LIBRARY OF THE University of California. GIFT OF I e^^^f^J^^^U, A^;a^^ APPLETOJfS' MATHEMATICAL SERIES MJMBERS SYMBOLIZED AN ELEMENTARY ALGEBRA BY DAVID M. SENSEMG, M.S. PROFESSOR OF MATHKMATICS, STATE NORMAL SCHOOL, WEST CHESTER, PA. NEW YORK, BOSTON, AND CHICAGO D. APPLETON AND COMPANY 1888 St Copyright, 1888, By D. APPLETON AND COMPANY. PREFACE. The aim of this volume is to lay the foundation for a more extensive and philosophical treatise soon to follow, and to aid in supplying the needs of the common, high, normal, and other preparatory schools and academies, where the time allotted to this department of knowledge is necessarily limited to an elementary treatise. In scope it includes all subjects essential to a study of higher arithmetic, elementary geometry, and the ele- ments of physics. All matter, however, is treated in an elementary manner, so that any ordinarily intelligent stu- dent, with a fair knowledge of the principles of common- school arithmetic, may master it. All broad generalizations and discussion of general problems have been purposely excluded. In the earlier lessons, fundamental ideas and principles are developed inductively, and then formulated into as simple and concise statements as is consistent with truth. Further on, definitions appear at the beginning of subjects, and principles are deduced from the solutions of character- istic examples. And still later, noticeably in proportion, propositions are first enunciated and then logically proved. Thus, the pupil is led by easy transition from the more elementary forms of reasoning to pure mathematical dem- onstration. t 83680 iv PREFACE, In numerous instances, after deducing one or more prin- ciples, I have introduced selections of easy examples to be worked at sight. These are intended to give opportunity for the application of the principles under which they appear, and to cultivate in the student a quick perception of letter, exponent, sign, and factor. An unusually large number of examples for written work are distributed throughout the book. These have been selected with special reference to the class of pupils for whom the work is intended. They are arranged for two readings. At the first reading it is recommended that all miscellaneous examples, which are generally more diffi- cult than the others, shall be omitted. These, in connec- tion with a review of the definitions and principles, will form a good second reading. Long, pointless examples, requiring much time and labor in their solution, have been generally avoided. The rather extensive treatment of factoring, and the preparation provided for it by the introduction of a partial treatment of involution, a treatise on composition, and one on exact division, it is believed will be commended by teachers generally. No one can expect to make much progress in the study of algebra who is not somewhat of an adept in factoring. The early introduction of the equation, and the frequent return to it, are features so well adapted to practical work that comment upon their merits is unnecessary. The simplicity of the treatment of generalization and specialization, negative solutions, inequalities, binomial surds, and limiting ratios, is a sufficient excuse for their introduction into an elementary treatise on algebra. These subjects may, however, be omitted where a shorter course PREFACE. y is desirable, without doing violence to the logic of other parts. In conclusion, I desire to express my deep obligations to my wife, Annie M. Sensenig, whose experience as teacher has been nearly coextensive with mine, and from whom I have received many practical helps and encouragements in the preparation of this work. I am also greatly indebted to Prof. A. J. Rickoff, of New York, for a careful examination of the manuscript before publication, and for many practical hints obtained through his criticisms. David M. Senseis'ig. Normal School, West Chester, Pa., | June 1, 1888. \ CONTENTS. INTRODUCTION. PAGE Literal quantities — Ideas and expression 1 Kinds of literal quantities 5 Concrete examples involving literal quantities . . . .7 Positive and negative quantities 10 Definitions of quantities 12 CHARTER I. INTEGRAL QUANTITIES. Algebraic addition 13 Definitions of addition 15 Principles of addition and applications 15 Addition of similar monomials 18 Addition of dissimilar monomials 19 Addition of monomials with common factor . . , .20 Algebraic subtraction 21 Definitions of subtraction 22 Principle of subtraction and applications 23 Subtraction of monomials 25 Algebraic multiplication 26 Definitions of multiplication 29 Multiplication of monomial by monomial 30 Multiplication of polynomial by monomial 31 Algebraic division 32 Definitions of division 35 Division of monomial by monomial 36 Division of polynomial by monomial 37 Simple numerical equations 39 Definitions of simple equations 41 Axioms of algebra 42 Concrete examples in simple equations 44 viii CONTENTS, PAGE Addition of polynomials 48 Subtraction of polynomials 49 Symbols of aggregation — Definitions and principles . . .50 Simplification of parenthetical expressions 53 Inclosing terms by symbols of aggregation 53 Multiplication by polynomials 54 Division by polynomials 55 Simultaneous equations of two unknown quantities . . .57 Definitions of simultaneous equations 58 Elimination by addition and subtraction 59 Concrete examples in simultaneous equations . . . .60 Partial treatment of algebraic involution — Definitions and prin- ciples 63 Involution of monomials 64 Squaring of binomials — Principles and applications . . .65 Cubing of binomials — Principles and applications . . .66 Composition — Definitions and general principles '. . . .67 Special principles and applications 69 Cross-multiplication — Principle and application . . . .71 Exact division — Definitions and principles 73 Application of principles 76 Factoring — Definitions and principles . . . . . .77 Factoring polynomials with common factor in terms . . .78 Factoring the difference of two squares 79 Factoring the sum or difference of equal odd powers . . .79 Factoring trinomials that are perfect squares . . . .80 Factoring trinomials composed of binomial factors having a common term 81 Factoring trinomials composed of any binomial factors . . 