. 
 
 22. a^-12a'b + Bi a' b^ - 108 aH^ + 81 b\ 
 
 „, 1 4 6 4 ] 
 
 24- -1 + — + -1-1 + — 3 + - • 
 a;' x"ff x'y' xy^ y* 
 
 223. To find any root of a polynomial. 
 
 Since, according to the Binomial Theorem, when the terms of a 
 power are arranged according to the power of some letter beginning 
 with its highest power, the first term contains the first term of the 
 root raised to the given power, therefore, if we take the required root 
 of the first term, we shall have the first term of the root. And since 
 the second term of the power contains the second term of the root 
 multiplied by the next inferior power of the first term of the root with 
 a coefficient equal to the index of the root, therefore, if we divide the 
 second term of the power by the first term of the root raised to the 
 next inferior power with a coefficient equal to the index of the root, 
 we shall have the second term of the root. In accordance with these 
 principles, to find any root of a polynomial we have the following 
 
EVOLUTION. 239 
 
 Kule. 
 
 Arrange the terms according to the powers of some letter. 
 
 Find the required root of the first terpi, and write it as the 
 first term of the root. 
 
 Divide the second term of the polynomial hy the first term 
 of the root raised to the next inferior power and multiplied 
 hy the index, of the, root. 
 
 Involve the whole of the root thus found to the given power, 
 and subtract it from the polynomial. 
 
 If there is any remainder, divide its first term by the 
 divi^sor first found, and the quotient will be the third term 
 of the root. 
 
 Proceed in this manner till the potver obtained by involv- 
 ing the root is equal to the given polynomial. 
 
 . Note 1 . This rule verifies itself. For the root, whenever a new term 
 is added to it, is involved to the given power; and whenever the root thus 
 involved is equal to the given polynomial, it is evident that the required^ 
 root is found. 
 
 Note 2. As powers and roots are correlative words, we have used the 
 phrase given power, meaning the power whose index is equal to the index 
 of the required root, and the phrase next inferior power, meaning that power 
 wTiose index is one less than the index of the required root. 
 
 1. Find the 
 Constant divisor, 
 
 cube root of 
 3a^)a8- 
 
 -3oS 
 •3a6 
 
 - 3 a* + 5 a= - 
 + 5 flS — 3 a - 
 + 3 a* — qS 
 
 - 3 a - 1. 
 
 _l(a2_a-l 
 
 
 o«- 
 
 •3a6 
 
 + 5as — 
 
 ist term of remainder. 
 3a — 1 
 
 The first term of the root is a% the cube root of a', a^ raised to 
 the next inferior power, that is, to the second power, with the' coeffi- 
 cient 3, the index of the root, gives 3 a*, which is the constant divisor. 
 — 3 a^, the second term of the polynomial, divided by 3 a*, gives ^— a, 
 the second term of the root, (a" — a)' = a' — 3 a^ + 3 a* — a^ ; 
 and subtracting this from the polynomial, we have — 8 a^ as the first 
 term of the remainder. — 3 a* divided by 3 a* gives — 1, the third 
 term of the root, (a^ — a — 1)' = the given polynomial, and there- 
 fore the correct root has been found. 
 
240 ALGEBUA. 
 
 2. rind the fourth root of 81a' - 108 a»a;» + 64a«a;» 
 
 — 12a^x^ + x^\ 
 
 81 a8_ 108 a»a8 + 54 a*a:6— 12 a2a;9 + a;i2 
 
 3. Find the cuhe root of x^ + 3 x' y + 3 xy^ + y'^ — 3 x^z 
 
 — 6xyz — 3y^z + 3xz^ + 3yz^ — ^. 
 
 4. Find the fourth root of 16 &« - 96 5° c + 216 V c^ 
 
 — 216 6^08 + 81c*. 
 
 rind the cube root of ; 
 
 5. a?x^ — 3a'^x'^y'' -r 3axy* — y'^. Ans. ax — y\ 
 
 6. 1 + 3 a; + 6 a;^ + 7 a^ + 6 a;* + 3 a;« + x'. 
 
 7. l-6a; + 21a;''-44a;»+ 63a;*-54a;« + 27a;«. 
 
 8. a= + 6 a'' 6 - 3 a^ c + 12 a J'^ - 12 a 6 c + 3 ac'' + 8 6» 
 
 — 12 5^ c + 6 J c^ — c». Ans. a + 2b — c. 
 
 9. 8 a;« + 12 a;^ - 30 a;* - 35 a;» + 45 a;^ + 27 a: - 27. 
 
 Ans. 2x^ + x — 3. 
 ._ x» 3 a;* , 3 a; ^ a ^ i 
 
 lO-g-^+T-^- Ans. 2-1. 
 
 11. 8a;»-4a;''2/»+|a;y<-|^. 
 
 12. ^'+64Vi^_4_^^+^'-^;. 
 
 JT y y X x^ a;' 
 
 Ans. - + 2-^. 
 
 a;» a;* 18 27 27 
 
 ^^- 27-3+^"'-^ + ^~^ + ^- 
 
 ^, a;» 12 a;^ 64a; ,^„ 108 a 48 a'' 8 a» 
 
 14. -: — — ^ 4 112 H ^ H ^ • 
 
 a' or a x x^ x' 
 
 ^„ 64a» 1920^^ , 240a ^-- 60a; 12a;'' a;= 
 
 15. — ^ -^ 1 160 H 5- + — • 
 
 x^ ar X a or a? 
 
 A 4a . X 
 Ans. 4-1 
 
 X a 
 
THEORY OF INDICES. ' 241 
 
 CHAPTER XVI. 
 THEORY OF INDICES. 
 
 224. In Art. 72 >ye have proved that a? y^ a? = a?, 
 and inferred that the same principle is true whatever 
 the indices may be. We now propose to prove that, if 
 m and n are positive integers, 
 
 I. or y, dJ" = »"■+"- 
 II. a^ -h-a' = a""-". 
 
 III. (a")" = «""•- 
 
 IV. (ahY = drlr. 
 
 225. To prove (I.), a*" X a" = «"+». 
 
 By Art. II, a" = a X o X a . . . to »n factors, 
 and o" = aXaXa...to n factors. 
 
 Therefore, 
 
 am X a" = (a X a X a ... to m factors) (a X a X a ... to n factors) 
 = aXaXa... to m + n factors 
 
 __ gm-Vv. 
 
 If p is a positive integer, then 
 
 a^ X a" X ar = a'» + " X a» = 0"" + "+?, 
 and so for any number of factors. 
 
 226. To prove (II.), a'^ -^ a" = a*""", when m > n. 
 
 a"" aXaXa... to m factors 
 
 a^ ^a* = — := 
 
 a" a X a X a ... to ra lactors 
 
 = aXaXa...tOTO — n factors 
 
 = a' 
 
 m — n 
 
 16 
 
242 ALGEBKA. 
 
 227. To prove (111.), (a")" = a*"". 
 
 (a'n)n = a"" X a™ X a™ • ■ • to n factors 
 = aw + ra + m ... to n terms 
 
 228. To prove (IV.), (a &)"" = a-'V. 
 
 (a li)"" = (a 6 X a 6 X • • . to m factors 
 
 ^ (a X a X ... to m factors) (i X 6 X • • ■ to m factors) 
 = a™ 6™ 
 
 229. It now remains to show that these laws (§§ 224-228) 
 hold for fractional and negative indices as well, and hence 
 are universally true. We assume that ««, a", a~ " conform 
 to the fundamental law, a" x «" = «"" + ", and accept the 
 meaning to which this assumption leads. 
 
 For example, to find the meaning of a^. According to our assump- 
 tion, we must tiave 
 
 ai X as X as = aJ ■*" 3 "•" 8 = a. 
 Hence, d^ must be such that its third power is a, that is, a* = Va. 
 
 m 
 
 230. To find the meaning of a ", wlien m and n are positive 
 integers. 
 
 m m m 
 
 By Art, 229, a« X a" X a" . ■ to n factors 
 = 0" " "... to n terms 
 
 m + Tti + m . . to n terma 
 
 = a n 
 
 m — I — 
 
 Hence, an must be such that its nth power is a , that is, a« = VO!^. 
 In particular, if m = 1, 
 
 • n 
 
THEORY OF INDICES. 243 
 
 m 
 
 Hence, a» means the nth root of the mth power of o; that is, in a 
 fractional index the numerator denotes a power, and the denominator 
 a root. 
 
 Thus, a^ = ('"a*, a^ = Vcfi. 
 
 231. To find the meaning of a**. 
 
 By Art. 229, a» X a™ = 0° + ^ = a™ 
 
 «*" 
 ... a» = — = 1 
 
 Hence, any number with zero for an exponent is equivalent to 1. 
 
 232. To find, the meaning of a~", wliere m has any 
 positive value. 
 
 By Art. 229, a-" X a" = a -•»+•" = a" = 1 
 
 1 , 1 
 
 Hence, °""' = ^' ^''"^ "" = ^^ 
 
 233. It follows that a factor may he transferred from the 
 numerator of a fraction to its denominator, or vice versa, 
 provided the sign of the exponent of the factor is changed 
 from + to —, or — to +. 
 
 Express with positive indices : 
 1. »-». 9. 
 
 a 
 
 d-^ 
 
 2. 
 
 Sa-n-'. 
 
 3. 
 
 4.x~^a^ 
 
 4. 
 
 1 
 
 4 a--'' 
 
 5. 
 
 1 
 
 6. 
 
 3a-»a;2 
 
 5y^o-' 
 
 7. 
 
 aH--" 
 
 8. 
 
 5x--'y-'z-' 
 
 6a-^6c- = 
 
 10. 
 
 11. 
 
 12. 
 13. 
 14. 
 15. 
 16. 
 
 x^ y~'^ . 
 
 x^y~^! 
 
 b(x — y) ^ 
 X -\- y 
 
 ax~^ z^ 
 b'^c^^y 
 
 (a + 5) (g — 5)- 
 (a-byia + by 
 
 5-^a-''b''x- 
 8' ^m'^ny 
 
244 ALGEBRA. 
 
 Express with radical signs and positive indices : 
 
 17. x's. 
 
 19. 
 
 a-t. 
 
 b-i 
 
 20. 
 
 2 ' 
 
 Pind the value of : 
 
 
 
 23. lei 
 
 25. 
 
 8-1 
 
 24. i-i. 
 
 26. 
 
 (A)-*. 
 
 21. 2a-i 
 
 22. — 1. 
 
 27. {U)l 
 
 28. (/A)-^- 
 
 234. To prove a"" -^ a" = tt'"~" for all values of in and n. 
 
 a"* 
 By Art. 232, a'»-^-o» = — ==o"'Xa-» = a""- » 
 
 Thus, a''.4-a6 = a^-6 = a-» = l 
 
 235. According to the principles estahlished, reduce the 
 following expressions to their simplest form : 
 
 1. a' X a-^ 14. x-^-^x-^. 
 
 2. «-« X ft~^ 15. a'-=-a-^. 
 
 3. a° X a-^. 16. a"* -^ a\ 
 
 4. a-' xa*. 17. 16h^c~i:Pc\ 
 
 5. a^xUfxa-'x-^y''- 18. Vo^-^Va"'"- 
 
 6. a^x^y-'^Xa-^'x-^f. 19. V^i -^ '-v^?- 
 
 7. Sz-iy-^s X 2a;2/=«-^. 20. -^^ X \/^^ \/a'\ 
 
 8. 15 aH'xX 4 a-Hx^y. ^^ x 's/f 
 
 9. 7a-^x3a \ ' -y^y-^ X -y^as" ' 
 
 10. Vxx^x^ 2«*xatx6a-*. 
 
 11. a;'-^»«. 22. 
 
 9 a^ X a^ 
 
 12. a;»-=-a;-''. . 4 
 
 Ans. -^ . 
 
 13. x-^-^x\ ^" 
 
THEORY OF INDICES. 245 
 
 236. To prove that (a")" = «"■"« true for all values of 
 m and n. 
 
 This has already been shown to be true when m and n, are positive 
 integers (§ 227). 
 
 I. Zrt n be a positive fraction,^, where p and q are 
 positim integers. Now, whatever be the value of m, 
 
 By Art. 230, (o™)" = (o™)' = V(a"»)'' 
 By Art. 227, = Vo^ 
 
 mp 
 
 By Art. 230, = a « 
 
 II. Let n be negative, and equal to —p, where p is positive 
 and m unrestricted, as before. Then, 
 
 1 
 
 By Art. 232, (a")" = (a™)-'' = • 
 By Art. 227, 
 
 1 
 
 a""" 
 
 By Art. 233, =a-'»i> 
 
 Hence, (a™)" = a""" for all values of m and n. 
 
 i i m 
 
 It follows that (a")" = (o")" = (a)» 
 
 That is, the nth root of the mth power of a is equivalent to the mth 
 power of the nth root of a. 
 
 Also, (a"*)" = (a")™ = a"" 
 
 That is, the nth root of the mth root of a, or the mth root of the nth 
 
 root of a, is equal to the mnth root of a. 
 
 Thus, (b^y = ji X f = 6^ 
 
 237. To prove that (a 6)" = a" b", whatever be the value 
 of n. 
 
 We have already proved this to be true where n is a positive 
 integer (§ 228). 
 
 Let n be a positive fraction, ", where/) and q are positive integers. 
 
246 ALGEBRA. 
 
 p 
 Then (aJ)»=(aJ)« 
 
 p 
 Now by Art 236, [(a 6)«]9 = (a ft)' 
 
 By Art. 228, = aPb^ 
 
 p p 
 = (a« i')« 
 p p p 
 .-. (a6)' = a«6' 
 
 Let »i have any negative value, say — m, where m is a positive 
 integer. Then, 
 
 (aJ)» = (a6)— »= ^ 
 
 = = O""" J"*" 
 
 238. It follows that 
 
 1 i i i 
 
 (a6c)'' = a"t"c" 
 .•. Toic = Va • rb • re 
 That is, the nth root of the product of several numbers is equal to 
 the product of nth roots of those numbers. 
 
 239. It should he observed that, in the proof above, the 
 numbers a and b are wholly unrestricted, and may them- 
 selves have indices. 
 
 Express in the simplest form, without negative indices, the 
 following examples : 
 
 (z'^^~^)3 -f- (x^y~^)~^ = x^ y~^ -r- a;~3ys = x' i/~^ 
 
 V b ^a-'' ' * b V^^V ' 
 
 = (a^)« = a2 
 
THEORY OF INDICES. 247 
 
 3. {x^y-^f X (x^if)-". 8. '^^^. 
 
 5 i'^-^W^ Ans^_ 10. (x-^v'-)- 
 
 6. (^) • Ans. -\^. 
 
 ■^^ T" 13. V^"^ X -v^^F^. 
 
 14. v^ai-'c-' X (a-iJ-'^c-')"'*. Ans. ai 
 
 15. v''S^^X (a^a:-i)-\ 
 
 16. v'a;- 1 4// -^ V^l^- 
 
 17. (a"^ Vx)-^ X VaJ-Va-*- 
 
 18. -v/»""^''*''"~*-^(*"* ") • '^°®" "*''■ 
 
 19. v^Ca + *)' X (« + &)"*• 
 
 20. {(a;-2/)-»}''^{(a' + y)"}''- 
 
 23. (a~^a;'V^aa;i t^a;^) . 
 
 24. ^/(a + *)° X {a'-jT^- 
 
 Ans. -=• 
 a* 
 
 25 
 26 
 
248 ALGEBRA. 
 
 ^'•(v^x^^'" 
 
 Ans. — - 
 
 28. 
 
 4^^£L. 29. («--)^ + :^" 
 
 30. (.==^0""^ + ^^". 
 
 a; 
 
 31. ■] X a^O'-fy ■ Ans, a*"('-»). 
 
 32. (a=^2/-0V (^)^ 
 
 33 C^lVN* ^ f 2^!^V\ 
 
 2'-x(2-y 1 
 
 2» + i X 2"-^ 4-"' 
 
 36. — ■- ~ ■■ Ans. i. 
 
 C2")"'~ " (2"~^)'''''^ * 
 
 240. Since the index laws are universally true, all the 
 operations of multiplication, division, involution, and evolu- 
 tion are applicable to expressions which contain fractional 
 and negative indices. 
 
 In working examples, orderly aiTangement must be ob- 
 served. The descending powers of x are 
 
 6 4 3 2 11111 
 
 X J X f X J X f Xy ) —^) —J —J) —If 
 
 X x^ x' X* Q^ 
 
 or (§ 232), 
 
 nA ™4 ~,8 ™2 /y, ™ — 1 -, — 'A ,j, — 3 A* — 4 /m — 5 
 
THEORY OF INDICES. 249 
 
 EXAMPLES. 
 
 1. Multiply 3 a;-^ + a; + 2 a;^ by a;* — 2. 
 Arrange in descending powers of x : 
 
 a;i — 2 
 
 a;^ + 2 a; +3 
 
 — 2x —4x^— 6x~'^ 
 
 2. Divide 16 a-o - 6 a-^ + 5 a-^ + 6 by 1 + 2 a'K 
 
 2a-i+l)16a-3— 6 a— ■' + 3a-i+ 6 ( Sa-" _ 7a-i + 6 
 16 a-' + 8a-2 
 
 — 14a-2 + 5a-i 
 
 — 14a-2 — 7a-i 
 
 12a-i+ 6 
 12a-i + 6 
 
 3. Find the square root of -' - ^ + ?( - 1 + — • 
 
 Arrange in descending powers of x : 
 
 x^y-^ + 2xy-i- — l — 2x-^y + x-^y'(xy-i- + l- x-'^y 
 
 2xy-'- + l)2xy-i — 1 — 2x-^y + x-'^y' 
 
 2xy-'^+ 1 
 
 2xy-^ + 2 — x-^y)—2 — 2x-^y + x-^y^ 
 
 — 2 — 2x-^y + x-^y^ 
 
 Multiply : ' 
 
 4. ai 4- 1 + a~^ by a^ — 1 + a~i- 
 
 5. 3a^ — 4ai — a~i by 3a«^ + a"^ — 6a~5- 
 
 Ans. 9 a* — 9 0.^ — 25 + 23 a~* + 6 a~i. 
 
 6. 5 + 3 a:*" + 3 x-^" by 4 a;" — 3 a;-". 
 
 7. a^ — Sa^J + 4a~^ — 2a* by 4a~^ + a* + 4a~i 
 
250 ALGEBRA. 
 
 8. l-2y'x-2xi hy 1- \/x. 
 
 9. 2v'^-ai-- by 2a-3y--a'~*. 
 
 Divide : 
 
 10. 21 a; + a;^ + a;i + 1 by 3 a;* + 1. 
 
 Ans. 7a;9 — 20!^ + 1. 
 
 11. 15 a - 3 a* - 2 a~4 + 8 a"* by 5 a^ + 4. 
 
 12. 16 a-' + 6 a-2 + 5 a-i - 6 by 2 d-i - 1. 
 
 13. 5b^-6bi — 4:b~^-ib~i-5hy &^-2 6"'i 
 
 Ans. 5b^ + ibi + 3b~i + 2b~i- 
 
 14. 8 C-" — 8 c" + 5 c'" — 3 c-*" by 5 c" — 3 c-". 
 
 15. -\^'»2+2a;i-16a;~t_?? by a!B^ + 4a;"* + -^■ 
 
 Ans. xi — 2 xi + 4 a;~^ — 8 x''^. 
 
 16. 1 - Va - -4i + 2 a^ by 1 - ak 
 
 17. 4v^^-8xi-5 + -^ + 3a;~*by2a;TV_^a!-^. 
 
 yx yx 
 
 Ans. 2 a;i — 3 a;~Tif _ a;~ A 
 Find the square root of : 
 
 18. 9 a; - 12 a;i + 10 - 4 a;"i + x'K 
 
 Ans. 3 x^ — 2 + w~i 
 
 19. 25 a^ + 16 - 30 a -24 a* + 49 J. 
 
 20. 4 a" + 9 a:-» + 28 - 24 x~^ — 16 a;^. 
 
 21. 9 x-^ - 18 x-'^y + ^ - 6 y/© + v'- 
 
 Ans. 3a;-= — 3cB-'?/^+y. 
 
RADICALS. 251 
 
 CHAPTER XVII. 
 RADICALS. 
 
 24L A Radical is the indicated root of a number, as 
 ^/x, a2, 's/21, 5^, etc. If the indicated root cannot be ex- 
 actly obtained, the radical is called an irrational expression, 
 or Surd ; otherwise it is called a rational expression. 
 
 242. The factor standing before the radical is the coeffi- 
 cient of the radical. Thus, 5 is the coefficient of -x/S in 
 the expression 5 V3. 
 
 243. Similar Radicals are those which do not differ, or 
 differ only in their coefficients. Thus, '^a, 5 \/a, and 6 ^/a 
 
 are similar radicals ; but 3 Va and 3 y/b, or 3 x^ and 3 x^, 
 are dissimilar radicals. 
 
 244. The degree of a radical is denoted by the root index, 
 or by the denominator of the fractional index. 
 
 The various operations in radicals are presented under 
 the following cases; 
 
 Case I. 
 
 245. To reduce a Radical to its Simplest Form. 
 
 Note 1. A radical is in its simplest form when it is integral, and con- 
 tains no factor whose indicated root can he found. 
 
 1. Reduce y&iaF^ to its simplest form. 
 
 We first resolve 64 a' b^ into two factors, one of which, 64 a', is 
 the greatest perfect cube in 64 o^ b^ ; then, as the root of the product 
 
252 ALGEBRA. 
 
 is equal to the product of the roots (§ 220, Note 2), we extract the 
 cube root of the perfect cube 64 a% and annex to this root the factor 
 remainius under the radical. Hence, 
 
 Resolve the expression U7ider the radical sign into two fac- 
 tors, one of which is the greatest perfect power of the same 
 name as the root. Extract the root of the perf^t power, 
 viultiply it hy the coejffiicient of the radical, and annex to 
 the result the other factor, with the radical sign before it. 
 
 Note 2. M^hen the greatest perfect power of the numerical part of 
 the expression cannot be readily determined by inspection, it should lie 
 resolved into its prime factors, and these factors treated in just the same 
 ■way as the literal part of the expression. 
 
 Thus, 5 ^MV.a'b'^c 
 
 = 5^28-3<a'A«c 
 = 5 slWWWh' X slZahe 
 = 5 ■ 2 ■ 3 (i2 ^iabc 
 
 Reduce the following expressions to their simplest form : 
 
 2. a/S"^*^. 10. v^-lOSx*/ 
 
 3. v^64^^. ^^^- -3xy^^. 
 
 4. V50x'y\ 11. v'gOOOOO. 
 
 -',,6 
 
 5. v'128m^m». 12. ^WoF^. 
 
 6. Sy'Ui^lF. 13. \/x"'tf + '. 
 
 7. 6 ^'260^^. An.. x'fvy' 
 
 Ans. 25 a S v^lO a\ 14. Va' - «' x. 
 
 8. 3 v^3125 c=. 15. {3a'b'-2a'>b<')k 
 
 1 
 
 9. ^^-2187.' 16. (3x^"'y'"'-5siry''"'y 
 
RADICALS. 253 
 
 I 1 
 
 17. a-"'c«(a'""c'"' — a^™»c")". Ans. c\<f — a"")". 
 
 18. Va' + 2a^i + ab\ 
 
 ■ 19. v^Co! + 1)' (^ - 1). 
 20. -^{x^ - a") {x - af. 
 
 21. {x-5)^/ax^+10ax + 2ba. 
 
 22. (8 a;«2/ - 24 a'j/^ + 24 a;^?/' - Sxif)^. 
 
 Ans. 2 (a; — J/) v'a^. 
 
 If the expression under the radical sign is a fraction, 
 multiply both terms of the fraction by the least number that 
 mil make its denominator a perfect power of the same name 
 as the root, and then remove a factor according to the Rule. 
 
 23. Reduce Vi to its simplest form. 
 
 t'l = -^'1 = y-f 1^6 = 4 l'6 Ans. 
 
 24. Reduce y ._ „ to its simplest form. 
 
 TV 1875 62 ~ 7 V 5* • 36" "■ 7 V 5«6» ^2'' • 5= • 3^6 
 
 2-2 
 
 "" 7^5^ V2^ • 52 • 32 6 = j^gj ^9006 Ans. 
 Reduce the following to their simplest forms : 
 25. jVJ- 01 2 
 
 3a! 
 
 .s/27 a:* 
 
 26. 3v^f. 
 
 28. -ft- V-V-- Ans. -r'r Vi^i. 
 
 29. 5'\/77l28. 
 
 
 30, ^*^ 
 
 i/MJ. 34. «C/— ■ 
 
 2c ^ 9a^& 
 
254 ALGEBRA. 
 
 35. 
 
 . 2/t/^!:^". 36. i^x + y)J^-^y. 
 
 Ans. _ V^y 
 
 y- " 37. ^^v/^ 
 
 — as V a 
 
 a 
 
 oo c < /a;" — 2ax^-\-<i^x . 1 ^ — 
 
 "'• S^:^V ?^-^ • ''"^- cl^T^)^^- 
 
 246. Conversely, the coefficient of a radical can be placed 
 under the radical sign, hy involving it to a 'power of the 
 same degree as the root indicated hy the radical sign, multi- 
 plying it ly the radical, and placing the given radical sign 
 over the product. 
 
 Note. Likewise a whole or fractional number may be expressed in the 
 form of a radical, by raising the number to a power denoted by the degree 
 of the radical, and placing the result under the corresponding radical sign. 
 
 In the following examples, place the coefficient under the 
 radical sign. 
 
 1. 4V3^. 
 
 4 V^x = y'le aJWx = /y/48^ Ans. 
 
 2. 5-v^2a;^y. 
 
 3. 3 a; "y/xy. 
 
 5. (x + y)^^^.. 
 \ X + y 
 
 ^ x —y f x"^ + xy y 
 X + y\x^ — 2xy + yV 
 
 4. 2a6v'l-3i. 
 
 Express as radicals of the 
 
 4th and wth degrees : 
 
 7. 2. 8. i. 
 
 9. x\ 10. x". 
 
 Which is greater : 
 
 
 11. 5 a/3 or 3 a/5 ? 
 
 13. 3 v5 or 2 ^Vt ? 
 
 12. 2 a/6 or 3 a/3 ? 
 
 14. 5v^ or 4-V/4? 
 
 15. Arrange in order of magnitude J a/2, J a/3, \ a/4- 
 
RADICALS. 255 
 
 Case II. 
 
 247. To reduce Radicals of different Degrees to equivalent 
 ones of tlie same Degree. 
 
 1. Eeduce ya and '\/h to equivalent radicals of the same 
 degree. 
 
 1 „ _ In this case we write the radicals with 
 
 a^ =^ ofi = y<i? their fractional indices; and then, as the 
 
 ji _. 1,1 __ ^j2 denominator is the iadex of the root, in 
 
 order that the two radicals may have the 
 
 same root index, we reduce the fractional indices to equivalent ones 
 
 having a common denominator. It is evident that we have not 
 
 changed the values of the given radicals by the process. Hence, 
 
 Bnle. 
 
 Meditce the fractional indices to equivalent ones having a 
 common denominator ; involve each numher to the power de- 
 noted by the numerator of tlie reduced index, and indicate 
 the root denoted hy the denominator. 
 
 Reduce the following to equivalent radicals having a com- 
 mon index : 
 
 2. V3, VS. 
 
 3^ = 3ff = 27^ = hi 1 j^^^ 
 5^ = 5^ = 25^= v-2Bf 
 
 3. VI, ^i. 6. V3, ^4, Vs. 
 
 6. 'y/x + y, ^/x — y, 's/x. 8. V^^"? V^^"*- 
 
 9. a^a^, W, Vc^'. Ans. "V^, ""V*^', "" 
 10. Which is greater, -\/2 or ^3 ? 
 
 1^ = 2^ = 2^ = 1^ 
 
 h: 
 
 :3* = 3t=^9 Ans. V3. 
 
256 
 
 ALGEBRA. 
 
 Which is greater : 
 
 11. V2 or 4^5? 12. -v^2 or v^? 13. -v^i or v3? 
 
 Arrange in the order of their magnitude : 
 
 14. V3, 'V'6, and \'12. 15. V5, v^, and -y^B, 
 
 Case III. 
 248. Addition and Subtraction of Radicals. 
 
 1. Find the sum of 2 -y/a and 3 Va. 
 
 It is evident that 2 times the Va and 3 times the Va make 5 times 
 the Va. Ans. 5 Va. 
 
 2. From 7 V* take 4 ^/a- 
 
 It is evident that 7 times the Va minus 4 times the Va is 3 times 
 the Va. Aus. SV^. 
 
 3. Find the sum of -v''500 and y^WS. 
 
 ■^500 = 5 V4 
 
 Vi08 = 3 V4 
 
 Sum = 8 ^ 
 
 4. From V200 take 
 ■V200= 10^2 
 ^128= 8 4^ 
 
 In this case we make the radical 
 parts similar by reducing them to 
 their simplest form (§ 245), and then 
 add their coefBcients as in Example 1. 
 
 -IM. 
 
 Remainder = 2V2 
 5. Simplify V50 + VSi - V24 + 2a/|, 
 
 V50 = V2F2 = 5V2 
 
 Vbi= VW^Q = 3V6 
 
 _V24=— V4^ =—2Vq 
 
 24/1 = 2y/0 = V6 
 
 We make the radical parts similar 
 by reducing them to their simplest 
 form_(§ 245). And 8 V% taken from 
 10 V2 evidently leaves 2 V^. 
 
 Algebraic sum = 2 V^ + 5 V^ 
 
RADICALS. 257 
 
 From the above examples we deduce the following 
 
 Kule. 
 
 Reduce each radical to its simplest form ; then find the 
 algebraic sum of the coefficients of similar radicals, prefix 
 it to the common radical, and indicate the addition or sub- 
 traction of the dissimilar radicals. 
 
 Simplify the following : 
 
 6. a/32 + VSO. 10. 3 'V^192«« + 2 x/Sla^ 
 
 7. 3 Vi2^ + 5 V75^. 11. 2v/405^* + 3v^l280p. 
 
 8. A/i25-V45: 12. V^+ V^^. 
 
 9. v'54-v^l6. 13. 5V24-2V54- V6. 
 
 _ Ans. 3V6. 
 
 14. v^Sl - 7 v^l92 + 4 v^648. 
 
 16. A/S-5V176 + 2V99. Ans. - 12 a/H. 
 
 16. 4 A/i28 + 4 a/75 - 5 Vi62. 
 
 17. 3 aJ'162 - 7 \^22 + v^l250. 
 
 18. 5 -^''I^Si - 2 -^^^I^e + 4 ^^686'. Ans. 17 \^. 
 
 19. 2 a/363 - 5 (243)* + (192)*. 
 
 20. a/252- a/294 --^. Ans. 6 a/7 -15 a/6. 
 
 a/6 
 
 21. 3 A/i47 - 5 a/4 - a/^. 
 
 22. :^-i^+6^2H Ans. 8-^. 
 
 23. A/20 + 3Vt + 3(S)-i-(M)^- 
 
 24. (^)-* + 4d + ^ '^^ + ^ (^)* + 3 ('^-)" ^ + '^*?- 
 
 a/s 
 
 Ans. 27| a/5- 
 17 
 
258 ALGEBEA. 
 
 25. 2a Vc^x — c'y — 3c \/d^x — a'y + 5 '\/d''c^x — a^c^y. 
 
 26. {a" + 2an + aPy^ - {a^ — 2aH + ab^)^. 
 
 27. 6(aa;»-Ja;')i-3(6a;»-aa;»)i Ans. %x{a-b)^. 
 
 28. J^±y + J^. 
 
 IX — y \ X + y 
 
 . ax + a — 2 S ,—. j- 
 
 Ans. 5 z va; — 1. 
 
 x" — 1 
 
 Case IV. 
 249. Multiplication of Radicals. 
 
 1. Multiply 2 V« by 5 \/b. 
 
 2VaX5V6 = 2x5X Va X Vb = 10 Vab 
 
 As it makes no difference in what order the factors are taken 
 (§ 66), we unite in one product the numerical coefficients ; and 
 Va-KVh= V^b (§ 220, Note 2). 
 
 2. Multiply 3 yTa by 4 -y^SoJ. 
 
 3 V2a = 3 tS a' We reduce the radical parts to 
 4 VZah = 4 \'Q~^¥ equivalent radicals of the same 
 degree (§ 247), and then multi- 
 Product = 12 V 72 a^ b^ ply as in the preceding example. 
 
 From these examples we deduce the following 
 
 Knle. 
 
 Reduce tlie, radical 'parts, if necessary, to equivalent radi- 
 cals of the same degree, and to the product of the radical 
 parts placed under the common radical sign prefix the 
 product of their coefficients. 
 
RADICALS. 
 
 259 
 
 7. a Vi' X h^ Va- 
 
 8. 5 -v^2 X 2 VS. 
 Ans. 10v^500. 
 
 9. V2 X -^3 X 'v'B. 
 Ans. "v'a'i". 
 
 Ans. 288v'72. 
 
 Find the value of : 
 
 3. 2 Vi5 X 3 VS. 
 
 4. 8 ^12 X 3 ^24. 
 
 5. 5 v^m X 2 'v''432. 
 
 6. 5 v'32 X 1/48 X 2 V54. 
 
 Ans. 2880. 
 
 10. V^ X ^/h. 
 
 11. V2^ X v'i^ X ^W^. 
 
 12. 3 a/8 X 2 v'g X 3 ^^54. 
 
 13. 2 V24 X 3 Vis X 4 v^24. 
 
 14. 2 Vf X 3 Vf. 
 
 1^- ^V^><3V2-^^' 
 16. ^V^X^'. 
 
 V (a + a;)* V a" V (a — a;)^ 
 
 18. Multiply 2 \/5 + 3 VS by VS — \/i. 
 
 2 V5 + 3 V^ 
 
 10 +3 Vb^ 
 
 — 2 V5x — 3 X 
 
 10 + V5k — 3x Ans. 
 
 19. (2 V« - 5) X 3 \/^- 
 
 20. (v^ + a/6) X a/oS. 
 
 Ans. 3 3. 
 
260 ALGEBRA. 
 
 21. (3 V5 - 4 V2) (2 VB + 3 V^). Ans. 6 + VlO. 
 
 22. (V2 + V3 - VS) (^2 + a/5 + -v/S)- 
 
 23. (VS + 3 V2 + V7) (VS + 3 ^2 - a/7). 
 
 Ans. 16+6 VlO. 
 
 24. (2 V3 + 3 's/2)\ 
 
 25. (Vaj + « — Va! — «) (VaJ + «)• 
 
 26. ( Va+^ - Va - ^)'- Ans. 2 a - 2 Va' - a;^ 
 
 27. (3 ^7 + 5 Vli) (3 V7 - 5 VlT)- 
 
 28. (3 Va + Va; - 9 a) (3 Va - Vai — 9 a). 
 
 Ans. 18 a — K. 
 
 29. (3 A/a' + «•' - 2 Va' - 5^)^ 
 
 30. (Va + a; + Va — !") (Va + ^c — '\/a — a;). 
 
 31. (3 ^2 + 2 a/S) (2 -v/3 - 3 V2) (3 V3 + 2 V2). 
 
 Ans. -18V3-12a/2. 
 
 Case V. 
 250. Division of Radicals. 
 
 1. Divide 15 a/6^ by 6 a/2». 
 
 1 K i/fi — 2 As division is finding a quotient 
 
 ^^ = 3 VSi wliich, multiplied by the divisor, will 
 
 ^ ^ ^ produce the dividend, the coefficient of 
 
 the quotient must be a number which, 
 multiplied by 5, will give 15, the coefficient of the dividend, that is, 3; 
 and the radical part of the quotient must be a number which, mul- 
 tiplied bj' V2^, will give V6 a^, that is, Vs x ; the quotient required, 
 therefore, is 3 ^/3 x. 
 
 2. Divide 8 ^/6a by 4 v^3 a. 
 8 V6"a 8 'V2W^ 
 
 4V3a 4 -f^^ 
 
 = 2 r24 a Ans. 
 
RADICALS. 261 
 
 We reduce the radical parts to equivalent radicals of the same 
 degree (§ 247), and then divide as in the preceding example. 
 
 From these examples we deduce the following 
 
 Bnle. 
 
 Reduce the radical parts, if necessary, to equivalejit radi- 
 cals of the same degree, and to the quotient of the radical 
 parts placed under the common radical sign prefix the 
 quotient of their coefficients. 
 
 Find the value of : 
 
 3. 5V27^3V2i. _ 3-V/48 . 6 V84 
 
 5 a/112 ^392 
 
 Asia, ^V V2. 
 
 */ 
 
 4. 2lV38i-=-8 V98. 
 
 Ans. 3 Vs. 
 
 5. -13Vl2-5-^5V65. '■ 6 ^^^8 ^ 3 ^1.44, 
 
 6. 4cV6^-^3c-^8^^ 10. v^90000 -^ -v^. 
 
 ^ Ans. 5 Vs. 
 
 7. 5 -=- v5. 
 
 ^ 7 V 6 5 V a 
 
 12 -J- l/^^- Sj^^^^. Ans. ^^* . 
 
 a-b\ a-b ' \ (a-b)^ x 
 
 Case VI. 
 
 251. Involution and Evolution of Radicals. 
 
 1. Find the cube of 2 -y/a. 
 
 (2 Vay = 2V^X2VSx2Va 
 
 In accordance with the definition of involution, we take the num- 
 ber three times as a factor. By Art. 249 the product is 8 a Va. 
 
262 ALGEBRA. 
 
 2. Find the cube root of 27 a;' ^/al). 
 
 As the root of the product is 
 
 'V/27 x^ Vab = Sxiab equal to the product of the roots 
 
 (§ 238), we prefix to the cube 
 root of the radical part the cube root of the rational part. The cube 
 root of the radical part must be a number which, taken three times 
 as a factor, will produce Vab ; that is, Vab. 
 
 3. Find the square of 3 '\/x. 
 
 In this case we have used the 
 (S l^xy = (d x^)^ = 9 x« fractional exponent, and found 
 
 the square of the given number 
 by multiplying its exponent by the index of the required power, 
 according to Art. 236. 
 
 Note 1. Dividing the index of the root is the same as multiplying the 
 fractional exponent. Thus, the square of y^a is y'a ; for (a^)^ = a*, 
 or i/a. 
 