83 Factoring trinomials composed of any trinomial factors . . 83 Factoring polynomials 84 Miscellaneous examples in factoring 85 Highest common divisor— Definitions and principles . . .86 Highest common divisor of monomials 87 Highest common divisor of polynomials 88 The lowest common multiple — Definitions 90 Lowest common multiple of monomials 91 Lowest common multiple of polynomials 93 Cancellation — Definitions and principles 93 Multiplication and division by cancellation 94 Simultaneous equations of three unknown quantities — Elimina- tion by addition and subtraction 96 Concrete examples in simultaneous equations . . . .97 CONTENTS, ix CHAPTER II. ALGEBRAIC FEACTIONS. PAQE Preliminary definitions 99 Reduction of fractions — Definition and principles . . . .101 Ileduction of fractions to lowest terms 105 Reduction of mixed quantities to improper fractions . . . IOC Reduction of fractions to whole or mixed quantities . . . 107 Reduction of fractions to similar forms 108 Addition and subtraction of fractions 110 Multiplication and division of fractions 113 Multiplication and division by fractions — Definitions and prin- ciples 115 Written examples 117 Complex fractions 119 Involution of fractions 120 Miscellaneous examples in fractions 121 CHAPTER III. GENERAL TREATMENT OF SIMPLE EQUATIONS. General definitions 124 Transformation of equations — Definition and principles . . 125 Simple equations of one unknown quantity — Solution of numeri- cal equations 126 Solution of literal equations 129 Miscellaneous equations 130 Concrete examples 131 Simple equations of two unknown quantities — Definitions and principles 140 Elimination by substitution 141 Elimination by comparison 142 Solution of numerical equations 143 Solution of literal equations 146 Concrete examples 147 Simple equations of three unknown quantities — Solution of ab- stract equations 152 Concrete examples 155 Generalization and specialization 157 CHAPTER IV. POWERS AND ROOTS. Involution of binomials — The binomial theorem .... 161 Involution of polynomials — Principles and applications . . 163 X CONTENTS. ^ PAGE Algebraic evolution — Definitions and principles . . . .165 Roots of numerical quantities by factoring 107 Roots of monomials 168 Square root of a polynomial 169 Square root of numbers 172 Cube root of polynomials 175 Cube root of numbers 177 Higher roots 180 Factoring with the aid of evolution 180 CHAPTER V. QUADRATIC EQUATIONS. Quadratic equations of one unknown quantity — Pure quadratics — Definitions and principles 182 Solution of pure quadratics . . . . . . . . 183 Affected quadratics — Definitions and principles .... 185 Solution of numerical affected quadratics 185 Solution of literal affected quadratics 189 Equations in the quadratic form 190 Solution of equations by factoring . , 192 Formation of quadratic equations 194 Formation of equations by composition 195 Miscellaneous equations 197 Concrete examples 198 Quadratic equations of two unknown quantities — Definitions . 201 Solvable classes — Illustrations 202 Solution of homogeuQous equations 207 Miscellaneous equations . . 208 Concrete examples 209 Negative solutions 211 CHAPTER VI. EXPONENTS, RADICALS, AND INEQUALITIES. Fractional and negative exponents — Principles and applications . 214 General principles of exponents 219 Miscellaneous examples in exponents . . . . . . 221 Radicals — Definitions and principles 222 Reduction of radicals — Mixed to pure 224 To lower degree . . 225 Rational to radical quantities ....... 226 To same degree 227 Addition and subtraction of radicals 228 COJ^TENTS. xi PAGE Multiplication of radicals 229 Division of radicals 230 Involution of radicals 231 Evolution of radicals 232 Rationalization 233 Imaginary quantities 234 Square roots of binomial surds 236 Miscellaneous examples in radicals 238 Radical equations 242 Character of the roots of equations 245 Inequalities — Definitions and principles 248 Examples 250 CHAPTER VII. RATIO, PROPORTION, AND PROGRESSION, Ratio — Definitions and principles ....... 251 Examples 253 Proportion — Definitions 255 Propositions 256 Solutions of equations by proportion 264 Examples involving proportion 265 Limiting ratios— Definitions and principles 267 Examples 272 Arithmetical progressions — Definitions and principles . . . 273 Examples involving arithmetical progression .... 274 Concrete examples in arithmetical progression .... 276 Geometrical progression — Definitions and principles . . . 279 Examples in geometrical progression 280 Concrete examples in geometrical progression .... 282 Infinite series — Definitions and principles 285 Examples involving infinite series 286 CHAPTER VIII. MISCELLANEOUS EXAMPLES. Abstract examples 288 Concrete examples 293 General definitions 298 Principles reviewed 300 APPENDIX 310 NUMBERS SYMBOLIZED. INTRODUCTION. LITERAL QUANTITIES— IDEAS AND EXPRESSION EXERCISE 1. 1. What is the sum of 2 units, 3 units, and 4 units? 2 tens, 3 tens, and 4 tens ? 2 fives, 3 fives, and 4 fives ? What, then, is the sum of 2 times any number, 3 times that number, and 4 times that number ? 2. If we let a stand for any number, what will be the sum of 2 times a, 3 times a, and 4 times a ? 3 times a, 4 times «, and 6 times a ? Two times a is written 2a, and is read two a; three times a is written 3a; etc. 3. What is the sum of 4 «, 5 a, and 6a? 8 a, 4 a, and 7a? 4. If we let b stand for any number, what will be the sum of 4:b, 3b, and 2b? 6b, ^b, and Qb? In algebra, any letter may stand for any number. 