 4. Find the fourth root of ^/x. 
 
 J In this case we have used the 
 
 'V Vx = (a;*)T = x^, or Vx fractional exponent, and found 
 
 the fourth root by dividing the 
 exponent of the given number by the index of the required root, 
 according to Art. 236. 
 
 Note 2. Multiplying the index of the root is the same as dividing the 
 fractional exponent. Thus, the square root of Vb is Vb; for (6")^ = i", 
 
 10/- 
 
 or Vh. 
 
 From these examples we deduce for Involution and 
 Evolution the following 
 
 Rule. 
 
 I. Involve or evolve, the radical as if it were rational, 
 and, placing it under its proper radical sign, prefix the re- 
 quired poiver or root of its coefficient. 
 
 II. A radical can he involved or evolved by multiply- 
 ing or dividing its fractional exponent by the index of the 
 required power or root. 
 
RADICALS. 263 
 
 Perform the operations indicated in the following examples : 
 
 5. (ix\^y. 9. -^(x^zVxfz). 
 
 6. {3a-v^xy. 10. ^{b^cd^^liF^). 
 
 7. VSa-'i'a 11. v^(16v'2a). Ans. 2v8«. 
 
 8. 'V/54V2. 12. a/{2v^(4'^)}. 
 
 Case VII. 
 
 Rationalization. 
 
 252. A fraction having a radical for its denominator 
 can be changed to an equivalent fraction with a rational , 
 denominator. This is called rationalizing the denominator 
 of the fraction. 
 
 1st. When the denominator is a monomial, 
 
 g 
 
 1. Rationalize the denominator of — =: • 
 
 -\/2 
 
 5 _ .5xV2 _5V2 
 \'2 V2 X V2 2 
 
 • 7 
 
 2. nationalize the denominator of -j-= • 
 
 V^ 
 
 7 _ 7 X 3? _ 7 h 
 
 3 — 1 2 q~ Ans. 
 
 \i 3* X 3« "* 
 
 From these examples it will be seen that, to rationalize 
 the denominator of a fraction when that denominator is a 
 monomial, we multiply ioth terms of the fraction by the 
 number that is under the radical in the denominator with an 
 index equal to 1 minus the fractional index of this number. 
 
 Eationalize the denominators of the following fractions : 
 
 5 's/Sx 4 c 8a 
 
 a/3 V2a 4^2 \/2 
 
 Ans. 
 
14 
 
 
 
 ALGEBRA. 
 
 
 7. 
 
 3 
 
 8. 
 
 1 
 
 \/8 
 
 9. 
 
 5 
 
 ^5 
 
 11. 
 
 Find the 
 
 square root of 
 
 5 
 
 8' 
 
 
 
 v/i= 
 
 _V5 
 
 V5 
 2V2 
 
 V5 X 
 2V2 > 
 
 V2 
 
 : V2 
 
 10. -^- 
 
 T' 
 
 .^ = ^(3.1623-) = 0.79054-- Ans. 
 
 By this method find the values of the following : 
 
 12. v1- 13. -V^. 14. Vi- 15. ^l 
 
 2d. JF/ieji <Ae denominator is a binomial involving radi- 
 cals of tJie second degree only. 
 
 g 
 16. Rationalize the denominator of 
 
 V3+ v^ 
 5(V3— V^) 5V3 — 5V2 
 
 V3+V2 {V& + V2)(VS— V2) '^~^ _ 
 
 = hVi — oV2 Ans. 
 
 Jj + ^^ 
 
 17. Eationalize the denominator of — ^ -= • 
 
 V7 - V3 
 
 V7 + V'3 _ ( I'Y + V3) ( V7 + V3) _ 7 + 2 V'2l + 3 
 V7 — -fS ~ ( V7 — ^'Z) ( f'7 + ^3) ~ 7—3 
 
 10 + 2 V'2l 5 . , ^.-T, . 
 
 From these examples it will be seen that, when the de- 
 nominator of a fraction is a binomial involving radicals of 
 the second degree only, the denominator can be rationalized 
 iy multiplying hath terms of the fraction hy the denominator, 
 with the opposite sign hctween the terms. 
 
 Eationalize the denominators in the following fractions : 
 
 18. --A-_. 19. "+^. 
 V4 — VS a— \/b 
 
20. 
 
 3 
 
 3+ -\/2 
 
 21. 
 
 3 
 
 3- V2 
 
 22. 
 
 2- V2 
 2+ V2 
 
 23. 
 
 V2+ V5 
 
 
 V'2 - VS ' 
 
 Ol 
 
 Vx— Vy 
 
 
 V^ + Vy 
 
 OK 
 
 3^/6 + V3 
 
 RADICALS 265 
 
 Va; + 1 + 3 
 
 26. 
 
 Va; + 1- 2 
 
 27 Va' — 2 + Va ; 
 -v/a; — 2 — Va: 
 
 2g Va''' + 1 — g ^ 
 Va2 + 1 + a' 
 
 a + Va" + 3 
 
 29. 
 
 a - V*^ + 3 
 
 g^ y/a — b — V« + 
 
 31. 
 
 V'2 a; — 1 + V'a:+ 1 
 
 2 Ve - V3 V'2«^^ - Va; + 1 
 
 32, 1« 
 
 V8 + V108 + VK - a/12 + VB ■ 
 
 33.^-. 34. —^—_. 35. ^- ^. 
 
 V3 - 1 V2 + V3 VS + a/3 
 
 3d. JFAew </ie denominator is a trinomial involving radi- 
 cals of, the second degree only. 
 
 2 
 
 36. Rationalize the denominator of = . 
 
 V7 + '\/S+ a/2 
 2 2iVl + VZ—V^) 
 
 _ 2 ( 1^ + Va — V2) 2 ( 4^ + V3 — -k^) 
 ~ ( l/V + ^3)2 — ( V^)2 ^ 8 — 2 121 
 
 ^ V7 + ^3 — V2 _ ( i^7 + V3— V2)(4 + ^21) 
 4 — V2r ~ (4 — i^)(4 + f^T) 
 
 7V7 + Jli'i$ — 4V^— V42 . 
 = — ^ — '— = Ans, 
 
266 ALGEBRA. 
 
 From this example it will be seen that in this case 
 rationalization is effected by twice applying the principle 
 stated in the preceding article. 
 
 Rationalize the denominators of the following fractions : 
 37. _ ^^ -_. 39. ^ 
 
 VS + V'3 - V2 y2 - V6 — VT 
 
 V" + V* + Vc V* — V^ — Vc 
 
 IMAGINARIES. 
 
 263. An Imaginary Nuinber is an indicated even root 
 of a negative number; as, V— T, V— 2, 's/—(t. In dis- 
 tinction from imaginaries, all other numbers (§ 39) are 
 called real. 
 
 ^SA. The symbol V— 1 is called the Imaginary Unit, 
 and may be defined as an expression which when multi- 
 plied by itself produces — 1. Therefore, 
 
 V^ X V=n: = {'sf-if = - 1 
 
 (V=^/ - (V=^)^ (V-^)^ = (- 1) (- 1) = 1 
 
 Thus, the first four powers of V— 1 are, in order, V— 1, 
 — 1, — V— 1, 1 ; and if we develop the higher powers, we 
 shall find this order continually repeated. 
 
 255. As (V^2 = -1, therefore, (V^ C'^^) « 
 = — a. Now, if a represents a line measured in one di- 
 rection, — a represents a line of equal length measured in 
 the opposite direction (§ 39) ; that is, a line which could 
 be obtained by turning on the zero point as a centre, the 
 first line through two right angles. Hence, the operation 
 hy V— 1 must indicate the revolution of the line a through 
 mve, right angle. 
 
IMAGINARIES. 267 
 
 Now all imaginary numbers may be reduced to the form 
 a V— 1, and hence may be represented by lengths set off 
 on a straight line passing through the zero point at right 
 angles to the line of real numbers. That is, if a represents 
 a certain number of units on the line of real numbers, 
 a V— 1 will represent the same number of units on a line 
 through the zero point at right angles to the first line. 
 
 ADDITION AND SUBTRACTION OF IMAGINARIES. 
 
 266. Addition and Subtraction of Imaginaries do not 
 differ materially from Addition aind Subtraction of ordi- 
 nary Eadicals. 
 
 1. Add V- 9 and 's/^^2§. 
 
 V— 9 = V9 (- 1) = 3 V— T 
 Vir25 = y25(— 1) = 5 V^^ 
 
 2. Simplify 'v'^Tlg _ 2 V^Tl + /^/Hie. 
 
 V^^^:^^ = 1^49 (—1) = 7 V;^ 
 
 — 2V^^ = — 2t'4(— 1) =_4V=^ 
 
 V~\Q= V16(-1)== 4V^rT 
 
 r ;, f ..; 
 
 Reduce the imaginaries to the form a V— 1, and then 
 affix the imaginary unit to the sum of the coefficients of these 
 units. 
 
 Simplify the following : 
 
 3. 3 v"^^ - 2 v^- 4096. • Ans. t/^T. 
 
 4. 6 a/^^25 + 3 V=^8i - 7 •/=1[9, 
 
 6. 6 a V- 25 - 5 a V^^ + 3a 'sf^. Ans, 21a'/^- 
 
2G8 ALGEBUA. 
 
 6. (5a + b V^ - (3 a + 5 J V^ + (8 - 9 i- V^^)- 
 
 7. 5aV^ + (2a + 66V^)-(3a-2iV^) + 7a'\/^- 
 
 8. (a + b V^^) + (.d.+ c V^^). 
 
 9. (a-b V^^) + (a + b) -/^l. 
 
 10. ( V2 + 3 V^ + (V2 - 3 V^^)- 
 
 MULTIPLICATION AND DIVISION OF IMAGINARIES. 
 
 257. As Iraaginaries are Eadicals, the rules already given 
 for Multiplication and Division of Eadicals (§§ 249, 250) 
 might seem sufficient for the Multiplication and Division 
 of Imaginaries. The only difficulty is in the signs. For 
 example, V— 3 X V— 2 is not VG, unless we call this 
 square root — ; the sqiiare root of this particular 6, the pro- 
 duct of — 3 and — 2, is not ambiguous, it is minus. 
 
 As^y^l) (a/=I) = -1, (V=^) (a/=1) = - 2, so 
 (V— 2) (V^^) = not VTQ (unless we call it — ), not 
 V— 16, but — Vl6, or —4. The true result can be se- 
 cured by reducing the imaginaries to the form a V— 1- 
 
 1. Multiply V=r3 by V=^. 
 
 ( ^'—3) ( ^ — ^) = ^'^ t^— 1 ^s ^'^ = ^^ ^5 ( ^^^y 
 
 = Vl5 (— 1) = — VT6 Ans. 
 
 2. Divide - VJ5 by V^^. 
 
 ( Viri5) -^ VITs = (—1 Via) -^ ( i''5 4^^) 
 
 = — ^ X^^ = 4^3 V^ = V—S Ans. 
 4.'5 V— 1 
 
 3. Multiply - ViO by V-10. 
 
 (_ 1/10) ( V^^lO) = (— 1 (liO) ( VITl ViO) = (_!)( Vli"]) ( VlOf 
 = — 10 4^^ Ans. 
 
IMAGINARIES. 269 
 
 4. Divide V-15 by V- 3. 
 
 ( V-15) -^ ( V_ 3) = ( V'15 i^- 1) H- ( V^ i'rT) 
 
 = — X = r a Ans. 
 
 f'3 V— 1 
 
 Bule. 
 
 Reduce the imaginary terms to the form a V— 1- Multi- 
 ply or divide the real parts and the imaginary units sepa- 
 rately ; the product or quotient of the two results will he the 
 answer required. 
 
 Note 1. The treatment of signs in roots may be better understood by 
 considering the folio tiring : 
 
 If we change — 3 to + 3, we change from minus to plus ; if we change 
 ( — 3)2 to (+ 3)*> we have changed the signs of two factors, and the result 
 is no change at all; (—3)2 and (+3)^ both give +9. So (a — b)^ 
 = (6 — of. In changing ( — 3)' to (+ 3)' we change the signs of three 
 factors, and the result is a change from minus to plus; and so on. That 
 is, changing the sign of the number to be raised to an odd power changes 
 the result; to an even power does not change the result; and changing the 
 sign of a number to be raised to the first power is one change, to the 
 second power two changes, to the third power three changes, and so on; 
 that is, the number of changes is equal to the index of the power. In 
 roots then, that is, when the index is a fraction, the same law would in 
 the square root make Jut!/ a change, in cube root a third of a change, and 
 so on. 
 
 Examples 3 and 4 may be more readily worked by the ordinary 
 method, thus : 
 
 3. (- a/IO) X (V^^W = — V^^lOO = - 10 V^ Ans. 
 
 4. (A^n5) -=- V- 3 = V5 Ans. 
 
 Note 2. A departure from' the ordinaiy method will only be necessaiy 
 then, in multiplication, when both tenns are imaginary, and in division 
 when one is imaginary and the other real. 
 
 258. The sum of a real and an imaginary is called a 
 Complex Number ; as, a + 6 V^^. 
 
270 
 
 ALGEBRA. 
 
 259, Two complex numbers which differ only in the 
 sign of their imaginary part are said to be conjugate; as, 
 
 - 3 - 2 V^^ and — 3 + 2 V^^ ; 
 
 a -\- b V — 1 and a — h \/ — 1. 
 
 260. The sum, and the product, of two conjugate complex 
 numbers are real. 
 
 For a + hV^^ + a — b's/—l = 'ia; 
 
 and {a-\-h V^^) (a-b y^^) = < 
 
 ■ - (- J2) = a= + 6^. 
 
 26L It follows that the sum of two squares can be fac- 
 tored by the introduction of the imaginary unit. TIius 
 the factors of a^ + V^ are a + 6 V— -1 and a — h V— 1. 
 
 Further, if the denominator of a fraction is of the form 
 a + b V— 1, it may be rationalized by multiplying the nu- 
 merator and the denominator by the conjugate expression 
 a — h V — 1. 
 
 262, Complete the work indicated in the following expres- 
 sions : 
 
 5. (V=^)(V^^). 
 
 6. (V^Tfi^) (V^^. 
 
 7. (_ 5 V^^) (3 V=^). 
 
 8. (v'=:2)(V=^)(v':i4). 
 
 1. (V^6)(aA^). 
 
 2. (V=^) (v^Tg). 
 
 3. (- V8) (V^Tl). 
 
 4. (3 V^^) (2 V^^)- 
 9. (V^ (V^^) (V=^) (V^. 
 
 10. (2 + 3 V^^) (3-2 V^^)- 
 
 11. (5 - V^^) (4 + 2 a/^TS). 
 
 12. (2 V^^ + 4 -/^TS) (3 V^^ - 5 V^^)- 
 
 13. (2 + a/=^) (2 ^ V'=2). 
 
IMAGINARIES. 
 
 14 (a; — V^) (a + \/^). 
 
 16. (V^^ + V^^ + V^^) (a/= 
 
 271 
 
 2 - V- 3 - V^^). 
 Ans. 5 + 4 \/3. 
 
 16. {1 - V(l - «')} {1 + V(l - e-")}. 
 
 17. (a V^^ + b V^^) (a V^^ — * V^)- 
 
 ,.(. 
 
 18. U 
 
 1 + 
 
 ^)(' 
 
 1- 
 
 19. (-Vi8)-H(v/=^). 
 
 20. (- V-15) -^ (- V5)- 
 
 21. ( V^^^^^) ^ (- V5). 
 
 22. (V15) -=- (V=^). 
 
 23. (- \/i5) -^ (V^rg). 
 
 24. (-v^Zl5)-^(-V=^. 
 
 25. (a/=^15) -^• ('Z^). 
 
 Ans. a;'' — a; + 1. 
 
 28. (3 + V=^)'. 
 
 29. (V^^ - 3 V^=T)». 
 
 30. (\/=n + 2 \/^^)«. 
 
 31. (i + v^r3)«. 
 
 32. (1-V^*. 
 
 33. (-l+^^/ITlf. 
 
 26. (6V^^10)^(2-v/-5). 
 
 34. (-i-f 
 
 27. 8A/-a'^-=-(2V^^). 
 
 Ans. 1. 
 
 35. (Vo + 40 V_ 1 4- Vg - 40 ^- 1)". 
 Express with rational denominator : 
 
 36. 
 
 39. 
 
 1 
 
 3- V2 
 
 37. 
 
 4 + V^^ 
 
 (a + V_i)s _ (g _ ^_^i). 
 
 38. 
 
 Ans. 
 
 a + aiV— 1 
 a — a; V — 1 
 
272 ALGEBRA. 
 
 Find the factors of : 
 40. a" + 9. 41. a + b. 
 
 42. a;^ — 6 a; + 13. 
 
 Ans. (a: - 3) + 2 V^^, (a; - 3) - 2 -v/=^- 
 
 43. a;'-8a!+ 17. 
 
 44. x^ + 3. Ans. x + V^^, ^ — V^^- 
 
 45. Find the value of (— V^^)*" + ', when w is a positive 
 integer. 
 
 BINOMIAL SURDS. 
 
 263. A Binomial in which one or both of the terms are 
 surds is called a Binomial Surd. 
 
 264. To explain the method of finding the square root of 
 a binomial surd, we square V^ + V3. 
 
 ( V5 + V3)2 = 5 + 2VT5 + 3 = 8 + 2Vl5 
 
 To find the square rorrt of 8 + 2 Vl5 it is evident, then, that we 
 must find two numbers whose sura is 8 and whose product is 15. 
 The only numbers that answer these conditions are 5 and 3. Hence, 
 
 V8 + 2Vl5= VE+ Vs. 
 
 1. Find the square root of 11 + 6 \/2. 
 
 11 is the sum of the two numbers sought, and 6 V2 is ttoice their 
 product, or 3 ^2 = vTs, their product. The only numbers whose 
 sum is 11 and whose product is 18 are 9 and 2. Hence, 
 
 Vn + QV2= V9 + V2 = Z+ Vo Ans, 
 Hence, to find the square root of a binomial surd. 
 
BINOMIAL SUKOS. 273 
 
 Rule. 
 
 Write the binomial surd so that the radical part shall 
 have 2 for its coefficient. Then by inspection find two num- 
 bers whose sum, is the rational term and whose product is 
 the number under the radical, and connect tlie square roots 
 of these two numbers vAth the sign of the radical part. 
 
 Find the square root of : 
 
 2. 9 + 4V5. 5. r-2ViO. 
 
 3. 7 + 4 V3. 6. 10 - ,6 VI. 
 
 4. 27 + 10 V2. 7. 23-4 VlS. 
 
 8. 4 + Vis. 
 
 V 2 ^ 
 
 9. V32 - V30. 
 
 V( V3J— vao) = V{ ^"2 (I — VIE)} 
 
 = V^ V(4: — Vl5) 
 
 _ V2(V^— V3) ^ V5— VS ^^g 
 
 ~ V2 " - 1^2 
 
 10. 4-2 V3.~ 14. 31 + 12 V^^. 
 
 11. O-^VIi. 15. -5+12 V^. 
 
 12. 10 + 2 V21. 16. V^ - V-^- 
 
 13. 6 - VSB. 17. a= - 1 + 2 a V^- 
 
 18, 4a6-2(a«-&'*) V^^. 
 
 Ans. (a + J) — (a — b) V^- 
 18 
 
274 ALGEBRA. 
 
 CHAPTER XVIII. 
 
 RADICAL EQUATIONS. PURE EQUATIONS. 
 
 265. Badical Equations, that is, equations having the un- 
 known number under the radical sign, require Involution 
 in their reduction. 
 
 1. Eeduce 4 V^ ■ 
 
 -8 = 8. 
 
 Transposing, 
 
 4t^=16 
 
 or, 
 
 V5 = 4 
 
 Squaring, 
 
 a; =16 
 
 2. Reduce V*^ - 
 
 - 3 + a; = 3. 
 
 Transposing, 
 
 Va;2 — 3 = 3 - 
 
 Squaring, 
 
 g;2 _ 3 = 9 - 
 
 or, 
 
 6a: = 12 
 
 Ans. 
 
 a; = ]2 
 
 3. Reduce ^/x + 5 — 's/x — 1 = 2. 
 
 Transposing, Vs + 5 = 2 + Va; — 1 
 
 Squaring, x-{-5 = i:-\-iVx — 1 + a; — 1 
 
 Transposing, uniting, and dividing by 2, 
 
 i = 2VS'^::ri 
 
 Squaring, 1 = 4 a; — 4 
 
 Whence, a; = ^ 
 
 Hence, to reduce radical equations, we deduce from these 
 examples the followjng general 
 
RADICAL EQUATIONS. 275 
 
 Bule. 
 
 Transpose the terms so that a radical part shall stand iy 
 itself; then involve each member of the equation to a power 
 of the same degree as the radical; if the unknown number is 
 still under the radical sign, transpose and involve as before; 
 finally reduce as usual. 
 
 Beduce the following equations : 
 
 4. 7+f + 5v^ = ^K J Vac _ je_ 
 
 x — cx~ y^' 
 
 Vc _ 8. Va; — 9 = V« - 1- 
 
 6. 0.2 + 0.5v/f = 2.2. 9.(^2^^2)^ = 2. 
 
 10. V«T8 = V«^=T + Vs. 
 
 11. Vi-Vx+a= ^ — . Ans. -^a. 
 
 yx + a 3 
 
 12. Va;" — U = x— y 2. 
 
 la Vx-Vx-9 + 17=a/x- 23. 
 
 > . . - 
 
 Vx + 3 _ yS — 18 
 Va: — 8 V* + 4 
 
 15. V^JJ^4V3^+10^ ^^^3 
 V3a! + 2 4v'3a:-2 
 
 16. V5a + 8 = 13. 
 
 17. VS X + 17 — V2x = V2 a; + 9. Ans. 8. 
 
 18. Va! + 3 + V« + 8 — ^4 a; + 21 = 0. 
 
 19. V^T^ + Vix+1. - a/9 a; + 7 = 0. 
 
 20. ^x + Vi^a + x = 2 Vb + x. Ans. ' _^ 
 
276 ALGEBUA. 
 
 5 
 
 ■ 22. V9 + 2x— V'^x- 
 
 23. ^x — \^x — 8 : 
 
 Va + 2x 
 2 
 
 Vx- 8 
 
 24. :i-^ + -*• + ^ = 0. 
 1— « Va; — 1 Vx — l 
 
 25. V7 x - 5 + a/4 a; - 1 = a/7 X - 4 + a/4 a; - 2. 
 (Square without transposing.) 
 
 PURE EQUATIONS. 
 
 266. A Pure Equation if? one that contains but one power 
 of the unknown number ; as, 
 
 X + ab = c, 5a;^+8 = 13, or 7 a,™ = ac. 
 
 267. A Pure Quadratic Equation is one that contains only 
 the second power of the unknown number ; as, 
 
 5 a;^ — 7 a = 6, af + S = d, or a ia;^ =. 4. 
 
 268. Equations containing the unknown number in- 
 volved to any power require Evolution in their reduction. 
 
 269. To reduce pure equations containing the unknown 
 number involved to any power. 
 
 1. 
 
 Keduce -. = 1. 
 
 4 
 
 
 4 
 
 
 
 3 a;2 ■ „ 
 4 -^ 
 
 
 a;2 = 4 
 
 
 x = ±2 
 
 Performing the division indicated 
 in the first member, transposing, and 
 imiting, we have f a;^ = 3 ; dividing 
 by |, we have x' = i; extracting the 
 square root of each member of x' = 4, 
 we have a: = ± 2 (§ 213). 
 
rURE KQUATIONS. 277 
 
 2. Reduce 1 + 50 = 1. 
 
 — 4- ISO ^ 1 Transposing and uniting, we have 
 
 s - — = — 49 ; clearing of fractions, we 
 
 |r = - 49 7 
 
 < havea;'^7( — 49); extracting the cube 
 
 x^ = 7 {-— 49) root of each member of k' = 7 ( — 49), 
 
 j; = 7 we have x = — 7. Hence, 
 
 Bule. 
 
 Meduce the equation so as to have as one mewher the un- 
 hnown number involved to any degree,, and then extract that 
 root of each memier which is of the same name as the power 
 of the unknown number. 
 
 Note. It appears from the solution of Example 1 that miery pure quad- 
 ratic equation has two roots numerically the same, hut vnth opposite signs. 
 
 Eeduce the following equations : 
 
 3. (x — 2y = 4: — 4=x. g ac — x^ _ bd-x'' 
 
 c d 
 
 4. 4 z'' — 7 = f a;2 + 14. 
 
 X — 3 3x — 1 
 
 5. fx^+f^^x^-W- ^- 3^rfl~l^T3- 
 
 270. Equations may require in their reduction both In- 
 volution and Evolution ; and in this case the rule in Art. 
 265, as well as that in Art. 269, must be applied. Which 
 rule is first to be applied depends upon whether the ex- 
 pression containing the unknown number is evolved or 
 involved. 
 
 a — b 
 
 8. .^/x — a = 
 
 ■\/x -f a 
 
 Clearing of fractions, Vx^ — a^ = a — b 
 Squaring, x^ — a^ = a^ — 2 a 6 + 6^ 
 
 Transposing and uniting, x^ = 2 a^ — 2ab -\- b^ 
 Extracting the square root, a; = ± V(2 a' — 2 ab -{- b^) Ans. 
 
278 
 
 ALGEBKA. 
 
 9. 15 + Va;' + 17 = 24. 11. (a/9 + «» - 2)» = 64. 
 Ans. 3. 
 
 lO' V ^^ = ^" 12. -V- - ^2(Sm:a) = ¥• 
 
 27L Simultaneous equations, containing two or more 
 unknown numbers, may require for their reduction Invo- 
 lution, or Evolution, or both. In these equations the elimi- 
 nation is effected by the same principles as in simple 
 equations (§§ 180-182). 
 
 13. 
 
 2x 
 
 y . 
 
 3 +5 = «-^- 
 3a;^-i/ = 25. 
 
 (1) 
 
 6 -I- I = 6.4 
 5 
 
 « = 2 
 
 (5) 
 (6) 
 
 3a; 
 
 »_j,= 
 
 25 
 
 (2) 
 
 10 a' 
 3 
 
 + 2/ = 
 
 32 
 
 
 
 19x2 
 3 ~ 
 
 57 
 
 (3) 
 
 
 a;2 = 
 
 9 
 
 
 
 X = 
 
 ±3 
 
 (4) 
 
 Adding five times (1) to (2), we obtain (3), which reduced gives 
 (4), or a; = ± 3 ; substituting this value of x in (1), we obtain (5), 
 which reduced gives (6), or ^ = 2. 
 
 ■a;=.vz = 48. 
 
 14. -^xfz = 72. 
 
 .xyz^ — 96. 
 
 15. 
 
 a;y = 20, 
 5 
 
 = 2(x-y) 
 
 4 
 
 Ans, 
 
 X = 5. 
 4. 
 
 16. 
 
 17. 
 
 18. 
 
 <x* 
 
 ( X 
 
 '* = 15. 
 + 2i/ — 6y — X. 
 
 |2a! + 3y = 10y. 
 U''-4y2=33. 
 
 3xy= 24. 
 4a:s= 48. 
 5 y « = 120. 
 
PURE EQUATIONS. 279 
 
 272. The following problems give equations that are to 
 be reduced according to the principles in Arts. 265-271. 
 
 1. What two numbers are as m : n, of whose squares the 
 sum is 9 ? 
 
 Let ma; = the first number, 
 
 then nx = the second number. 
 
 Then m^ x^ + n" x^ = 9 
 
 9 
 
 or. 
 
 m2+ n" 
 
 3 
 
 a; = ± - 
 
 Ans. 
 
 mx = ± — ^1::=:^ , the first number. 
 nx = ± -— ^==^:^ , the second number. 
 
 2. There is a rectangular field whose area is 6 acres, and 
 whose length is to its breadth as 5:3. Find the length and 
 breadth. 
 
 3. The sum of two numbers is 25, and the less divided by 
 the greater is to the greater divided by the less as 4 : 9. 
 Find the numbers. Ans. 15 and 10. 
 
 4. A teacher, being asked how many pupils he had, said, 
 "If you subtract 17 fropi the number, add 4 to the square root 
 of this remainder, and multiply this sum by 10, I shall have 
 100." How many had he ? 
 
 5. A lumber-dealer bought two wood-lots, containing to- 
 gether 210 acres. For each he paid as many dollars an acre 
 as there were acres in the field, and what he paid for the 
 greater was to what he paid for the less as 16 : 9. Find the 
 area of each. Ans. Greater, 120 ; less, 90 acres. 
 
 6. Find three numbers such that the square of the first 
 multiplied by the second is 45, the square of the second mul- 
 tiplied by the third is 175, and the square of the third multi- 
 plied by the first is 147. 
 
280 ALGEBRA. 
 
 CHAPTEE XIX. 
 AFFECTED QUADRATIC EQUATIONS. 
 
 273. An Affected Quadratic £q[uatiou is one that contains 
 only the first and second powers of the unknown numbei'; 
 
 ' 2x^ — 7 X = 15; 01 ax — bx^ = c. 
 
 274 Every affected quadratic equation can be reduced 
 
 to the form „ , ^ 
 
 x^ + ox = c, 
 
 in which b and c represent any numbers whatever, posi- 
 tive or negative, integral or fractional. 
 
 For all the terms containing x^ can be collected into one term 
 ■whose coefficient we will represent by a ; all the terms containing x 
 can be collected into one term whose coefficient we will represent 
 by d ; and all the other terms can be united, whose aggregate we 
 will represent by e. Therefore every affected quadratic equation 
 can be reduced to the form 
 
 a a;^ + rf X = e (1) 
 
 Dividing (]) by a, x^ + - x = 1 (2) 
 
 d e 
 
 Letting - = b, and _ := c, we have x'^ + hx = c (3) 
 
 a a 
 
 275. The first member of the equation x'.+ bx = c can- 
 not be a perfect square (§ 222, Note 2). But we know 
 that the square of a binomial is the square of the first term, 
 plus or minus twice the product of the two terms plus the 
 square of the last term ; and if we can find the third term 
 which will m.ake x'^ + bx a perfect square of a binomial, 
 we can then reduce the equation x^ + bx = c. 
 
AFFECTED QUADKATIC EQUATIONS. 281 
 
 Since bx has in it as a factor the square root of x% x^ can be 
 the first term of the square of a binomial, and b x the second term 
 of the same square ; and since the second term of the square is twice 
 the product of the two terms of the binomial, the last term of the 
 binomial must be the quotient found by dividing the second term 
 of the square by twice the square root of the first term of the square 
 
 of the binomial; that 
 a;'^ + i r := c (1) is, the last term of the 
 
 x>' + 6^+|' = ^ + c (2) binomial is || = ^; 
 
 , I— — '■ — and therefore the third 
 
 a; + - =: ± i/ — + c (3) term of the square must 
 
 - to each member, we 
 4 
 
 have (2), an equation whose first member is a perfect square. Ex- 
 tracting the square root of each member of (2), and transposing, we 
 
 obtain (4), or a; = — o * V/T "I" '^' ''^^i'^'i i^ ^ general expression 
 for the value of x in any equation in the form of a;^ + ft a; = c. 
 
 Hence, as every affected quadratic equation can be re- 
 duced to the form x^ ■\-hx = c, in which & and c repre- 
 sent any numbers whatever, positive or negative, integral 
 or fractional, every affected quadratic equation can be re- 
 duced by the following 
 
 Bnle. 
 
 Reduce the equation to the form .x^ + hx = c, and add to 
 each member the square of half the coefficient of x. 
 
 Extract the square root of each member, and then reduce as 
 in simple equations. 
 
 Note 1. Reducing an equation to the form x^ -\- bx = c means, not 
 only that all the terms enntainiug x'' are to be imited into one term, and 
 all those containing x into one term, and all the other terms transposed to 
 the right-hand member of the equation, but also that the coefficient pi x^ 
 must be one, and its sign -|-. (See Art. 213, last section of the Rule.) 
 
282 ALGEBRA. 
 
 Reduce the following equations : 
 1. 5 a;'' - 7 = 10 a; + 68. 
 
 Transposing and uniting, ox^ — 10 x = 75 
 
 Dividing by 5, . x^ — 2 a; = 16 
 
 Completing the square, x^ — 2 a; + 1 = 16 
 
 Evolving, X — 1^+4 
 
 Transposing, x — X ±4 = 5, or — 3. 
 
 Note 2. Since, in reducing the general equation a;^ + J a = c, we find 
 
 x = — Q ± 1 / h c, every affected quadratic equ ation m ust have two 
 
 V * . IT^ 
 
 roots ; one obtained by considering the expression \/ "2 "r " positive , the 
 
 other by considering this expression negative. Whenever 1/ — + c = 0, 
 these two roots will be equal. 
 
 6 
 
 ^" 4 "*■ 6 6 12 12 
 
 X' , X 
 
 o 
 
 Transposing and uniting, -|- - = . 
 
 4 o 
 
 2x 8 
 Multiplying by —4, x^ — - = - 
 
 Completing the square, a2 __ + - = - + -= — 
 
 Evolving, a: — J = ± f 
 
 Transposing, a; = J±| = 2, or— | 
 
 • 3. 5 a:^ — 27 — 10 a; = 13. Ans. a; =: 4, or — 2. 
 
 5 2 (a; + 5) . „ „ 
 
 4. X = — ^ • Ans. a; = 5, or — 6. 
 
 X X 
 
 5. X — 5 = ^ ^ ■ Ans. x = 9, or 3. 
 
 X — 6 
 
 4 41 
 
 6. 5a;- 23 + -=7 Ans. a; = 3. 
 
 X X 
 
 Note 3. In this example both roots are 3. 
 
 25 
 
 7. 3a! H .,-p; = 10. Ans. x = 8L or 5. 
 
 a; — 10 
 
AFFECTED QUADRATIC EQUATIONS. 283 
 
 _a;64 — a;a;.„ ^j- x 
 
 «• 8-^34=4-1^- 11-^ = 4^^3-2 + 2. 
 
 7'a; — 3 ' " a! + 3 3x + 9 
 
 a; +8 4 1x1 
 
 10. ., =-• 13. a; = 4 — Tt + j-— ■ 
 
 x^+la: a; 22a: 
 
 14. 3a;=-6aa: = 6a + 3. 
 
 Dividing by 3, z^ — 2ai = 2a + l 
 
 Completing the square, x'^ — 2aa; + a^ = a^+2a+l 
 Evolving, X — a = ± (a + 1) 
 
 Whence, a; = a ± (a + 1) 
 
 = 2a + l, or — 1 
 
 15. a:= — m'' =3 2 w a; - «^. 21. a;" + i" = 2 a a;. 
 
 1 111. 
 Id. ; ; — = 1 1 Ans. a; = —n, or —m- 
 
 m + X + n m x n 
 
 17. x — b = a. 22. bdx^ + ac — bcx + adx. 
 
 X 
 
 a; + l a + 1 x^ + 1 m ^ n 
 
 or a^ X n m 
 
 19. (a; + 2 a)» - (x + a)' = 37 a». 
 
 20. ~ = pr—j r . 24. 2a! + 3a = -5-H 
 
 a; — 3c9(« — c) 3 x 
 
 276. Whenever an equation has been reduced to the 
 form x^ + bx = c, its roots can be written at once ; for 
 
 this equation reduced (§ 275) gives x = — ^± 1/ — + c. 
 Hence, 
 
 The roots of an equation reduced to the form a^ + hx — c 
 are equal to one half the coefficient of x with the opposite sign, 
 plus or mimis tlie square root of the sum of the square of one 
 half this coefficient and the second member of the equation. 
 
284 ALGEBRA. 
 
 In accordance with this, find the roots of x in the following 
 equations : 
 
 25. x^ + lQx = 56. 
 
 Ans. a; = - 5 ± V25 + 56 = — 5 ± 9 = 4, or — 14. 
 
 26. x^ — i.x = 12. 
 
 Ans. a; = 2 ± v'4 + 12 = 2 ± 4 = 6, or — 2. 
 
 27. a;" + 8 a; = 20, 
 
 Ans. X = iV ± V^k + tV = iV ± A = ^, or - f . 
 29. a;= + 6 a; = 50. 30. a;^ - 4 aa; = - 4 a^. 
 
 31. a;^— (a + S)a; = — a6. 
 
 Ans. ^ = -II- ±y/ ^_^j _ ab 
 
 vT-^T 
 
 ' 2 * 
 
 (1 + 6 a — ft 
 
 ± — - — = a, or 
 
 2 2 
 
 32. a;= + (m — m) .T = m re. 35. 7 — 2 a; = | x^. 
 
 33. a5'=-2(c + cZ)a; = -4frf. 36. 5a;2 — 39 = 2a;. 
 
 34. x"" = a^(l-a)x .+ a\ 37. 10 x^ + 30 = 40 x. 
 38. cx = ax^ — h. Ans. a; = jr— (e ± -y/c^ + 4 a 4). 
 
 39. 3 a;^- 7 a; = 13. 
 
 
 
 
 Ans. a; = J (7 ± V205) 
 
 .- 7a;2 a 4 .r 
 ^'- 5a 7--5-- 
 
 
 
 
 . 5a a 
 Ans. « = — , or --. 
 
 41. 3a;^+^^'^_2a;'' 
 
 — 
 
 4. 
 
 
 Ans. a; = — ^, or — 8. 
 
 2x x — Z_ 
 
 x 
 
 — 
 
 1 
 
 
 14 -a; 
 
AFFECTED QUAUBATIC EQUATIONS. 285 
 
 ,„ 2a!'-8 ^ „ a: -3 
 
 43. ^ 3 aj + 7 = — g- . 
 
 44. a;= - 2 Sa; = (a + 6) (a - J). 
 
 45. f £«!^ — a; = ^. 
 
 48. 
 
 a;2 + f = V- «• 
 
 16 ^ - "^ 
 
 49. 
 
 1 2 _ .2 
 Sa; a;+l 
 
 ■ 586 88 a;-^- 30a; 
 
 *7- i9+.? =1- 
 
 a; + 2 5a! 
 
 50. 
 
 3 13 2 
 
 a; + 2~ 5 7-2a: 
 
 51. 5-2a;-2 = 3a;-». (Multiply by a;^) 
 
 52. 2 V^ — 10 v5 = 3 Va"2. (Divide by V^.) 
 
 3 7 -33 a; . „ a;«-2a;= + 5 
 
 ^^- "^ c r~n = 1- 55. -5-— 75 ^7 = a; — 1. 
 
 a;'' — 6a; + 9 a5'' + 2a; — 8 
 
 54. l-22x-2 3=|a;-i. 56. (a; - 2) (a; - 5) = 13|. 
 
 g a; — 2 a; + 1 _ 2 a; +20 
 a; — 1 X — 3 33 + 3 
 
 NoTK. There are other methods of completing the square, but nothing 
 is gained by their use. The method of finding the roots of an affected 
 quadratic equation given in this article is the shortest method, and it will 
 be used in all the examples that follow. 
 
 277. The rule which has been given for the solution of 
 afi'ected quadratic equations applies equally well to any 
 equation containing but two powers of the unknown num- 
 ber whenever the index of one po^uer is exactly twice that of 
 the other. By the same reasoning as in Art. 274, it can be 
 shown thiat all such equations can be reduced to the form 
 
 a x^" + d x" — e, 
 or 
 
286 ALGEBRA. 
 
 It will be seen that the first member is composed of two 
 terms so related that they may be the first two terms of a 
 binomial square, and we can supply the third by the rule 
 already given for completing the square. 
 
 Reduce the following equations : 
 1. JC* - 4 «'' = 45. 
 
 X* — 4 x'-2 = 45 (1) Since the square root 
 
 a;4_43;2_|_ 4 = 4_^ 45 = 49 (g) of a:* is a^jt is evident 
 
 2 2 _. .7 fQ\ that the second tenn con- 
 
 „ „ , ,,. tains as one of its factors 
 
 a;^ = 9, or — 5 (4) , „ , „ 
 
 , the square root of the iirst 
 
 x= ±3, or ± V^^ (5) . *i, 4. • 4.1, c 4. 
 
 ' *• ■' term ; that is, the first 
 
 member of the equation 
 
 is composed of two terms so related that they may be the first two 
 
 terms of the square of a binomial. Completing the square, we have 
 
 (2); extracting the square root of each member of (2), we obtain (3); 
 
 transposing, we have (4) ; and extracting the square root of (4), we 
 
 have a; = ± 3, or ± V — 5. 
 
 Or, reducing the equation by the rule in Art. 276, 
 
 X* — 4 a;2 = 45 
 
 x2=2±V4.+ 45 = 2±7 = 9, or —5 
 x = -±3, or ± V^^5 
 
 2. 5x^ + 3a:*=344. 
 
 5 a;t + 3 x^ = 344 
 
 xi= — 0.3 ± VO.09 + 08,8 = _ 0.3 ± 8.3 = 8, or — S.6 
 a; = 16, or (—8.6)* 
 
 3. x*+ -^ = j^. Ans. a; = ± |, or ± iV—S. 
 
 4. a;« - 2x« = 48. Ans. a; = 2, or - v6. 
 
 5. 2 a;*- 3 a;* = 2. Ans. a: = 8, or - ^. 
 
AFFECTED QUADRATIC EQUATIONS. 287 
 
 6. a;« — I a;» = 58. Ans. a; = 2, or - ■^T^. 
 
 \/x + 6 2 Vai+ 6 
 
 Verify both answers. 
 
 ^/x 's/x + 1 
 
 8. x — iVx = 22. 10. x' - « a;' = 6". 
 
 9. fa;''»-iz» = 57i. 11. a;-< + 4a:-'' = 32. 
 
 278. A polynomial may take the place of the unknown 
 number in an affected quadratic equation. In this case 
 the equation can be reduced by considering the polynomial 
 as a single term. 
 
 Reduce the following -equations : 
 
 1. (x + 5)'' - 4 (a; + 5) = 32. 
 
 Considering a: + 5 
 (x -\- oy — 4 (x -f- 5) := 32 (1) iis a single term and 
 
 (a; + 5)2 — 4 (x + 5) + 4 = 36 (2) completing the square, 
 
 1+5 — 2:=±6 (3) we have (2); extracting 
 
 s = 3, or 9 (4) the square root, trans- 
 posing, &c., we have 
 (4), or a; = 3, or —9. 
 Or, by Art. 276, from (1), we have at once 
 
 a: + 5 = 2 ± i'4 + 32 = 2 ± 6 = 8, or —4 
 x = S, or — 9 
 
 Note. We might put {x + 5) =y; then (x + 5)^ = !/'•', and the 
 equation becomes y" — iy = S2. After finding the value of y in this 
 equation, a; + 5 must be substituted for y. 
 
 2. (2 X + 3)* - (2 a; + 3) = 42. Ans. a; = 2, or - 4^. 
 
 3. V19 — X — \/l9 — a; = 2. Verify both answers. 
 
 4. a;2 — a: + 4 + \/x^ — a; + 4 = 2. 
 
 Ans. a; = ^ (1 ± V— H), or 1, or 0. 
 Verify these answers. 
 
288 ALGEBRA. 
 
 3 
 
 5. {2x- Sf + 2 V2a;-3 
 
 ' 2x-S 
 Ans. a; = 2, or 1(3 + -y^). 
 
 6. a;= + 2 a; - Va;' + 2 a; - 6 = 12. 
 (Subtract 6 from both sides.) 
 
 7. 4 X + 7 + V4: + 2 a; - a^2 = 2 x\ 
 o 3 + a:2 
 
 9. 1 + ^ Va;' + ^ + 5 
 
 a; + Ve — 2 X + a;''. 
 10 
 
 ^/x^ + a; + 5 
 
 MISCELLANEOUS EXAMPLES. 
 
 279. In the following examples, when the answers are 
 not given, the answers obtained should be proved by sub- 
 stituting them in the original equation. 
 
 Reduce the following equations : 
 
 4 
 
 1. X -\ f- 5 = 0. Ans. X = — 1, or — 4. 
 
 X 
 
 ^- -2-x + ^^ = l- Ans. a.= l, or-f 
 
 a;2 25 X 
 
 3. 2" ~ "6" ^ 9 ■ ■^"®" ^ = ^> or - 25. 
 
 4. (a; + 3) (a; — 3) = 6 X — 14. Ans. x — b, or 1. 
 
 5. (x + 2y + 61 = (x + 3)'. 
 
 6. 5 a;^ + 8 = 2 x^ + 8 a; + 24. 
 
 7. (2x + 3)'' = (.r + 8)'' 
 
 _ 9— a; x + 3_ 6 — x 
 
 ~2 5~ " 2 a; -13 ' 
 
AFFECTED QUADRATIC EQUATIONS. 289 
 
 9. (a; + 2Y-{2x + If = {x + If. 
 
 10. ^_^ = 4f. 
 x — 4t a; + 4 ' 
 
 11. (10 x + iy=2{lQx + 17). Ans. a; = 2, or - 0.76. 
 
 19 «-. 4a;-l 3a;2 + 9 
 
 1.^. o a; — = 5 a; H 3— • 
 
 a; — 2 a; + 1 
 
 13 3a: + 2 ^ 4a: + 6 ^^ a;-3« _ 9(&-a) 
 
 'aj + l 2a; + 5' ' b ~ x 
 
 Hg _4_^ 5 - 2j: ^ 21 2 + a; 9 _ 1 - a; 
 
 'a;+10 2a; + 4 82-a: 5~l + a;' 
 
 a; + 3 _ 4 _ 3a:+l ^°«- a^ = §> or ^3. 
 
 a;-2 3~2a; + 2' 
 
 17. 
 
 18.^-^ = 12. 21.^ + 4 = 
 
 X 
 
 1 — x X ' ■ 2 + 0!^ 10 2a; 
 
 ^gl20_m^_36 22. 5+1 2a; + l 
 
 X a; + l X + 2 x 5 
 
 3^-2 _ 2^-3 _ 15 
 
 2 a; -3 3a;-2~ 4 ■ 
 
 X 
 
 24. --fa; = 3(a; + l). 
 
 25. (a; - 1) (x - 2) - 6 (a; - 3) (a; - 4) = 0. 
 
 -IK ^ 
 
 26. 5a;-i^--p = 15. 27. a!-i + a:-^ = 6. 
 
 a- + 4 
 
 28. (x + 2) (a; + 3) = 3a; (a; - 2) + 2 (2a; + 1). 
 
 19 
 
290 ALGEBRA. 
 
 ^^•5^«-^« + T = «-T' 
 
 30. 5 V^ - 3 v'^^^= 13. Alls, x = 25, or 9||. 
 
 31. 4 Vie + a; = r Vie + a; - 2 a; + 3. 
 
 32. 3 V^^^ + 2 a/IO^ = ■^^ " "*" ^ "" 
 
 Ans. X = 10 a, or — J a. 
 
 33. V3a; + 1 + — == = 3 yx. 
 V3a; + 1 
 
 34. V^x — l — Vx — l = 1. 
 
 35. V3x + 4t + V2x + 1 = Vll x + 5. 
 
 36. a: - 1 = Vl + V5 a:» - 4 a;^ 
 
 37. i-^^ + 2^-H = ft- 
 
 3S. 5 -4a:-i + 7a;-'' = 19a;-'2. Ans. a; = 2, or - 1.2. 
 
 39. V2a; + 1— -^2 a; + 1 = 6. 
 
 4 3 7~3'^7 4' 
 
 Ans. a; = ± 3, or ± V— 1- 
 
 41. 4 a;« - T^ = ^ - 2 a;'. Ans. a; = J, or ^ \/^5. 
 
 42. 3v^-t = V-4v'^- 
 
 .o X _ n + X ±in 
 
 *'*• i2 — 7 — ; — N— — 2 — • Ans. a; = 
 
 0'' c (w + a;) a' ex a ^ b 
 
 44. a;^ — a^ = -5 ; • 
 
 a'' a;'' 
 
AFFECTED QUADRATIC EQUATIONS. 291 
 
 ,. Ab~x x — 2a + 2b 7b — 4ca 
 
 45. -^ J 4 = J— • 
 
 a — ao ax ax — ox 
 
 Ans. X = 2 a + b, or —a + 2b. 
 
 X x — bb _ 1 a; — 2a + 196 
 
 a^-U^ (a + 2b)x~ a + 2b'^ (a-2b)x 
 
 Ans. a; = 2a + 6 6, or a — 8b. 
 
 . » 18 a . _ x + a 
 
 5a-x~3b x + 2b' 
 
 Ans. a; = ^ — 3 5, or — 5 a — 5. 
 
 48. x* - 5 (a^ + 5^) a; = - (2 a= + 3 a J - 2 S'')^. 
 
 Ans. a; = ± (2 a — 6), or ± (a + 2 6). 
 
 49. .^ + . = ^^±4f + 2. 
 
 a — 6 
 
 Ans. X :=: ; — • , Or — 2. 
 
 a — b 
 
 M/^ 1 -I 1 1 a ± 6 
 
 50. a; + - = 1 + — . ,2 Ans. x = — — -r- 
 
 X ■] _ ^" a ^ b 
 
 a^ + 3 b' 
 ^^6^ + 3 a;2 + a! + l . 1±6 
 
 ^^- WTT = «3^-x + i • ^"^- ^ = IT"*- 
 
 52. a;^ - VS a;' - 2 a: + 3 = ^-^3^ • 
 
 Ans. a: = 1, or — J, or J (1 ± V— 5)- 
 
 53. (a; + 4) (x - 1) = 4 V^; (a: + 3) - 2 - 2. 
 
 Ans. a; = 3, or - 4, or i (- 3 ± \^). 
 
 64. 2 Vfl!' - 3 X + 4 = 5 a;^ - 15 a; + 4. 
 
 Ans. a; = 3, or 0, or 2.4, or 0.6. 
 
 , 3a^x 6a'' + ab-2b^ Px 
 55. a bx^ H = -2 — • 
 
292 ALGEBRA. 
 
 X 
 
 + Vx' - 1 X- \/x^ - 1 
 
 66. ■^^-v^-^_"--v "---^ ^s ^/x^zri: 
 
 X — -x/x^ — i X + /y/as^ — 1 
 
 Ans. a; = ± 2, or ± 1. 
 
 57. a;" — 6 a;^' + 9 = 0. 
 
 58. 
 
 a i (a + 6) Vi + 6 / (a — 
 
 by 
 
 . a(a + b) b(a + b) 
 Ans. X = — ^^ -^, or ^ 
 
 a-b ' b- 
 
 a 
 
 59. 
 
 60. 
 
 1 
 
 (a''- If a;n(a + l)^ a;>'' - 1)^ ^ (a - 1) V 
 
 Ans. a; = ± :7 • 
 
 a T 1 
 a;" 6 a; 
 
 («'' + J'') (4 a - 36)2 (5 ^2 ^ 52^ (^2 + 5 ^2^ (4 a _ 3 5) 
 
 1 
 
 ~ (5 a^ + J2) ia' + 5 6') (a'' + b") ' 
 
 PROBLEMS 
 
 PRODUCING AFFECTED QUADRATIC EQUATIONS WITH BUT 
 ONE UNKNOWN NUMBER. 
 
 280. The negative values obtained in solving the follow- 
 ing Problems always satisfy the equations formed in ac- 
 cordance with the given conditions, and can generally be 
 explained, and the words of the Problem so changed as to 
 reverse the signs (§§ 189, 190). In all cases where it is 
 possible, the student should be required to explain the 
 negative answers, and change the problem so as to change 
 the negative to positive answers. 
 
 1. A man who had travelled 400 miles found that if he .had 
 travelled 10 miles less each day he would have taken 2 days 
 more to complete his journey. How many days was he on the 
 road, and what was his daily rate ? 
 
AFFECTED QUADRATIC EQUATIONS, 293 
 
 Let a: = number of days on the road ; 
 
 *i, 400 , . , ., 
 
 tnen — = his daily rate. 
 
 Then 4_00_J00__^„ 
 
 X x+2 
 40 40 
 
 = 1 
 
 x 
 
 x-{-2 
 Clearing of fractions, 
 
 40 a; + 80 — 40 a; = z2 + 2 a; 
 or, a» + 2 a; = 80 
 
 Whence, a; = — 1 ± 9 = 8, or — 10, number of days, 
 
 and — = 50, or — 40, daily rate. 
 
 The negative answers verify in the equation. In the problem, if 
 more and less change places, the answers 10 days and 40 miles a day 
 ■will be found correct. 
 
 2. Divide 60 into two parts, such that the sum of their 
 squares shall be 2138. Ans. 43 and 17. 
 
 Notice that, in solving this example, if we consider the square root 
 plus, we get the greater part ; if minus, the less. 
 
 3. A merchant who sold a piece of cloth for $72 found 
 that, if the price which he paid for it were multiplied by his 
 gain, the product would be equal to the cube of the gain. 
 Find the gain. Ans. $8. 
 
 If the word " gain " were changed to loss, the other answer, — 9, 
 or as it would then become + 9, would be correct. 
 
 4. The sum of the squares of two consecutive numbers is 
 145. What are the numbers ? 
 
 5. A gentleman distributed among some boys f 10. If . he 
 had begun by giving each 5 cents more, 10 of the boys would 
 have received nothing. How many boys were there, and how 
 much did he give to each ? 
 
294 ALGEBRA. 
 
 6. A man who had travelled 224 miles found that, in order 
 to return the same road in 1 day less, he must travel 4 miles 
 more each day. Find how many days he had travelled, and 
 how many miles a day. 
 
 7. Find two numbers whose difference is 6, and the sum of 
 whose fourth powers is 2402. 
 
 . Let X — 3 and a; + 3 be the numbers. 
 
 8. A man hires a piece of land for $330. He lets all but 
 25 acres for $ 5 an acre more than he pays for it, receiving 
 enough to pay for the whole. How many acres does he hire ? 
 
 9. A cistern is filled by two pipes in 2 h. 55 m. By the 
 greater alone it can be filled in 2 hours less time than by the 
 smaller alone. Find the time for each pipe. 
 
 10. Find two numbers whose sum is 9, and the sum of 
 whose cubes is 189. 
 
 11. A drover sold a cow for $56, and found that if the 
 price which he paid for it were multiplied by his gain, the 
 product would be equal to the cube of the gain. What- was 
 his gain ? Ans. $ 7. 
 
 12. A farmer, having built 42 rods of fence, found that, if 
 he had built one more rod a day, he would have completed 
 the work in one day less time. How many rods did he build 
 a day, and how many days did he work ? 
 
 13. The length of a rectangular field exceeds the breadth 
 1 rod, and the area is 3| acres. Find the dimensions. 
 
 14. Find a number such that 4 times its square less 6 times 
 the number itself is 270. 
 
 15. A man sold a horse for $275. gaining a per cent ex- 
 pressed by a twenty-fifth of the cost of the horse. How much 
 did the horse cost him ? Ans. $250. 
 
AFFECTED QUADRATIC EQUATIONS. 295 
 
 16. From a cask containing 50 gallons of white syrup 
 enough was drawn to fill a small keg, and the same quan- 
 tity of darker colored syrup was put in ; then the same keg 
 was filled again from the cask, and then there were 32 gallons 
 of the original white syrup left in the cask. How many gal- 
 lons did the keg hold ? 
 
 17. A merchant bought a piece of cloth for $54. He used 
 for himself 8 yards, and sold the rest for 20 cents more a yard 
 than he gave, receiving for what he sold $51.80. How many 
 yards did he buy ? Ans. 45 yards. 
 
 18. There is a rectangular piece of land 84 rods long and 
 52 rods wide, and just within the boundaries is a ditch of 
 uniform breadth running entirely round the land. Within 
 the ditch the area is 26 acres and 73 square rods. Find the 
 width of the ditch. 
 
 19. The sum of the two figures of a certain number of two 
 figures is 11, and their product is 19 less than the number 
 expressed by the figures in reverse order. What is the 
 number ? 
 
 20. What are eggs a dozen, when 4 more in 40 cents' worth 
 lowers the price 4 cents a dozen ? Ans. 24 cents. 
 
 21. Two numbers are as 4 : 3, and their product plus their 
 sum is 62. Find the numbers. 
 
 22. Three students. A, B, and 0, began on the same day to 
 solve a number of problems. A solved 10 a day, and finished 
 them 4 days before B. B solved 2 more a day than C, and 
 finished them 5 days before C. How many problems were 
 there, and how many days did each work ? 
 
 23. A broker sells some railroad shares for $3150. He 
 afterward buys for the same sum 7 more shares at $5 less a 
 share. Find the number of shares he sold, and the price. 
 
 Ans. 63 shares at $50 a share. 
 
296 ALGEBRA. 
 
 24. Two horsemen start at the same time from two places 
 18 miles apart. At the end of 12 hours the second horseman 
 overtakes the first, and on comparing their rates they find 
 that there has been a difference in their rates of 2 minutes 
 in each mile. Find their rates and the distance each has 
 travelled. 
 
 25. A rectangular piece of land has an area of 3 acres, 
 and if its length is decreased 5 rods, and its breadth in- 
 creased 4 rods, its area is increased 20 square rods. Find 
 the dimensions. 
 
 26. A man bought a number of $100 railway shares, at a 
 certain rate per cent discount a share, for $8100, and after- 
 wards sold for $8250 all but 15 at the same rate per cent 
 premium a share. How many shares did he buy, and what 
 did he give a share ? 
 
 27. A man walks at a uniform rate on a road which passes 
 over a bridge distant 18 miles from the point which the man 
 has reached at noon. If his rate were half a mile an hour 
 less than it is, the time at which he crosses the bridge would 
 be 1 h. 12 m. later than it is. Find his rate, and the time at 
 which he crosses the bridge. 
 
 Ans. Rate, 3 miles an hour ; time, 6 P. m. 
 
 28. A man bought a number of sheep for 1400 ; he sold all 
 but 20 for $352, gaining $0.40 a head. How many sheep did 
 he buy, and at what price ? 
 
 29. Several boys on an excursion spent each the same 
 amount of money. If there had been 6 more boys in the 
 party, and each had spent 25 cents less, the sum spent would 
 have been $ 39. If there had been 4 less, and each had spent 
 25 cents more, the sum spent would have been $32. Find 
 the number of boys, and the amount each spent. 
 
 Ans. 20 boys; amount spent by each, $1.75, 
 
SIMULTANEOUS QUADRATIC EQUATIONS. 297 
 
 CHAPTEE XX. 
 
 SIMULTANEOUS QUADRATIC EQUATIONS 
 CONTAINING TWO UNXfNOWN NUMBERS. 
 
 28L The Degree of an equation is shown by the sum 
 of the indices of the unknown numbers in that term in 
 which this sum is the greatest. Thus, 
 
 7 X — 3a;y=;9 is an equation of the second degree, 
 a3?y + x'^y — be^ " " " fourth « 
 
 8 a; — 7y* = a!'y» " " « fifth 
 
 Note. Before deciding of what degree an equation is, if the unknown 
 numbers appear both in the denominators and in the numerators or inte- 
 gral tei-ms, it must be cleared of fractions, and also from negative and 
 fractional exponents. 
 
 282. A Homogeneous Equation is one in which the sum 
 of the exponents of the unknown numbers in each term 
 containiiig unknown numbers is the same. Thus, 
 
 x^ + ^xy +y^^ 25 
 or a;' — 3 a; ?/'' + 3 sb" ?/ — 2/' = 64 
 
 or ai^ + ix'y+Qx^y^ + 'ixy^ + y* — 625 
 
 is a homogeneous equation. 
 
 283. Two numbers enter Symmetrically into an equation 
 when, whatever their values, they can exchange places 
 without destroying the equation. Thus, 
 
298 ALGEBKA. 
 
 x^-2xy ^if^ 36 
 or a:' + 3a;^2/ + 3a;/ + /= 64 
 
 or a;^ — 2 X y + 2/'' + 6 a; + 5 2/ = 100 
 
 284. Simultaneous quadratic equations containing two 
 unknown numbers can generally be solved by the rules 
 already given, if they come under one of the three follow- 
 ing cases : 
 
 I. When one of the equations is simple and the other 
 quadratic. 
 
 II. When the unknown numbers enter symmetrically 
 into each equation. 
 
 III. When each equation is quadratic and homogeneous. 
 
 Case I. 
 285. When one of the Equations is Simple and the other Quadratic. 
 
 1. Solve } ,"=1^ = ^ 
 
 a; + y=5 (1) 
 
 
 x-^ — 2y^=l (2) 
 
 y = h-x (3) 
 
 x^- 
 
 -50 +20 a; — 2x2=1 
 
 a:2 — 20a; = -51 (4) 
 
 2, = -12, or 2 (6) 
 
 
 X = 17, or 3 (5) 
 
 From (1) we obtain (3), or p ^ 5 — x. Substituting this vahie 
 of y in (2), we obtain (4), an affected quadratic equation, which 
 reduced gives (5); and substituting these values of x in (3), we 
 obtain (6). 
 
 In this Case the values of the unknown numbers can 
 generally be foimd by substituting in the quadratic equation 
 the value of OTie unknown number found by reducing the 
 simple equation. 
 
SIMULTANEOUS QUADRATIC EQUATIONS. 299 
 
 2. Solve I 
 xy = 20 
 
 xy = 20. 
 
 
 + 2/= 9. 
 
 
 (1) x^y^ 9 
 
 (2) 
 
 x-' + 2xy + y"' = ?,\ 
 
 (3) 
 
 ixy =80 
 
 (4) 
 
 x^-2xy + y-'== 1 
 
 (5) 
 
 x — y = ±l 
 
 (6) 
 
 2 a; = 10, or 8 
 
 (7) 
 
 2y=. 8, or 10 
 
 (8) 
 
 x= 5, or 4 
 
 (9) 
 
 ^ = 4, or 5 
 
 (10) 
 
 Subtracting four times (1) from the square of (2), we obtain (5) ; 
 extracting the square root of each member of (5), we obtain (6) ; 
 adding (6) to (2), we obtain (7) ; subtracting (6) from (2), we ob- 
 tain (8) ; and reducing (7) and (8), we obtain (9) and (10). 
 
 Note. Though Example 2 can lie solved by the same method as Ex- 
 ample 1, the method given is preferable. 
 
 Solve the following equations : 
 
 „ { xy = 15. n ( xy= 5. 
 
 'Xx — y= 2. ' \Zx-2y = lZ. 
 
 ■ \x^-y^ = S3. 
 g ( a: + 2/ = 13. 
 
 -y= 2. 
 164. 
 
 „ ( a;y = 24. 
 
 ■ l5a;-w = 37. 
 
300 ALGEBRA. 
 
 Case II. 
 
 286. When the Unknown Numbers enter symmetrically into each 
 
 Equation. 
 
 1. Solve \''+'^=J- 
 ix" + / = 243. 
 
 x + y= 9 (1) a;8 + 2/' = 243 (2) 
 
 x-' + 2xy + f = Sl (3) 
 
 a:2_ xy + y^ = 27 (4) 
 
 Zxy =54 (5) 
 
 xy = 18 (6) 
 
 x'^-^xy + y^^ 9 (7) 
 
 x -y = ±S (8) 
 
 2 a; = 12, or 6 (9) 
 
 2 y = 6, or 12 (10) 
 
 x= 6, or 3 (11) 
 
 y= 3, or 6 (12) 
 
 Squaring (1), we obtain (3) ; dividing (2) by (1), we obtain (4); 
 subtracting (4) from (3), we obtain (5), from which we obtain (6) ; 
 subtracting (6) from (4), we obtain (7) ; extracting the square root 
 of each member of (7), we obtain (8) ; adding (8) to (1), we ob- 
 tain (9) ; subtracting (8) from (1), we obtain (10) ; and reducing (9) 
 and (10), we obtain (11) and (12). 
 
 Note 1 . It must not be inferred that a; and y are equal to eaeh other 
 in these equations; for when sc = 6, y = 3; and when x = S, y = 6. In 
 all the equations under this Case the values of the two unknown numbers 
 are intei-changeable. 
 
 Note 2. Although 2^ + 3^ = 243 is not a quadratic equation, yet, as 
 we can combine the two given equations in such a manner as to produce 
 at once a quadratic equation, we introduce it here. 
 
SIMULTANEOUS QUADRATIC EQUATIONS. 301 
 
 a;;/ = 6 (1) x^ + y^ — 2 x — 2 y = S (2) 
 2xy =12 
 
 (x + yy-2(x + yy=15 (3) 
 
 a; + 2/ = 1 ± 4 = 5, or — 3 (4) 
 
 „ „ — 3±'»''=^T5 ,., 
 
 a; = 3, or2,or (o) 
 
 y = 2, or 3, or (6) 
 
 Adding twice (I) to (2), we obtain (3) ; from (3) we obtain (4) ; 
 and combining (4) and (1) as the sum and product are combined in 
 Ex. 2, Art. 285, we obtain (5) and (6). 
 
 in Case II. the process varies as the given eqnatious 
 vary. In general the equations are reduced by a proper 
 combination of the sum of the squares, or the square of the 
 sum or of the difference, with multiples of the product of 
 the two unknown numbers; and finally, of the sum,, with the 
 difference of the two unknown numbers. 
 
 Note 3. "When the unknown numbers enter into each equation sym- 
 metrically in all respects except their signs, the equations can be reduced 
 by this same method; e. g. x — y — 7, and a? — ifi = 511. In such 
 equations the values of the unknown numbers are not interchangeable. 
 
 Note 4. The signs ± T standing before any number taken indepen- 
 dently are equivalent to each other; but when one of two numbers is 
 equal to ±a while the other is equal to T6, the meaning is that the first 
 is equal to -f a, when the second is equal to — J ; and the first to — a, 
 when the second is equal to +6. 
 
 By this method solve the following equations : 
 
 3 ( 6a; — 5?/= 35. . (a; = 8, or-1. 
 
 ■ l2a!«-2/ = 1022. ' (2^ = l,or-8. 
 
 X + y = 10. 
 x' 4-f = 370. 
 
 g (x -y = 9. 
 
 x' + y^= 91. 
 xy + y^ = 13. 
 
302 ALGEBRA. 
 
 '■ l:c»-/ = 316. (2/ = 3, or -7. 
 
 g I «» - 3/S = 875. 9 5 a; + V«2/ + 2/ = 7. 
 
 la;'' + a;^' + / = 175. ' 1 x^ + xy + y^^ 21. 
 
 ■ I x^ + xy + y^ = 4:81. 
 
 Case III. 
 287. When each Equation is Quadratic and Homogeneous. 
 
 (2) 
 
 (4) 
 (6) 
 
 v^ — 2v~'6v — l ^^ 
 
 45 u — 15 = 14 w2 — 28 w (8) 
 
 «2_l|„ = _^| (9) 
 
 I' = 5, or T^ (10) 
 
 ~ 14 14 
 
 1. Solve {f-^'^^^J!- 
 lSxy — y^ = 14. 
 
 
 a;2 — 2 xy = 15 (1) 
 
 3x!/ — 3/2= 14 
 
 Let X = 
 
 = K!/ 
 
 K^y" — 2j;3/2=15 (3) 
 
 3 J, 2,2 _ j,2 = 14 
 
 * 3«— 1 
 
 15 
 
 14 
 
 " ~ 15 - 1 ' " T^ - 1 ^"^ 
 
 2r = ± 1, or ± iji V:r5 (12) 
 
 x = vy= ±5, or ±| i''^^ (13) 
 
 Substituting vy for a; in (1) and (2), we obtain (3) and (4) ; from 
 (3) and (4) we obtain (5) and (6) ; putting these two values of y'^ 
 equal to each other, we obtain (7), which reduced gives (10) ; sub- 
 stituting this v^lue of w in (6), we obtain (11), which reduced gives 
 (12); and substituting ra. x = vy the values of v and y from (10) 
 and (12), we obtain (13). 
 
SIMULTANEOUS QUADRATIC EQUATIONS. 303 
 
 Examples under Case III. can generally be reduced best 
 by substituting for one of the unknown numbers the ^product 
 of the other by some unknown number, and then finding the 
 value of this third unknown number. When the value of 
 this third number becomes known, the values of the given 
 unknown numbers can be readily found by substitution. 
 
 By this method solve the following equations : 
 
 „ SZxy + tf^l. . ( a: = ± 2, or ± 0.3 a/IO. 
 
 2. i „ / •' „ Ans. i 
 
 (.2x- — xy = G. (y= ±1, or Tl.^-v/iO. 
 
 <x' + 2xy^'ll. { x^ — xy = 10. 
 
 la;'' + 2/' =13. X^xy -Zy"" = A?,. 
 
 g Sx^-%xy^2y''-&%. ^^ r a; = ± 6, or ± §f ^83. 
 Xxy-5y''^x^-^2. ^' 1 y = ± 4, or ± 4»V83. 
 
 (2a:^-l=/+3a;2/. 
 1 &-x^=^y^+ 3. 
 
 MISCELLANEOUS EXAMPLES. 
 
 288. Solve the following equations: 
 
 Note. Some of the examples given below belong at the same time to 
 two Cases. Thus in Example 1 both the equations ai-e symmetrical, and 
 both are quadratic and homogeneous, and therefore it belongs both to 
 Case II. and Case III. Example 2 belongs both to Case I. and Case II. 
 
 ( xy = l?,. S x + y= 1%. 
 
 (.a;2 + y2 = 45. ^- \x^ + y^==lU. 
 
 (x^ + xy = 126. . (a;= ±7\/2, or ±9. 
 
 ■ Xxy-y''= 20. "^' ly= ±2V2, or ±5. 
 
 ( a; 2/ = 24. (3x'-5y^ = -5. 
 
 *• la;'' +y^ = x-y + 50. I x^ + 3xy=69 + y\ 
 
304 
 
 ALGEBRA. 
 
 6. < 
 
 x-y rj \ X+tJ= 5. 
 
 5a;y + 3 = 13. ' Xx" y" - 4.xy = 12. 
 
 (In Ex. 7, considering xy a. single number, find its value in the 
 second ec[uation.) 
 
 Wy-xy'' = 2(i. 
 
 (Subtract from the second equation three times the first, and ex- 
 ,ct the cube root of each member of the resulting equation.) 
 
 tract 
 
 {x'^y + xy^ = 
 ^- \ »» + «» = 
 
 330. j a; = 6, or 5. 
 
 ( y = 5, or 6. 
 
 2/» = 341. 
 
 10. ( a;y = 14. 
 
 \27{^ + f) = lS{x + yy. 
 
 \ x + y = 12. ( xy= 2. 
 
 la;»+2/8 = 18a;y. U* + «« = 17. . 
 
 11 
 
 ( 4.x^y + ^xy^= 80. f xy= 12, 
 
 ■ l3a:'y''+3a;==2/' = 240. ( a;* - y* = 175. 
 
 5xy ^ 45. 
 a;2 + 5 y = 24 + 5 a; — 2/". 
 
 f X = H3 ± a/69), or 5, or - 3. 
 ly = - J (3 T ^69), or 3, or - 5. 
 
 Ans. 
 
 16. 
 
 17. 
 
 18. 
 
 a-l y t _ 9 ^2. 
 4 a;4 + 3 yi = 15. 
 
 5a;= + 3a;y = 270. 
 2y= — a;y= 20. 
 
 a; + y = 25. 
 \'x+ '\/y = x— y. 
 
 19 U^ + y*= 5. 
 a; + y = 35. 
 
 { 
 
 Iv: 
 
 20. 
 
 x^ — y^ = 1. 
 a; - y = 37. 
 
 21. (^-' + y-'= h 
 
22. 
 
 SIMULTANEOUS QUADRATIC EQUATIONS. 305 
 
 fa;»-2a;=2/ + 2a;2/''-2,»= 7. 
 I x^ — x^y + xf — y'^ = 13. 
 
 23 
 
 Si=ii. 
 
 25. 
 26. 
 
 27. 
 
 . f X = 3 or — 2. 
 ^"M^ = 2,or-3. 
 
 24. -1 ^ + 2^= 4- '29. I *-2'= 1- 
 
 Ui + 2/« = 82. U^ + 2/^ = 337. 
 
 ( a;»-2/» = 208. 3^ | a; - y = a. 
 
 (x'' + a;y + 2/''=: 52. ' la;» + y^ = 5 a". 
 
 { x — y= 1. 3|_(a: — y = m + re. 
 
 ( a;5 _ y6 __ 211. ■ ( 4:xy = 15in + 10mn. 
 
 ( X + y= 6. 32i* + y = ^'^- 
 
 1 a;5 + 3/6 = 3126. " ( a:' + 2/« = 35 m». 
 
 C a;+y = 10. 
 33. < X* y* 1225 „ 
 
 y^x^=-^-^^y- 
 
 iX — &, or 4, or 5 ± T«i- V— 143. 
 -*-°^- iy = 4, or 6, or 5 :^^^^14^. 
 
 PROBLEMS 
 
 PRODUCING SIMULTANEOUS QUADRATIC EQUATIONS CONTAINING 
 TWO UNKNOWN NUMBERS. 
 
 289. Some of the following problems may be solved 
 with one unknown number. In the answers given, the 
 negative values are omitted. 
 
 1. There is a certain mimber of two figures, whose product 
 is 10 ; and if 56 is added to the number, and the sum of the 
 squares of its figures subtracted from the sum, the order of 
 the figures will be reversed. Find the number. 
 
 20 
 
306 ALGEBRA. 
 
 2. The area of a rectangular field is 975 square rods, and if 
 the length were decreased 5 rods, and the breadth increased 
 5 rods, the area would be 1020 square rods. What are the 
 length and breadth ? 
 
 Ans. Length, 39 rods ; breadth, 25 rods. 
 
 3. The difference of the cubes of two numbers is 316, and 
 the sum of their squares plus their product is 79. What are 
 the numbers ? 
 
 4. The area of a rectangular field whose perimeter is 88 rods 
 is 363 square rods. Find the length and breadth. 
 
 5. The fore wheels of a carriage make 8 more revolutions 
 than the hind wheels in going 160 yards, but if the circum- 
 ference of each wheel is increased 3 feet the carriage must 
 pass over 240 yards in order that the fore wheels may make 
 8 revolutions more than the hind wheels. What is the cir- 
 cumference of the wheels ? 
 
 Ans. Fore wheels, 12 feet ; hind wheels, 15 feet. 
 
 6. There are two pieces of cloth of different lengths. The 
 difference of the squares of the number of yards in each is 76 ; 
 and the product of the numbers representing the number of 
 yards in each plus one half the square of the number of yards 
 in the longer is 560. Find the length of each. 
 
 7. Find two numbers such that 4 times the square of the 
 greater minus 5 times the square of the less shall be 20, and 
 5 times the square of the less plus their product shall be 100. 
 
 8. A drover bought 10 oxen and 18 cows for $ 1040, buying 
 one more ox for $ 150 than cows for $ 60. Find the price of 
 an ox, and the price of a cow. Ans. Ox, 150; cow, $30. 
 
 9. Find two numbers such that the sum, the product, and 
 the sum of their squares shall be equal to one another. 
 
 10. A and B, talking of their ages, find that the square of 
 A's age minus the product of the ages of both is 385, and 
 
SIMULTANEOUS QUADRATIC EQUATIONS. 307 
 
 3 times this product plus the square of B's age is 3096. 
 Find the age of each. 
 
 11. A and B purchased a wood-lot containing 100 acres, 
 each agreeing to pay 15000. Before paying for the lot, A 
 offered to pay $ 10 an acre more than B, if B would allow A 
 to have his choice in the division of the lot. How many acres 
 should each receive, and at what price an acre ? 
 
 . (A, 47.5 acres, at $ 105.25— an acre. 
 
 ( B, 52.5 acres, at $ 95.25+ an acre. 
 
 12. Two sums of money, amounting to $9000, are loaned 
 at such a rate of interest that the income from each is the 
 same. But if the first part had heen at the same rate as the 
 second, the income from it would have been 1 160 ; while if 
 the second had been at the same rate as the first, the income 
 from it would have been $250. Find the rate of each. 
 
 Ans. 1st, 5% ; 2d, 4%, 
 
 13. A boat's crew, rowing at half their tisual rate, row 
 
 4 miles down a river and back in 3 hours and 20 minutes. 
 At their usual rate they can go over the same course in 1 
 hour and 20 minutes. Find the usual rate of the crew, and 
 the rate of the current. 
 
 14. A, working alone, built 8 rods of wall ; then he hired B 
 to work with him, and at the end of 3 days from the time A 
 began they had completed 24 rods. Again, A worked 3 days 
 and B 1 day, building 21.6 rods. Find the number of rods 
 each can build a day, and the number of days B worked. 
 