5. What is the sum of 3 J, 4i, and 2by it b stands for 3 ? lib stands for 4 ? The symbol of addition is + , read plus, 6. What is the sum of 2m4-3m-|-5w? What when m equals 2 ? When m equals 5 ? The symbol = is read equals. 2 ELEMENTARY ALGEBRA. 7. What is the value of ^x-\-4:X-\-Qx^ What when a; == 3 ? When a; = 6 ? 8. bx-\-^x-\-^x= what ? 4^ + 5?z + ^ = what ? 9. What is the difference between 8 tens and 3 tens? 8 20's and 3 20's ? 8 times any number and 3 times that number? What, then, is the difference between 8a and 3a? 8m and 3w? The symbol of subtraction is — , read minus. 10. What is the value of 15a; — 7a; ? 12y — 5y? 11. What is the value of 12 a — 7 a ? What when a = 3 ? When a = 7 ? 12. What is the value of 6a-\-5a—7 a? What when « = 5 ? When a = 8 ? 13. What is the value of a times b when a = 3 and 5 = 4? When a = 6 and ^> = 7 ? The symbol of multiplication is x , read times. 14. What is the value ot x X y when x = 6 and y = S? When a; = 10 and y = 9? The product of two or more letters is expressed by writing them together without any symbol between them. Thus, a x b = ab, and a X b X c = abc. 15. What is the product otmXn?pXq? xXy Xz? 16. What is the product of p X q X r? What when p = 2y q = 3, and r = 4 ? When J9 = 3, q = 4:, and r = 5 ? 17. What is the value of 2 a X 3 J, when a=6 and ^ zz: 7 ? When a = 6 and Z* = 3 ? 18. What is the quotient of x divided by 5 when rr = 10 ? When a; = 15 ? When ic = 30 ? 19. What is the value of a divided by J^when a = 15 and ^> = 5 ? When a = 24 and ^> = 6 ? LITERAL QUANTITIES. g The symbol of division is -f-, read divided hy. Division is also expressed by writing the dividend over the divisor with a line between d them. Thus, a divided by b is written a-i-b, or y . 20. What is the value oi x-~y when x=lb and y = 6? When x = (j3 and y = 9? 771 21. What is the value of — when m = 12 and 7i = 3? n When m = 18 and w = 6 ? 22. What is the value of — when a = 12, 5 = 0, and c c = 9 ? When ff = 10, 2> = 7, and c = 5 ? 23. What two numbers multiplied together will produce 10? 15? 21? da? 5x? ay? xz? 24. What three numbers multiplied together will pro- duce 12 ? 18? 30? 10a? 5ab? xyz? The numbers multiplied together to produce a given number are the factors of that number. 25. Name the two factors of 14. 21. 6 m, c d. 26. Name the three factors of 10 2;. 6 ay. pq r. 27. What number is produced by using 2 twice as a factor ? Three times ? Four times ? The result obtained by using a number two or more times as a factor is a power of the number. When tlie number is used twice as a factor the result is called the square of the number. When used three times, the cube of the number. When used four times, the fourth power of the number, etc. • 28. What is the square of 3 ? The cube of 4 ? The fourth power of 2 ? 29. What is the square of a when a = 4 ? The cube of X when a; = 3 ? The symbol of power is a number called an exponent, written on the right hand above the number whose power is to be obtained. Thus, a squared is written a* ; a cubed, a^ ; a fourth power, a*, etc. 4 ELEMENTARY ALGEBRA. 30. What is the value of x^ when a; = 2 ? When :c = 3 ? When ^ = 4 ? 31. What is the value of a? y^ when ic = 2 and ?/ = 3 ? When a; = 1 and ?/ = 4 ? 32. What are the factors of a^ ? -^^s p ^4 p ^,2^3 p 33. What is one of the two equal factors of 4 ? 9 ? 16? a^p ^2p 34. What is one of the three equal factors of 8 ? x^Y 21! a^? One of the equal factors of which a number is composed is a root of the number. One of the two equal factors is the square root ; one of the three equal factors, the cube root ; one of the four equal factors, the fourth root, etc. 35. What is the square root of 16 ? 25 ? a^ ? a^x^ ? 36. What is the cube root of 27 ? 64 ? ^^ ? a^a^? The symbol of root is a/, called the radical sign. A number called the index is written in the angle of the sign to show the kind of root. When no index is used the square root is expressed. Thus, y/x is the square root of x, and l/y is the cube root of y. 37. What is the value of V^ when x = 16? When a; = 49 ? 38. What is the value of V« when a = 21! ? When a = 64 ? 39. What is the value of V^ ? VaF ? V^? 40. What is the value of Va^ when « = 4 and x = 9 ? When a = 2 and x = 8? 41. Write the square of m ; the cube of n ; the product of m and 71 ; tlie quotient of m and n ; the square of m divided hj the square of n. In algebra, numbers expressed by figures only are called numerical quantities ; and those expressed by letters only, or by both figures and letters, literal quantities. Thus, 24 is a numerical quantity, and a, x^, and 3 h are literal quantities. LITERAL QUANTITIES. 6 Kinds of Literal Quantities. EXERCISE 2. 1. What is the sum of a and h when a = 3 and 5 = 5? When a = 8 and 6 = 9? The sum of different literal quantities, when their values are not given, is expressed by simply writing plus between them. Thus, the sum of a and 6 is a + 6, and of a, h, and c is a + b + c. 2. What is the sum of x and y? x, y, and ;? ? 2 « and 35? 4a:, 5y, and 6;z? 3. What is the difference of x and y when a; = 10 and y = 5 ? When ic = 12 and 2^ = 6 ? The difference of different literal quantities, when their values are not given, is expressed by simply writing minus between them. Thus, the difference of a and h is written a — h. 4. What is the difference of m and w ? 2 a and 3 5? 5 7? and 7 y^ ? a:^ and ?/^ ? The different parts of which a sum or difference is composed are called terms. Thus, 2 a, 3 6, and 4 c are the terms of 2a + 3& + 4c. 5. Name the terms in x-\-y -\-z, 2a-\-ZJ)-\-4:Z, X — y. x-\-y — z. X — 2y-\-3z. When a quantity consists of only one term it is called a monomial ; when of two or more terms, a polynomial. Thus, 3 a 6 is a monomial, and 2 a + 3 6 and 4a — 26 + c — d! are polynomials. A polynomial of two terms is a binotnial, and one of three terms a trinomial. 6. What are the values of the following binomials, when a = Q and 5 = 3? 