 15. A reservoir is filled in 10 hours by means of several 
 pipes, through which the water flows at a uniform rate. If 
 there were 2 less pipes, and each pipe discharged 50 gallons 
 more an hoiir, the reservoir would be filled in 12 hours and 
 30 minutes. If there were 3 more pipes, and each pipe dis- 
 charged 25 gallons less an hour, the reservoir would be filled 
 in 7 hours and 30 minutes. Find the number of pipes, and 
 the capacity of the reservoir. 
 
308 ALGEBRA. 
 
 16. A tailor bought two pieces of cloth for $ 100. For the 
 first he paid ^ as many dollars a yard as there were yards in 
 both ; and for the second | as many dollars a yard as there 
 were yards in the first more than in the second ; and the first 
 piece cost 3 times as much as the second. Find the number 
 of yards, and the cost a yard of each. 
 
 17. A man walks 3 hours at the rate of 3 miles an hour, 
 and then takes a different rate. After a number of hours he 
 finds that, if he had kept his first rate, he would have been 
 2 miles less distant from his starting point, while if he had 
 walked at his first rate 2 hours, and at his second rate 5 hours, 
 he would be half a mile farther from his starting point. Find 
 the whole time of the walk, and the entire distance travelled. 
 
 Ans. 7 hours, and 23 miles. 
 
 18. The soldiers of a regiment can be arranged so as to have 
 twice as many men in a line as there are lines. But 16 men 
 must be added to the regiment in order that the men may be 
 arranged in a hollow square six deep, having the same number 
 of men in each outer side of the square as there were in the 
 lines before. Find the number of men in the regiment. 
 
 19. The area of a certain rectangle is equal to the area of a 
 square whose side is 6 meters less than one of the sides of the 
 rectangle, and also equal to the area of another rectangle whose 
 length is 2 meters less, and width 1 meter more. Find the 
 length of the sides of the rectangle. 
 
 20. At 7 o'clock A. M., A and B set out in opposite direc- 
 tions on their bicycles, from the same point. A's hourly rate 
 was 5 miles. B, riding at a fixed rate, after a while turned 
 and followed A. Three hours after he turned, B passed the 
 point where A was when B turned, and at 12 o'clock he had 
 reduced the distance between them at the time of turning 
 one half. Find B's rate, the time when he turned, the dis- 
 tance between A and B at that time, and the time when B 
 will overtake A if both continue at the same rate of speed. 
 
PROPEKTIES OF QUADKATIC EQUATIONS. 309 
 
 CHAPTEE XXI. 
 PROPERTIES OF QUADRATIC EQUATIONS. 
 
 290. Every affected quadratic equation can be reduced 
 (§ 274) to the form 
 
 x^+ bx+ c=zO. 
 
 This reduced (§ 275) gives 
 
 b J¥ 
 ^ = -2±V4-'^- 
 
 Thus, there are two roots of a quadratic equation, and the sum of 
 
 these two roots, ~-^+K c and — ?r— v/:r — c, is —b; and 
 
 2 V 4 2 V 4 
 
 their product is c. 
 
 That is, the sum of the roots is equal to the coefficient of the second 
 term with its sign changed; the product of the roots is equal to the 
 last term. 
 
 291. A quadratic equation, x^ + hx + c=^0, will have 
 two real and unequal roots, two real but equal roots, or 
 two imaginary roots, according as S^ is > , =^ or < 4 c. 
 
 For the roots are 
 
 -b + ^b'- 
 
 -4c 
 
 2 
 
 
 -h-^Vb^- 
 
 -4fl 
 
 and 
 
 Now if 6^ > ic, the expression under the radical is positive, and 
 the roots are real and unequal 
 
 If J2 __ 4 p^ tijg roots are real and equaL 
 
 If 6^ < 4 c, the expression under the radical is negative, and both 
 roots are imaginary (§ 213). 
 
310 ALGEBRA. 
 
 292. To illustrate this subject still further, we solve the 
 following example. 
 
 1. a;'^ + 7a: + 12 = 0. , 
 
 Transposing, x^ -\- 7 x = — 12 
 
 Whence x = _ J ± y/if^^l2 =— |±J = — 3, or — 4 
 
 Now, since a: = — 3 
 
 and also x = — 4 
 
 then a; + 3 = (1) 
 
 and a; + 4 = (2) 
 
 Multiplying (1) by (2), a;2 + 7 x + 12 = (3) 
 
 Now (3) is the equation given in the example. 
 
 In general, if a; = a or & 
 
 then x — a = (1) 
 
 and X— b = (2) 
 
 Multiplying (1) by (2), x^ — (a + b) x + ab = (3) 
 
 Equation x^ — (a + b)x + ab = 
 
 reduced, gives x = a, or 6 
 
 Thus, it will be seen that a quadratic equation having 
 any two given roots can be formed by subtracting each of 
 the roots from x, multiplyi'ng these two expressions together, 
 and raahing the product equal to zero. 
 
 2. Write an equation whose roots are — 3 and 5. 
 
 Ans. x'' — 2 a; - 15 = 0. 
 
 Form the equations whose roots are : 
 3. -1, 3. 8. 5, -3. 13. 6, -^. 
 
 4 2, -5. 9. 4, -1. 14. 5, -2f. 
 
 6. -3, -1. 10. i, -i. 15. ±4. 
 
 6. -2, 4 11. 2, 'Z^^. 16. ±(2+\/3). 
 
 7. 1, -1. 12. 6, -4 17. ±{a-h). 
 
PROPERTIES OF QUADRATIC EQUATIONS. 311 
 
 293. . If m w = 0, either m = 0, or w = 0. If we kuow 
 that m is not equal to 0, then we linow that n is equal to 
 ; and if we know that n is not equal to 0, then we know 
 that m is equal to 0. So, if Imn — 0, at least one of the 
 factors must be equal to ; and so on, for any number of 
 factors. 
 
 In the following examples find the roots: 
 
 1. a:^ + 3 a; - 10 = 0. 
 
 Or, (a; — 2) (r + 5) = 
 
 Then a; — 2 = 0, or 1+5 = 
 
 that is, X = 2, or — 5 
 
 2. a;" - a; — 20 = 0. 3. 2 a;" + 16 a; + 24 = 0. 
 
 4. a;« — 27 a;^ + 50 a; = 0. 
 
 Or, x{x — 2)(x — 2o) = 
 
 Whence a; = 0, or x — 2^0, or .t — 25 = 
 
 Then a; = 0, or 2, or 25 
 
 5. K»-14a!=-61a; = 0. 
 
 6. a;«-l = 0. 
 
 Factoring, (a; — 1) (a;^ + a; + 1) = 
 
 Then a; — 1 = (1), or a;' + a: + 1 = (2) 
 
 From (1), a; = 1 
 
 From (2), a;= + a; = — 1 
 
 Whence a: = — ^ ± v| = -^ (1 ± 1/3) 
 
 7. a;' = 8. 
 
 Or, a;8— 8 = 
 
 Factoring, (a; — 2) (a;'' + 2 a; + 4) = 
 
 Then a; — 2 = (1), ora;2+2a;+4 = 
 
 From (1), a; = 2 
 
 From (2), a;'' + 2a;= — 4 
 
 Whence i = _ 1 ± t/_ 3 
 
 8. a:« — 7 a:^ + 10 a; = 0. 9. x" = 27. 
 
312 ALGEBRA. 
 
 294. From these examples it will be seen that, if the first 
 member of an equation whose second member is can be 
 divided by the unknown number plus any number, the 
 negative of this second number is one of the roots of the 
 equation. 
 
 10. x*-7x'-6x = 0. 
 
 Or, x{x+l){x+2){x — S) = 
 
 Then (§ 293), a; = 0, or — 1, or — 2, or 3 
 
 Note. The difficulty iu this example is in finding the factors. That 
 a; is a, factor, is apparent. Then the part x" — 7 x — 6 can be written 
 
 a!S + 2!E2 — 2a;2— 4a; — 3a; — 6 = a-^ (a; + 2) — 2 a; (a; + 2) — 3 (a; + 2) 
 
 = {x+2)(a^ — 2x — 3) 
 
 295. Prom these examples it might be inferred that the 
 number of roots in an equation containing only one un- 
 known number is the same as the index of the highest 
 power of the unknown number in the equation. But it 
 can be proved that, in an equation containing only one 
 unknown number, the number of different roots cannot 
 exceed the index of the highest power of the unknown 
 number. Thus, 
 
 Let X — a ^ 0, a form to which all equations containing only the 
 first power of an unknown number can be reduced. 
 
 Then x = a 
 
 Now suppose 6 and c are the roots of this equation. Substituting 
 
 for X, b and c, we have 
 
 b = a (1) 
 
 c = a (2) 
 
 Therefore, 6 = c ; that is, the equation x — o = does not have 
 two different roots, that is, it has only one root. 
 
 Again : 
 Let x^ + mx ■]- n = 
 
 Suppose it has three different roots, a, b, and e. 
 
PROPERTIES OF QUADRATIC EQUATIONS. 313 
 
 Then, substituting these values for x, we have 
 
 a^ + am + n = (1) 
 
 62 + 6 m + n = (2) 
 
 c^+cm + n = Q (3) 
 
 Subtracting (2) from (1), 
 
 a2 — 6= + (a — 6) m = 
 
 or (a — 6) (a + 6 + m) = (4) 
 
 Also, subtracting (3) from (1), 
 
 a2 — c^ + (a — c) m = 
 or (a — c) (a + c + m) = (5) 
 
 Now, since a, b, and c are different roots, neither a — b nor a — c 
 can equal 0. 
 
 Therefore, o + 6 + m = (6) 
 
 and a + c -{- m = (7) 
 
 Subtracting (7) from (6), b — c = 
 
 or b = c 
 
 That is, 6 and c are not different roots ; that is, there are only two 
 different roots. 
 
 Again : 
 Let x^ + mx^-\-nx-\-p = 
 
 Suppose this equation has four different roots, a, b, c, and d. 
 Then, substituting these values for x, we have 
 
 0.'^ + a''m+ an+ p = (1) 
 
 b' + b^m + bn+p^O (2) 
 
 c» + c" TO + c n + ^ = (3) 
 
 d» + d^m + dn+p = (4) 
 
 Subtracting successively (2), (3), and (4) from (1), 
 
 aS — J5 + (a^ — J^) m + (a — 6) n = (5) 
 
 a' — c« + (a" — c=) m + (a — c) n = (6) 
 
 a8 — rfs + (a2 _ ^2) TO + (a — d) n = (7) 
 
 These factored become 
 
 (a—b){a^+ab+b''+(a+b)m + n]=0 (8) 
 
 (a — c) {a2 + a c + c' + (a + c) m + «} = (9) 
 
 (a — d){a^ + ad + d^+ia + d')m + n}=0 (10) 
 
314 ALGEBRA. 
 
 Now, since a is not equal to b, or c, or d, a — h, a — c, and a — d 
 are not 0. Therefore, from (8), (9), and (10), we have 
 
 a2 + a6+ h^ + am + bm + n = 
 
 (11) 
 
 a^ + ac-{-c''+am+ cm + n = 
 
 (12) 
 
 a^ -\- ad -\- d^ + am + dm -\- n ^0 
 
 (13) 
 
 Subtracting (12) and (13) from (11), we have 
 
 
 a 6 — a c + 62 — c2 + (ft — c) m = 
 
 (14) 
 
 ab — ad + b'' — d^+(b—d)m = Q 
 
 (15) 
 
 These factored become 
 
 
 (6 — c) (a + ft + c + m) = 
 
 (16) 
 
 (ft— d) (a + 6 + d + ro) = 
 
 (17) 
 
 As before, ft — c and b — d are not ; therefore. 
 
 from (16) 
 
 (17), we have 
 
 a+6+c+m=0 
 
 (18) 
 
 a + b +d + m = Q 
 
 (19) 
 
 Subtracting (19) from (18), we have 
 
 c_d=0 
 
 (20) 
 
 or c = d 
 
 
 That is, c and d are not different roots ; that is, there are only 
 three different roots. 
 
 In like manner, for higher powers it can be shown that 
 the number of different roots cannot exceed the index 
 of the highest power of the unknown number in the 
 equation. 
 
 It must not be inferred, however, that an equation 
 cannot have two, or more, equal roots. The equation 
 x^ — 2 ax + a^ — 0, that is, (x — a) (x — a) = 0, has two 
 roots, both of which are a. 
 
 It follows, too, that if any expression containing only 
 one unknown number is multiplied by this unknown num- 
 ber minus any number, and if this last number is substi- 
 tuted for the unknown number in the product, this product 
 will prove to be 0. For example, multiplying x^~5x + ^ 
 by x — 2, we have 3? — 7 x^ + ^-x ~1. Substituting 2 
 
PROPERTIES OF QUADRATIC EQUATIOMS. 315 
 
 for X, we have 8 — 28 + 21 — 1 = 0. Or, again, if a is a 
 root of any equation containing only one unknown num- 
 ber, as X, then, if the numbers are transposed so as to 
 make the second member 0, the first member is divisible 
 by X — a. 
 
 296. This principle gives another method of factoring 
 an algebraic expression. For if, putting the expression 
 equal to 0, we can then reduce the equation thus formed 
 so as to obtain its roots, the expression formed by subtract- 
 ing any one of these roots from the number wliose value 
 we have found will be a factor of the given expression. 
 For example, 
 
 Find the factors of : 
 
 1. 13 cb'' -t- 221 a; - 234 = 0. 
 
 Put 13 a;2 + 221 X — 234 = 
 
 Then a;^.^ 17 a; =18 
 
 "Whence a: = V" ±V^F+18 = — V- ± ¥ = 1. or — 18 
 
 Then the factors of 13 x^ -f- 221 x — 234 are 13, a; — 1, and 
 x+ 18. 
 
 2. 6 a;=- 102 a; -504 = 0. 
 
 3. 2x''-5a! + 3 = 0. 
 
 4. a;' - 2 z^ - 5 a; + 6 = 0. 
 
 By trial we find that 1 will satisfy this equation, therefore 1 is one 
 of its three roots. 
 
 Dividing a;' — 2a;^ — 5a;-|-6 = by x — 1 
 gives %''■ — X — 6 = 
 
 Or, a:2 — a; = 6 
 
 Whence a; = Ji /y/j -f- 6 = ^±1 = 3, or — 2 
 
316 ALGEBUA. 
 
 Hence the three roots are 1, — 2, and 3. Therefore the required 
 factors are x — 1, x — 3, and a; + 2. 
 
 When from a cubic equation one of the roots is thus removed, 
 the equation becomes quadratic, and the other two roots can be found 
 as previously shown. 
 
 If the roots are integral, it is clear that they are among the prime 
 factors of the last term of the first member. 
 
 5. a:» + 5 a:^ - 2 K - 24 = 0. 
 
 6. a;=-6a!='-a; + 30 = 0. 
 
 7. a;' + 6 a;^ - 37 a; + 30 = 0. 
 
 8. 8 a:» - 66 x" - 520 a; + 1848 = 0. 
 
 9. 9 a;' - 9 a;'' — 2 a; + 2 = 0. 
 
 10. 3a;2 + 9a; - 54 = 0. 11. llx'' + 77a; - 198 = 0. 
 
 12. x' + 13 a;^ - 68 = 0. 
 X* = 4, or — 17 
 
 a;2 = ± 2, or ± V^^^^ 
 
 x = ±V±2, or ±-v/± V— 17. 
 
 13. x^ + 15a x^ + 44 a^ = 0. 
 
 a; = — -v/4 a, or — -v/ll a. 
 
 14. a;* - 18 S a!= + 65 S^ = 0. 
 
 297. From what has gone before it is evident that, if we 
 can find one root of any equation, another equation can 
 be derived from it in which the highest power of the un- 
 known number will be one less. Ey various artifices of 
 this nature, equations involving higher powers of the un- 
 known number can be reduced. 
 
 1. Find the three cube roots of 64. 
 
 2. Find the four roots in x* — 1 = 0. 
 
 3. Find the six roots in a;^ — 1 = 0. 
 
PROPERTIES OF QUADRATIC EQUATIONS. 317 
 
 298. If a rational and integral polynomial in terms of x 
 is divided hy x — a until a remainder independent of x is 
 obtained, this remainder is tlie value of the polynomial vjlien 
 x= a. 
 
 If U is the polynomial, Q the quotient, and B the 
 remainder, 
 
 U=Q(x-a) + R. 
 
 Now this equation is true for all values of x. If, then, 
 x = a, ir= B, since B is independent of x. 
 For example, 
 
 2a;« + 3a;-8 „ ^ , , , ,. , 14 
 
 x-2 
 
 2 a;» + 3 X — 8 = 14, if a; = 2 
 or, 2 a;' + 3 a; — 22 = 0, if a; = 2 
 
 If, in 2 0.-^ + 3 J5 — 8, we substitute 2 for x, we have 
 2- 23 +3. 2-8 = 14; and 14 is the remainder if we 
 divide 2a^+3a;-8bya;-2. 
 
 The proposition in this article includes the particular 
 case, viz., that if any rational and integral polynomial in 
 terms of x vanishes when we substitute a for x, then 
 x — a is a factor of the expression. (See Art. 295, last 
 paragraph.) 
 
 EXAMPLES. 
 
 1. Find the remainder when 3x^ — 5x^ + 3x — i is 
 divided by x — 3. 
 
 2. Find the remainder when 2 x* + 3 x^ — 4:X -\- 12 is 
 divided by a; + 4. 
 
 3. Prove that a; + 3 is a factor of 2 a;* + 5 a:" — 3 a; — 56. 
 
 4. Prove that a; — 1 is a factor of any rational and integral 
 polynomial in terms of x, if the sum of the coefficients (includ- 
 ing the coefficient of a;") is zero. 
 
318 ALGEBRA. 
 
 CHAPTEE XXII. 
 RATIO, PROPORTION, AND VARIATION. 
 
 299. Ratio is the relation of one number to another ; or 
 it is the quotient obtained by dividing one number by 
 another. 
 
 Eatio is indicated by writing the two numbers one after 
 the other with two dots between, or by expressing the di- 
 vision in the form of a fraction. Thus, the ratio of a to 6 is 
 
 written, a:h, or - ; read, a is to 6, or a divided by 6. 
 
 300. The Terms of a ratio are the numbers compared, 
 whether simple or compound. 
 
 The first term of a ratio is called the antecedent, the 
 other the consequent ; the two terms together are called a 
 couplet. 
 
 301. An Inverse or Beciprocal Batio of any two numbers 
 is the ratio of their reciprocals. Thus, the direct ratio of 
 
 a to & is a : 6, that is, - ; the inverse ratio of a to & is - : -, 
 h ah 
 
 that is, - -^ - = - , or o:a. 
 aha 
 
 302. Two numbers are commensurable if there is a third 
 number of the same kind which is contained an exact 
 number of times in each. This third number is called the 
 common measure of these two numbers. Thus, m and n 
 are commensurable if there is a third number, d, that is 
 contained an exact number of times in eacli; as, for exam- 
 
RATIO AND PROPORTION. 319 
 
 pie, 0.7 times in m, and 0.5 times iu n ; and d is the com- 
 mon measure of m and n. Then m = 0.7 d, and n = Q.bd, 
 
 and m : TO = 0.7 6^ : 0.5 <^ = 7 : 5, or - :== - . 
 
 n 5 
 
 Two numbers are incommensurable if they have no com- 
 mon measure. 
 
 The ratio of two numbers, as m and n, whether commen- 
 surable or not, is expressed by — . If m and n are incom- 
 
 n 
 
 mensurable, — is called an incommensurable ratio, 
 n 
 
 A constant ratio is a ratio which remains the same, 
 though its terms may vary. Thus, the ratio of 3 : 4, 6 : 8, 
 9 : 12, is constant ; also the ratio oi A:B and mA -.m B. 
 
 303. Proportion is an equality of ratios. Four numbers 
 are in proportion when the ratio of the first to the second 
 is equal to the ratio of the third to the fourth. 
 
 The equality of two ratios is indicated by the sign of 
 equality ( = ), or by four dots (: :). 
 
 Thus, a-.h — c-.d, or a:b::c:.d, or - = - ; read, « to & 
 
 d 
 
 equals c to d, or a is to 6 as c is to d, or a divided by 6 
 equals c divided by d. 
 
 In a proportion the antecedents and consequents of the 
 two ratios are respectively the antecedents and consequents 
 of the proportion. The first and fourth terms are called 
 the extremes, and the second and third the means. 
 
 304. A Continued Proportion is a series of equal ratios ; 
 
 , 7 j> T a c e q 
 
 as, a:b = c:d = e:f=g:li, or _ = - = - = ^. 
 
 305. When three numbers are in proportion, for exam- 
 ple, a:b = b:c, the second is called a mean proportional 
 between the other two ; and the third, a third proportional 
 to the first and second. 
 
320 ALGEBRA. 
 
 306. A proportion is transformed by Alternation when 
 antecedent is compared with antecedent, and consequent 
 with consequent. 
 
 307. A proportion is transformed by Inversion when the 
 antecedents are made consequents, and the consequents 
 antecedents. 
 
 308. A proportion is transformed by Composition when 
 in each couplet the sum of the antecedent and consequent 
 is compared with the antecedent or with the consequent. 
 
 309. A proportion is transformed by Division when in 
 each couplet the difference of the antecedent and con- 
 sequent is compared with the antecedent or with the 
 consequent. 
 
 THEOREMS. 
 
 310. In a proportion the product of the extremes is equal 
 to the product of the means. 
 
 Let a : b = c : d 
 
 , . a c 
 
 that IS, -7 = - 
 
 a 
 
 Clearing of fractions, a d ^ be 
 
 311. If the product of two numhers is equal to the product 
 of two others, the factors of either product may he made the 
 extremes, and the factors of the other the means, of a pro- 
 portion. 
 
 Let ad = b c 
 
 Dividing \ij bd, 7i ~ rf 
 
 that is, a:hz=c:d 
 
 312. If four numhers are in proportion, they will he in 
 proportion hy alternation, and, hy inversion. 
 
RATIO AND PEOPOUTION. 321 
 
 Let a : b — : d 
 
 By Art. 310, ad^bc 
 
 By Art. 311, a : o = b : d, alternation. 
 
 or, b : a — d : c, inversion. 
 
 313. If four numbers are in proportion, they will be in 
 proportion by composition and by division. 
 
 Let a : b = c : d 
 
 that IS, ~ = ~ 
 
 b a 
 
 Adding ±1 to each member, -r ± 1 = -^ ± 1 
 
 a ±b c ± d 
 
 that is, o, -\- b : b ^= c + d : d, composition, 
 
 or, a — 6 : 6 = c — d : d, division. 
 
 314. Corollary. Since, by Art. 813, if a:b — c:d 
 
 a + b:b = c + d:d 
 and also, a — b:b = c — d:d 
 
 by Art. 312, and Art. 36, Ax. 8, 
 
 a + b:a — b = c + d:c — d 
 
 316. Equimultiples of two numbers have the same ratio as 
 
 the numbers themselves. 
 
 _ a ma 
 
 For A = ~T 
 
 b mb 
 
 that is, a : b ^= m a : m,h 
 
 316. Corollary. It follows that either couplet of a pro- 
 portion may be multiplied or divided by any number, and 
 the resulting numbers will be in proportion. And since, 
 by Art. 312, if a : b = m a : mb, a -.ma — b -.mb, ov 
 ma : a = mb :b, it follows that both consequents, or both 
 antecedents, may be multiplied or divided by any number, 
 and the resulting numbers will be in proportion. 
 
 21 
 
322 ALGEBRA. 
 
 317. If fimr numbers are in proportion, like powers or 
 like roots of these numbers will be in proportion. 
 
 Let a : b =^ c : d 
 
 . a c 
 
 that IS, V = - 
 
 a 
 
 a" c» 
 -Hence, tz =^ -^ 
 
 b" d" 
 
 that is, a" : b" = d" : d" 
 
 Since n may be either integral or fractional, the theorem 
 is proved. 
 
 318. In a continued proportion any antecedent is to its 
 consequent as the sum, of any number of the antecedents is 
 to the sum of the corresponding consequents. 
 
 Let a : b ^= c : d ^ e '.f 
 
 Now, ab = ab (1) 
 
 and by Art. 310, ad — be (2) 
 
 and also, af=be (3) 
 
 Adding (1), (2), (3), a(b+d+f) = b(a + c + e ) 
 
 Hence, by Art. 312 , a: b = a + c+e: b + d+f 
 
 319. If there are two sets of numbers in proportion, their 
 products, or quotients, term by term, loill be in proportion. 
 
 Let a : b = c : d 
 
 and e :f— g : h 
 
 By Art. 310, ad = bc (1) 
 
 and eh=fg (2) 
 
 Multiplying (1) by (2), adeh = b cfg (3) 
 
 Dividing (1) by (2), ~n=Tg ^^^ 
 
 From (3) by Art. 311, a e : bf^ eg -.dh 
 
 and from (4), -:- = -:- 
 
 e f 9 h 
 
RATIO AND PROPORTION., 323 
 
 320, The proofs which have been given for commensura- 
 ble numbers are also true for incommensurable numbers. 
 For the ratio of two incommensurable numbers can be ex- 
 pressed to any required degree of accuracy. 
 
 Suppose, for example, it is required to find the ratio of 
 two in eom mensurable numbers, a and 6, to a degree of 
 
 accuracy within — -. Let the less, h, be divided into 100 
 
 equal parts, and suppose a contains 217 such parts with a 
 remainder less than one of the parts, then we have 
 
 a 217 .^, . 1 
 - = - within-, 
 
 that is, — is the approximate ratio of a to 6 to the re- 
 quired degree of accuracy. 
 
 Or, to make the reasoning general, let b be divided into 
 n equal parts, and suppose a contains m such parts with a 
 remainder less than one of the parts, then we have 
 
 a m ... . 1 
 - = — within - . 
 on n 
 
 As n may be taken as great as we please, - may be 
 
 Tit 
 
 made as small as we please, and — will be the ratio of 
 a to & to any required degree of accuracy. 
 
 A good example of the ratio of incommensurable numbers is the 
 ratio of V^ to 1. The V5 is a surd (§ 241), but its value can be 
 found to any required degree of accuracy. Thus, V^ > 1.4142 but 
 
 < 1.4143; hence — = 1.4142 within 0.0001. 
 
 321. Two incommoisurahle ratios are equal, if their ap- 
 proximate numerical values are always equal when both 
 ratios are expressed to the same degree of accuracy. 
 
324 ALGEBRA. 
 
 Let - and -; be two incommensurable ratios, wliose 
 a 
 
 approximate numerical values are always the same when 
 
 expressed to the same degree of accuracy ; then 
 
 a _ c 
 
 Let the numerical ratio — = v accurate within - : 
 n n 
 
 then, by hypothesis, "^1 ^^^^'^^'^^ within - . 
 
 £t c 1 
 
 That is, Y and -^ differ by a quantity less than - . But as 
 
 n may be taken as great as we please, - may be made as 
 
 small as we please. Now, if it is possible for the ratios to 
 
 differ at all, - can be made less than that difference, unless 
 n 
 
 that difference is actually zero ; that is, they do not differ, 
 
 1 a c 
 
 '"•^ i = d- 
 
 322. The laws that have been discussed in Eatio aud 
 Proportion of numbers apply also to quantities. But it 
 must be understood that only quantities of the same kind 
 can have a ratio one to another, and the ratio itself, that 
 is, the quotient of one quantity divided by another of the 
 same kind is an abstract number. The two ratios that 
 form a proportion need not be of the same kind. Thus, if 
 A receives $ 70 for 35 days' worlc, and $14 for 7 days' 
 work, we can say, 
 
 $70 :$ 14 = 35 days : 7 days. 
 
 In this case the ratio of $ 70 to $ 14 is 5, an abstract num- 
 ber, and equal to the ratio of 35 days to 7 days. 
 
RATIO AND PROPOllTION. 325 
 
 323. A good example of two incommensurable quantities 
 is the ratio of the diagonal to the side of a square. If the 
 numbers, s and d, represent the- measurement of the side 
 and diagonal, respectively, then 
 
 and this, as in Art. 320, can be found to any required 
 degree of accuracy. 
 
 324. EXERCISES IN PROPORTION. 
 
 ■'■• ^^ ^1. I 7. „ ~ TS^i — 2 ' P^°'^^ t^^* a-b = b:c. 
 
 «' + b^ _ ab + be 
 ab + bc ~ P + o^ 
 Clearing the given equation of fractions, 
 
 a'i 62 + a'^c^ + b* + b' c^ = a^b^ + 2 ab^c + b^c^ 
 a^c^ — 2ab^c + b* = 
 ac — b^ — O 
 ac = b^ 
 Or (§311), a:6 = 6:c 
 
 2. If ^ = -, prove that- = Y/^-,-^,. 
 
 3. If -^^ = -^, prove that -^ = - • 
 
 a+b+c+d ^ a-b+c-d ae 
 
 ^■^'■a + b-c-d a-b-c + d'^ b d 
 
 5. If 10a + b:10c + d = 8a + b:8c + d, prove that 
 a : b =: c : d. 
 
 6. If a : 6 = c : c? = e : /, prove that ma — no: mb — nd 
 = «:/. 
 
 7. Prove that, if the first term of a proportion is greatest, 
 the last term is least. 
 
 8. If a+b: o + d = c — d: a — b, and a + 6 is the 
 greatest, prove that b y d. 
 
326 ALGEBRA. 
 
 Find the ralue of x in 
 
 Solve the following equations : 
 
 j2 j 3x'y=81. (1) 
 
 < a:» + / : a;' - 2/8 = 14 : 13. (2) 
 
 From (2), by Art. 315, 2 x» : 2 y^ =^ 27 : 1 
 By Art. 317, x^ -.1/^ = 27 -.1 
 
 By Art. 318, x : y = 3 : 1 
 
 By Art. 310, x = Sy (3) 
 
 From (1) and (3) we find x = S and y = 1. 
 
 13. |^-3' = y = 200:a;. (1) 
 ' lx-y:x= 8:y (2) 
 
 From (1) and (2), by Art. 320, (x — y)^ : xy = 1600 : xy 
 
 By Arts. 317 and 318, x — y : I = 4:0 : 1 
 
 Oi"' x — y = 40 (3) 
 
 Substituting x — y = 40 in (2), 40 ; a; = 8 : .y 
 
 By Art. 317, ^ . ^ =. i ■_ y 
 
 Or, x = 5y (5) 
 
 From (3) and (5) we find x = 50,and y = 10. 
 
 14. The square of the difference of two numbers is 9 ; and 
 the difference of their cubes is to the cube of their difference 
 as 43 : 3. What are the numbers ? 
 
 15. The mean proportional between two numbers is 6 ; and 
 the cube of the sum is to the sum of the cubes as 100 : 73. 
 What are the numbers ? 
 
VARIATION. 327 
 
 16. Find two numbers whose sum is 14, and whose product 
 is to tlie sum of their squares as 10 : 29. 
 
 17. The difference of two numbers is 5 ; and the square of 
 their difference is to the difference of their squares as 6 : 11. 
 Find the two numbers. 
 
 18. As two boys were talking of their ages, they found that 
 the product of the numbers representing their ages in years 
 was 378 ; and the difference of the cubes of these same num- 
 bers was to the cube of their difference as 127 : 1. What was 
 the age of each ? 
 
 VARIATION. 
 
 325. A Variable is a number which may take a series of 
 different values. 
 
 326. A Constant is a number whose value is fixed. 
 
 327. When the successive values of a variable constantly 
 approach, by some fixed law, a constant, so that the differ- 
 ence between the variable and the constant may become 
 as small as we please without actually reaching it, the 
 constant is called the limit of the variable. 
 
 Thus, the continuous series, 
 
 0.6, 0.66, 0.666, 0.6666, , 
 
 is approaching its limit, the constant |. 
 
 328. A number is said to vary as another, if the two 
 numbers are so related that, if one changes, the other 
 changes in the same ratio. If they botli increase in the 
 same ratio, the variation is direct ; but if one decreases in 
 the same ratio as the other increases, the variation is m- 
 verse. For example, the number of dollars paid for a piece 
 of work varies directly as the amount of work; but tlie 
 number of days it takes to do a piece of work varies in- 
 versely as the number of men employed. 
 
328 ALGEliKA. 
 
 329. A number varies jointly as two or more others, if 
 it changes as the product of these others change. For 
 example, the number of dollars paid for a piece of woik 
 varies as the product of the number of men by the number 
 of days they work. 
 
 330. Variation is denoted by the sign x (read, varies 
 as); thus, a cc b signifies that a varies directly as b; and 
 
 a oc =- signifies that a varies inversely as 6. 
 
 Note. The word directly is usually omitted. 
 
 331. A variation is a proportion, and is the same as 
 the statement in Art. 315. Thus, a x b means that 
 a -.b — m a : mb; that is, any multiple of .a is to the 
 same multiple of & as a is to &; or, a oc 6 means that a, 
 or any multiple of a, is a definite number of times b, or 
 the same multiple of b ; that is, if a x b, a = nb, and 
 ma = n (vi b). 
 
 332. The statement that a = nb if a cc &, is not true 
 of quantities, face 6 days, must mean %a:%ma = b 
 days : m b days, and it is not true that %a = nb days. It 
 is true that the number of dollars varies as the number of 
 days. 
 
 THEOREMS. 
 
 In the Theorems, m and n are constants. 
 
 333. If a <x. b, and b cc c, then axe. 
 For a = mb, and b = no 
 
 .'. ar= mnc 
 .'. axe 
 
 334. If a X c, and b x c, then a ± b x e, and V«& °c c. 
 For a = mo, and b = no 
 
 .'. a ± b — (m ± n) c 
 ' .'. a ± b X c 
 
VARIATION. 329 
 
 Also, .-. ab = mnc^ 
 
 .'. '^ab cc e 
 
 335. If a (X he, then b cc -, and c oc -. 
 
 c 
 
 For a = m & c 
 
 . . - = m 6, and - = »« c 
 c 6 
 
 . . cc - , and c oc -7 
 c 6 
 
 336. If a cc b, and c cc d, then a c cc bd. 
 
 For a = mb, and e := nd 
 
 ,'. ae = mnbd 
 .'. ac cc bd 
 
 337. i/ a oc h, then a" oc J". 
 
 For a = mb 
 
 .-. a" = m"b'' 
 .-. a" oc b" 
 
 338. If a cc b when c is constant, and a cc c when b is 
 constant, then a cc be when both b and c are variable. 
 
 Suppose, while c is constant, a changes to a' and b to b' ; 
 
 «=» (1) 
 
 Then, suppose c changes to c', causing a' to change to a"; 
 then ^-' (2) 
 
 Multiplying (1) 
 
 and 
 
 (2) together, 
 
 
 
 
 a a' 
 
 be 
 ¥ci 
 
 
 
 « 
 
 be 
 
 z" b' c' 
 a cc bo 
 
330 ALGEBRA. 
 
 339. EXERCISIIS IN VARIATION- 
 
 1. If X oc y, and, when x = h, y = 20, what is the value 
 of X when y — 100 ? Ans. x — 25. 
 
 2. If a oc J, then a 6 is constant. 
 
 3. If a oc -, then i oc - • 
 
 b a 
 
 4. If a (X ^, and b x -, what is the relation of a to c ? 
 
 c 
 
 a b 
 
 5. If d oc 5, then ax cc bx, and - oc - • 
 
 a; 03 
 
 6. If a; oc -, and, when a; = 25, y = 4, what is x when 
 
 y 
 
 2/ = 20? 
 
 7. If a oc 6 when c is constant, and a cc — when b is con- 
 stall t, prove that a oc - . 
 
 8. If 4, 3, and 2 are simultaneous values of a, S, and c in 
 Example 7, what is the value of a when 5 = 3 and = 1? 
 
 9. If 8 men earn $400 in 5 weeks, find bj'^ Example 7 how 
 many weeks it will take 6 men to earn $ 240 ? 
 
 10. The area of a triangle is measured by the product of its 
 base and altitude. Prove that in triangles of equal area the 
 bases vary inversely as the altitudes. 
 
 11. If a;' oc y^, and a; = 3 when y = 4, find the equation 
 between x and y. 
 
 12. If a: oc y, and x — ^ when 2/ = |, what is x when 
 2/ = 25? 
 
 13. If a' — b^oz c% and, when a = 5 and J = 3, n = 2, find 
 the equation between a, b, and c, and prove that S is a mean 
 proportional between a. + 2 c and a — 2 c. 
 
PROGRESSIONS. 331 
 
 CHAPTER XXIII. • 
 PROGRESSIONS. 
 
 340. A Progression is a series of numbers which increase 
 or decrease according to some fixed law. 
 
 341. The Terms of a series are the successive numbers 
 that form the series. The first and last terms are called 
 the extremes, and the others the means. 
 
 ARITHMETIC PROGRESSION". 
 
 342. An Arithmetic Progression (A. P.) is a series in which 
 each term, except the first, is obtained from the preceding 
 by the addition of a constant number called the common 
 difference. Thus, each of the following series is an arith- 
 metic progression. 
 
 3, 6, 9, 12, 15, . . . 
 
 27, 23, 19, 15, 11, . . . 
 
 -2, -4, -6, -8, -10, . . . 
 
 a, a ■\- d, a + 2d, a + 3d, a + 4:d, . . . 
 
 The first is an ascending progression, the second a de- 
 scending progression. 
 
 The common differences of the series are, respectively, 
 3, —4, -2, and d. 
 
 The common difference of an arithmetic progression can 
 be found by subtracting any term from that which imme- 
 diately follows it. 
 
332 ALGEBRA. 
 
 343. In Arithmetic Progression there are five elements, 
 any three of which being given, the other two can be 
 
 found. 
 
 1. The first term. 
 
 2. The last term. 
 
 3. The common difference. 
 
 4. The number of terms. 
 
 5. The sum of all the terms. 
 
 344. In Arithmetic Progression there are twenty possi- 
 ble cases. In discussing this subject we shall let 
 
 a = the first term, 
 I = the last term, 
 d = tlie common difference, 
 n = the number of terms, 
 s = the sum of all the terms. 
 
 Case I. 
 
 345. The First Term, Common Difference, and Number of Terms 
 given, to find the Last Term. 
 
 In this Case, a, d, and n ane given, and I is required. 
 The successive terms of the series are 
 
 a, a + d, a -j- 2d, a -\- S d, a + id, . . . 
 
 that is, the coefficient of d in each term is one less than the number 
 of that term, counting from the left; therefore, the last or nth term in 
 the series is 
 
 a+ (n—l)d 
 
 or I = a -\- (n — 1) d 
 
 in which the series is ascending or descending according as <f is posi- 
 tive or negative. Hence, 
 
 Rule. 
 
 To the first term add the product formed by multiplying 
 tJie common difference hy the number of terms less one. 
 
AKITHMETIC PROGUESSION. 833 
 
 1. Given a = 2, d = 3, and n = 17, to find I. 
 
 l = a+ {n-l)d = 2+ {11 -1)3 = 50 Ans. 
 
 2. Give-n .a = — 7, d = 3, and m = 8, to find I. 
 
 3. Given a = — |, rf ;= — J, and n = 25, to find I. 
 
 4. Given a = ^, «^ = i, and n = 20, to find I. 
 
 Case II. 
 
 346. The Extremes and the Number of Terms given, to find the 
 Sum of the Series. 
 
 In this Case, a, I, and n are given, and s is required. 
 
 Now s = a + (a + d) + {a + 2d) + (a + 3d)-\ \- I 
 
 or, reversiBg the series, « = !+(J — <f) + (2 — 2(Q + (Z — 3d)4-''- + " 
 
 Adding these together, 2« = (a + + (« + ') + («+ + (« + -| 1-(« + 
 
 And since (a + Z) is to be taken as many times as there are terms, 
 hence 2s = n(a + I) 
 
 or, s = - (a + 0- Hence, 
 
 Rule. 
 
 ^i?ic? owe half the product of the sum of the extremes and 
 the number of terms. 
 
 Note. If in place of the last term the common difference is given, the 
 last term must first be found by the Rule in Case I. 
 
 1. Given a = 2, I — 50, and n = 17, to find s. 
 
 s = |(a + 0=Y(2 + 50) = 442 Ans. . 
 
 2. Given a = 5, 1 = 19, and n = 7, to find s. 
 
 Ans. s = 84. 
 
 3. Given a = — 10, «? = — 2^ and w = C, to find s. 
 
 4. Given a = 5^, d= 1\, and n = 17, to find s 
 
 5. Given a = \, d= — |, and .« = 21, to find s. 
 
334 ALGEBRA. 
 
 Case III. 
 
 347. The Extremes and Number of Terms given, to find the 
 Common Difference. 
 
 Ill this Case, a, I, and n are given, and d is required. 
 From Case I. we have I = a + (n — 1) d 
 
 Transposing and reducing, d = — — — . Hence, 
 
 Kule. 
 
 Divide the last term minus the first term ly the number of 
 terms less one, and the qiwtient will he the common difference. 
 
 1. Given a—i, 1 — 23, and n = 6, to find d. 
 
 ^ I- a 23-3 . . 
 
 d = r = -^ J- = 4 Ans. 
 
 n—1 6—1 
 
 2. Given a = 7, 1 — 37, and n = 11, to find d. 
 
 3. Given a = 9, I = — 6, and n = 16, to find d. 
 
 4. Given a = J^, I = i, and n = 5, to find d. 
 
 Note 1. This rule enables us to insert any number of arithmetic means 
 between two given numbers ; for the number of terms is two greater than 
 the number of means. Hence, if m ^ the number of means, m -^ 2 = n, 
 
 I — a 
 or m 4- 1 = n — 1, and d = r-z ■ Having found the common differ- 
 
 ence, the means are found by adding the common difference once, twice, 
 &c., to the first term. 
 
 5. Find 6 arithmetic means between 4 and 39. 
 
 6. Find 4 arithmetic means between 37 and 52. 
 
 7. Find 6 arithmetic means between 2 and — 26. 
 
 Note 2. When m = 1, the formula becomes d= ' "' . Adding a 
 
 2 
 to each member, / „ I ■\- a 
 
 a + (i=— 2— + a = — g— 
 
 But (i + (i is the second term of a series whose first term is a and common 
 difference d, or the arithmetic mean of the series a, a -\- d, a -\- 2d. 
 Hence, the arithmetic mean between two numbers is one Tialf of their sum. 
 
AlUTHMKTIC PROGRESSION. 335 
 
 8. Find the arithmetic mean between 8 and 10. Ans. 9. 
 
 9. Find the arithmetic mean between J/- and f . 
 10. Find the arithmetic mean between 7 and — 7. 
 
 348. From the formulas established iu Arts. 345 and 
 346, viz.: 
 
 lz=a + (n — l)d (1) 
 
 s = l(a + l) (2) 
 
 can be derived formulas for all the Cases in Arithmetic 
 Progression. 
 
 From (1) we can obtain the value of any one of the four numbers, 
 I, a, n, or d, when the other three are given ; and from (2), the value 
 of s, n, a, or I, when the other three are given. Formulas for the 
 remaining twelve Cases are obtained by combining the two formulas 
 (1) and (2), so as to eliminate that one of the two unknown numbers 
 whose vahie is not sought. 
 
 1. Find the formula for the value of n, when a, d, and s are 
 given. 
 
 l=a + {n-l)d (1) s = 'i(a + (2) 
 
 In := an -\- dn' — dn (3) 2s — an=ln (4) 
 
 an -{- dn'' — dn = 2 s — an (.5) 
 
 2 (d-2a\ 9s .„^ 
 
 d — 2 a ± V(d — 2 ay + B d s „, 
 
 n = ;^ — (() 
 
 2d ^ ^ 
 
 To obtain the formula required in this example, I must he elim- 
 inated from (1) and (2). From (1) and (2) we obtain (3) and (-1). 
 Placing these two values oi In equal to each other, we form (5), 
 which reduced gives (7), or the value of n in known numbers. 
 
336 
 
 ALGEBHA. 
 
 349. The twenty Cases appear in the following table. 
 
 From Nos. 1 and 9 let the pupil obtain the other formulas. 
 
 No. 
 
 Given. 
 
 REQOraED. 
 
 Kesdits. 
 
 1 
 2 
 3 
 
 4 
 
 a dn 
 ads 
 an s 
 
 dns 
 
 I 
 
 / = a+(n-l)d. 
 
 I = -id±V2ds+ {a-idf. 
 
 1 = ^-1 -a. 
 
 l-s (n-l)d 
 n^ 2 
 
 5 
 6 
 
 7 
 
 8 
 
 dnl 
 dis 
 nls 
 
 dns 
 
 a 
 
 a = ;- (n-l)rf. 
 
 a = id±V(id + lf-2ds. 
 
 „^s _(n-l)d 
 n 2 
 
 9 
 10 
 
 11 
 12 
 
 adn 
 adl 
 
 a n I 
 dnl 
 
 s 
 
 s = in[2a+ {n-l)d]. 
 2 ^ 2d ' 
 
 s = in[2/-(n-l)(f]. 
 
 13 
 14 
 15 
 16 
 
 anl 
 an s 
 a I s 
 nls 
 
 d 
 
 n — 1 
 
 , 2{s-an) 
 n(n-l)' 
 
 d- ^^-«^ . 
 2s-l-a 
 
 ^ _2{nl-s) 
 
 n{n-l) 
 
 17 
 18 
 19 
 20 
 
 adl 
 a ds 
 al s 
 dls 
 
 u 
 
 l-a., 
 "" d +'■ 
 
 ^ _ d-2a±V{d-2a}^ + 8ds. 
 2d 
 2s 
 
 „ _ 2l + d±V(2l + df-Sds 
 2d 
 
ARITHMETIC PROGEESSION. 337 
 
 350. To find any one of the five elements when three 
 others are given, we substitute the given values in that 
 formula whose first memher is the required term, and whose 
 second contains the three given terms. 
 
 1. Given rf = 2, 1 = 21, and s = 120, to find a. 
 
 « = |±\/(| + ')-2<i* (1) 
 
 a = 3, or — 1 ' (3) 
 
 In the table, Art. 349, we find (1), the required formula ; substi- 
 tuting the given values of d, I, and s, we obtain (2), which reduced 
 gives (3), or a = 3, or — 1. 
 
 Note 1. If a = 3, 7t = 10 ; but if a = — 1, n = 12. 
 
 2. Given d = 4, Z = 36, and s = 120, to find n. 
 
 Aus. m = 4, or 15. 
 
 Note 2. When m = 4, a = 24 ; but when re = 15, a^ — 20. 
 
 3. Given d = j^^, w = 8, and s = 3, to find l. 
 
 Ans. I = f . 
 
 4. Given d = 3, ra = 9, and a = — 12, to find s. 
 
 5. Given a = 5J, Z = 25^, and s = 263^, to find d. 
 
 351. MISCELLANEOUS EXAMPLES. 
 
 Find the lOOtli term of the following series : 
 
 1. 3, 5, 7, &c. 2. i, i, i, &c. 3. a + b,a,a-b, &c. 
 
 Find the sum of the following series : 
 
 4. — 3, 1, 5, . . . to 17 terms. 
 
 5. a, 3 a, 5 a, ... to a terms. Ans. a'. 
 
 22 
 
338 ALGEBRA. 
 
 6. — , v'2, — = .... to 7 terms. 
 
 ^2 + 1 V2-1 
 
 Ans. 7 (V2 + 2). 
 
 „?i — In — 2ji — 3 ^ ^ . M 
 
 7. , , .... to w terms. Ans. — 
 
 2 
 
 8. The 3d term of an A. P. is 10, and the 14th term is 54. 
 Find the 20th term. Ans. 78. 
 
 9. Which term of the series J, ^, §, etc. is 18 ? 
 
 Ans. 102d. 
 
 10. Insert 10 arithmetic means between 5 a — 6 &, and 
 5 6 — 6 a. Ans. 4a — 5 6, 3a — 4 5, etc. 
 
 11. Show that the sum of any number of consecutive odd 
 numbers, beginning with unity, is a square number. 
 
 12. Find the sum of 15 terms of an A. P. of which the 8th 
 is 6. 
 
 13. Find the sum of all the numbers between 100 and 500 
 which are divisible by 3. 
 
 14. How many terms of the series 24, 21, 18, etc. must be 
 taken in order that the sum may be 105 ? Ans. 7 or 10. 
 
 15. How many strokes do the clocks of Italy, which strike 
 up to 24, make in one revolution of the index ? 
 
 16. One hundred stones are placed on the ground at a dis- 
 tance of 2 feet from each other, the first 20 yards from the 
 basket. How far does a person travel who starts from the 
 basket and brings them one by one to it ? Ans. 6jij miles. 
 
 17. If the nth term is — , and s = j^ (3 re + 1), find 
 
 the series. Ans. ^, g, |, etc. 
 
ARITHMETIC PUOGRESSION. 339 
 
 18. An. A. P. consists of 21 terms ; the sum of the three 
 terms in the middle is 129, and of the last three is 237. Find 
 the series. Ans. 3, 7, 11, . , . 83. 
 
 19. If the sum of the second and eleventh terms is equal to 
 three times that of the fourth and fifth, and the eighth term 
 is 12, find the series. Ans. — Y-j — V") — 1> it etc 
 
 20. The sum of three numhers in A. P. is 9, and the sum 
 of their squares is 29. What are the numbers ? 
 
 (Let X — y, ji, X + y, represent the numbers.) 
 
 21. The sum of three numbers in A. P. is 33, and their 
 product is 792. What are the numbers. 
 
 22. There are four numbers in A. P. ; the sum of the 
 squares of the extremes is 272, and the sum of the squares of 
 the means is 208. W^hat are the numbers ? 
 
 (Let X — ^y,x — y, X + y, X -\-iiy, represent the numbers.) 
 
 23. The base of a right triangle is 12, and its sides are in 
 A. P. Find the other sides. 
 
 24. The product of five numbers in A. P. is 10395, and 
 their sum is 35. What are the numbers ? 
 
 Ans. 3, 5, 7, 9, 11. 
 
 25. The three digits of a certain number are in A. P. ; if 
 the number is divided by the sum of the digits in the units' 
 and tens' place, the quotient is 107. If 396 is subtracted from 
 the number, the order of its digits will be reversed. Required 
 the number. 
 
 26. A and B have to walk a distance of 27 miles ; A starts 
 at 2^ miles an hour, and increases his pace by a quarter of a 
 mile every hour ; B starts at 5 miles an hour, but falls off at 
 the rate of half a mile every hour. Find which will finish the 
 distance first, and by what length of time. 
 
 Ans. A, by an hour. 
 
340 ALGEBUA. 
 
 27. A body falls through a space of 4.9 meters the first 
 second, and in each succeeding second 9.8 meters more than 
 in the next preceding one. How far will it fall in 20 
 seconds ? 
 
 28. A man saves every year $25, which he puts at interest 
 at the rate of 4 per cent a year. How long will it take for 
 the interest to amount to $ 91 ? 
 
 29. Divide unity into four parts in A. P., of which the sum 
 of the cubes shall be •^. 
 
 30. A and B jointly owe $ 23661, of which A offers to pay 
 every day $ 20, if B will pay $ 1 the first day, $ 2 the second, 
 and so on in A. P. How many days will it take to pay the 
 debt, and how much will each pay ? 
 
 Ans. 198 days; A $3960; B $19701. 
 
 GEOMETRIC PROGEESSION. 
 
 352. A Geometric Progression (G. P.) is a series in which 
 each term, except the first, is obtained by multiplying the 
 preceding term by a constant number, called the ratio. 
 
 Thus, each of the following series forms a Geometric 
 
 Progression. 
 
 
 
 
 3, 
 
 6, 
 
 12. 
 
 24, 
 
 72, 
 
 36, 
 
 18, 
 
 9, 
 
 1, 
 
 -h 
 
 *, 
 
 -^, 
 
 a, 
 
 ar, 
 
 ar\ 
 
 ar\ 
 
 The first is an ascending progression, the second a de- 
 scending progression. 
 
 The ratios of the series are respectively 2, J, — -J, and r. 
 
 The ratio of a geometric progression can be found by 
 dividing anij term by that which immediately precedes it. 
 
GEOMETRIC PROGRESSION. 341 
 
 363. In Geometric Progression there are five elements, 
 any three of which being given, the other two can he 
 found. These elements are the same as in Arithmetic 
 Progression, except that in place of the common difference 
 we have the ratio. 
 
 354. In Geometric Progression there are twenty possible 
 cases. In discussing these cases we shall preserve the 
 same notation as in Arithmetic Progression, except that, 
 instead oi d — the common difference, we shall use r = the 
 ratio. 
 
 Case I. 
 
 355. The First Term, Ratio, and Number of Terms given, to 
 find tlie Last Term. 
 
 In this Case, a, r, and n are given, and I required. 
 
 The successive terms of the series are 
 
 a, ar, a r'^, a r'*, a r*, etc. 
 That is, each term is the product of the first terra and that power of 
 the ratio which is one less than the number of the term ; therefore 
 the last or nth term in the series is 
 
 ar^-', or l = at^-\ Hence, 
 
 Bule. 
 
 Multiply the first term ly that power of the ratio whose 
 index is one less than the number of terms. 
 
 1. Given a = l, r = 2, and w = 8, to find I. 
 
 I — ar"-^ = 7 X 2' = 896 Ans. 
 
 2. Given a = 3, r = 2, and n = 10, to find I. 
 
 Ans. Z = 1536. 
 
 3. Given a = 3, r = i, and re = 6, to find I. 
 
 4. Given a = — 5, r = — 3, and n = 7, to find I. 
 
 5. Given a= —i, r = |, and w = 8, to find i. 
 
342 ALGEBRA. 
 
 Case II. 
 
 356. The Extremes and the Ratio given, to find the Sum of 
 the Series. 
 
 In this Case, a, I, and r are given, and s is required. 
 Now s = a + ar + ar^ + ar^ + ■ ■ ■ + I (1) 
 
 Multiplying (1) by r, rs = ar + ar^ + ar» -[ \-l+lr (2) 
 
 Subtracting (1) from (2), rs — s = lr — a 
 
 Whence, s= '"~° . Hence, 
 
 ' r — 1 
 
 Bule. 
 
 Multiply the last term ly the ratio, from the product sub- 
 tract the first term, and divide the remainder by the ratio 
 less one. 
 
 1. Given a = 5, I = 320, and r = 2, to find s. 
 Ir — a 320 X 2 — 5 
 
 r-1 
 
 = 635 Aus. 
 
 2. Given a = 4, 1 = 78732, and r = 3, to find s. 
 
 Ans. s = 118096. 
 
 3. Given a = 5, I — — 640, and r = — 2, to find s. 
 
 Case III. 
 
 357. The First Term, Ratio, and Number of Terms given, to 
 find the Sum of the Series. 
 
 In this Case, a, r, and n are given, and s required. 
 The last terra can be found by Case I., and then the sum of the 
 series by Case II. Or better, since 
 
 Ir ^ ar" 
 
 Substituting this value of Z r in the formula in Case II., we have 
 
 r" — 1 
 
 s = X a. Hence, 
 
 r — 1 
 
GEOMETRIC PROGRESSION. 343 
 
 Bule. 
 
 From the ratio raised to a power whose index is equal to 
 the number of terms subtract one, divide the remainder by the 
 ratio less one, and multiply the quotient by the first term. 
 
 1. Given a = 7, r = 2, and w. = 6, to find s. 
 
 ^» _ 1 2« — 1 
 
 s = 5" X a = -^ 5- X 7 = 441 Ans. 
 
 r—1 2—1 
 
 2. Given a = 5, r = i, and w = 9, to find s. 
 
 Ans. s = 436905. 
 
 3. Given a = 2, r = — |, and w = 8, to find s. 
 
 4. Given a = §, r = — |, and w = 7, to find s. 
 
 5. Given a = 2, r = — |, and w = 6, to find s. 
 
 358. In a geometric series whose ratio is a proper frac- 
 tion, the greater the numher of terms, the less numeri- 
 cally the last term. If the number of terms is infinite, the 
 last term must be infinitesimal; and in finding the sum 
 of such a series the last term may be considered as noth- 
 ing. Therefore, when the number of terms is infinite, the 
 
 formula Ir — a ^ 
 
 s = -T- becomes 
 
 1 — 1 
 
 r — 1 1 — r 
 Hence, to find the sum of a geometric series whose ratio is 
 a proper fraction and number of terms infinite, 
 
 Bnle. 
 
 Divide the first term by one minus the ratio. 
 
 1. Pind the sum of the series 12, 6, 3, etc. to infinity. 
 
 s = 5 = 5 r = 24 Ans. 
 
 1—r 1 — i 
 
 2. Find the sum of the series 5, 1, ^, etc. to infinity. 
 
 Aris. V-- 
 
344 ALGEBRA. 
 
 3. Find the sum of the series a, h, -, etc., ii b < a, to 
 
 ii^fi^i*y- Ans. -^. 
 
 a — b 
 
 4. Find the sum of the series 0.9, 0.03, 0.001, etc. to 
 infinity. 
 
 6. Find the value of the decimal 0.3333, etc. to infinity. 
 (This decimal can be written ^ + ^f^ + yAtj ^^'^•) 
 
 Case IV. 
 359. The Extremes and Number of Terms given, to find the Ratio. 
 
 In this Case, a, I, and n are given, and r is required. 
 
 From Case I. / = ar^-^ 
 
 " — '^11 
 Whence, r= »/-. Hence, 
 
 Knle. 
 
 Divide the last term ly the first, and extract that root of 
 the quotient whose index is one less than the number of terms. 
 
 1. Given a = 2, 1= 162, and re = 5, to find r. 
 
 "TTi 4/162 , , 
 '=\a = \^ = ^ ^°^- 
 
 2. Given a = 2, I =: lOj, and w = 5, to find r. 
 
 Ans. r = f . 
 
 3. Given a = — J, I = j^\-s, and re = 6, to find r. 
 
 Note 1. This rule enables us to insert any number of geometric means 
 between two numbers ; for the number of terms is two gi'eater than the 
 number of means. Hence, if m = the number of means, m -\- 2 = n, or 
 
 m -\- 1 = 11 — 1 ; and r = 4 /- . Having found the ratio, the means 
 
 are found by multiplying the first term by the ratio, by its square, its 
 cube, &c. 
 
 4. Find three geometric means between 5 and 405. 
 
 Ans. 15, 45, 135. 
 
GEOMETRIC PROGRESSION. 345 
 
 5. Find four geometric means between 160 and 5. 
 
 6. Find three geometric means between 7 and 35/^. 
 Note 2. When m = 1, the formula becomes 
 
 Multiplying by a, ar ^= akl - = \fal. 
 
 Y a 
 
 But ffi r is the second term of a series whose first term is a and ratio r ; 
 
 or the geometric mean of the series a, ar, ar"^. Hence, the geometric 
 
 mean between two numbers is the square root of their product. 
 
 7. Find the geometric mean between 3 and 27. Ans. 9. 
 
 8. Find the geometric mean between J and 2187. 
 
 9. Find the geometric mean between — ^ and — xxjy 
 
 360. From the formulas established in Arts. 355 and 356, 
 l-ar"-^ (1) 
 
 s = ^JL^ (2) 
 
 r — 1 
 
 can be derived formulas for all the Cases in Geometric 
 
 Progression. 
 
 From (1) we can obtain the value of any one of the four numbers, 
 I, a, 71, or r, when the other three are given ; from (2), the value of 
 s, I, r, or a, when the other three are given. Formulas for the re- 
 maining twelve Cases are obtained by combining the formulas (1) 
 and (2) so as to eliminate that one of the two unknown numbers 
 whose value is not sought. 
 
 1. Find the formula for the value of s, when I, n, and r are 
 given. 
 From (1), 
 
 Substituting this value of a in. (2), s 
 
 r"- 
 
 -1 
 
 = a 
 
 ; 
 
 
 C2"» 
 
 .» 
 
 Ir- 
 
 rn-i 
 
 
 K^J 
 
 r ■ 
 
 -1 
 
 
 
 s 
 
 _('•»- 
 
 -1)1 
 
 
 
 (r- 
 
 ■ 1) »•»- 
 
 -1 
 
346 
 
 ALGEBRA. 
 
 36L The twenty Cases appear in the following table. 
 
 From Nos. 1 and 9 let the pupil obtain the other formulas, except those 
 named in the note on the next page. 
 
 No. 
 
 OlTEN. 
 
 Required. 
 
 Results. 
 
 1 
 
 am 
 
 
 / = ar»-l. 
 
 I _a+ {r-l)s 
 
 2 
 
 a r s 
 
 
 
 
 
 
 8 
 
 
 I 
 
 J _ (r-lja)"-! 
 
 
 
 
 r»-l 
 
 4 
 
 ans 
 
 
 i(s-/)"-^-a(s-a)"-^ = 0. 
 
 5 
 
 ml 
 
 
 a- ' . 
 
 
 
 
 ,«-! 
 
 6 
 
 r n s 
 
 
 „-{r-l)s 
 
 
 
 
 r»-l 
 
 7 
 
 rls 
 
 
 a = rl-{r-l)s. 
 
 8 
 
 n I s 
 
 
 a(s-a)"-'-/(s-?)"~* = 0. 
 
 9 
 
 arn 
 
 
 ,_a(r"-l) 
 r — 1 
 
 10 
 
 arl 
 
 
 • = ^-- 
 
 
 
 s 
 
 V^- I/a" 
 
 11 
 
 anl 
 
 
 " — n-l . n-1 - 
 
 1//- Va 
 
 12 
 
 ml 
 
 
 (r-l)r"-i 
 
 
 
 
 n-l/7 
 
 13 
 
 anl 
 
 
 '=\/i- 
 
 14 
 
 ans 
 
 
 r»_V+«-°-0. 
 
 
 
 r 
 
 a a 
 
 15 
 
 ats 
 
 
 r-'-°. 
 
 s - / 
 
 16 
 
 nls 
 
 
 J-"- * r— 1+ ', - 0. 
 s - / s — I 
 
 17 
 
 arl 
 
 
 ^ _ log/- logo _^^ 
 logr 
 
 18 
 
 ar s 
 
 
 „ _ log[« + (r-l)s]-log(i 
 
 
 
 n 
 
 logr 
 
 19 
 
 als 
 
 
 logi-loga ^^ 
 
 log (s — a) — log (s — /) 
 
 20 
 
 rls 
 
 
 „_log/-log[;r-{r-l)s]_^i 
 logr 
 
GKOMETRIC PROGEESSION. 347 
 
 Note. The four formulas for the value of n cannot be obtained or used 
 without a knowledge of logarithms ; .and four others, viz. 4, 8, 14, and 16, 
 when n exceeds 2, cannot be reduced without a knowledge of equations 
 that cannot be reduced by any rules given in this book. 
 
 362. To find any one of the five elements when three 
 others are given, we substitute the given values in that for- 
 mula whose first member is the required term, and whose 
 second contains the three given terms. 
 
 1. Given »■ = 3, w = 5, and s = 726, to find I. 
 
 Z=(jl^il^' (1) 
 
 Z = (l^iKZM)!l! = 486 (2) 
 
 In the table, Art. 361, we find (1), the required formula, and, 
 substituting the given values of r, n, and «, we obtain (2), or Z = 486. 
 
 2. Given a = 1, m = 8, and I = 128, to find s. 
 
 Ans. s — 256. 
 
 3. Find the 7th and 11th terms of the series 64, — 32, . . . 
 
 4. rind the 4th and 8th terms of the series 0.008, 0*04, . . . 
 
 363. MISCELLANEOUS EXAMPLES. 
 
 Find the last term in the following series : 
 
 1 • .1 
 
 1. a;, 1, - , ... to 30 terms. Ans. -^' 
 
 X •*■ 
 
 2. X, x\ x\ to p terms. Ans. x^^-\ 
 Find the sum of : 
 
 3. 16.2, 5.4, 1.8, ... to 7 terms. 
 
 4. 1, 5, 25, ... to j3 terms. Ans. J (5' — 1). 
 
 5. a, - . ^ , . . . to M terms, 
 
 ' r r 
 
348 ALGEBRA. 
 
 6. f , - 1, t, . . . to infinity. 
 
 7. 3-S 3-^ 3-^ ... to infinity. 
 
 8. VI, i V2, f VI ... to infinity^^^ ,(3^e^,^,) _ 
 
 46 
 
 9. Insert 3 geometric means between 486 and 6. 
 
 10. Insert 5 geometric means between |J and 4|. 
 
 11. The 5th term of a G. P. is f, and the 7th term |f 
 Determine the series. Ans. |, ± 3, 2, ± J, . . . 
 
 12. Find a G. P. whose first term is unity, and whose tliird 
 term is ^ij. 
 
 13. The sum of an infinite series in G. P. is 12, and the 
 second term is 27 times the fifth term. Find the series. 
 
 14. The sum of an infinite series in G. P. is f, and the 
 sum of the first two terms is ^|. What is the series ? 
 
 15. What is the amount at compound interest of S500 for 
 5 years at 6 % ? 
 
 16. Pind the amount at compound interest of $p for n 
 years at r per cent. 
 
 17. Prom a vessel containing 182^ centiliters of alcohol a 
 chemist draws daily a fixed quantity, and replaces it with 
 water. At the end of 6 days there are only 16 centiliters of 
 alcohol in the vessel. How many centiliters does he draw a 
 day. Ans. 60J. 
 
 18. A certain number is formed of three digits that are in 
 G. P. Now twice the digit in the hundreds' place is equal to 
 the difference between the digits in the tens' and the units' 
 place ; and if 594 is added to the number, the order of the 
 digits will be reversed. What is the number ? 
 
 Let X, xy, xy"^, represent the digits. 
 
HAEMONIC PROGRESSION. 349 
 
 Note. It will often be best to represent a G. P. as follows : 
 ^% ^ ijt iPt for three terms ; 
 
 — , X, y, ^, for four terms; 
 y "' 
 
 — , x^, xy, «2 ?_ for Ave terms. 
 y X 
 
 19. There are three numbers in geometric progression 
 whose sum is 63 ; and the sum of the extremes is to tlie 
 mean as 17 : 4. What are the numbers ? 
 
 20. There are five numbers in geometric progression ; the 
 sum of the first four is 40, and the sum of the last four 120. 
 What are the numbers ? 
 
 21. The sum of the squares of three numbers in geometric 
 progression is 819 ; and the sum of the extremes is 21 more 
 than the mean. What are the numbers ? 
 
 22. There are four numbers in geometric progression whose 
 continued product is 729 ; and the sum of the series is to the 
 sum of the means as 10 : 3. What are the numbers ? 
 
 23. Of four numbers in geometric progression the sum of 
 the first and third is 61 ; and the difference of the means is 
 to the difference of the extremes as 4 : 21. What are the 
 numbers ? 
 
 HARMONIC PROGRESSION. 
 
 361 An Harmonic Progression (H. P.) is a series of num- 
 bers whose reciprocals are in arithmetic progression. 
 Thus, a, b, c, . . . are in H. P. 
 
 if _,_,_,.,. are in A. P. 
 
 a c 
 
 365. There is no general formula for the suvi of an H. P., 
 but^many problems with respect to such a series may be 
 solved by inverting the terms and proceeding as in an 
 A, P., and then inverting back again. 
 
350 ALGEBRA. 
 
 THEOREMS. 
 
 366. If a, b, c, are in H. P., then a : c = a — b : b — c. 
 Since a, b, o, are in H. P., 
 
 i, -, -, arein A. P. 
 a b c 
 
 b a c b 
 a — h b — c 
 
 a c 
 
 or, a: c-=a — b:b — c 
 
 367. The harmonic mean between two numbers is twice 
 their product divided by their sum. 
 
 Let a, b, be the two numbers, and H their harmonic mean. 
 
 Then - , - ,, y, are in A. P, 
 
 a H b 
 
 1 
 
 111 
 
 h' 
 
 a~ b H 
 
 
 H a^ b 
 
 
 jj- 2 ab 
 
 a + b 
 
 368. The geometric mean between two numbers is the geo- 
 metric mean between the arithmetic and harmonic means of 
 the same numbers. 
 
 Let A, G, H, represent, respectively, the arithmetic, geo- 
 metric, and harmonic means between a and b ; then 
 
 Art. 347, Note 2, A= — "^-^ 
 
 Art. 359, Note 2, G= Vab 
 
 Art. 367, ^=1^ 
 
 ' a + b 
 
 A TT "•+ b 2ab 
 
 2 a + b 
 
 and G^ = ab 
 
HAEMONIC PROGRESSION. 351 
 
 369. EXAMPLES. 
 
 1. The second term of an H. P. is 2, and the fourth term 
 is 6. Find the series. 
 
 The 2d and 4th terms of the corresponding A. P. are ^ and J, 
 respectively. Therefore a + d = i, and a + 3 d = ^ ; whence 
 a = §, and d = _ J. Therefore the A. P. is f, ^, ^, ^, . . . , 
 and the H. P. is f , 2, 3, 6, . . . 
 
 2. If a, b, c, are in H. P., prove that -, , , ;, 
 
 are also iu H. P. * + " « + '^ '^ + * 
 
 Since -, -, 1 are in A. P. 
 
 a b c ' 
 
 a + b + c a-\-b + c a 4- b + c . . _ 
 
 , — -^ , — ■ ■ — , are in A. P. 
 
 a b c 
 
 ... l + *±f, l + ?+f, l + ^±_*, areinA.P. 
 a b c 
 
 b + c a + c a + b ..-d 
 
 , — i — , , are in A. P. 
 
 a c 
 
 Hence, —^ , — -— , — f_ , are in H. P. 
 
 b -j- c a + c a -\- b 
 
 Find the arithmetic, harmonic, and geometric means be- 
 tween : 
 
 3. i and tV- r 1 ,1 
 
 and 
 
 4. X + y and x — y. ^ + y 
 
 6. Insert three harmonic means between 2§ and 12. 
 
 Ans. 3, 4, 6. 
 
 7. Continue to five terms the H. P. If, 1, ^. 
 
 8. The first term of an H. P. is unity, the third term is J. 
 Find the tenth term. Ans. ■^. 
 
 9. If b + c, a + c, a + b, are in H. P., prove that a\ 
 h\ c\ are in A. P. 
 
 10. The difference of the arithmetic and harmonic means 
 between two numbers is If. Find the numbers, one being 
 four times the other. Ans. 2, 8. 
 
352 ALGEBRA. 
 
 CHAPTEE XXIV. 
 THE BINOMIAL THEOREM. 
 
 370. The laws for the expansion of a binomial, as illus- 
 trated by actual multiplication in Art. 208, can be proved 
 to be true for any index. The following is the proof, when 
 the index is any positive integer. Following the laws of 
 Art. 208, we have 
 
 (a+br = a-'+ma'^%+ "'('»-l) am-26.+ '"("'-l)('»-2) „^^.+^ etc. 
 1 ■ 2 1 * J • o 
 
 Now, multiplying this equation by a + b, for the first member we 
 have (a + 6)"'+i ; and for the second, 
 
 ,, , m(m — 1) „ „,„ , m(m — l)(m — 2) „ ,1. , . 
 
 a™ + m a"'-'6 -| 5^ a"'-^b^ + — ^^ — 5 — ^ ^ a'^-^b' +, etc. 
 
 1*2 x ' 2 ' o 
 
 a+b 
 
 „_Li 1 ™i , >n(m, — 1) „ , , „ . m(m — 1) (m — 2) _ ,,„ . . 
 a'"+^+ma"'b-\ \ ° ^^ + — ^ — ToT ^aP'-'b'+, etc. 
 
 a'-b+ ma»-i6''+ !!lil!!—}la"'-ib'+, etc. 
 
 1-2 
 
 0"+'+ (m+l)a''b-\- ^ ' a"*-' 6^+ '- , „ q ^a"~ 6°+, etc. 
 
 1*2 1*2*3 
 
 Now let m + 1 = «, or m = n — 1, and this product becomes 
 
 a- + na''-^b + "("~^> a«-= J'' + n(n-l)(n-2) „„_3 js , etc. 
 1*2 1-2-3 ^ 
 
 That is, (a + ft)" x (a + 6) = (a + ft)"' + i = (a + 6)» 
 
 ^an+„an-i6 + !i(!^:zl)a»-2a2_^''("-l)("-2)„„-36.^,etc. 
 1*2 1*2*0 
 
 and this is exactly the form that is obtained in applying the laws of 
 Art. 208 directly to expanding a + 5 to the nth power. Therefore, 
 if the laws are true for the power whose index is m, they are true for 
 the power whose index is (m + 1). These laws have been proved to 
 
THE BINOMIAL THEOREM. 353 
 
 be true for the 5th power (§ 208); therefore they are true for the 6th; 
 and therefore for the 7th ; and so on. 
 
 Note. This method of proof is called mathematical induction. 
 
 371. If — & is substituted for b, in the binomial a +6, the 
 signs of the terms containing the odd powers of 6 will be — 
 (§ 208) ; that is, the second, fourth, sixth, etc. terms. 
 
 372. To find any term in the, expansion of a binomial. 
 
 The indices in any term in the expansion of (a + 6)" can he 
 writteli at once from the last formula in Art. 370. 
 
 Thus, the indices in the rth form are, for a, re — (r — 1) = n + 1 — r, 
 and for b, r — 1 ; and if the sign of 6 is +, the sign of the rth term 
 is +, but if b is — , and r — 1 is odd, that is, if r is even, the sign 
 is — , otherwise + . 
 
 The coefficient of the 2d term is re 
 
 n(n — 1) 
 
 it it It OJ tt ^ £ 
 
 « " 4th ' 
 
 « « tt 5tjj . 
 
 u tt tt fifVi * 
 
 °^'^ 1 ■ 2 • 3 • 4 ■ 5 
 
 and so on. Hence, 
 
 The rth term of the nth power of a + 6 is 
 
 re (re — 1) (re — 2) (re — r + 2) ^„_r+i j,._i 
 
 1 •2-3 . .(r— 1) 
 
 1. Find the 6th term in (a — by. 
 
 In this example n = 18, and r = 6. Hence, the index of a in the 
 6th term is 18 — (6 — 1) = 13 ; and the index of b is 5. The sign 
 of the 6th term is — ; re — r + 2 = 14 ; r — 1 = 5 ; and the coeffi- 
 cient of the 6th term is 
 
 HjlIii±l±«lJi = 8568 
 1 . a- . * ■ J^ . -5- 
 
 Hence, the 6th term in (a — ft)" = —8568 a"&^ 
 
 23 
 
 1-2 
 
 
 n(re-l)(»-2) 
 
 
 1-2-3 
 
 
 n (re — 1) (re — 2) (n — 3) 
 
 
 1 •2-3-4 
 
 
 n (re — 1) (re — 2) (n — 3) (re ■ 
 
 -4) 
 
354 ALGEBRA. 
 
 Note 1. In writing the formula for the coefficient, write for the denom- 
 inator 1 • 2 • 3 ■ 4 . . . etc., with the last number = r — 1; then, for the 
 numei-ator, over each, number in the denominator write a series beginning 
 with a number = n, and decreasing by 1. 
 
 Find the 
 
 2. 8th term in (x - yY' 5. 96th term of (m - ny^- 
 
 NoTE 2. Notice that the 96th term of the 100th power is the 6th from 
 the last term. 
 
 3. 7th term of (a + &)" 6. Middle term of (a; +y)"- 
 
 4. 8th term of (k — 2/)'"'. 7. Middle terms of (m—«)". 
 
 8. 7th term in (2 a; + 3 r/)". 
 
 11 ■ 10 • 9 • 8 ■ 7 • 6 /2.j;)5(3„)6 := 11 ■ 7 ■ 2° ■ 3'' a:°»° Ans. 
 
 fa bV^ 
 
 9. Middle term of ( 5 — 7 ) • 
 
 10. Middle terms of f 2 a; — |j . 
 
 11. 6th term of (a^ - &^)". 
 
 Note 3. Whether the index is positive or negative, integral or frac- 
 tional, the Binomial Theorem can be api)lied equally well. 
 
 12. Expand (a + b)-\ 
 
 („ + i) -1 = a-i + (— 1) a-2 b + (1) a-«b^ + (—l)a-n» +, etc. 
 = a-^ — a-^ b + a-^ b" — a-* b" + . etc. 
 
 1 b b^ b" , 
 
 = ^ A — ; 7+- etc. 
 
 13. Expand {a — b)-\ 
 
 (a — 6)-i = a-i — (— l)a-25-f (l)a-3js_(_i)a-4j8+^ etc. 
 = a-i + a-2 6+a-n2 + a— '6' + , etc. 
 