1. a + 5 3. a-\-ab 5. ab-b- 7. a^b-ab- 2. a-b 4. a2_^2 g. a^-b"" 8. 2a + 35 7. What are the values of the following trinomials, when a; = 10, y = 6, and z = 4 ? l'^ + y + 2^ ^' X — y — z , 1. 2x — y -\-z % x — y-\-z 5. x-\-2y — z 8. 2x-\-dy + 2z 3. a; + ?^ — ;z 6. 3a; — 2«/4-2 9. 5a; — 3?/ + 2^ 6 ELEMENTARY ALGEBRA. When terms have the same letters affected by the same exponents they are similar. Thus, 3 a^ b\ 5 a^ b^, and 6 a* b^ are similar terms. 8. Arrange the following terms into groups, placing similar terms into one group : 2ab^ 3aH, 4.al)\ ^aH, ha^W, ^aH, 1 ah^, ^a^W, The numerical factor in a term is generally called the coefficient of the term, but any factor may be taken as the coefficient. When no numerical coefficient is expressed the factor 1 is understood to be the numerical coefficient. 9. Name the numerical coefficients of Zax, ^hc, Qmn, ax, 4:cd, 6x^y^, a^a^, 2hxy, m^n. Sometimes terms have factors that are alike and some that are unlike ; then the unlike ones are taken as the coefficients of the terms, and the terms are considered similar with respect to the like terms. Thus, axy,bxy, and cxy sue similar with respect to xy. a, b, and c are the coefficients. 10. With respect to what letters are the following terms similar, and what are the coefficients of the terms ? 1. ax, hx, and ex 3. 2 ax, 3b x, and 4= ex 2. cxy, dxy, and exy 4. 2my, 3ny, and 4:sy The symbol ( ), called a parenthesis, is used to inclose two or more terras that are to be taken together as one factor or one term. Thus, a + b multiplied by c is written {a + b)c, and b + c subtracted from a is written a — {b + c). 11. Find the value of (4 a — 2 1))c when a = 3, b = 2, and c = 4. If a and b represent two numbers, what will represent 12. Their sum ? 16. The square of their sum ? 13. Their difference ? 17. The sum of their squares ? 14. Their product ? 18. The cube of their sum ? 15. Their quotient ? 19. The sum of their cubes ? 20. The product of their sum and difference ? 21. Their product times their difference ? 22. The quotient of their sum and difference ? LITERAL QUANTITIES. 7 Concrete Examples involving Literal Quantities. EXERCISE 3. 1. A boy paid a cents for a slate and h cents for a book. What did he pay for both ? Solution. — lie paid for both the sum of a cents and h cents, which is a + & cents. 2. I paid 2a^ cents for an apple and ^a cents for an orange. What was the cost of both ? 3. A man had 5 a dollars and spent 2 a dollars. How much money had he left ? 4. Mary bought a lemon for 3 a cents and an orange for h cents. What did she pay for both ? 5. Thomas rode Qx miles and then walked 4ic miles. How far did he go in all ? 6. Mary had 15 a quarts of berries and sold 9 a quarts. How many quarts had she remaining ? 7. A boy bought an apple for c cents and handed over a 10-cent piece. How much change should he receive ? 8. I bought a horse for 160 and sold it for y dollars. How much did I gain ? 9. What will be the cost of 3 chairs at x dollars apiece, and 4 tables at y dollars apiece ? 10. I bought m sheep at $6 apiece and sold them at 19 apiece. What did I gain ? 11. If 8 ropes, each h feet long, be cut from a coil con- taining a yards, how many feet will remain ? 12. If X acres of land are worth $1000, what is the value per acre ? 13. If 6 horses are worth 5 y dollars, what are 10 horses worth at the same price per head ? 14. At h dollars a head, how many horses will c sheep at d dollars apiece buy ? 8 ELEMENTARY ALGEBRA. 15. At m cents apiece, how many apples will $1 buy ? 16. At $6 apiece, how many pigs will x dollars buy ? 17. A bought a farm of m acres at n dollars an acre, and sold it at r dollars an acre. What was his gain ? 18. If a bushels of wheat cost $60, what will x bushels cost at the same price ? 19. If a men can do a piece of work in m days, in how many days can b men do it ? 20. A man bought a farm of a acres at x dollars an acre, and sold it at y dollars an acre. How much did he gain ? 21. At m cents a pound, how many pounds of sugar are worth as much as c pounds of coffee at d cents a pound ? 22. A is a years old and B is twice as old. What will be B's age 20 years hence ? What was it 10 years ago ? 23. A and B start from the same place at the same time and travel in the same direction. If A travels m miles a day and B n miles a day, how far apart will they be in c days ? 24. What will be the cost of a rectangular piece of land X rods long and y rods wide at c dollars an acre ? 25. What is the interest of a dollars for t years at r per cent ? 26. A man bought a horse for p dollars and sold him at a gain of r per cent. What did he receive for him ? 27. What will it cost to plaster a room a feet long, h feet wide, and c feet high at d cents a square yard ? 28. A bought A:X bushels of clover-seed at c dollars a bushel, and sold one half of it at d dollars a bushel and the rest at cost. What did he gain ? 29. A miller mixed a bushels of corn worth m cents a bushel with c bushels of oats worth n cents a bushel. What was the value per bushel of the mixture ? LITERAL QUANTITIES. 9 30. In what time will p dollars at r per cent amount to a dollars ? 31. How many board feet in a plank m feet long, n inches wide, and c inches thick ? 32. If A can do a piece of work in a days, what part of it can he do in c days ? 33. I bought some goods at a cents a yard and sold them at b cents a yard. What was my gain or loss per cent ? 34. A is a rods ahead of B, and goes c rods while B goes d rods. How many rods must B go to overtake A ? 35. If A can go a mile in a minutes and B a mile in h minutes, how much will A gain on B in one hour ? In c hours ? 36. What is the value of a square field x rods long at m dollars an acre ? 37. How much larger is a rectangular tract of land x rods long and y rods wide than a square tract z rods long ? 38. What is the weight of a cubical stone a feet long if c cubic feet weigh a ton ? 