 = - + -5 + -; + -i+' etc. 
 
 n n* n^ n* 
 
 Note 4. The same results will be obtaiued in these examples if we 
 
 1 
 a-\-b' 
 
 write {a + *)""' and (a — J)— i in their other forms, and — 
 
 and perform the division. 
 
THE BINOMIAL THEOREM. 355 
 
 Expand 
 
 14. (m + «) ^' to five terms. 
 
 15. (a — i)~' to four terms, 
 
 16. (1 + 3 y) ~ ' to five terms. 
 
 17. (^ — x^ ] to five terms. 
 
 18. {«— t) to four terms. 
 
 19. Expand (a + J)i 
 
 (a + 6)i = ai + (1) a-^b + (-|) a-^i^ + (i^) a"^ 6» +, etc. 
 = ai + l a~i b — l a~H^ + -^ a~^6» — , etc. 
 
 ^ b h^ , b^ 
 
 ^=va-^ — ^ ;= H ;= — , etc. 
 
 20. Expand (a - 6)*. 
 
 (a _ 6)* = a^ - (I) a-H + (-"i) a"^^^ _ (J^) a"! J8 + , etc. 
 = ai — ^a^ifi— ^a~t62_ J-Q-f i8_^ etc. 
 
 = ■f a — o — ?^ ;^^ := — i etc. 
 
 i^Va 8Va» 16Va6 ' 
 
 21. Expand (a + 6)~i 
 
 (a + 6)-^ = a-i + (-i)a-*ft + (|)a-t62+(-^6-)a-^6s+. etc. 
 
 1 " b 3i2 56= 
 
 — • ^ -^ 4- etc 
 
 \a 2Va» 8V~a^ 16 Va' ' 
 
 22. Expand (a - b) ~*. 
 
 (a _ J) -i- = a-^-i-l) o-t4+ (Da-^fia-C-^^e) o~^ *'+> etc. 
 = a~^+^a~^6 + |a~^i-^+l^a~^6s+, etc. 
 
 = H — -^= + — =^ T ;= + , etc. 
 
 Va 2Va8 8Va5 16 Va' 
 
356 ALGEBUA. 
 
 Expand to five terms 
 
 23. (m - n)k ■ 25. (x - 1) ~i 27. (a + 3 a^k 
 
 24. (a-b)i. 26. (l + 3a)"i 28. (m-%n)^. 
 
 29. Find the 8th term in (a + b)- ^ 
 
 In the 8th term of {a -\- b)-^ the index of a is —10, and the in- 
 dex of b is 7. The signs of all the terms (disregarding the sign of 
 the coefficient) are +. In this example n = — 3 and r = 8. Hence, 
 n — r + 2:= — 9, and r — 1=7; and the coetficient is 
 
 -3 (-4) (-5) C-6) (-7) (-8) (-9) _ _ 
 1-2-3-4-5-6-7 ~ 
 
 36 i' 
 Hence, the 8th term in (a + 6) ~' = jq- • 
 
 30. Find the 7th term in (x — y)~*. 
 
 The index of x in the 7th term will be — ^-, and the index of y 
 is 6. The sign of the 7th term (disregarding the sign of the co- 
 efficient) is + . In this example re = — f , and r = 7. Hence, 
 n — r-|-2= — ^, and r — 1 = 6, and the coefficient is 
 
 - 1 (- 1) (- ^) (- ¥) (- ¥) (- ¥) 100947 
 1.2--8--4--6-6 12 X 4'' 
 
 Hence, the 7th term in (x — y) ~* = =-^ — . 
 
 ^ "^ 12X4'!'^ 
 
 Find the 
 
 31. 10th term in (ot — 1)-''. 35. 7th term in (m -f w)~ ^ 
 
 32. 9th terra in (x + y^) ' \ 36. 9th term in (a + 1)" ^. 
 
 33. 11th term in (a - 3) -^. 37. 6th term in (2 — 6)"i 
 
 34. 12th term in (1 + b)-\ 38. 8th term in {a-P)~^. 
 39. Find V^ ; that is, expand (1 + 1)4. 
 
LOGARITHMS. 357 
 
 CHAPTEK XXV. 
 
 LOGARITHMS. 
 
 373. Logaxithms are exponents of the powers* of some 
 number which is taken as a base. In the tables of loga- 
 rithms in common use the number 10 is taken as the base, 
 and all numbers are considered as powers of 10. 
 
 By Arts. 224, 231, 
 
 10° — 1, that is, the logarithm of 1 is 
 10^ = 10, " " « 10 « 1 
 
 lO'' = 100, " " " 100 " 2 
 
 10» = 1000, " " " 1000 « 3 
 
 &c., &c. 
 
 Therefore, the logarithm of any number between 1 and 10 
 is between and 1, that is, is a fraction ; the logarithm of 
 any number between 10 and 100 is between 1 and 2, that 
 is, is 1 plus a fraction ; and the logarithm of any number 
 between 100 and 1000 is 2 plus a fraction ; and so on. 
 
 By Arts. 231, 232, 
 
 10° = 1, that is, the logarithm of 1. is 
 
 10-1 = 0.1, " " " 0.1 " — 1 
 
 10-2=0.01, " " " 0.01 "-2 
 
 10-'= 0.001, « « " 0.001 « —3 
 
 &c., &c. 
 
 Therefore, the logarithm of any number between 1 and 0.1 
 is between and — 1, that is, is — 1 plus a fraction ; the 
 logarithm of any number between 0.1 and 0.01 is between 
 —1 and —2, that is, is —2 plus a fraction ; and so on. 
 
 * The word power is used here to denote a number with any real index whatever. 
 
358 ALGEBRA. 
 
 The logarithm of a number, therefore, is either au in- 
 teger (which may be 0) positive or negative, or an in- 
 teger positive or negative and a fraction, which is always 
 positive. 
 
 The representation of the logarithms of all numbers less 
 than a unit by a negative integer and a positive fraction is 
 merely a matter of convenience. The integral part of a 
 logarithm is called the characteristic, and the decimal part 
 the mantissa. Thus, the characteristic of the logarithm 
 3.1784 is 3, and the mantissa .1784. 
 
 374, The characteristic of the logarithm of a number 
 is not given in the tables, but can be supplied by the 
 following 
 
 Bule. 
 
 Tlie characteristic of the logarithm of any number is equal 
 to the number of places by which its first significant figure 
 on the left is removed from units' place, the characteristic 
 being positive when this figure is to the left, and negative 
 when it is to the right of units' place. 
 
 Thus, the logarithm of 59 is 1 plus a fraction ; that is, 
 the characteristic of the logarithm of 59 is 1. The loga- 
 rithm of 5417.7 is 3 plus a fraction ; that is, the charac- 
 teristic of the logarithm of 5417.7 is 3. The logarithm of 
 0.3 is —1 plus a fraction; that is, the characteristic of the 
 logarithm of 0.3 is —1. The logarithm of 0.00017 is —4 
 plus a fraction ; that is, the characteristic of the logarithm 
 of 0.00017 is -4. 
 
 375. Since the base of this system of logarithms is 10, if 
 any number is multiplied by 10, its logarithm will be in- 
 creased by a unit (§ 72) ; if divided by 10, diminished by 
 a unit (§ 78). 
 

 
 LOGARITHMS. 
 
 
 is, the 
 
 log of 5549. 
 
 being 
 
 3.7442 
 
 (I 
 
 
 554.9 
 
 is 
 
 2.7442 
 
 (I 
 
 
 55.49 
 
 ii 
 
 1.7442 
 
 a 
 
 
 5.549 
 
 a 
 
 0.7442 
 
 a 
 
 
 0.5549 
 
 (( 
 
 T.7442 
 
 a 
 
 
 0.05549 
 
 » 
 
 2.7442 
 
 a 
 
 
 0.005549 
 
 a 
 
 3.7442 
 
 359 
 
 Hence, the, mantissa of the logarithm of any set of figures 
 is tlie same, wherever the decimal point may he. 
 
 As only the characteristic is negative, the minus sign is 
 written over the characteristic. 
 
 TABLE OF LOGARITHMS. 
 
 376. To find the Logarithm of a Number of Two Figures. 
 
 Disregarding the decimal point, find the given number 
 
 in the column N (pp. 373, 374), and directly opposite, in 
 
 the column 0, is the mantissa of the logarithm, to which 
 
 must be prefixed the characteristic, according to the Eule 
 
 in Art. 374. 
 
 Thus, the log of 85 is 1.9294 
 
 « " 26 " 1.4150 
 
 The first figure of the mantissa, remaining the same for 
 several successive numbers, is not repeated, but left to be 
 
 supplied. Thus,. the log of 83 is 1.9191 
 
 As, according to Art. 375, multiplying a number by 10 
 increases its logarithm by a unit, therefore, to find the log- 
 arithm of any number containing only two significant fig- 
 ures with one or more ciphers annexed, we use the same 
 rule as above. 
 
 Thus, the log of 850 is 2.9294 
 « " 750000 « 5.8751 
 
 The principle just stated is applicable also in the cases 
 that follow. 
 
360 ALGEBRA. 
 
 377. To find the Logarithm of a Number of Three Figures. 
 
 Disregarding the decimal point, find the first two figures 
 in the column N, and the third figure at the top of one of 
 the columns. Opposite the first two figures, and in the 
 column under the third figure, will be the last three figures 
 of the decimal part of the logarithm, to which the first fig- 
 ure in the column is to be prefixed, and the character- 
 istic, according to the Eule in Art. 374. 
 
 Thus, the log of 695 is 2.7745 
 " " 249 " 2.3962 
 
 In the columns 1, 2, 3, etc., a small cipher Q or figure 
 (i) is sometimes placed below the first figure, to show that 
 the figure which is to be prefixed from the column has 
 changed to the next larger number, and is to be found in 
 the horizontal line directly below. 
 
 Thus, the log of 7960 is 3.9009 
 " " 25900 " 4.4133 
 
 378. To find the Logarithm of a Number of more than 
 Three Figures. 
 
 On the right half of pages 373 and 374 are tables of 
 Proportional Parts. The figures in any column of these 
 tables are as many tenths of the average difference of the 
 ten logarithms in the same horizontal line as is denoted by 
 the number at the top of the column. The decimal point 
 in these differences is placed as though the mantissas were 
 integral. 
 
 1. To find the logarithm of a number of four figures, 
 find as before the logarithm of the first three figures; to 
 this, from the table of Proportional Parts, add the number 
 standing on the same horizontal line and directly under 
 the fourth figure of the given number. 
 
LOGARITHMS. 361 
 
 Thus, to find the log of 5743. 
 
 The log of 5740 is 3.7589 
 In Proportional Parts, in the same line, under 3 " 2.3 
 
 Therefore, the log of 5743 « 3.7591 
 
 It is always best to find the logarithm of the near- 
 est tabulated number, and add or subtract, as the case 
 may be, the correction from the table of Proportional 
 Parts. 
 
 Thus, to find the log of 6377. 
 
 6377 = 6380 - 3 
 The log of 6380 is 3.8048 
 
 correction for 3 " 2 
 
 Therefore, the log of 6377 " 3.8046 
 
 Whenever the fractional part omitted is larger than half the unit 
 in the next place to the left, one is added to that figure. 
 
 2. For a fifth or sixth figure the correction is made in 
 the same manner, only the point must be moved one place 
 to the left for the fifth, two for the sixth figure. 
 
 Thus, to find the log of 3.6825. 
 
 The log of 3.68 is 0.5658 
 
 correction for 2 " 2.4 
 
 « « 5 « .59 
 
 Therefore, the log of 3.6825 " 0.5661 
 
 To find the log of 112.82. 
 
 112.82 = 113 - 0.18 
 The log of 113 is 2.0531 
 
 correction for 0.18 is (3.8 + 3.02) " 6.82 
 
 Therefore, the log of 112.82 « 2.0524 
 
362 ALGEBllA. 
 
 379. To find the Number corresponding to a Given Logarithm. 
 
 Find in the table, if possible, the mantissa of the given 
 logarithm. The two figures opposite in the column N, 
 -with the number at the head of the column in which the 
 logarithm is found, affixed, and the decimal point so placed 
 as to make the number of integral figures correspond to 
 the characteristic of the given logarithm, as taught in 
 Art. 374, will be the number required. Tlius, 
 
 The number corresponding to log 6.6378 is 345000 
 " " " " 1.8745 " 74.9 
 
 If the mantissa of the logarithm cannot be exactly found, 
 take the number corresponding to the mantissa nearest the 
 given mantissa ; in the same horizontal line in the table of 
 Proportional Parts find the figures which express tlie dif- 
 ference between this and the given mantissa; at the top of 
 the page, in the same vertical column, is the correction 
 that belongs one place to the right of the number already 
 taken, to be added if the given mantissa is greater, sub- 
 tracted if less. Thus, 
 
 1. To find the number corresponding to 
 log 2.7660 
 
 next less log, 2.7667, and number corresponding, 583. 
 
 difference, 3 correction, 0.4 
 
 Number required, 583.4 
 
 2. To find the number corresponding to 
 log 3.8052 
 
 next greater log, 3.8055, and number corresponding, 0.00639 
 
 difference, 3 correction, 44 
 
 Number required, 0.0063856 
 
 The nearest number in the table of Proportional Parts to 3 is 2.7 ; 
 corresponding to this at the top is 4, which belongs as a correction 
 one place to the right of the number (0.00639) already taken, but 
 
LOGARITHMS. 363 
 
 3 — 2.7 =^ 0.3 ; this, in like manner, gives a still further correction 
 of 4, one place farther still to the right. The whole correction, there- 
 fore, is 44, to be deducted as shown above. 
 
 Find the logarithms of the following numbers : 
 
 3. 365. 4. 34700. 5. 83.24. 6. 0.00018. 
 
 Find the antilogarithms of the following numbers : 
 
 7. 2.096. 8. 1.346. 9. 3.62004. 10. 5.83156. 
 
 380. The great utility of logarithms in arithmetical op- 
 erations is that addition takes the place of multiplication, 
 and subtraction of division, multiplication of involution, 
 and division of evolution. That is, to multiply numbers, we 
 add their logarithms; to divide, we subtract the logarithm 
 of the divisor from that of the dividend; to raise a number 
 to any power, we multiply its logarithm by the exponent 
 of that power; and to extract the root of any number, we 
 divide its logarithm by the number expressing the root to 
 be found. 
 
 This is the same as multiplication and division of differ- 
 ent powers of the same letter by each other, and involving 
 and evolving powers of a single letter; the number 10 
 takes the place of the given letter, and the logarithms 
 are the exponents of 10. 
 
 MULTIPLICATION BY LOGARITHMS. 
 
 Kule. 
 
 381. Add the, logarithms of the factors, and the sum will 
 he tJie logarithm of the product (§ 72). 
 
 1. Multiply 246.5 by 0.003574. Ans. 0.881. 
 
 2. Find the product of 9.4478, 0.397626, 16.784. 
 
 Ans. 63.06. 
 Note. It must be carefully borne in mind that the mantissa of the 
 logarithm is always positive. 
 
364 ALGEBRA. 
 
 3. Multiply 0.00456 by 2.57. 
 
 0.00456 log 3.6590 
 
 2.57 " 1.4082 
 
 Product, 0.1367 " T.0672 
 
 4. Multiply 36.75 by 0.003725. 
 
 Since the nwnerieal product is the same whether the 
 factors are positive or negative, we can use logarithms in 
 multiplying when one or more of the factors are negative, 
 taking care to prefix to the product the proper sign accord- 
 ing to Art. 70. When a factor is negative, to the logarithm 
 which is used n is appended. 
 
 5. Multiply -0.7546 by 0.00545. 
 
 -0.7546 logT,8777ji 
 
 0.00545 " 5.7364 
 
 Product, -0.004113 " S.6141w 
 
 6. Find the product of — 0.025, 625, and - 12.125. 
 
 7. rind the product of -16, -67.23, and -0.008. 
 
 Ans. —8.606. 
 
 DIVISION BY LOGARITHMS. 
 
 Kule. 
 
 382. From the logarithm of the dividend subtract the 
 logarithm of the divisor, and the remainder will he the 
 logarithm of the quotient (§ 78). 
 
 1. Divide 34.66 by 0.0123. 
 
 34.56 log 1.5386 
 
 0.0123 " 2.0899 
 
 Quotient, 2810 " 3.4487 
 
Logarithms. 365 
 
 2. Divide 18.5741 by 0.009496. 
 
 18.5741 log 1.2689 
 
 0.009496 « 5.9776 
 
 Quotient, 1956 " 3.2913 
 
 Negative numbers can be divided in the same manner 
 as positive, taking care to prefix to the quotient the proper 
 sign, according to Art. 77. 
 
 3. Divide 84.52 by 3.514. Ans. 24.05. 
 
 4. Divide 5672 by 0.0037. Ans. 1533000. 
 
 5. Divide 0.053 by 797. Ans. 0.0000665. 
 
 6. Divide -16.54 by 345. Ans. -0.04794. 
 
 7. Divide - 0.2456 by 25.05. Ans. -0.009806. 
 
 383. Instead of subtracting one logarithm from another, 
 
 it is sometimes more convenient to add what it lacks of 10, 
 
 and from the sum reject 10. The result is evidently the 
 
 same. For , ,^„ . .„ 
 
 X — y = X + (10 — 2/) — 10 
 
 The remainder found by subtracting a logarithm from 
 10 is called the arithmetical complement of the logarithm, 
 or the cologarithm. The cologarithm is easiest found by 
 beginning at the left of the logarithm, and subtracting each 
 figure from 9, except the last significant figure, which must 
 be subtracted from 10. 
 
 By this method, Ex. 2 will appear as follows : 
 18.5741 log 1.2689 
 
 0.009496 colog 12.0224 
 
 Quotient, 1956 log 3.2913 
 
 DIVISION AND MULTIPLICATION BY LOGARITHMS. 
 
 384. In working examples combining multiplication and 
 division, the use of cologarithms is of great advantage. 
 
366 ALGEBRA. 
 
 Kule. 
 
 Find the sum of the logarithms of the multipliers a^nd the 
 cologarithms of the divisors ; reject as many tens as there are 
 cologarithms {divisors); the result will be the logarithm of 
 the number sought. 
 
 ^ -o^- ^.^ , . (39.74) (0.0861) (-470) 
 1. Fmd the value of ^^684) (1.2475) " 
 
 39.74 log 1.6992 
 . 0.0861 " 2.9350 
 
 -470 " 2,6721 w 
 
 — 684 colog 7.1649% 
 -1.2475 « 9.9039 ?» 
 
 Ans. -1.8846 log 0.2751 
 
 An even number of negatives gives a positive result ; an 
 odd number, a negative (§ 70). 
 
 PROPOETION BY LOGARITHMS. 
 
 Kule. 
 
 385. Add the cologarithm of the first term to the logarithms 
 of the second and third terms, and from the sum reject 10. 
 
 1. Given 44 : 240 = 4522 : x, to find x. 
 
 44 colog 8.3565 
 
 240 log 2.3802 
 
 4522 " 3.6553 
 
 Ans. 24662 " 4.3920 
 
 2. Given 324 : 672 = 125 : x, to find x. Ans. 259.2. 
 
 3. Given x : 9.426 = 908.4 : 15.42, to find x. Ans. 555.3. 
 
LOGARITHMS. 367 
 
 INVOLUTION BY LOGARITHMS. 
 
 Bule. 
 
 386. Multiply the logarithm of the number hy the expo- 
 nent of the pvwer required (§ 206). 
 
 In involution, as the error in the logarithm is multiplied 
 hy the index of the power, the results with logarithms of 
 only four decimal places cannot be relied on for more than 
 three significant figures. 
 
 1. Find the 6th power of 2.34. 
 
 2.34 log 0.3692 
 
 6 
 
 Ans. 164.1 " 2.2152 
 
 2. Find the 3d power of 0.000961. 
 
 0.000961 log 3.9827 
 
 3 
 
 Ans. 0.000000000748 « TD.9481 
 
 3. Find the 6th power of 2.74119. Ans. 424.5. 
 
 4. Find the 4th power of 0.8724. Ans. 0.5791. 
 
 Negative numbers are involved in the same manner, 
 taking care to prefix to the power the proper sign, accord- 
 ing to Art. 205. 
 
 5. Find the 5th power of —0.225. Ans. —0.0005767. 
 
 6. FindtheSthpowerof — 12.3. Ans. 623875000. 
 
 EVOLUTION BY LOGARITHMS. 
 
 Kule. 
 
 387. Divide tlie logarithm of the number by the index of 
 the root required (§ 220). 
 
368 ALGEBRA. 
 
 When the characteristic is negative, and not divisible by 
 the index of the root, we increase the negative character- 
 istic so as to make it divisible, and to the mantissa prefix 
 an equal positive number. 
 
 1. Find the 4th root of 0.254. 
 
 0.254 log 5.4048 
 
 JL 
 4)1 + 2.4048 
 
 Ans. 0.3992 « T.6012 
 
 2. Find the 3d root of 0.7589. Ans. 0.9121. 
 
 3. Find the 4th root of 0.0019. Ans. 0.2088. 
 
 Negative numbers are evolved in the same manner, tak- 
 ing care to prefix to the root the proper sign, according to 
 Art. 213. 
 
 4. Find the 6th root of -0.037. Ans. - 0.5172. 
 
 5. Find the 7th root of -0.000257. Ans. -0.307. 
 
 388. An Exponential Equation, that is, an equation hav- 
 ing the unknown number as an exponent, may be solved 
 by means of logarithms. 
 
 For, if a' — n, by Art. 386, 
 
 X log a = log n 
 
 log a 
 
 1. Solve 5" = 625. 
 
 X ■ log 5 =: log 625 
 log 625 2.7959 , . 
 log 5 0.699 
 
 2. Solve 4913" = 17. 3. Solve (-^—Y^iS. 
 
LOGAKITHMS. 369 
 
 SYSTEMS OF LOGARITHMS. 
 
 389. The system of logarithms which has 10 for its base 
 is the one in common use. It was first introduced in 
 1615, by Briggs, a contemporary of Napier, the inventor of 
 logarithms. As in this system the mantissa of the loga- 
 rithm of any set of figures is the same, wherever the deci- 
 mal point may be (§ 375), which (in the Arabic notation of 
 numbers) would not be the case with any other base, it is 
 far the most convenient system. The number of possible 
 systems, however, is infinite. 
 
 In general, if of = n, then x is tlie logarithm of n to the 
 base a ; and n is the number called the antilogarithm, cor- 
 responding to the logarithm x, in a system whose base is a. 
 
 390. The logarithm of 1 is 0, whatever the base may be. 
 For the power of every number is 1, or a" = 1 (§ 231). 
 
 391. TJie logarithm of the base itself is 1. 
 
 For the first power of any number is the number itself, 
 or «■' = a. 
 
 392. Neither nor 1 can be the base of a system of 
 logarithms. 
 
 For all the powers and roots of are 0, and of 1 are 1. 
 
 393. The logarithm of the reciprocal of any number is the 
 negative of the logarithm of the 7iumber itself. 
 
 For the reciprocal of any immber is 1' divided by that 
 number (§ 9) ; that is, it is the logarithm of 1 minus the 
 logarithm of the number, or minus the logarithm of the 
 number (§ 382). 
 
 394. In a system whose base is between 1 and 0, the less 
 the number the greater its logarithm. 
 
 24 
 
370 ALGEBRA. 
 
 For the greater the power of a proper fraction, the less 
 its value. With such a base, the logarithms of numbers 
 greater than 1 will be negative, less than 1 positive. 
 
 Thus, with J as the base, 
 
 the log of ^ is 2 ; of ^^ is 3 
 " , " 9 " — 2; " 81 " —3 
 
 395. The logarithms of numbers which form a geometric 
 series form an arithmetic series. 
 
 For, if a series increased or decreased by a constant ratio, 
 its logarithms would increase or decrease by a constant 
 difference equal to the logarithm of the constant ratio. 
 
 For an example see Art. 375; here the numbers decrease 
 by the constant ratio 10, and the logarithms by the constant 
 difference 1. 
 
 396. From the principles of the previous articles it will 
 be easy to find the logarithms of the perfect roots and 
 powers of any number. Thus, 
 
 In a system whose base is 8, 
 
 8^ = 2, that is, the log of 2 = 0.3 
 
 8*= 4, " 
 
 
 4 = 0.6 
 
 8' = 8, « " 
 
 
 8 = 1. 
 
 8^ = ]6, " « 
 
 
 16 = 1.3 
 
 &c.. 
 
 
 &c. 
 
 Then, according to Art. 393, 
 
 
 
 the log of 1 = — 0.3 
 
 = 
 
 T.6 
 
 « 1 = - 0.6 
 
 = 
 
 T.3 
 
 " " i = -1. 
 
 = 
 
 T. 
 
 « « tV= -1.3 
 
 = 
 
 2.6 
 
 &C., I 
 
 &c 
 
 , 
 
LOGARITHMS. 371 
 
 397. MISCBLLANEOUS EXAMPLES. 
 
 1. In a system whose base is 3, what is the logarithm of 
 81? of 3? of 27? of 1? of i? of ^? of 0? 
 
 2. Find the logarithms of ^^, ^\-g, 25, \, 5, to base ^. 
 Note. Log4 256 means the logarithm of 256 to hase i. 
 
 3. Find the value of 
 
 loge siff. logs 128, log„ (/' a "^-, log„ -=■ . 
 
 Ans. -3, I, -j, -i. 
 
 4. Find the logarithm of 1000 to base 0.01, and of 0.0001 
 to base 0.001. Ans. — |, |. 
 
 5. The logarithm of 0.5 is T.6, what is the base ? 
 
 Ans. 8. 
 
 6. Simplify log y 729V^9~' • 27"^- -^-ns. log 3. 
 
 7. Simplify log H - 2 log I + log ^^2^. Ans. log. 2. 
 
 8. What logarithms would j'ou need to find to reduce 
 
 ^ Vf X 0.012 a!^ ? 
 
 Ans. logs of 7, 2, 3, 11. 
 
 9. Find the product of 37.2, 3.72, 0.000372, and 37200. 
 
 Ans. 1914.5. 
 
 10. Find the number of digits in S^^ X 2'. Ans. 9. 
 
 11. Find the number of digits in (876)". Ans. 48. 
 
 12. Solve (1.2)'- = 1.1. Ans. x = 0.5227. 
 
 13. Find log, 16.345, where e = 2.71828. Ans. 2.794-. 
 
372 ALGEBRA. 
 
 Verify the following equations : 
 
 (213) (7.655) _ 
 ^*- (3145) (718) - "•"""7^^- 
 
 (47) (0.653) (12g) _ 
 ^''- (3576) (1520) - 0-00007247. 
 
 IG. (1)^^1 = 11.83. 
 
 17. -v^ = 0.9592. 
 
 18. ^'^' = 15.84. 
 * V2 
 
 19. VXW ■ (^y = 0.01063. 
 
 20. .3/ K294)(125) r 9o^e 
 V I (42) (32) I - ^•"'''• 
 
 21. /o.43 p^8 v^OTT = 0.9434 
 
 22. Solve (12.9) (7»"^) = y ^^- Ans. x = 0.1987. 
 
 24. Given a = 25, r = 5, Z = 78125, to find n. Ans. 6. 
 
 25. Given a = -^j r = 3, s = 364f, to find n. Ans. 8. 
 
 26. What is the amount of $1000 for 25 years at 5% 
 compound interest? Ans. $3388. 
 
 27. Find the amount at 4% interest of $500 for 10 years, 
 compounded semiannually. Ans. 1 748. 
 
 28. In how many j'ears will a sum of money be doubled at 
 5% compound interest ? Ans. 14.2 years. 
 
LOGARITHMS OF NUMBERS. 
 
 378 
 
 N 
 10 
 
 
 
 1 
 043 
 
 2 
 
 086 
 
 3 
 
 128 
 
 4 
 
 no 
 
 5 
 
 6 
 
 7 
 
 294 
 
 8 
 
 334 
 
 9 
 
 374 
 
 PROPORTIONAL PARTS. 
 
 _l_ 
 4-1 
 
 2 
 
 8.3 
 
 3 
 12.4 
 
 4 
 i5.6 
 
 5 
 20.7 
 
 6 
 
 24.8 
 
 7 
 29.0 
 
 8 
 33.1 
 
 9 
 37-3 
 
 0000 
 
 212 253 
 
 11 
 
 414 
 
 453 
 
 492 531 
 
 569 
 
 607 645 
 
 682 
 
 719 
 
 755 
 
 3.8 
 
 7.6 
 
 II-3 
 
 15.1 
 
 18.9 
 
 22.7 
 
 26.5 
 
 30.2 
 
 340 
 
 12 
 
 79-2 
 
 828 
 
 864 899 
 
 934 
 
 969 „04 
 
 „S8 
 
 oT^ 
 
 ,06 
 
 3-5 
 
 7.0 
 
 10.4 
 
 13-9 
 
 17.4 
 
 20.9 
 
 24.3 
 
 27.8 
 
 31.3 
 
 13 
 
 1139 
 
 173 
 
 206 
 
 239 
 
 271 
 
 303 335 
 
 367 
 
 399 
 
 430 
 
 3-2 
 
 6.4 
 
 9-7 
 
 12.9 
 
 161 
 
 19.3 
 
 22.5 
 
 25-7 
 
 29.0 
 
 14 
 
 461 
 
 492 
 
 5-23 
 
 313 
 
 563 
 
 847 
 
 684 
 875 
 
 614 644 
 
 673 
 959 
 
 703 
 
 987 
 
 739 
 
 7* 
 
 3.0 
 
 2.8 
 
 6.0 
 76 
 
 9.0 
 
 8.4 
 
 12.0 
 II. 2 
 
 140 
 
 18.0 
 
 21.0 
 19.6 
 
 240 
 22.4 
 
 27.0 
 25.2 
 
 15 
 
 nei 
 
 790 
 
 903 
 
 931 
 
 16 
 
 -2041 
 
 068 
 
 095 
 
 122 
 
 148 
 
 175 
 
 201 
 
 227 
 
 353 
 
 279 
 
 2.6 
 
 53 
 
 7-9 
 
 10.5 
 
 132 
 
 IS. 8 
 
 .8.4 
 
 21.1 
 
 237 
 
 17 
 
 304 
 
 330 
 
 356 
 
 380 
 
 405 
 
 430 
 
 456 
 
 480 
 
 504 
 
 629 
 
 2.5 
 
 S-o 
 
 7-4 
 
 9-9 
 
 12.4 
 
 14.9 
 
 17.4 
 
 19.9 
 
 22.3 
 
 18 
 
 553 
 
 577 
 
 601 
 
 625 
 
 648 
 
 672 
 
 695 
 
 718 
 
 742 
 
 765 
 
 2.3 
 
 4-7 
 
 7.0 
 
 9-4 
 
 II. 7 
 
 141 
 
 ■5.4 
 
 18.8 
 
 21,1 
 
 19 
 
 788 
 
 810 
 
 833 
 
 856 
 
 878 
 
 900 
 
 923 
 
 946 
 
 967 
 
 989 
 
 2.2 
 
 4-5 
 
 _67 
 
 8.9 
 
 11. 1 
 
 13.4 
 
 .5.6 
 
 17^8 
 
 20.0 
 
 20 
 
 3010 
 
 032 
 
 054 
 
 075 
 
 096 
 
 118 
 
 139 
 
 160 
 
 131 
 
 301 
 
 2.1 
 
 4-2 
 
 6.4 
 
 "8.5 
 
 10.6 
 
 12.7 
 
 148 
 
 17.0 
 
 19. 1 
 
 21 
 
 223 
 
 243 
 
 263 
 
 284 
 
 304 
 
 324 346 
 
 366 
 
 385 
 
 404 
 
 2.0 
 
 4.0 
 
 6.1 
 
 8.1 
 
 10,1 
 
 12. 1 
 
 141 
 
 16.2 
 
 18.2 
 
 22 
 
 424 
 
 444 
 
 464 
 
 483 
 
 502 
 
 622 641 
 
 660 
 
 679 
 
 598 
 
 1.9 
 
 3.9 
 
 S-8 
 
 77 
 
 97 
 
 1 1.6 
 
 13.5 
 
 15.4 
 
 17.4 
 
 23 
 
 617 
 
 636 
 
 655 
 
 674 
 
 692 
 
 711 729 
 
 747 
 
 766 
 
 784 
 
 1.8 
 
 3-7 
 
 SS 
 
 7-4 
 
 9.2 
 
 11. 1 
 
 12.9 
 
 148 
 
 16.6 
 
 21 
 
 80-2 
 
 820 
 
 838 
 
 856 
 
 874 
 
 892 909 
 
 927 
 
 946 
 
 963 
 
 1.8 
 
 35 
 
 53 
 
 7-1 
 
 8.9 
 
 10.6 
 
 12.4 
 
 14.2 
 
 160 
 
 26 
 
 3979 
 
 997 
 
 0^ 
 
 Ji 
 
 ^ 
 
 ^te 
 
 ^ 
 
 1^ 
 
 733 
 
 1-7 
 
 3-4 
 
 5-1 
 
 6.8 
 
 8.5 
 
 10.2 
 
 11.9 
 
 ■36 
 
 I5-3 
 
 26 
 
 4150 
 
 166 
 
 183 
 
 200 
 
 216 
 
 232 249 
 
 266 
 
 281 
 
 398 
 
 1.6 
 
 3-3 
 
 4.9 
 
 6.6 
 
 8.2 
 
 9.8 
 
 II. 5 
 
 13-1 
 
 .4.8 
 
 27 
 
 ai4 
 
 330 
 
 346 
 
 362 
 
 373 
 
 393 409 
 
 426 
 
 440 
 
 466 
 
 1.6 
 
 3.2 
 
 4-7 
 
 63 
 
 7.9 
 
 9-5 
 
 11. 1 
 
 126 
 
 14 z 
 
 28 
 
 472 
 
 487 
 
 502 
 
 518 
 
 633 
 
 548^664 
 
 679 
 
 594 
 
 609 
 
 1-5 
 
 3.0 
 
 4.6 
 
 6.1 
 
 7.6 
 
 91 
 
 10.7 
 
 12.2 
 
 I3-7 
 
 29 
 SO 
 
 624 
 
 639 
 
 786 
 
 654 
 300 
 
 669 
 814 
 
 683 
 829 
 
 698713 
 843|867 
 
 728 
 871 
 
 743 
 886 
 
 787 
 900 
 
 ■■5 
 1.4 
 
 2.9 
 2.8 
 
 4.4 
 4-3 
 
 5-9 
 5-7 
 
 7-4 
 7-1 
 
 8.8 
 8.5 
 
 10.3 
 10.0 
 
 II.8 
 11.4 
 
 13.3 
 12.8 
 
 4771 
 
 31 
 
 914 
 
 928 
 
 942 
 
 955 
 
 969 
 
 983 997 
 
 o" 
 
 024 
 
 „38 
 
 1.4 
 
 2.8 
 
 41 
 
 5-5 
 
 6.9 
 
 8.3 
 
 97 
 
 11. 
 
 124 
 
 32 
 
 S051 
 
 065 
 
 079 
 
 092 
 
 106 
 
 119132 
 
 146 
 
 169 
 
 172 
 
 "•3 
 
 2-7 
 
 4.0 
 
 5-3 
 
 6.7 
 
 8.0 
 
 9.4 
 
 10.7 
 
 12.0 
 
 33 
 
 185 
 
 193 
 
 211 
 
 224 
 
 237 
 
 250|263 
 
 276 
 
 389 
 
 302 
 
 1-3 
 
 2.6 
 
 3.9 
 
 5-2 
 
 6.5 
 
 7.8 
 
 91 
 
 10.4 
 
 11.7 
 
 31 
 
 315 
 
 328 
 
 340 
 
 353 
 
 366 
 
 378,391 
 
 403 
 
 416 
 
 438 
 
 '■3 
 
 2-5 
 
 _3_8 
 
 il° 
 
 6.3 
 
 7.6 
 
 8.8 
 
 10. 1 
 
 "■3 
 
 35 
 
 5441 
 
 453 
 
 465 
 
 478 
 
 490 
 
 
 
 502614 
 
 527 
 
 539 
 
 551 
 
 1.2 
 
 2.4 
 
 37 
 
 4.9 
 
 6.1 
 
 7-3 
 
 T6 
 
 "^ 
 
 II. 
 
 36 
 
 563 
 
 S75 
 
 587 
 
 599 
 
 611 
 
 623 636 
 
 647 
 
 668 
 
 670 
 
 1.2 
 
 2.4 
 
 3.6 
 
 4.8 
 
 5-9 
 
 7." 
 