39. A has a garden m feet long and n feet wide. What would be the Side of a square garden of equal area ? 40. How many cubical blocks x inches long are equiva- lent to one block p feet long, q feet wide, and r feet high ? 41. A rectangular field is a yards long and h yards wide. How far is it across it from corner to corner ? 42. A ladder c feet long reaches to the top of a tower a feet high. How far is the foot of the ladder from the base of the tower ? 43. A bought a horse for x dollars and sold him to B at a gain of x per cent, who again sold him to C at a gain of X per cent. Wfiat did B gain ? 10 ELEMENTARY ALGEBRA. Positive and Negative Quantities. EXERCISE 4. 1. Does money gained in business increase or diminish one's capital ? Money lost has what effect ? 2. Distance traveled in the direction of one's destina- tion has what effect upon one's journey ? Distance trav- eled in the opposite direction has what effect ? 3. Power applied to assist a moving cart has what effect upon the moving force of the cart ? Power applied to retard it has what effect ? Quantities that have directly opposite tendencies in a mathematical calculation are called positive and negative. Illustrations. — 1. If gains be considered positive, then losses will be negative. If losses be considered positive, then gains will be negative. 2. If past time be considered positive, then future time will be negative. If future time be considered positive, then past time will be negative. 3. If distance in any direction be considered positive, distance in the opposite direction will be negative. It is customary, but not essential, to consider quantities that ex- press favorable conditions in an example positive, and those that ex- press unfavorable conditions negative. 4. Tell which of the following quantities are positive and which negative : John earns 110, spends $8, finds $9, loses 112, gives a poor man 15, receives a reward of $6. 5. Tell which of the following quantities are positive and which negative : A man deposits 150 in bank, then *' checks out" $30, then deposits 120, then deposits $40, then *' checks out" $50, then deposits $10, then "checks out" $12. A quantity is marked positive by writing the symbol + (plus) be- fore it, and negative by writing the symbol — (minus) before it. 6. Write the following quantities with their proper signs : Thomas buys 8 sheep, sells 7, buys 9, sells 6, buys 5, kills 10, buys 12. POSITIVE AND NEGATIVE QUANTITIES. H 7. Write 12 positive units, 3 negative units, 5 a positive units, x-\-y negative units, a-\-h positive units. 8. Write the following quantities with their proper signs : A Philadelphian bound for California travels west 2 a miles on Monday, east 3 a miles on Tuesday, west 5 a miles on Wednesday, west 4 a miles on Thursday, east 6 a miles on Friday, west 7 a miles on Saturday, and rests on Sunday. 9. If a man walks 10 miles in the direction of his des- tination, and then walks 5 miles in the opposite direction, what effect do the last 5 miles have upon the first 10 ? 10. If one boy pulls at a cart with a force of 20 pounds, and another holds back with a force of 12 pounds, what effect does the 12-pound force which the second boy exerts have upon the 20-pound force exerted by the first boy ? 11. If a man gains $15 in one transaction and loses $25 in another, what effect does the gain have upon the loss ? Positive and negative quantities tend to destroy each other when combined in an operation, and hence are said to be opposed to each other in character, 12. Which is the more favorable condition, to be merely penniless or to be in debt $10 ? To rest or to go 6 miles in the opposite direction from one's destination ? To be idle or to lose $20 in business ? A negative quantity is sometimes regarded as less than zero. 13. Which is the more favorable condition, to owe $5 or to owe $10 ? To lose 10 sheep or to lose 20 sheep ? To go 8 miles or 15 miles in a wrong direction ? Of two negative quantities, that is considered the greater which has the less number of units. 14. One boy helps a cart along with a force of 12 pounds and another retards it with a force of 8 pounds. Write the combined effect of these forces upon that of the cart. 15. How many and what kind of units are there in + 7 ? -6? -fa? -^>? +3a? -2Z^? -f«^? -//? 12 ELEMENTARY ALGEBRA. 16. A miller bought 80 bushels of oats and sold 95 bushels in one day. Write the combined effect of these transactions upon the amount of oats on hand. 17. A earns a dollars and spends h dollars. Write the combined effect of these transactions upon his finances 1. When a is greater than h. 2. When a is less than h, 18. A man earned x dollars one day and y dollars an- other. Write the combined effect of the two days' wages upon his finances. 19. A man spent a dollars at one time and J dollars at another time. Write the combined effect of these trans- actions upon his finances. 20. A land-holder buys a tract of land a rods long and h rods wide. Write the effect of this transaction upon the amount of land he owns. Definitions. 1. A Unit is a single thing. 2. One or more units of a kind is a Number. 3. A definite number of units is a Specific Quantity; as, seven birds. 4. An indefinite number of units is a General Quan- tity; as, a jioch of birds. 5. A number expressed by figures only is a Numerical Quantity; as, 125. 6. A number expressed by letters, or figures and letters, is a Literal Quantity; as, x and 6x. 7. Numbers opposed to each other in character are dis- tinguished by the symbols + (plus) and — (minus), and are called Positive and Negative Quantities. Note.— For complete definitions, see pages 298 and 299. CHAPTER I. INTEGRAL QUAJ^TITIES, Algebraic Addition. EXERCISE B. 1. A man earned 15 one day, $4 the next, and $7 the next. What was the combined effect of these earnings upon his finances ? Form. Solution. — Since earnings increase his _\ ^k money, we mark each earning positive. •" The whole increase is evidently the sum ~r 4 of $5, $4, and |7, which is $16, which + '^ we mark positive. -|- $16 2. One boy helps a cart along with a force of 16 pounds, another with a force of 20 pounds, and another with a force of 25 pounds. What is the combined effect of these forces upon that of the cart ? 3. A miller sold 5 bushels of oats to one man, 6 bushels to another, and 9 bushels to another. What was the com- bined effect of these transactions upon the amount of oats on hand ? ForiDi Solution. — Since oats sold diminishes p, , the amount on hand, we mark each quan- tity negative. The whole decrease is evi- " dently the sum of 5 bu., G bu., and 9 bu., — 9 *^ or 20 bu., which we mark negative. — 20 bu. 4. A man sold 10 cows one day, 15 the next, and 20 the next. What was the combined effect of these transactions upon the number in his herd ? + 15 - $3 + 7 - 6 + 8 - 9 + $20 -$18 - 18 + $2 14 ELEMENTARY ALGEBRA, 5. A man earns $5, then spends $3, then earns $7, then spends $6, then earns $8, then spends $9. What is the combined effect of these transactions upon his finances ? Solution. — We mark all incomes posi- ponn. tive, and all outlays negative. The sum of the incomes is $20, which we mark positive. The sum of the outlays is $18, which we mark negative. Now, an outlay of $18 will destroy an income of $18, or — $18 will destroy + $18, and there will remain an income of $2, which we mark positive. Eemark. — A negative quantity will destroy a positive quantity of the same number of units when combined with it. 6. Six men push at a moving car. A pushes forward 80 pounds, B backward 90 pounds, forward 100 pounds, D backward 95 pounds, E backward 110 pounds, and F forward 85 pounds. What is the combined effect of these forces upon that of the car ? 7. A drover adds 2 a sheep to his flock, then sells 3 a, then buys 4 a, then sells 3 a, then buys 5 a, then sells 3 a, then buys 6 a. What is the combined effect of these trans- actions upon the number in his flock ? Combining algebraic quantities is called adding them. 8. Find the sum of + 3 «, — 4:a, -{-Qa, —6 a, — 3 a, and -\-2a. Solution. — The sum of the positive quantities is + 11a, and the sum of the negative quantities is — 12 a. If we com- bine 11 a positive units with 12 a nega- tive units, they will destroy 11a negative units, and a negative units, or — a, will remain. Eemark. — Equal positive and nega- tive quantities may be omitted in addi- — tion, since they destroy each other. 9. Find the sum of -f 2, - 3, -f 4, - 5, and + 7. ] Form. -f da - 4a -1- 6a - 5a -f 2a - da + 11 a -12 a + 11 a ALGEBRAIC ADDITION, 15 10. Find the sum of + 2^> + 3 o^, — 4 «, — 5 a, + 7 a, — 6 a, and + 3 fl. When no sign is written before an algebraic quantity, + is under- stood. 11. Find the sum of 3 a;, — 4 a;, 7 a;, --5 a;, and 3 x. 12. Find the value of +2m + (+3?^) + (-2^0 + Definitions. 8. The result obtained by combining two or more quantities without regard to their character as positive or negative, is the Arithmetical Sum of the quantities. 9. The result obtained by combining two or more quantities with regard to their character as positive or negative, is the Algebraic Sum of the quantities. niostration. — If a man goes 10 miles in the direction of his destination and 4 miles in the opposite direction, the entire distance traveled, the arithmetical sum, is 14 miles ; but the distance he advanced on his journey, the algebraic sum, is only 6 miles. 10. The process of finding the algebraic sum of two or more quantities is Algebraic Addition. Principles and Applications. 1, Find the sum oi -\-'Za, + 3 a, and + 4a ; also the sum of — 2 a, — 3 a, and — 4 a. Solutioii. — 1. The sum of 2 a positive units, 3 a positive units, and 4 a positive Forms. units is evidently 9 a positive units. Therefore, the sum of -h 2 a, -I- 3 a, and + 2 « "~ f ^ + 4ais+9a. 4-3a 2. The sura of 2 a negative units, 3 a -]- 4 a negative units, and 4 a negative units is 19^ 9 a negative units. Therefore, the sum of — 2 a, — 3 a, and — 4 a is — 9 a. le ELEMENTAR Y ALGEBRA. Therefore, Principle 1, — The algebraic sum of two or more similar terms with like signs equals their arithmetical sum with the same sign. SIGHT EXERCISES. Name at sight the sum of the following quantities : 1. 2. 3. 4. 6. 6. -\-2a -5x -6y -]-dz + 11 a^ -9ab -j-Sa -7x -8y -\-Sz + 7 a^ -6ab 7. 8. 9. 10. 11. 12. + 2«2 -6x^ -4.ah + ^b -bax^ -]-6bx + 3 a' -5a^ -Qab + 8& -lax^ + 8bx + 5a2 -7x' -lab + 10^> -9ax^ + 7bx 13. +3a + (+4a) + {+6a) 14. -6ic + {-5x) + {-2x) 15. -\-6x'-{- {-{-6x') + i+Sa^) 16. - 5 m^ + (- 7 m^) + (- 3 m^) 2. Find the sum of -{-6a and —2a; also the sum of — 6a and + ^ ^• Solution. — 1. If 2 a negative units be Forms. combined with 5 a positive units, they _|_ 5 // 5 a will destroy 2 a positive units, and da ^ positive units will remain. Therefore, ZL, Jl the sum of + 5 a and — 2a is +3 a. -\- 3 a -^ 3 a 2. If 2 a positive units be combined with 5 a negative units, they will destroy 2 a negative units, and 3 a negative units will remain. Therefore, the sum of — 5 a and + 2 a is —3 a. Therefore, I*rin, 2, — The algebraic sum of two similar terms with unlike signs equals their arithmetical difference with the sign of the greater. ALGEBRAIC ADDITION, 17 SIGHT EXERCISES. Name the sum of the following quantities : 1. 2. 3. 4. 6. 6. + %a +6a -5a -Ix ~9a^ +3aJ -5a -3a +6« -^2x -{-7a^ -Sab 7. 8. 9. 10. 11. 