 8.3 
 
 9-5 
 
 10.7 
 
 37 
 
 682 
 
 694 
 
 706 
 
 717 
 
 729 
 
 740 762 
 
 763 
 
 776 
 
 786 
 
 1.3 
 
 2-3 
 
 3-5 
 
 4.6 
 
 58 
 
 69 
 
 8.1 
 
 9-3 
 
 10.4 
 
 38 
 
 798 
 
 809 
 
 821 
 
 832 
 
 843 
 
 865 866 
 
 877 
 
 888 
 
 899 
 
 l-I 
 
 2.3 
 
 3-4 
 
 4-5 
 
 S6 
 
 6.8 
 
 79 
 
 9.0 
 
 10.2 
 
 39 
 
 911 
 
 922 
 
 031 
 
 933 
 
 042 
 
 944 
 063 
 
 956 
 064 
 
 966 977 
 075 085 
 
 988 
 096 
 
 999 
 107 
 
 ol£ 
 117 
 
 I.I 
 I.I 
 
 2.2 
 
 2.1 
 
 3-3 
 
 3-2 
 
 44 
 4-3 
 
 5-5 
 5-4 
 
 6.6 
 ~4 
 
 7.7 
 
 7-5 
 
 8.8 
 8.6 
 
 99 
 9-7 
 
 10 
 
 6021 
 
 11 
 
 128 
 
 138 
 
 149 
 
 160 
 
 170 
 
 180 191 
 
 301 
 
 212 
 
 222 
 
 I.O 
 
 2.1 
 
 3-1 
 
 4.2 
 
 5-2 
 
 6.3 
 
 7-3 
 
 8.4 
 
 9.4 
 
 12 
 
 232 
 
 343 
 
 263 
 
 263 
 
 274 
 
 284 294 
 
 304 
 
 314 
 
 328 
 
 I-O 
 
 2.0 
 
 3-1 
 
 41 
 
 5-1 
 
 61 
 
 7-2 
 
 8.2 
 
 92 
 
 13 
 
 335 
 
 345 
 
 355 
 
 365 
 
 375 
 
 385 395 
 
 405 
 
 416 
 
 426 
 
 1.0 
 
 2.0 
 
 3.0 
 
 40 
 
 S-o 
 
 6.0 
 
 7.0 
 
 80 
 
 9.6 
 
 11 
 15 
 
 435 
 
 444 
 542 
 
 454 
 551 
 
 464 
 561 
 
 474 
 571 
 
 484 
 580 
 
 493 
 590 
 
 803 
 599 
 
 613 
 609 
 
 622 
 618 
 
 I.O 
 
 1.0 
 
 2.0 
 1.9 
 
 2.9 
 2.9 
 
 39 
 .1.8 
 
 49 
 4.8 
 
 5.9 
 5-7 
 
 6.8 
 
 _7^ 
 7.6 
 
 8.8 
 8.6 
 
 6532 
 
 16 
 
 628 
 
 637 
 
 646 
 
 656 
 
 665 
 
 676 
 
 684 
 
 693 
 
 702 
 
 712 
 
 o.g 
 
 "■9 
 
 2.G 
 
 3-7| 4 7 
 
 5-6 
 
 6.5 
 
 75 
 
 8.4 
 
 17 
 
 721 
 
 730 
 
 739 
 
 749 
 
 758 
 
 767 
 
 776 
 
 786 
 
 794 
 
 803 
 
 0.9 
 
 1.8 
 
 27 
 
 3.7| 46 
 
 5-5 
 
 6.4 
 
 7.3 
 
 8.2 
 
 18 
 
 812 
 
 821 
 
 830 
 
 839 
 
 848 
 
 857 
 
 866 
 
 875 
 
 884 
 
 893 
 
 0.9 
 
 1.8 
 
 27 
 
 3-6 4-5 
 
 5-4 
 
 6.3 
 
 7-2 
 
 8.1 
 
 19 
 
 902 
 6990 
 
 911 
 
 998 
 
 9-20 
 
 9-28 
 
 937 
 
 946 
 
 955 
 
 964 
 
 972 
 
 981 
 
 0.9 
 
 ,1.8 
 
 2.6 
 2.6 
 
 3-s' 4-+ 
 
 5 3 
 
 S.2 
 
 6.1 
 6.0 
 
 7.0 
 6.9 
 
 7-9 
 7-7 
 
 50 
 
 ^ 
 
 ^ 
 
 Ja 
 
 ^ 
 
 ^ 
 
 ^ 
 
 ^ 
 
 0.9 
 
 '•7 
 
 3.4 
 
 4-3 
 
 61 
 
 7076 
 
 084 
 
 093 
 
 101 
 
 110 
 
 118 
 
 126 
 
 136 
 
 143 
 
 162 
 
 a.8 
 
 '■7 
 
 2.5 
 
 3.4 
 
 4-2 
 
 5.1 
 
 5-9 
 
 6.7 
 
 76 
 
 62 
 
 160 
 
 168 
 
 177 
 
 185 
 
 193 
 
 202 
 
 210 
 
 218 
 
 226 
 
 236 
 
 0.8 
 
 1.7 
 
 2.5 
 
 3-3 
 
 41 
 
 5.0 
 
 s-s 
 
 6.6 
 
 7-4 
 
 63 
 
 243 
 
 251 
 
 259 
 
 267 
 
 276 
 
 284 
 
 292 
 
 300 
 
 308 
 
 316 
 
 0.8 
 
 1.6 
 
 2.4 
 
 3-2 
 
 41 
 
 49 
 
 5-7 
 
 6-5 7-3 
 
 51 
 
 324 
 
 332 
 
 340 
 
 348 '366 
 
 364,372 
 
 380|388 
 
 396 
 
 0.8 
 
 1.6 
 
 2-4 
 
 3-2 
 
 4.0 
 
 48 
 
 56 
 
 6.4 7.2 
 
374 
 
 LOGAKITHMS OF NUMBERS. 
 
 N 
 56 
 
 
 
 7404 
 
 X 
 412 
 
 2 
 
 3 
 
 4 
 
 436 
 
 5 
 
 6 
 
 7 
 459 
 
 8 
 
 9 
 
 PROPORTIONAL PARIS. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 3-1 
 
 5 
 3-9 
 
 6 
 4-7 
 
 7 
 S-S 
 
 
 6-3 
 
 _9^ 
 7.0 
 
 419J427 
 
 443'451 
 
 466 
 
 474 
 
 0.8 
 
 1.6 
 
 2.3 
 
 66 
 
 48a 
 
 490 
 
 497 
 
 505 
 
 613 
 
 520 528 
 
 536 
 
 643 
 
 551 
 
 0.8 
 
 i-S 
 
 2.3 
 
 3-1 
 
 3-8 
 
 4.6 
 
 S-4 
 
 6-1 
 
 6.9 
 
 67 
 
 539 
 
 566 
 
 574 
 
 582 
 
 589 
 
 597 604 
 
 612 
 
 619 
 
 627 
 
 0.8 
 
 i-S 
 
 2-3 
 
 30 
 
 3-8 
 
 4.5 
 
 5-3 
 
 6.0 
 
 6.8 
 
 6S 
 
 634 
 
 642 
 
 649 
 
 657 
 
 664 
 
 672 679 
 
 686 
 
 694 
 
 701 
 
 0.7 
 
 IS 
 
 2.2 
 
 ■3-0 
 
 37 
 
 4-S 
 
 5-2 
 
 S-9 
 
 6.7 
 
 59 
 60 
 
 709 
 
 716 
 
 789 
 
 723 
 
 731 
 
 738 
 
 745 752 
 
 760 
 
 767 
 
 774 
 
 0^ 
 
 i-S 
 
 1.4 
 
 2.2 
 2.2 
 
 2-9 
 2.9 
 
 3-6 
 
 44 
 4-3 
 
 S-i 
 
 S° 
 
 _£_8 
 5-7 
 
 6.6 
 
 778-J 
 
 7961803 
 
 810 
 
 818 825 
 
 832 
 
 839 
 
 846 
 
 0.7 
 
 61 
 
 863 
 
 860 
 
 868 875 
 
 882 
 
 889 896 
 
 903 
 
 910 
 
 917 
 
 0.7 
 
 14 
 
 2.1 
 
 2.8 
 
 3S 
 
 4-2 
 
 49 
 
 S.6 
 
 64 
 
 62 
 
 924 
 
 931 
 
 938i945 
 
 962 
 
 959 966 
 
 973 
 
 980 
 
 987 
 
 0.7 
 
 1.4 
 
 2-1 
 
 2-8 
 
 3S 
 
 4-2 
 
 4.9 
 
 S-6 
 
 6.3 
 
 63 
 
 993 
 
 „oo 
 
 0»7 
 
 0" 
 
 o21 
 
 „28„35 
 
 o"! 
 
 0« 
 
 .,68 
 
 0.7 
 
 1.4 
 
 2.1 
 
 2-7 
 
 3-4 
 
 4-1 
 
 4.8 
 
 S-5 
 
 6.2 
 
 61 
 
 8062 
 
 069 
 
 075 
 
 032 
 
 089 
 
 096 102 
 
 109 
 
 116 
 
 123 
 
 0.7 
 
 1-3 
 
 20 
 
 2.7 
 
 3-4 
 
 4.0 
 
 47 
 
 5-4 
 
 6.1 
 
 65 
 
 8139 
 
 136 
 
 142 
 
 I« 
 
 156 
 
 162 169 
 
 176 
 
 182 
 
 189 
 
 0.7 
 
 1.3 
 
 2.0 
 
 2-7 
 
 -3-3 
 
 4.0 
 
 4.6 
 
 S-3 
 
 6.0 
 
 66 
 
 i95 
 
 202 
 
 209215 
 
 222 
 
 228 236 
 
 241 
 
 248 
 
 254 
 
 0.7 
 
 1.3 
 
 2.0 
 
 2.6 
 
 3-3 
 
 39 
 
 46 
 
 5-2 
 
 5-9 
 
 67 
 
 261 
 
 207 
 
 274280 
 
 ■287 
 
 293 299 
 
 306 
 
 312 
 
 319 
 
 0.6 
 
 1-3 
 
 1.9 
 
 2.6 
 
 3-2 
 
 3-9 
 
 4-5 
 
 S-i 
 
 5-8 
 
 6S 
 
 326 
 
 331 
 
 3381344 
 
 361 
 
 367 363 
 
 370 
 
 376 
 
 332 
 
 0.6 
 
 . 1-3 
 
 1.9 
 
 2.5 
 
 3.2 
 
 3-8 
 
 44 
 
 S-i 
 
 ■S-7 
 
 69 
 
 383 
 
 395 
 
 401 
 
 407 
 
 414 
 
 420 [426 
 
 432 
 
 439 
 
 445 
 
 06 
 
 1.2 
 
 1-9 
 
 2.5 
 
 3.1 
 
 37 
 
 4-4 
 
 S-o 
 
 JS 
 
 70 
 
 8461 
 
 467 
 
 463 
 
 470 
 
 476 
 
 482J488 
 
 494 
 
 600 
 
 506 
 
 'ZI 
 
 1.2 
 
 1.8 
 
 2.5 
 
 3-1 
 
 3-7 
 
 4.3 
 
 4-9 
 
 S5 
 
 71 
 
 813 
 
 519 
 
 525 
 
 631 
 
 537 
 
 543 1 649 553 
 
 561 
 
 667 
 
 0.6 
 
 1.2 
 
 1.8 
 
 2-4 
 
 3.0 
 
 3-6 
 
 4-3 
 
 4-9 
 
 S-5 
 
 72 
 
 673 
 
 579 
 
 585 
 
 591 
 
 597 
 
 603 609 
 
 616 
 
 621 
 
 627 
 
 06 
 
 1.2 
 
 1-8 
 
 2.4 
 
 3.0 
 
 3.6 
 
 4-2 
 
 4.8 
 
 S-4. 
 
 73 
 
 633 
 
 639 
 
 645 
 
 661 
 
 667 
 
 663 669 
 
 676 
 
 631 
 
 686 
 
 0.6 
 
 1.2 
 
 1.8 
 
 24 
 
 3-0 
 
 3-5 
 
 4-1 
 
 4-7 
 
 S3 
 
 74 
 
 692 
 
 698 
 
 704 
 
 710 
 
 716 
 
 722 727 
 
 733 
 
 739 
 
 746 
 
 0.6 
 
 1.2 
 
 1-7 
 
 2.3 
 
 2-9 
 
 JJ 
 
 4-1 
 
 4 7 
 
 _5f 
 
 75 
 
 8751 
 
 766 
 
 762 
 
 768 
 
 774 
 
 779 786 
 
 ™ 
 
 797 
 
 803 
 
 0.6 
 
 1.2 
 
 >-7 
 
 2-3 
 
 2-9 
 
 3-5 
 
 4.0 
 
 4.6 
 
 5.2 
 
 76 
 
 808 
 
 814 
 
 820 
 
 825 
 
 831 
 
 837 842 
 
 848 
 
 854 
 
 869 
 
 0.6 
 
 I.I 
 
 •-7 
 
 2-3 
 
 2.8 
 
 3-4 
 
 4.0 
 
 4S 
 
 S-I 
 
 77 
 
 866 
 
 871 
 
 876 
 
 882 
 
 887 
 
 893 899 
 
 904 
 
 910 
 
 916 
 
 06 
 
 I.I 
 
 »-7 
 
 2-2 
 
 2.8 
 
 3-4 
 
 3-9 
 
 4S 
 
 S-o 
 
 78 
 
 921 
 
 927 
 
 932 
 
 933 
 
 943 
 
 949 954 
 
 960 
 
 965 
 
 971 
 
 0.6 
 
 I.I 
 
 1-7 
 
 2-2 
 
 2.8 
 
 3-3 
 
 3-9 
 
 4-4 
 
 S-o 
 
 79 
 
 976 
 
 982 
 
 987 
 
 993 
 
 998 
 
 o04 
 
 ofl? 
 
 ol£ 
 
 0^ 
 
 o25 
 
 _o^S 
 
 I.I 
 
 1.6 
 
 2.2 
 
 _i7 
 
 33 
 
 Jf 
 
 4-4 
 
 4-9 
 
 To 
 
 9031 
 
 036 
 
 0T2 
 
 047 
 
 063 
 
 058 
 
 063 
 
 069 
 
 074 
 
 079 
 
 OS 
 
 1.1 
 
 Te 
 
 2-2 
 
 2-7 
 
 3.2 
 
 3-8 
 
 4-3 
 
 4-9 
 
 81 
 
 085 
 
 090 
 
 096 
 
 101 
 
 106 
 
 112 
 
 117 
 
 122 
 
 128 
 
 133 
 
 O-S 
 
 I.I 
 
 1.6 
 
 2.1 
 
 2.7 
 
 3-2 
 
 .3.7 
 
 4-3 
 
 4.8 
 
 82 
 
 138 
 
 143 
 
 149 
 
 154 
 
 159 
 
 165 
 
 170 
 
 175 
 
 180 
 
 186 
 
 0.5 
 
 1.1 
 
 1.5 
 
 2.1 
 
 2.6 
 
 32 
 
 3-7 
 
 4.2 
 
 4 7 
 
 83 
 
 191 
 
 196 
 
 201 
 
 206 
 
 212 
 
 217 
 
 322 
 
 227 
 
 233 
 
 238 
 
 OS 
 
 1.0 
 
 1.6 
 
 2.1 
 
 2.6 
 
 3-1 
 
 36 
 
 4-2 
 
 4-7 
 
 84 
 
 343 
 
 248 
 299 
 
 253 
 304 
 
 258 
 309 
 
 263 
 315 
 
 269 
 320 
 
 274 
 325 
 
 279 
 330 
 
 384 
 336 
 
 289 
 340 
 
 0.5 
 
 I.O 
 
 1.0 
 
 1-5 
 1-5 
 
 21 
 2.0 
 
 2.6 
 
 2-S 
 
 3 I 
 3-° 
 
 ±5 
 3-6 
 
 4.1 
 4.1 
 
 46 
 4.6 
 
 85 
 
 9294 
 
 86 
 
 345 
 
 360 
 
 356 
 
 360 
 
 365 
 
 370 
 
 376 
 
 380 
 
 386 
 
 390 
 
 0.5 
 
 1.0 
 
 "S 
 
 2-0 
 
 2-5 
 
 3.0 
 
 3-S 
 
 4-0 
 
 4-S 
 
 87 
 
 395 
 
 400 
 
 405 
 
 410 
 
 416 
 
 420 
 
 426 
 
 430 
 
 436 
 
 440 
 
 o-S 
 
 1.0 
 
 l-S 
 
 2-0 
 
 2-S 
 
 3-0 
 
 3-5 
 
 4.0 
 
 4S 
 
 88 
 
 445 
 
 450 
 
 455 
 
 460 
 
 465 
 
 469 
 
 474 
 
 479 
 
 484 
 
 489 
 
 o-S 
 
 1.0 
 
 l-S 
 
 2-0 
 
 2-S 
 
 2.9 
 
 3-4 
 
 39 
 
 4.4 
 
 89 
 
 494 
 
 499 
 
 504 
 
 509 
 
 513 
 
 518 
 
 523 
 
 628 
 
 533 
 
 533 
 
 o.S 
 
 1.0 
 
 i-S 
 
 1-9 
 
 2-4 
 
 29 
 
 3-4 
 
 39 
 
 4.4 
 
 90 
 
 9542 
 
 547 
 
 552 
 
 557 
 
 662 
 
 666 
 
 571 
 
 676 
 
 581 
 
 686 
 
 o-S 
 
 1.0 
 
 1.4 
 
 1-9 
 
 2.4 
 
 2-9 
 
 3-4 
 
 3-8 
 
 4-3 
 
 91 
 
 590 
 
 696 
 
 600 
 
 605 
 
 609 
 
 614 
 
 619 
 
 624 
 
 638 
 
 633 
 
 0.5 
 
 0.9 
 
 1-4 
 
 1.9 
 
 2.4 
 
 2.8 
 
 3-3 
 
 3-8 
 
 4-3 
 
 92 
 
 638 
 
 643 
 
 647 662 
 
 657 
 
 661 
 
 666 
 
 671 
 
 675 
 
 680 
 
 o-S 
 
 0.9 
 
 '■4 
 
 1.9 
 
 2.3 
 
 2.8 
 
 3-3 
 
 3-8 
 
 42 
 
 93 
 
 685 
 
 689 
 
 694 699 
 
 703 
 
 708 
 
 713 
 
 717 
 
 722 
 
 727 
 
 o-S 
 
 0.9 
 
 1-4 
 
 i.g 
 
 2-3 
 
 2.8 
 
 3-3 
 
 3-7 
 
 4.2 
 
 94 
 
 731 
 
 736 
 
 741,745 
 
 760 
 
 764 
 
 759 
 
 763 
 
 768 
 
 773 
 
 °;5 
 
 °-9 
 
 '■4 
 
 1.8 
 
 2.3 
 
 2.8 
 
 3-2 
 
 3-7 
 
 4-1 
 
 96 
 
 9777 
 
 782 
 
 786,791 
 
 795 
 
 800 '806 
 
 309'814 
 
 818 
 
 0.5 
 
 0.9 
 
 1.4 
 
 1-8 T^ 
 
 2-7 
 
 3-2 
 
 3.6 
 
 4.1 
 
 96 
 
 823 
 
 827 
 
 832 836 841 
 
 345J850 
 
 854 859 
 
 863 
 
 0.5 
 
 0.9 
 
 1.4 
 
 1.8 2.3 
 
 2-7 
 
 3-2 
 
 3.6 
 
 4-1 
 
 97 
 
 868 
 
 872 
 
 877'88r886 
 
 890 894 
 
 899 903 908 
 
 0.4 
 
 0.9 
 
 1-3 
 
 1.8 2.2 
 
 2-7 
 
 3-1 
 
 3.6 
 
 4.0 
 
 98 
 
 912 
 
 917 
 
 92l'926'930 
 
 934 939 
 
 943 948 952 
 
 0.4 
 
 0.9 
 
 1-3 
 
 1.8 2.2 
 
 2.6 
 
 3-1 
 
 3S 
 
 4.0 
 
 99 
 
 956 
 
 961 
 
 966 969 974 
 
 978 983:987 991 '996! 
 
 0.4 
 
 0.9 1.3I 
 
 1.7 2.2 
 
 2.6 
 
 31 
 
 3-5 3-9'| 
 
HARVARD EXAMINATION PAPERS. 375 
 
 EXAMINATION PAPEES FOR ADMISSION TO 
 HARVARD COLLEGE. 
 
 September, 1884. 
 
 (Time allowed, IJ hours.) 
 
 1. Solve the equations 
 
 2a!= + 3a!y-3y''+12 = 0, 
 3a; + 5y + l = 0, 
 and state what values of x and y belong together. 
 
 2. Solve the equation ^^J' _ ^Zlf ^ 5b(x-c) 
 
 X — a a — 0^ {a — c) {x — a) 
 reducing the results to their simplest form. 
 
 3. Find the sixth term of the 19th power of (^x^ — — ) , 
 reducing the result to its simplest form. ^^ 
 
 4. Find the greatest common divisor of 
 
 2 a;* - 3a;' + 2 a;^- 2a; -3 and 4 2:^ + 3x2 + 4 a: -3. 
 
 6. Solve the equation 
 
 VT - X + VS a; + 10 + a/« + 3 = 0. 
 
 6. A vessel is half full of a mixture of wine and water. If 
 filled up with wine, the ratio of the quantity of wine to that 
 of water is ten times what it would be if the vessel were filled 
 up with water. Find the ratio of the original quantity of 
 wine to that of water. 
 
 JoNE, 1885. 
 (Time allowed, 1^ hours.) 
 
 1. Three students, A, B, and C, agree to work out a series 
 of difficult problems, in preparation for an examination ; and 
 each student determines to solve a fixed number every day. 
 A solves 9 problems per day, and finishes the series 4 days 
 before B ; B solves 2 more problems per day than C, and 
 
376 ALGEBRA. 
 
 finishes the series 6 days before C. Find the number of 
 problems, and the number of days given to them by each 
 student. 
 
 2. Solve the following equation, reducing the answers to 
 their simplest form : 
 
 2 _ / a(l + 2x) _ 5(3a;-l) \ _ 
 l + 3a: V6(l + 3x) a{2x + l)J~ ' 
 
 V3 1 
 
 3. Solve the equation — -= 
 
 V2a;-1 — Vx — 2 Vx-l 
 
 4. A certain whole number, composed of three digits, has 
 the following properties : 10 times the middle digit exceeds 
 the square of half the sum of the digits by 21 ; if 99 be added 
 to the number, the order of the digits is inverted ; and if the 
 number be divided by 11, the quotient is a whole number of 
 two digits, which are the same as the first and last digits of 
 the original number. Find the number. 
 
 5. Given ^ + ^I' == 8 ; find the value of "^^"'"^^ . 
 
 1 x — 2y 2x — y 
 
 6. Find the greatest common divisor of 
 
 3 a;* — x'— 2a;* + 2 a; — 8 and 6 x' + 13 x^ + 3 a; + 20. 
 
 7. Find the square root of 
 
 4 - 12 a; + 5 a;'' + 26 a;= - 29 x* - 10 a;= + 25 x\ 
 
 September, 1885. 
 
 (Time allowed, IJ hours.) 
 
 1. A certain manuscript is divided between A and B to be 
 copied. At A's rate of work, he would copy the whole manu- 
 script in 18 hours ; B copies 9 pages per hour. A finishes his 
 portion in as many hours as he copies pages per hour ; B is 
 occupied 2 hours more than A upon his portion. Find the 
 number of pages in the manuscript, and the numbers of pages 
 in the two portions. 
 
HARVARD EXAMINATION PAPERS. 377 
 
 2. Solve the following equation, reducing the answers to 
 their simplest form : / « , . 
 
 5 _ f a, _ —] 
 
 H^ — «) _i_2 ___L__iLi. 
 
 h {x + a) a (x + a)^ 
 
 3. Solve the equations 
 
 3 Va; + 2Vy ^ ^ a:'^ + 1 _ j/'' - 64 
 
 4:^x-2^y ' 16 a;'^ ' 
 
 finding all the values of x and y, and showing which values 
 helong together. 
 
 4. Two casks, of which the capacities are in the ratio of 
 a to b, are filled with mixtures of water and alcohol. If the 
 ratio of water to alcohol is that of w to »i in the first cask, and 
 that of p to q in the second cask, what will be the ratio of 
 water to alcohol in a mixture composed of the whole contents 
 of the two casks ? Eeduce the answer to its simplest form. 
 
 What does the answer (in its simplest form) become, if 
 m = q = ? and what is the simplest statement of the 
 question in this case ? 
 
 6. rind the 10th term of (x - yf ; of {-^ - — j • 
 
 The numerical coefficients are not to be computed, but expressed 
 in terms of their prime factors ; the literal parts are to be reduced to 
 the simplest form. 
 
 Note. The above five questions constitute the paper; and all applicants 
 are expected to do them if possible. The following question is not required, 
 and is not necessary to make a perfect exercise; but it may be added, at 
 the discretion of the student, and will be counted to improve the quality 
 of an imperfect exercise. 
 
 6. Eeduce to its lowest terms 
 
 6x«— 9x* + Ua^+6x^ — 10x 
 4a:= + 10 a;^ + 10 »» + 4 a;'' + 60 a; ■ 
 
378 ALGEBRA. 
 
 June, 1886. 
 
 (Time allowed, 1^ hours.) 
 
 1. A boat's crew, rowing at half their usual speed, row 
 three miles down a certain river and back again, in the middle 
 of the stream, accomplishing the whole distance in 2 hours 
 and 40 minutes. When rowing at full speed, they go over 
 the same course in 1 hour and 4 minutes. Find (in miles per 
 hour) the rate of the crew when rowing at full speed, and the 
 rate of the current. 
 
 (Notice both solutions of this problem.) 
 
 2. Solve the equation 
 
 3 Va;' + 17 + Va;' + 1 + 2 VS «» + 41 = 0. 
 
 Substitute the answers, when found, in the equation, and show 
 in what manner the equation is satisfied. 
 
 3. 
 
 Solve the equations 
 
 
 
 
 
 " + :. + 2y = 2 
 
 {y + 1)> 
 
 X 
 
 + 32 
 
 4. 
 
 Solve the equation 
 
 
 
 
 
 fa + 2 i) ; 
 a -2b 
 
 X a 
 
 2 
 
 4J'^ 
 
 
 a — 
 
 2h 
 
 X 
 
 and reduce the answers to their simplest form. 
 
 6. Find the greatest common divisor and the least comrrion 
 multiple of 4 a;' — 4 a;^ — 5 a; + 3 and 10 a;^ — 19 a; + 6. 
 
 6. Find the 6th and the 25th terras of the 29th power of 
 (x — y) ; reducing the numerical coefficients to their prime 
 factors, and not performing the multiplications. 
 
 Find the 6th term of the 29th power of f— — ^ ; 
 
 \ b 2 al 
 
 reducing exponents to their simplest form, and combining 
 similar factors. 
 
HARVARD EXAMINATION PAPERS. 379 
 
 Septembee, 1886. 
 
 t (Time allowed, 1} hours.) 
 
 1. Solve the equation 
 
 i[2b(x + 1)Y _ /I _ 5ax-ib \ _ 
 4:bx^ + 5ax "U 4ia;2 + 5J~ ' 
 and reduce the answers to their simplest forms. 
 
 2. Solve the equation x~' — x' = 7 (a;* +1). 
 
 3. A and B have 4800 circulars to stamp for the mail ; and 
 mean to do them in two days, 2400 each day. The first day, 
 A, working alone, stamps 800 circulars, and then A and B 
 together stamp the remaining 1600 ; the whole job occupying 
 3 hours. The second day, A works 3 hours, and B 1 hour ; 
 hut they accomplish only -^^ of their task for that day. Find 
 the number of circulars which each stamps per minute, and 
 the length of time that B works on the first day. 
 
 4. Find the value of x from the proportion 
 
 Sac s,—rT, .4/9^ 3a\ /3c 
 
 and express the answer with the use of only one radical sign. 
 
 6. Given the three expressions 
 
 2x*+ x^- Sx^- x+ 6, 
 4a;* + 12a;'- a;'' - 27 a; - 18, 
 4 a;* + 4 a;» - 17 a;^ - 9 a; + 18 ; 
 find the greatest common divisor and the least common mul- 
 tiple of the first two of these expressions ; also those of the 
 whole group of three. 
 
 Jdne, 1887. 
 
 (Time allowed, 1 hour.) 
 
 1. Solve the following equation : 
 
 V* — 3+V3a: + 4+ Vx + 2 = 0. 
 Find two answers, and verify the positive answer by showing 
 that it satisfies the equation. 
 
380 ALGEBRA. 
 
 2. A broker sells certain railway shares for $3240. A few 
 days later, the price having fallen $ 9 per share, he buys, for 
 the same sum, 5 more shares than he had sold. Find the 
 price and the number of shares transferred on each day. 
 
 3. Solve the following equation, finding four values of x : 
 
 a;* + (2 a^ + 3 a J - 2 b^y = 5{a^ + h^) x\ 
 
 4. Eeduce the following expression to its simplest form as 
 a single fraction : < « i 
 
 1 + tc' ~ 1 + a; 
 1 + a;^ 1 + a; 
 
 September, 1887. 
 
 (Time allowed, 1 hoar. ) 
 
 1. Solve the following equation, finding four values of x : 
 
 / , N/ .^ a\x + a) h\x-b) 3aH^ 
 
 (^ + «)(«,_ 5) --^-p^ ___=____. 
 
 2. At 6 o'clock on a certain morning, A and B set out on 
 their bicycles from the same place, A going north and B 
 south, to ride until 1^ p. m. A moved constantly northwards 
 at the rate of 6 miles per hour. B also moved always at a 
 fixed rate ; but, after a while, he turned back to join A. Four 
 hours after he turned, B passed the point at which A was 
 when B turned; and, at 1^ p.m., when he stopped, he had 
 reduced, by one half, the distance that was between them at 
 the time of turning. 
 
 Find B's rate, the time at which he turned, the distance 
 between A and B at that time, and the time at which B would 
 have joined A if the ride had been continued at the same rates 
 of speed. Find the answers for both solutions. 
 
 3. Find the sixth term of each of the following powers : . 
 
 ,2 
 
 (-^)'^ i^.-M 
 
 a/6 
 
HARVARD EXAMINATION PAPERS. 881 
 
 4. Eeduce the following fraction to its lowest term : 
 
 6a!^-13a:' + 3a;''+2a: 
 6x* - 9 x' + 15 x^ - 27 X - 9' 
 
 June, 1888. 
 
 (Time allowed, 1 hour.) 
 
 1. Eeduce the following expression to its simplest form as 
 
 a single fraction : 
 
 1 — x^fx \ 
 
 _ ( 1 _ x^ + f — x + y \ 
 ■ \l-y 1-y' I 
 
 1-f 
 
 2. Solve the following equations, finding, and reducing to 
 their simplest forms, two sets of values of x and y : 
 
 (. + 3,):(2._,) = (i^-^J:f, 
 
 a'' = H^2/ + 3ay + 18a=). 
 What are the answers, when a = 2 and b — —S? 
 
 3. Two travellers, A and B, go from P to Q at uniform but 
 unequal rates of speed. A sets out first, travelling on foot at 
 the rate of 20 minutes for every mile. B follows, going 1 
 
 P Q 
 mile while A traverses the distance -^^ . B overtakes and 
 
 oO 
 
 passes A, 8 miles from P; and when B reaches Q, he is 9 
 miles ahead of A. Find the distance P Q, and B's rate of 
 speed in minutes to the mile. 
 
 (Obtain two solution,s.) 
 
 4. Two men, working separately, can do a piece of work in 
 X days and y days, respectively; find an expression for the 
 time in which both can do it, working together. 
 
 A is 20 years old, and B is — 2 years older ; what is the 
 age of B ? 
 
 What are the values of x which satisfy the equation a;^ = 3a; ? 
 
382 ALGEBRA. 
 
 5. Write out (x — yY^- 
 Find the square root of 
 
 4 «= - 12 K^ + 5 a;* + 26 a;' - 29 a;'' - 10 a; + 25. 
 
 September, 1888. 
 
 (Time allowed, 1 hour.) 
 
 1. Eeduce the following expression to its lowest terms as a 
 single fraction : 2 x 
 
 J] 
 
 1 2 a:^ + 11 a;^ - 43 a: - 24 
 X 14 a;' — 31 a;2 — 31 a; — 6 
 
 2. Solve the following equations, finding, and reducing to 
 their simplest forms, two sets of values of x and y : 
 
 a 26 
 
 y + 46 X —y 
 
 {b — a)x \(a + b)y a" — Pj 
 What are the values of x and y, if a = 3 and J = — 1 ? 
 
 3. Tristram is ten years younger than Launcelot ; and the 
 product of the ages they attained in 1870 is 96. Find the 
 ages they attain in 1888. 
 
 (Two solutions.) 
 
 4. A sum of $ 100 is put at compound interest at 4 per cent 
 per annum for x years ; find a formula for the amount. 
 
 5. Write out the first five terms and the last five terms 
 of (x — yY^. 
 
 Find and reduce to its simplest form the fifth term of 
 
YALE EXAMINATION PAPERS. 383 
 
 EXAMINATION PAPERS FOR ADMISSION TO 
 YALE COLLEGE. 
 
 June, 1885. 
 
 l.Giv»5£+_2-(3-^-^) = ?4i5_(^,3), 
 
 to find X. 
 
 b — y ca + cy S^ + ?/ b 
 
 a^ + f >"^^' ¥+i/" a' 
 
 3. Multiply a; - 1(1 — V^ 3) by a; — J (1 + V^^). 
 
 4. Divide x^y~^ — 2 + x~^i/'-' by x'^y~^ — x~^ y^. 
 
 5. Given 91 a;'' — 2 a; = 45, to find both values of x. 
 
 7 4 
 
 6. Given -| = 4, 
 
 1 2 
 1 =: 1, to find X and y. 
 
 's/x Ajy 
 
 7. Expand by the Binomial Theorem to five terms (1 + a)"". 
 
 8. In Arithmetical Progression, given d = the common 
 difference, a = the first term, and s = the sum of series ; 
 derive the formula for I := the last term. 
 
 - _ . V* — bx + \/c — mx Va — bx — V" — ^-^ 
 
 y. If — =^= = . , prove 
 
 ya — bx + Y ra x — d ya — bx — y « x — d 
 
 ft j^ jjj 
 
 by using the principles of proportion that = 1. 
 
 nx — a 
 
384 ALGEBRA. 
 
 Septembeh, 1885. 
 
 ^ 1 
 
 ¥ a 
 
 1. Reduce to a simple fraction* 
 
 all ^ 
 
 h h a 
 
 2. Find the greatest common divisor oi x^ — 6 x^ — 8 a; — 3 
 and 4 a;^ - 12 a; - 8. 
 
 3. Given Vis + x + \/ld — a; = 6, to find x. 
 
 4. Given x* — 21 a;^ = 100, to find four values for x. 
 
 5. Find the value of a^ + ah^ + b^ when a — 8 and b = 64. 
 
 6. Given ] „ , ~ ,„ ?■ , to find x and «. 
 
 Lx^ — y^ = b^ ) " 
 
 7. Given (x^ — ax) : ^x : : ^x : x, to find values of x. 
 
 8. Expand =- into a series. 
 
 (2 a - 3)2 
 
 9. Compute the value of the continued fraction 
 1 _ 
 
 r 
 
 12 + 
 
 1 + i 
 
 2 + i 
 3 
 
 June, 1886. 
 
 . _,. . , c-b c«-b' , e + b c^ + b^ 
 
 1. Dmde ^^^ - ^^^3 l>y J^ + ^^^.• 
 
 2. Divide a;>"t _ 2 + ai-^j/^ by x^y~i — a;"^yi 
 
YALE EXAMINATION PAPERS. 385 
 
 3. Multiply VHrH + c'^b hj V^^ — « ^'*- 
 
 4. In — , make the denominator rational, and com- 
 pute the value of the expression to three places of decimals. 
 
 5. Given a + x= Va^ + x '^W + x'; to find x. 
 
 6. Solve the equations \ „ .,""_,' 
 
 ^ ( x^ + 2/ = 74. 
 
 7. U A : B = C : I), prove hy the principles of proportion 
 that A^ — B^:B^=C^-I>^: D\ 
 
 8. Find the sum of the infinite series ^ + 5V + Tsr + ^t"- 
 
 1 
 
 9. Expand to four terms by the Binomial Theorem 
 
 Vi+«' 
 
 September, 1886. 
 
 1. Divide -, TT-^, — 2 ^y --^-^ 
 
 x^ — 2xy -\- y^ x — y 
 
 2. Multiply a^ — aH^ + a^h^ — ab^a^lfi—b^ by ai+ 6^ 
 
 1 — a;~*2/~^ + a; 
 exponents 
 
 3. Free the fraction :j „-8..-2 . „-2 from negative 
 
 ,^. , , 7a; + 9 / 2x-l\ ^ 
 
 4. Find a: from — j la; ^ I = 7. 
 
 r a = 2/ + s, 
 
 5. Find x, y, and s from -j 6 = a; + s, 
 
 ( c = X -\- y. 
 
 6. Multiply a; - 5 + 2 V^^ by a; — 5 - 2 \/^^. 
 
 25 
 
386 ALGEBRA. 
 
 7. Make the denominator of the following fraction rational : 
 
 Va — Va: + y 
 ^Jx + Va: + 2/ 
 
 1 2 4 
 
 8. Solve the equation -t + = ^ • 
 
 9. If a : 6 = e : cZ, prove by the principles of proportion 
 
 that 
 
 a + 5 + c + <^_» — h -\- c — d 
 
 a + b ~ — d a — b — c + d 
 
 10. In a geometrical progression, having given first term, 
 ratio, and sum of series, write formula for last term. 
 
 11. Expand to 4 terms (a + x) ,*. 
 
 June, 1887. 
 
 1. Resolve each of the following expressions into three 
 factors : 
 
 a*b + 8 ac^b m% 4 c° a;^ + ic^xy + cy^- 
 
 2. Divide = r by • 
 
 a — b a + b a — b a + b 
 
 4. Solve V« + 40 = 10 — ^/x. 
 
 6. Solve mx^ -{- mn ^ 2 m '\/n x -\- nx'^. 
 
 15 21 
 
 6. Given — : — : : 3 : 7, and x^ — y^ — 9, to find x and y. 
 
 X y ' J ' f 
 
 7. Expand by the Binomial Theorem 3 & (2 a; — yp. 
 
YALE EXAMINATION PAl'EES. 387 
 
 June, 1888. 
 
 1. Remove the parentheses from the following expression 
 and reduce it to its simplest form. 
 
 5 a: - (3 a; - 4) - [7 a; + (2 - 9 a;)]. 
 
 2. Resolve each of the following expressions into as many 
 factors as possible. 
 
 (a.) a!« - 1. 
 
 (6.) (x^ + y^- zy -ix^ tf. 
 
 „ ^. ., 1 1 , 1 1 
 
 3. Divide :i- — by + 
 
 1 — X 1 + X 1 — X 1 + X 
 
 4. Solve the equations 
 
 3 1_5 
 X y 4: 
 
 X y 
 
 5. Solve the equation ^/x — 3 — VSa; + 8 = — 3. 
 
 6. Solve the equation x^ — a;* = 256. 
 
 7. Multiply a; + 3 - 2 V^ by x + 3 + 2 V^^- 
 
 8. Expand (a;* + b)~^ to four terms. 
 
 CT or X 
 
 9. Given the series y = x — -^ '^ 1 ~ 'r '^' ^*'*''' *° ^^^ 
 
 the value of x in terms of y. 
 
ALGEBRA. 
 
 EXAMINATION PAPERS FOR ADMISSION TO 
 AMHERST COLLEGE. 
 
 September, 1884. 
 
 1. Find the greatest common divisor of x^ — 2x'^ — x + 2 
 and a:^ - 3 3!= + 3 x^ - 3 a: + 2. 
 
 2. The sum of the digits of a numher of two figures is 9 ; 
 and if 9 be subtracted from the number the digits are re- 
 versed. What is the number? 
 