12. -23^ -bxy -lOa^ +122:=^ - bm7i -\-3{a-\-b) + 8ar^ ±Sxy +10a^ - 7a:^ -j-nmn -6(a + b) 13. -5arH- (+2a:2) ^g, _|_a;2^3 ^ (_ ^^3) 1^ +7xy -{- (-3xy) 16. - 3/^^ _|. (_|_6y ^2) 5. Find the sum of + ^> + ^> and — c. Solution. — If b positive units be added to a positive units, the sum will he a + b •^™™' positive units ; if now to « + & positive -J- a units c negative units be added, they will _1_ j^ destroy c positive units, and a + b — c __ ^ positive units will remain. Therefore, the sum of + a, + b, and _ c is + (a + -f- {a -[^ — c) Or b — c), or simply a + b — c, the positive a -\- C sign being understood. Bemark. — If c were numerically greater than a + b, the sum would be c — {a + b) negative units, which, as will be learned in subtraction, would still be a + & — c. Therefore, Prin, 3, — The algebraic sum of two or more dissimilar terms equals a polynomial composed of those terms, SIGHT EXERCISES. Name the sum of the following quantities : 1. a, +3^, and —2c 4. 2 a; +(— 3 «/) + (— 42;) 2. 2x, -^y, and +3^ 5. 1 z^ -\- (-\- 2 z) + (- b) 3. bz\ -7/, and -(Jar 6. S p^ -\- (^ ^ q^) -^ {-1 r") 18 ELEMENTARY ALGEBRA. Problem 1. To add similar monomials. Illustration.— Find the sum of + 3 «, — 4 «, + 6 «, — 5 «, —3 a, and -\-2a. Solution. — The sum of the positive Form, quantities is + 11 a [P. 1], and the sum of the negative quantities is — 2 a [P. 1]. Now, the sum of + 11 a and — 12 a is - a [P. 2]. Eemark.— If preferred, explain as on page 14. Suggestion to Teacher.— Require pu- — a . pils to recite principles whenever refer- ence is made to them in solutions. Do not demand the numbers of principles. EXERCISE 6. Find the sum of the following columns : + Sa - 4.a 4- 6a — 6a + 2a - da + 11 a — 12 a + 11 a 1. 2. 3. 4. -\-2a -6x + 2xy -Sab + 5a -Ix -"Ixy + 4:ab + 6« -%x -\-bxy -6ab + 7a. -4.x -^xy + ab 6. 6= 7. 8. Sax -6x^y — 6mn -Spq — 6ax ^7?y — Omn 7pq — 4,ax -%x^y Imn Spq -\-Qax -^7?y dmn -9pq + 2ax 10. — Smn - pq 9. 11. 12. d{a + b) — 3 (m — n) H^ + f) - Hx+yy -^{a + b) 7(m-n) S{x^ + f) - nx + yf Q{a + b) — 6{m — 71) -n^+/) + e(x + yr 2{a-{-b) 8 (m — n) -6{ar-j-f) + S{x + yY -.6{a + h) — 4:(m — n) -3(^ + /) - 10 (x + yY ALGEBRAIC ADDITION, 19 13. Add 4 a" b\ -lar W, 8 a^ W, - 6 «« W, and 12 a^ b\ 14. Add —6xyZf —9xyz, Ixyz, —^:xyz, and ^xyz. 16. Add 3 (a — m), — 7 (a — wi), 6 (« — ?^i)> and — 8 (« — m). Note. — When quantities are written in succession, separated by positive and negative signs, their sum is intended. Thus, 3 a - 5 a + 7a — 4a = (3a) + (-5a) + (+ 7a) + (-4a). 16. Collect into one quantity 7a — 4a + 5a — 6a4-'^^ — a. 17. Collect 9J-7*+6J-95 + 85-i-2J + 35 -lb, 18. Collect -^ab-\-'iab-10ab-{-dab-6ab^ 7 ab — Sab, 19. Collect 9 m^n^ — S m^ n^ — 12 m^ 7i^ + 7 m^ n^ — 20. Collect 3 (a + ^) - 5 (a + ^) + 'J' (« + *) - 6 (« + J) + 4(a + Z')-8(a + J) + 6(a + ^)-5(a + &). Problem 2. To add dissimilar monomials. Illustration.— Find the sum of 3 a, — 5 &, and -f- 2 c. Solution. — Since the algebraic sura of dissimilar terms equals a polynomial composed of those terms [P. 3], the sum of 3 a, — 5 6, and +2cis3a — 5&H-2c. EXERCISE 7. Add the following columns : 1. 2. 3. X 2x 5a y -3y -3b z 4:Z —2c Form. 3a -5b + 2c da -5b + 2c 6. Add 7 a ^, — 4 c ^, 5ac, 4. 5. — 5m -ab ■i-6n ■\-cd + 7r -\-4:h 6bd, and 4 a 771, 20 ELEMENTARY ALGEBRA. 7. Add —xy, -\-ltjz, — 4:Xz, -\-9my, and — Gnx. 8. Collect —4:al) -{- 7xy — Sab — 2xy -{- 4:ab and -xy. Suggestion. — Collect first the similar quantities, then combine the dissimilar sums. 9. Collect 6a-\-4:b — 3a-\-2I) — 5a-^l!b-\-6a — 5h. 10. Collect 6x-{-4:y — dz + '7z — 4:X-\-6y — 8y + 10z. 11. Collect 3m^ -\- 4:71^ — 5mn-\-7 771^ — n^-\-6mn — 4 m^ — 6 71^. 12. Collect 6ax — 4:by-^'7ax — Sax-\-4cby-{-6ax — 15 a X. Form. a xy I xy — c xy (a-{-b — c)xy Problem 3. To add monomials having a common factor. Illustration. — Find the sum of axy, bxy, and —cxy. Solution. — The common factor is xy. a times xy plus h times xy is {a + b) times X y ; and {a -\- h) times x y added to — c times xy is {a + h — c) times x y [P. 3]. EXERCISE 8. Add the following columns : 1. 2. 3. ax ayz 2ay hx —dyz Siy c X -\-myz — 4:cy 4. 2xz — axz — hxz 5. Add 2 axy, Zhxy, ^cxy, and — dxy, 6. Collect anl) — amh-\-aph — aqh-^ari, 7. Collect axy — l)xy-\-cxy — 2axy-\-3hxy — 4:Cxy, 8. Collect SaiTny — 4:hcmy -\- Gcdmy — badmy, 9. Collect a(c-^d) -\-l{c-{- d). XO. Collect a{x-\-y-Yz) -\-b{x-\-y-\-z) — c{x-\-y-\-z). ALGEBRAIC SUBTRACTION. 21 Algebraic Subtraction. EXERCISE 9. 1. A gain of how many dollars must be added to a gain of 3 dollars to make a gain of 7 dollars ? What then must be added to + $3 to make -j- $7 ? 2. A loss of how many dollars must be added to a loss of 3 dollars to make a loss of 7 dollars ? What then must be added to — $3 to make — $7 ? 3. A loss of how many dollars must be added to a gain of 7 dollars to make a gain of only 3 dollars ? What then must be added to + ^7 to make + ^3 ? 4. A gain of how many dollars must be added to a loss of 7 dollars to make a loss of only 3 dollars ? What then must be added to — $7 to make — $3 ? 5. A gain of how many dollars must be added to a loss of 3 dollars to make a gain of 7 dollars ? What then must be added to — $3 to make + $7 ? 6. A loss of how many dollars must be added to a gain of 7 dollars to make a loss of 3 dollars ? What then must be added to -f $7 to make — $3 ? 7. A loss of how many dollars must be added to a gain of 3 dollars to make a loss of 7 dollars ? What then must be added to + $3 to make — $7 ? 8. A gain of how many dollars must be added to a loss of 7 dollars to make a gain of 3 dollars ? What then must be added to — $7 to make + $3 ? 9. A loss of how many dollars must be added to a gain of 5 dollars to make neither a gain nor a loss? What then must be added to + $5 to make ? 10. A gain of how many dollars must be added to a loss of 5 dollars to make neither a gain nor a loss ? What then must be added to — $5 to make ? The quantity that must be added to one of two given quantities to make the other is the difference of the quantities. The process of f ^ OF THE ! ! M 1 \/ ERSiTY 1 22 ELEMENTARY ALGEBRA. finding the difference is subtraction. The quantity formed of the difference and one of the given quantities is the minuend. The quan- tity added to the difference to form the minuend is the subtrahend. 11. What must be added to + 3 a to make + 7 « ? What then is the difference of + 7 a and + 3 a ? W^hich quan- tity is the minuend, and which the subtrahend ? 