 „ ax + b y = c } „, , 
 
 ^- a'xl/y = A'^^^^^^^'^ 
 
 +b'y 
 
 4. The sum of the squares of two consecutive numbers is 
 545. What are the numbers ? 
 
 „ -, , ( a; — ?/ = 2. 
 S- Solve I ^,_^,^ 20. 
 
 6. Eaise 2 ySx^y to the 4th power and reduce the result 
 to its simplest form. 
 
 7. What number added to 2, 20, 9, 34, will make the result 
 proportional ? 
 
 8. Given d, the difference, n, the number of terms, and s, 
 the sum of an arithmetical progression ; find the formula for 
 I, the last term. 
 
 9. Find the sum of the infinite series 1 — J+J — i + etc. 
 
 10. Find the 4th term of (x — 2 yY" by the Binomial 
 Theorem. 
 
AMHEUST EXAMINATION PAPERS. 389 
 
 June, 1885. 
 
 1. Divide a ^ „ ^, a fey — r^ • 
 
 2. Factor a;^ - 2 a;''/ + ij^. 
 
 3. Eeduce the product (3a;-'^y'z-*) (5a;'y-*a») to its 
 simplest form, freed from negative expoueuts. 
 
 4. Solve Va: + 13 = 1 + ^/x. 
 
 5. Solve a;'' + 4 a a; = 5. 
 
 6. Given ] „ / „.' find a; and y. 
 
 ix- — y" =21; " 
 
 __. 3a; a; — 4 a; — 10 „ „. 
 
 7. G-iven —r- -^ jr — = a: — 6, to find x. 
 
 4 2 2 
 
 8. What is the sum of the first twenty odd numbers ? 
 
 9. The first term of a geometrical progression is \, the 
 ratio ^, the number of terms 7. Find the sum. 
 
 10. Give the fourth term of (2 a: — 3 y)-^ 
 
 September, 1885. 
 
 . _, S — c ^ , a — J , 
 
 1. From a ;r — take — ^ — — x. 
 
 u O 
 
 2. Divide a" — 1 into its prime factors. 
 
 4 a;' — 5 a;" + SB 
 
 3. Reduce to its lowest terms 
 
 4. Solve 
 
 8a;2-6a; + 1 
 
 3a; — 4y= — 6, 
 10 a; + 2?/ = 26. 
 
 6. Solve 2a;'' — 5a; + 2 = 0. 
 
390 ALGEBRA. 
 
 6. Solve^= + ^^ = \4' 
 
 a; + 2/ — 8. 
 
 7. What two numbers whose difference is d are to each 
 other as a to b. 
 
 8. Insert 5 arithmetical means between 2 and — 3. 
 
 9. "What is the sum of the first 10 terms of the series 1, 2, 
 4, 8, etc. ? 
 
 10. Expand {x^ - 2 hf. 
 
 June, 1886. 
 
 1. From „ take j 
 
 3 4 
 
 2. Eeduce --—5 — ^ , „ „ to its lowest terms. 
 
 3a;^+ Gxy -\- 3y^ 
 
 „ „. „ £B — 4 . 5a:+14 1 , ^j j 
 
 3. Given 3a; 4 = ^ 12' ^' 
 
 4. Solve X + \/^ + 3 = 4 a; — 1. 
 
 6. Given 20,-^ = 8, 4^-^ = 24^-^^; 
 find a; and y. 
 
 6. Find V24 + a/M - V6. 
 
 7. A person sets out from a certain place, and goes at the 
 rate of 11 miles in 5 hours ; and, 8 hours after, another person 
 sets out from the same place, and goes after him at the rate of 
 13 miles in 3 hours. How far must the latter travel to over- 
 take the former ? 
 
 8. The 1st and 9th terms of an arithmetical progression 
 are 5 and 22. Find the sum of 21 terms. 
 
AMHERST KXAMINATION PAPERS. 391 
 
 9. Find the 12tli term of the geometrical progression 
 
 V2, -2, +2 V2, -4, etc. 
 
 10. Find the first four terms of (2 a: — 3 «/)* by the Bino- 
 mial Formula. 
 
 September, 1886. 
 
 1. Reduce {a + b — cy + (a — b + c)^ to its simplest form. 
 
 2. Reduce ^r—z ^r-. ^ to its lowest terms. 
 
 3 a:* — 24 a: — 9 
 
 „^. 3a; + 4 7a;-3 x - 16 . . 
 
 3. Given p — = -. ; find x. 
 
 5 2 4 ' 
 
 4. Find X and y from the equations 
 
 x-2 10 - a: _ y — 10 
 ~"5 3 ~^ ' 
 
 2y + 4 2x + y__ x + 13 
 3 8 ~ 4~" 
 
 5. Solve 2a;-V'2a;-l = a; + 2. 
 
 fa;'' + if = 60, 
 
 ^- ®°^"^ "i9. + 7y = 70. 
 
 7. Reduce v'45c' — V^O c" + '/^vFc to its simplest form. 
 
 8. Find the sum of the first 90 odd numbers by arithmet- 
 ical progression. 
 
 9. Find the sum of the geometrical progression 20, 19, 
 I82V, etc. 
 
 10. Find the first four terms of (1 - a;)" by the Binomial 
 Formula. 
 
392 ALGEBRA. 
 
 Jdne, 1887. 
 
 1. Reduce (a + b — c) 's/x + »/ — (a + 6 + c) (a; + y)2 to 
 its simplest form. 
 
 2. Eesolve a" — J° into its prime factors. 
 
 3. Divide ^^ by ^",-^- 
 
 , 3x + 2 a X — 5a _ „, 
 
 4. ^ = 5 a ; find x. 
 
 5. What number multiplied by m gives a product a less 
 than n times the number ? 
 
 7. Find the square root of a -\- 2 d^ x^ + x. 
 
 8. Pind the square root of 81 a*x~ ySz~^. 
 
 9. Find the roots of ax^ + bx -\- c = Q. 
 
 10. x^ + xy ^^10, xy — y^ = —Z; find x and y. 
 
 September, 1887. 
 
 1. Eeduce a — [2 6— (3c+25 — a)] to its simplest form. 
 
 2. Divide a" 6"—" by a"-"'«Z-". 
 
 3. Eeduce to its simplest form. 
 
 c + 1 
 
 4 Given ■]_ ,„ Hnrjto find a; and y. 
 (.2a! + 3y = 12)' ^ 
 
AMHERST EXAMINATION PAPERS. 393 
 
 5. Reduce a.'\/48rt'«? and ■\/|6 to simpler forms. 
 
 6. Multiply Sy/^by 2y/g. 
 
 7. Given ^^ + | = 12 " ^^. *» find x. 
 
 8. Find two numbers whose sum equals s, and whose dif- 
 ference equals d. 
 
 9. Solve the equation 3 a;' — 4 a; = 119. 
 10. Find the first four terms of (a; — 2 y)'. 
 
 June, 1888. 
 
 1. Eesolve 16 a* 6'' w^ — 8 a' 6^ m + 1 into its factors. 
 
 2. Find the greatest common divisor of 6 x' — 6 a;^ y + 2 a;^'' 
 — 2f and 12 a;" - 15 a; y + 3 y''. 
 
 _ a! + 3 x-2 Zx-h ,1 ^ - 
 
 3. -^ g- = -j2— + 4; find a:. 
 
 a; + y ^—y_c,. x + y x-y .. „ , , 
 
 4. — 2 3~ ° ' — 3 1 T~ — 11 J find ar and j/. 
 
 5. x — 2y + 3z=:2; 2x — By + z = l; 8a; — y + 2z = 9; 
 find X and y. 
 
 12 
 
 6. Solve the equation Vx +'5 = 
 
 Va; + 12 
 
 7. Solve the equations x + y — a; x^ ^ y^ — V. 
 
 8. Find the sum | \/§ and | y^. 
 
394 ALGEBRA. 
 
 9. Demonstrate the fundamental formulae used in Arith- 
 metical Progression. Find the sum of the first n terms of the 
 progression 1, 3, 5, 7, etc. 
 
 10. Find the sum of the first n terms of a geometrical pro- 
 gression whose first term is a, and third term c, 
 
 September, 1888. 
 
 ^ T^. 1, , .a; + 2a £b + 26, 4aJ 
 
 1. Find .the value of pr— H prv when x = — -—, ■ 
 
 X — Ja X — Jo a + o 
 
 2. Resolve 1 — c* into its prime factors. 
 
 3. Multiply together , -^ , and 1 + 
 
 l + y'a; + a;'" ^1-a; 
 
 . o , , . a; + 3 .-r - 3 
 
 4. bolve the equation ^ ■ — ^ = a. 
 
 X — O 'X -\- o 
 
 5. ^ + 1=18; |-f = 21; findceandy. 
 
 6. x^ + y^ ^ SA; a; y = 16 ; find x and y. 
 
 a; 
 
 7, Reduce -== to an equivalent fraction having 
 
 a — -y/a^ — x'^ 
 
 a rational denominator. 
 
 8 Find the ratio of an infinite decreasing geometrical pro- 
 gression of which the first term is 1, and the sum of the 
 terms is |. 
 
 9. Find the sum of the terms of an arithmetical progression 
 formed by inserting 9 arithmetical means between 9 and 109. 
 
 10. Expand (a - by by the Binomial Formula. 
 
DAKTMOUTH EXAMINATION PAPERS. 395 
 
 EXAMINATION PAPERS FOR ADMISSION TO 
 DARTMOUTH COLLEGE. 
 
 1885. 
 
 1. Factor a;^ ^ 16 a^, a;^ + a», x^ — lax — ¥■ \- «=, 
 x^ - X — 12. 
 
 2. Find the greatest common divisor and least common 
 multiple of a;' — 3 a; — 2 and x^ — 2 x* ~ x + 2. 
 
 3. Simplify l—j. and -^ - (1 - ^^y. 
 
 ■I ^ + t V — " — 1 
 
 X ~i 
 
 , _, , I'l — X 1 + a;\ a; + 3 
 
 4. Solve X — 
 
 0- 
 
 6. Solve j + 2i' = --S + *- 
 [2x — Sy = t/ — x + i. 
 
 6 Solve Va;H- 2 — Va: — 2 = ^2 a;. 
 
 7. Find the value of 9^ X 8"* X 7° X 4~^ -=- (8"' x 3)~'. 
 
 8. Multiply X — a;-i by a; - x-\ 2 + \/~3 by 2 VS, 
 and a;* — y^ by a;^ + x^i/^ + y^. 
 
 1886. 
 
 1. Factor a;^ — 9 a*, a;' + j/', a;" + 4 a: y — 4 + 4 y'^, 
 x^ + Sax + 2 a\ 
 
 x^ + 1 
 2. Reduce i a , -i 
 
 X + 1 
 
 ' [1 + as (a; — 1)] to its simplest form. 
 
396 
 
 ALGEBUA. 
 
 — 1 by x — x-^ + 1, and x^ + y^ 
 
 3. Multiply aj + a;-' 
 by x* — xy^ -\- x^ y — y^. 
 
 4. Divide x — y hy x^ + y^, and a;* + a;-" by x+ a;"'- 
 
 5. Write the value of 8 ^ X 9"* X 2"' xS^xl^'x v81. 
 'x — 4 . X — 6 
 
 6. Solve X 
 
 -S-Q 
 
 7. Solve < 
 
 ■ = a; — 4 
 
 and ■> 
 
 y-^-y = y-^k 
 
 X 
 
 
 
 
 ¥ 
 
 ' 
 
 
 
 r1 
 
 
 1 
 
 
 
 + 
 
 
 b. 
 
 X 
 
 
 y 
 
 
 9. 
 
 
 8 
 
 
 
 + 
 
 
 13. 
 
 La; 
 
 
 y 
 
 
 1887. 
 
 1. Give all the theorems used in factoring binomials. 
 
 2. Find the prime factors of 1 + a', a« — J», a* + 4 4" 
 + 4 aH"" - cS a;^ - a; - 20, a;* + EB^ - 8 a; - 12. 
 
 3. Find the G. C. D. of a;* + 4a;' + 12a;2 + 16 a; + 16 and 
 4 x» + 12 a;^ + 24 a; + 16. 
 
 4. Solve - + - = 7, ?-- = 2. 
 
 y X y ^ 
 
 5. Write the values of 27 ~9, 27*, 27°, L(27~V*J • 
 
 6. Reduce to simplest form 
 
 (11 + 4V6)i, (- 1 - V^^)". 
 
 7. Eeduce to equivalent fractions having rational denomi- 
 nators . 1 
 
 3aib^, 
 
 a^ + b^ 
 
 8. Solve yST^ -^ V^ ^ ^ = Vs. 
 
DARTMOUTH EXAMINATION PAPEKS. 397 
 
 1888. 
 1. Eemove the parentheses from 
 
 3a — {3a — [3a — (3a -3a — 3a) — 3a] — 3a} -3a, 
 and simplify the result. 
 
 2. Give the three theorems used in factoring binomials. 
 
 3. Factor ia^x* — 9 b* c", Sb^c^ + AS h^ c^ + 72 V^ c, 
 X* - Zx'' -Ux" + 4S,x -32. 
 
 4. Eesolve 1 — a' into six factors. 
 
 5. Give two methods of finding the G. C. D. of two 
 
 quantities. 
 
 a' + 5" 
 
 6. Reduce — it. 3 X -= i. to a simple fraction and 
 
 11 «' + 6° 
 
 a h 
 lowest terms. 
 
 7. Solve ] _.j •^_, „' 
 
 (.ox ^ — 3 2/ ' = 18?/. 
 
 8. A and B can do J of a piece of work in 2 days. B can 
 do J of it in 6 days. How long would it take A to do J of it ? 
 
 9. ■ Reduce to simplest form 
 
 \/^-X^» y/6 + V=^13x^6-V=13, (36-12A/6)i. 
 
 10. Reduce to equivalent fractions having rational denomi- 
 nators 
 
 ac 1 
 
 c*(ai + fi4) ai + hi 
 
 11. Solve ^ + ^ = 12. 
 
 X + V a:^ — 1 X — V a;^ — 1 
 
398 ALGEBRA. 
 
 EXAMINATION PAPERS FOR ADMISSION' TO 
 BROWN UNIVERSITY. • 
 
 June, 1885. 
 
 1. Find highest common divisor of 16 a? x^ — 20 a^ x^ 
 ■65 a'' a; — 30 a'' and 12 ba? + 2Qbx^ - l&bx - l&b. 
 
 k z k{z — k) a {z -{■ Jc) _ k» 
 z^ k^ z {?,-]- k)~ k(z — k) ~ k^ — : 
 
 3. A numher is compounded of three figures whose sum is 
 17. The figure of the hundreds is douhle that of the units. 
 When 396 is subtracted the order of the figures is reversed. 
 What is the number? 
 
 4. Multiply 2 V^^ — 3 •v/^2 and 4 V^^ + 6 V^^- 
 
 5. Reduce to an equivalent fraction with a rational denomi- 
 nator 
 
 Va: — 4 '\Jx — 2 
 
 
 2 Va; + 3 Va; - 2 
 
 
 6. V2a;-3-'\/8a; + l + Vl8x-92 = 0. 
 of X. 
 
 7. ix'-^xy - if = 3; a:^ + Sxy + / = 
 values of X and y. 
 
 Rnd value 
 = 11. Find 
 
 8. In an arithmetical progression, given the last term, — 47; 
 the common difference, — 1 ; and the sum of the terms, — 1118; 
 find the first term and the number of terms. 
 
 9. In a geometrical progression, given the first term, i|; the 
 ratio, — ^ ; and the number of terms, 7 ; find the sum of the 
 terms. 
 
 10. Develop by Binomial Formula {a^b — \x a'^y. 
 
BROWN EXAMINATION PAPERS. 399 
 
 September, 1885. 
 
 1. Find the least common multiple of 2x'' — 3x'' — x + 1 
 and 6 a:8 - a;" + 3 a; - 2. 
 
 ^a; a — bcx x ac — ibx -r,. , , j. 
 
 2. ^ = ^ S-, ■ I^ind value of x. 
 
 2 2oc be obo 
 
 3. A number is compounded of three figures whose sum is 
 17. The figure of the units is two thirds that of the hundreds. 
 When 297 is subtracted the order of the figures is reversed. 
 What is the number ? 
 
 4. Multiply 3 V^^ - 2 V^^ and 4 V^^ - 2 V^l. 
 
 5. Reduce to an equivalent fraction with a rational denom- 
 inator 
 
 v^i^^n: - Va " + 1 
 
 91 
 
 6. V3 a: + VS a; + 13 = — z^=:^= ■ Find value of x. 
 
 V 3 a; + 13 
 
 7. (2 a; - 5)^^ - (2 a: - I)'' = 8 a: - 5 a;'' ~ 5. 
 
 8. In an arithmetical progression, given the first term, — f ; 
 the number of terms, I84 and the last term, 5 ; find the com- 
 mon difference and sum of terms. 
 
 9. In a geometrical progression, given last term, —12; sum 
 of terms, — 255 ; and ratio, 2 ; find first term and number of 
 terms. 
 
 10. Develop by Binomial Formula (^ a 6' — f a^ b-'^f. 
 
 June, 1886. 
 1. Multiply 5a;''-=2/'-+« - 2x"-i2/'-+i - a;''-^y'-+2 by 
 
400 ALGEBRA. 
 
 2. Simplify 
 
 blabc + a + c'] 
 
 1 ■ 1 
 
 a + -^ « + J 
 + - 
 
 
 
 3. Given "'a; + 2ny=^; -^j^^ ^j^^ ^^^j^^^g ^f ^ ^^g^ ^^ 
 
 2sx+ty = q. 
 
 4. The smaller of two numbers divided by the larger is 
 .21 with a remainder of .04162. The greater divided by 
 the smaller is 4 with .742 for a remainder. What are these 
 numbers ? 
 
 5. Given j^ ~ ' — '2 "^ '^ — 3 '^ i^^-^)> 
 
 to find value of x. 
 
 + 
 
 a; + V2 — a;"^ x — ^/2 — x^ 
 
 7. Expand (2x^ + Zx^yy. 
 
 8. Find sum of terms in a geometrical progression. 
 
 1. X 
 
 September, 1886. 
 2,y — 3 3a;-6) 2r — 3?/ — 1, 
 
 | 2y-3 3a:- 5 ) _ 
 (4 "^ 6 ) ~ 
 
 12 
 
 3-2* 
 
 6 
 Find values of x and y. 
 
 2. V* + X + -v/a — a; = VB. Find values of ar. 
 
 3. A number is compounded of three figures whose sum is 
 17. The figure of the hundreds is double that of the units. 
 When 396 is subtracted the order of the figures is reversed. 
 What is the number ? 
 
BROWN EXAMINATION PAPERS. 401 
 
 4. 3 x'^ — 4 a; — 4 = 0. Find values of x. 
 
 5. rind sum of six terms of the geometrical progression of 
 which f is the first term and f the second term. 
 
 June, 1887. 
 
 . „ Zx + I .-, 
 1- 5 y 2 — = l^, 
 
 Ax — 3 2x + Sy „„ -r^. ■, ■, , , 
 
 ^ , — - = — oft. imd values of x and ?/. 
 
 3 4^ -^ 
 
 2. 2a: — 3y = 8. 
 y-3s=-ll. 
 
 a: — 22/ + 4« = 17. Find values of x, y, and «. 
 
 3. A boy spent his money in oranges. If he had bought 5 
 more, each orange would have cost a half-cent less ; if 3 less, 
 a half-cent more. How much did he spend, and how many 
 did he buy? 
 
 4. Multiply -v/p + ? + Vp — 2 ^y Vp + 2 ~ Vp^-2- 
 
 5. Multiply V— b + a hy V— f> — V— «• 
 
 6. 7 a; — 3 a:^ -I- 14 = 0. Complete the square and find the 
 value of X. 
 
 1. -y/a: + 3 + a/S a; — 3 = 10. Find the value of x. 
 
 8. In an arithmetical progression there are given the first 
 term, 4; the number of terms, 10; and the sum of the terms, 
 175. Find common difference and the last term. 
 
 9. Expand by the Binomial Formula (2 a^ — 3 b^K 
 
 20 
 
402 ALGEBRA. 
 
 September, 1887. 
 
 . ^. - 4?/ -6 . 2x-4: 
 1. Given 3x --: = 4 
 
 3 " 5 ' 
 
 to find values of x and y. 
 
 2. Add -v'lea;*/, -y^faVj and 6a! j/ v8«». 
 
 3. A man bought a certain number of eggs for 2 dollars. 
 If he had paid 5 cents more per dozen, he would have received 
 two dozens less for the same money. How many dozens did 
 he buy, and what did he pay per dozen ? 
 
 4. 2 ^2 a; — 3 — V^x — 7 = V'4a; — 11. Find values of x. 
 
 5. 3x^ — 4 X = 55. Find values of x. 
 
 6. In an arithmetical progression, given the first term, 3 ; 
 the number of terms, 15 ; the sum of the terms, — 165 ; to find 
 the common difference and last term. 
 
 7. Expand (2 x — 3 y^y by the Binomial Formula. 
 
 
 
 
 June, 1888. 
 
 
 
 
 (Omit one from each set.) 
 
 
 
 
 
 I. 
 
 . Resolve 
 
 64 x' 
 
 — 
 
 xy" 
 
 into five factors, 
 
 '. Simplify 
 
 x' 
 
 y 
 
 a; + 
 
 + 
 
 X 
 
 t 
 
 x 
 
 -y 
 
 H^-'} 
 
 a ' b a' b' 
 
 6. (jiven — h = c, 1 — = c', to find values of x and v. 
 
 X y X y " 
 
BROWN EXAMINATION PAPERS. 403 
 
 II. 
 
 1. Add 3 X Va' — «'' a;, — 4 « -y/^ « as^ — 4 k", and 
 5 V''" ^^ — *'^ »:'. 
 
 2. Multiply 2 'V'" — V^^ l^y 3 a/^~« + 2 v'i. 
 
 3. Given : — = - , to find values of x. 
 
 X — 1 2x 3 
 
 III. 
 
 1. Given 2 a;* — 3 a; y + y'^ =13 35, 2 a; - 3 ?/ = 13, to find 
 values of x and y. 
 
 2. Expand (2 a — 3 i)^ by Binomial Formula. 
 
 3. In an arithmetical progression, given the first term, 14 ; 
 the number of terms, 7 ; the sum of the terms, 59^ ; find the 
 common difference and last term. 
 
 September, 1888. 
 
 1. Find value of 
 
 a: (2/ + ») + 2/ [« — (y + 2^)3 — « [y — a; (z — a;)] 
 when a: = 3, 2/ = 2, « = 1. 
 
 2. A and B set out at the same time from the same spot to 
 walk to a place 6 miles distant and back again. After walk- 
 ing for 2 hours, A meets B coming back. Supposing B to 
 walk twice as fast as A, and each to maintain uniform speed 
 throughout, find their respective rates of walking. 
 
 3. Solve the equation ^/x + V^ + x = —-= • 
 
 Va; 
 
 4. Solve the equation — — = — p-- -^ = — 5 ■ 
 
 ^ a; + 2 2 (a; — 2) 6 
 
 5. Find the sum of 10 terms of the geometrical progression 
 in which the fourth term is 1 and the ninth term is ^Jj. 
 
404 ALGEBRA. 
 
 June, 1889. 
 
 (Omit one from each set.) 
 I. 
 
 1. Find the lowest common multiple of 6 a;' + 11 x^ — 4c6x 
 
 + 24, and 12 k' + 37 a;^ - 42 a; + 8. 
 
 2. Simplify 
 
 x — y y — z z — x 
 
 {X + z) (y + s) (a; + ?/) (a; + z) {x -\- y) {y -\- z) 
 
 3. Solve 
 
 3 a! y X 2 y 
 
 T~ 3 2 "'"X 7 
 
 1 13 =~6" 
 
 2 4 
 
 2 y - 3 a; = 23. 
 
 II. 
 
 1. A and B run a race of 480 feet. The first heat, A gives 
 B a start of 48 feet, and beats him by 6 seconds ; the second 
 heat, A gives B a start of 144 feet, and is beaten by 2 seconds. 
 How many feet can each run in a second ? 
 
 2. Solve \/3x + 10- VSa; + 26 = -3. 
 2a;= + 3a;-5 2a:'^-a:-l 
 
 3. Solve 
 
 3a;'' + 4a!-l 3a;^-2a; + 7 
 
 III. 
 
 1. In an arithmetical progression, given the first term, — 3 ; 
 the common difference, 2^ ; and the sum of the terms, 143 ; 
 to find the last term and the number of terms. 
 
 2. In a geometrical progression, prove the formula for the 
 sum of n terms. 
 
 3. Solve 2a;''' - 3/ = 60, and 3a;^ — 4 a:y + / = 64. 
 
INST. OF TECHNOLOGY EXAMINATION PAPEES. 405 
 
 EXAMINATION PAPERS FOR ADMISSION TO MASSA- 
 CHUSETTS INSTITUTE or TECHNOLOGY. 
 
 September, 1884. 
 
 2 y + 4 a:" 
 
 f 
 
 1. Simplify ^ a , after substituting 1 — a;' for y. 
 
 2. Eesolve a}^ — b^^ into six factors. 
 Solve the following equations : 
 
 3.^+1 = 1, - y=i. 
 
 a b a 
 
 , x' + 2x x^ — x + 2,x-2 „ 
 4. 5 , + — ^— = 0. 
 
 5. VTx + Z — -v/ic + 1 = a/S a; — 8. 
 
 6. Prove that the square of half the sum of any two unequal 
 numbers is less than half the sum of their squares. 
 
 / 2b^\* 
 
 7. Expand by the Binomial Theorem I a ) • 
 
 8. Insert two arithmetical means between 24 and 81. Also 
 insert two geometrical means between the same numbers. 
 
 June, 1885. 
 
 1. Divide a' — a' by a^ — a*. 
 
 2. Factor a" - a; - 30, {x - yf — f, a;*"+' - x. 
 
 ^. -, ■, t ,^^ — V^ , a+b , a — b 
 
 3. Find the value of - , '^„, when x = and y = — — j- 
 
 x^ + y^ a — b a + 6 
 
406 ALGEBRA. 
 
 Solve the following equations : 
 
 5. . „ — , = 3 — X. 
 
 1 + 2x-' 
 
 G. X- Vx^ - 2 x' = 2. 
 
 7. (z + 2)(y-3) = 10, X2/ = 15. 
 
 8. Find the cube of 1 + \/^3. 
 
 9. The sum of three terms in arithmetical progression 
 beginning with | is equal to the sum of three terms in geo- 
 metrical progression beginning with J, and the common 
 difference is equal to the ratio. What are the two series ? 
 
 September, 1885. 
 
 1. Factor a;^ — 6 a; — 16 and 1 — 9 a + 8 o^. 
 
 2. Find highest common factor of x' -{- x — 6 and 
 2 a;^ - 11 a; + 14. 
 
 3. Simphfy + -—^ . 
 
 X — y X + y x^ — y' 
 
 Solve the following equations : 
 
 A X 1 — 2ax 2x — 1 
 
 4. K + — s + 2— = 0. 
 
 7 3 22 
 
 a;2 - 4 a; + 2 5 
 6. Vx - 32 + a/^ = 16. 
 
INST. OF TECHNOLOGY KXAMINATION I'APEKS. 407 
 
 7. Find the sum of 16 terms of the arithmetical progres- 
 sion f, f, 5, . . . 
 
 8. Find the sum to infinity of the geometrical series 
 -'■' 2' ~j ' ■ ■ ■ 
 
 9. Expand (3 a: — 2 yY by the Binomial Theorem. 
 
 JoNE, 1886. 
 
 1. Find the value of — ; , when x 
 
 a—b 
 
 oAjjj- ii, sp — a X -{■ a 2x{3a — x) 
 
 2. Add together - — ; — r^ , rj , -. ^7 — , — r-, 
 
 ° (x + ay [x — ay {x — a) (x + ay 
 
 3. Solve „ ^ , + ^ 
 
 2a; — 1 ix — 3 x — 1 
 
 4. Solve x + \/x^ — a^ = I 
 
 5. Show that ^ is the reciprocal of 
 
 / Va:" + 2+\/a;" — 2 y 
 
 6. Show that (-1 + -Z^)' + (- 1 - V^^)' = 16. 
 
 „ „ , 6 , a; 5 (a; — 1) 
 
 7. Solve - + „ = -^^—. ■ 
 
 X K) 4 
 
 8. Solve x^ ^ xy — \h, xy — y^ — 2, 
 
 9. Find the 4th term of (a — 25)". 
 
 30. How many terms of 16 + 24 + 32 + 40 + . . . amount 
 to 1840 ? 
 
408 ALGEBRA. 
 
 September, 1886. 
 
 1. Simplify ^^-2^ + ^._^_2 " ^T+^ " 
 
 2. Resolve a^^ — &^^ into its prime factors. 
 
 „ „, x — 5 5a: — 7 3a; — 7 5 — a; 
 
 3. Solve -g-+-9 ^ = _^. 
 
 4. Find the continued product of 
 
 Va^, \/^r^^b, and \/(a^ - by. 
 
 5. Extract the square root of 41 + 12 ■\/5. 
 Solve the following equations : 
 
 6. ^ + l + Uo. 
 X a 
 
 7. Vx+ i=\/x+ J. 
 
 g J V2 a; — y = 's/x — y + 1, 
 a;2 + 4y = 17. 
 
 9. There are two numbers whose geometrical mean is J of 
 their arithmetical mean ; and if the two numbers be taken for 
 the first two terms of an arithmetical progression, the sum of 
 its first three terms is 36. What are the numbers ? 
 
 June, 1887. 
 
 a;* + 3 a; - 2 
 
 1. Reduce to its lowest terms 
 
 2. Solve 
 
 a;^+ 3a;" + 4 
 a h 
 
 b — X a — X 
 
 [/--M-nXm + n /_n^n + p 
 
INST. OF TECHNOLOGY EXAMINATION PAPERS. 409 
 
 4. Solve V2 a; + 1 — V* + 3 = '^x. 
 6. Solve a:^ + / = 20, x^ — ai^^^S. 
 
 6. Solve 2a;» + 8a:-« =17. 
 
 7. Eeduce ^t — 'l~_- to the form 4+5 V^. 
 
 1 + V- 1 
 
 8. The 1st term of an arithmetical progression is 2, and 
 the difference! hetween the 3d and 7th terms is 6. Find the 
 sum of the first 12 terms. 
 
 September, 1887. 
 
 1. Divide a;^ + a;~^ by x^ + x~^. 
 
 2. Eesolve into two factors a? -\- V^ — (? — d^ ■\- 2 {ah ■\- c d). 
 
 3. Solve X — a + 's/i? — 2 ax = b. 
 
 4. Solve z=^^ = 33 + ya;'' — 8. 
 
 a; - Va;' - 8 
 
 5. Solve (a; - y) (a; - 3 y) == 24, a; - 2 y = 5. 
 
 6. Form the quadratic equation whose roots are a + h — c 
 and a — b + c. 
 
 7. Insert three geometrical means between 3| and 18 
 
 8. Give the first, third, and fifth terras ( 
 
 the Binomial Theorem of (x \/i/ + -~] 
 
 \ 2 Vx I 
 
 8. Give the first, third, and fifth terras of the expansion by 
 
 ,2 , 10 
 
410 ALGEBRA. 
 
 June, 1888. 
 PRELIMINARY. 
 
 1. Factor 8cx — 12ci/ + 2ax — 3ai/, 
 
 and 2 am — b^ + m^ + 2 bn + a' — n^. 
 
 2. Find the G. C. D. & L. CM. of 2 a;^- 11a;' + 3 a;^ 10a; 
 and 3 a;* — 14 a;' — 6 a;2 + 5 a;. 
 
 a; + 2 w X 
 
 \ +- 1 
 
 X ■}- y y i- 
 
 y ^ + 2/ 1 + 1 
 
 a; 
 
 4. Solve the equations 
 
 5-2a; 3-2a: 
 
 (a.) 
 
 a; + 1 a; + 4 
 
 5. At what time between 4 and 5 o'clock is the minute 
 hand of a watch exactly 5 minutes in advance of the hour 
 hand? 
 
 6. Solve the simultaneous equations 
 
 5a;-3y + 2s = 41. 
 2 a; + y — z — VI. 
 5x + iy — 2z = 36. 
 
 7. Extract the square root of 
 
 x'^ + Af+9s^ — ixy+6xs — 12yz. 
 
 8. Reduce to an equivalent fraction having a rational de- 
 nominator 
 
 V^ — iV^ — 2 
 2^/^ + 3 Vx — 2 ' 
 
INST. OP TECHNOLOGY EXAMINATION PAPERS. 411 
 
 FINAL. 
 
 1. Solve the equation 2a;^ + 3a; — 6 V'2a;2 + 3a; + 9 = —3. 
 
 2. Solve the simultaneous equations 
 
 (x^ + xy + iy^= 6. 
 
 3. Factor x* — 7x^f + if. 
 
 4. A person saves $270 the first year, $210 the second, and 
 so on. In how many years will a person who saves every year 
 $ 180 have saved as much as he ? 
 
 6. Expand (m~* + 2m°)'. 
 
 Find 5th term of (cc"^ — 2z/^)". 
 
 6. Form the equation whose roots are — § and f . 
 
 7. Derive the formula for the sum of a series in geometrical 
 progression. 
 
 8. Find three numbers in geometrical progression such that 
 their sum shall be 14 and the sum of their squares 84. 
 
 COMPLETE. 
 
 . _. ... a°-6 ° a-b If a + b 1 \ 
 
 1. Simplify ^,-^, - ^^-^, - 2 1^^,^, - ^^) ■ 
 
 2. Solve — ^ + -^—^ = 2, x+y = 2a. 
 
 a + a — b 
 
 3. Extract the square root of 
 
 4 a^ - 12 a= 6 + 29 aH'' - 30 a i' + 25 J* 
 
 . o- n-. fS + 2V3y /2-V3\' 
 
412 ALGEBRA. 
 
 5. Solve ^^ — = -r- . 
 
 1 D 
 
 X 
 
 X 
 
 6. Solve ya + x + \^a — x = 2 ■y'x. 
 
 7. Solve a; ^^ = 4, y - ^qjg "" 
 
 8. Find the sum of 18 terms of the series §, —1, — 2|, . . . 
 
 September, 1888. 
 
 4 x^ + 3 a; - 10 
 
 1. Reduce to its lowest terms 
 
 4 a;8 + 7 a;^ - 3 a; - 16 
 
 2 
 
 ..s,™p,i„(v/!i^-v/^y-(v/?-v/i 
 
 3. A fraction becomes | by the addition of 3 to the numer- 
 ator and 1 to the denominator. If 1 be subtracted from the 
 numerator and 3 from the denominator it becomes \. Find 
 the fraction. 
 
 4. Solve {x - a)2 = (a; - 2 a) (x^ + 4 a=)i 
 
 5. Form the quadratic equation w.hose roots are 
 
 (a + b f , , 
 
 1— and h — a. 
 
 a — b 
 
 '6. Expand by the Binomial Theorem \.x-\ — ) ■ 
 
 7. Divide 111 into three parts such that the products of 
 each pair may be in the ratios 4:6:6. 
 
 8. Find the sum to infinity of the geometrical progression 
 
INST. OF TECHNOLOGY EXAMINATION PAPERS. 413 
 
 Mat, 1889. 
 PEELIMINAUY. 
 
 1. Find the greatest common divisor of 2 a;* — 12 a;' + 19 x''' 
 -6a; + 9 and 4a;'- 18a;2+ 19a; -3. 
 
 2. Simplify ^^-p^3 X ^^^j X ^-,-^— ^-^, • 
 
 3. One tap will empty a vessel in 80 minutes, a second in 
 200 minutes, and a third in 5 hours. How long will it take 
 to empty the vessel if all the taps are opened ? 
 
 4. Solve -^-. + -^ ^2 a, "^ = 1. 
 
 x^ 4 
 
 5 Extract the square root of a;^ — a;' + - + 4 a; — 2 H — ^ • 
 
 6. Factor 2 rt m — 6^ + m^ + 2 J « + a^ — ^^^ and 2 c' m 
 + 8 c?'m, — 42 c m. 
 
 7. Which is the greater, •y/lO or ^46, and why ? 
 
 8. Extract the square root of 75 — 12 ^21- 
 
 FINAL. 
 
 1. Write out the first four terms, the last four terms, and 
 the middle term of (a; — 2 y)". 
 
 2. Find the sum of the first n terms of the series 1, 2, 3, . . . 
 
 3. Find three geometrical means between 2 and 162. 
 
 4. Show that in the equation a;^ + ^ a; + g- = 0, the sum 
 of the roots is —p, and the product of the roots q. 
 
 5. Find the four roots of the equation x^ — 3a;'^a'' + a* = 0. 
 
414 ALGEBRA. 
 
 6. A number consists of two figures whose product is 21 ; 
 and if 22 is subtracted from the number and the sum of the 
 squares of its figures added to the remainder, the order of the 
 figures will be inverted. What is the number ? 
 
 7. Solve 3 a;^ + 15 a; - 2 Va:' + 5 X + 1 = 2. 
 
 8. Form the equation whose roots are (a — |), (b + f). 
 
 COMPLETE. 
 1. Simplify 
 
 / xy-y^ \f xy^-i/\ ^ / xy-y^ \ 
 \ ^ + y A ~^ + f ) ' V ^' )' 
 
 a;*-2x« + 2a; — 1 
 
 2. Keduce to its lowest terms 
 
 15 a;2 + 24 a; - 10 
 
 3. Simplify (^ + ^)l + (^~^K(- + ^)l-(a-^)l 
 
 (a + b)^ - (a - b)i (a + b)i + (a - b)^ 
 
 4. A certain number when divided by a second gives a 
 quotient 3 and a remainder 2 ; if 9 times the second number 
 be divided by the first, the quotient is 2 and the remainder 11. 
 Find the two numbers. 
 
 5. Solve x^ — ai = (a; — b)i. 
 
 6. Solve a;* + 4 « 5 a;^ = (a'' - b^f. 
 
 7. If A is the sum of the odd terms, and B of the even terms, 
 in the expansion of (x + a)*, show that A^ — B^ ^ (x^ — a^Y- 
 
 8. If X — y is a mean proportional between y and ?/ + 2 — 2a;. 
 show that X is a mean proportional between y and z. 
 
 9. The second term of a geometrical progression is 64, and 
 the fifth term 16. Find the series.