12. What must be added to — 3 « to make — 7 « ? What then is the difference between — 7 « and —3a? Which quantity is the minuend, and which the subtrahend ? 13. What must be added to -\-^a to make ? What then is the difference between and + 4 a ? Which quan- tity is the minuend, and which the subtrahend ? 14. What must be added to — 5 a to make ? What then is the difference between and —5a? Which quan- tity is the minuend, and which the subtrahend ? Definitions. 11. The Difference of two quantities is such a quantity as added to one of them will produce the other. 12. The difference of two quantities without regard to their character as positive or negative is their Arithmetical Difference, 13. The difference of two quantities when regard is had to their character as positive or negative is their Algebraic Difference. Ulustration. — The difference between traveling 7 miles and 4 miles, irrespective of direction, is 3 miles. This is the arithmetical difference. But the difference made in one's journey between traveling 7 miles in the direction of one's destination and 4 miles in the opposite direction, is an increase of 11 miles. This is the algebraic difference. 14. The process of finding the algebraic difference of two quantities is Algebraic Subtraction. ALGEBRAIC SUBTRACTION, 23 15. The general problem of algebraic subtraction is : ''Given the algebraic sum of two quantities and one of them, to find the other,'* Principle and Applications. 16. Since the difference of two quantities is such a quan- tity as added to the subtrahend will produce the minuend, it may readily be found in three steps, as follows : 1. Find what quantity added to the subtrahend will produce zero. This is evidently the subtrahend with the sign changed. 2. Find what quantity added to ze7'0 will produce the minuend. This is evidently the minuend. 3. The sum of the two quantities thus added is evi- dently the difference. Therefore, JPWn. 4, — The algebraic difference of two quantities equals the algebraic sum obtained by adding to the minuend the subtrahend with the sign changed. Illustration.— Find the difference oi -\-^a and H-8«; that is, find what quantity added to + 8 a will produce Solution. — 1. If we add — 8 a to + 8 a, we will have zero, 2. If we add + 5 a to zero, we will have + 5 a. 3. Therefore, if we add the sum of — 8 a and + 5 a, or — 3 a, to + 8 a, we will have + 5 a. Hence, Difference of S „ v = sum of ■< !-, ?■ { +8a ) ( —Sa \ — 3a Exercise. — Prove as in the illustration the truth of the following examples : 1. 2. 3. 4. 6. Minuend, -\-Sa — 8a —5a -\- Sa — Sa Subtrahend, +5a —5a —Sa —3a -^ Sa Difference, +3a —3a -\-3a —11 a —11a 24 ELEMENTARY ALGEBRA. 17. The principle of algebraic subtraction may also be illustrated as follows : 1. Arrange positive and negative numbers as in the following scale : -10 -9 -8 -7 -6 -5-4 -3 -2 -1 +1 +2 +3 +4 +5 +6 +7 +8 +9 +lo "1 I 1 \ 1 1 \ \ 1 I \ \ I I i I 1)1 \ T" 2. Consider the difference of two numbers, the number of units passed over in going on the scale from one of them to the other. 3. Consider units passed over in going from left to right positive, and from right to left negative, 4. To find the difference, pass from the subtrahend to zerOy then from zero to the minuend, and show that the algebraic sum of these distances equals the number of units that must be passed over in going directly from the subtrahend to the minuend. niustration. — 1. Find the difference of + 3 and + 8. Solution.— From +8 to is — 8, and from to + 3 is + 3 ; hence, from + 8 to + 3 would seem to be the sum of — 8 and + 3, or — 5. This we see on the scale is true. 2. Find the difference of — 3 and + 8. Solution.— From + 8 to is - 8, and from to — 3 is — 3 ; hence, from + 8 to — 3 would seem to be the sum of — 8 and — 3, or — 11. This we see on the scale is true. SIGHT EXERCISE. Name the difference of the following quantities : 1. 2. 3. 4. 5. 6. + 5a +3a -ba -3a +6a -3a + 3« +5« —Sa —5a —3a +5a 7. 8. 9. 10. 11. 12. — 5a + da -xy -^x^ -9xy -9a;3 -f 3a -5a -\-3xy -\-9a? — 7 a:?/ -{-9x^ ALGEBRAIC SUBTRACTION, 25 13. 7Z>-(+3J) 17. -42;2-(_2z2) 14. —Gx—{'-%x) 18. —Qxy — {-\-lxy) 16. 8ar^--(-9 2;2) ^g _2a;-(+8a;) 16. 3 3/2- (+6 y2) 20. +5a;-(-8a;) 21. -5.^"^ -(-10 3^3) Problem 1. To subtract monomials. Ulnstration. — 1. Find the difference of -\-Zah and — 5ah, Solution : DifEcrence of ) ^^ ( =sum of { + ^«* [ [P. 4] = + Bab. Formt 2. From — Sa Solution : Minuend = —3a take — 2 b Subtrahend with sign changed = -{-2 b 2b — 3a Difference [P. 4] = 2b — 3a 3. From axy Solution: Minuend = a take — bxy Subtrahend with sign changed = -}- b xy xji (a'^b)xy Difference [P. 4] = {a-\-b)xy EXERCISE lO. I. From + 7 fl take -\-3a 2. From + 6 a take + 9 a 3. From — 9 a 4. From —3x 5. From -\-lb take —ba take —%x take — 6 5 6. From +3^ 7. From —^ab 8. From — %xy take - 11 J take + 7 a 6 take -\-l2xy 9. From + 5a 10. From -\-b 11. From —xy take — 12 d take — a take — 3 wi?t 12. Find the value of 3a^x^- (4 a^'x') 13. Find the value of —3m^n^—{—2m^ n^) 14. 7a,'«/-- (~ 6^2/) = what ? 16. 3a:«/-(+7ww)= ? 17. 4-c2-(- J«) = ? 16. — ?ri2 ,^ _ (__ (5 ^ ^) _ p 18. — 7V — (— m^) = ? 26 ELEMENTARY ALGEBRA. 19. From 3(« + .'r) take 4:{a-\-x) 20. From 5 {x^ — y'^) take — 4 (a;^ — ?/^) 21. From 8 {x — yf take — 8 (a; — yf 22. From o^?/^ take — hy"^ 24. From ma; take n^x 23. -ca:2-(-^a;2)^ p 25. 2aa;- (+3 5a;) = ? Algebraic Multiplication. Principles of Signs. EXERCISE 11. 1. Five times 4 a positive units are how many positive units ? Then, 5 (+ 4 «) = what ? 2. Five times ^a negative units are how many negative units ? Then, 5 (- 4 a) = what ? What is the value of 3. 3(+5«)? 5. 3(+6rr)? 7. 8(+5y)? 4. 4(-2a;)? 6. 5(-7«/)? 8. 6 (-3^)? 9. What is the meaning of the expression a: + 3 (+ 2 J) ? Solution : cc + 3(+ 2 6) denotes that 3 times 2& positive units are to be added to x. 10. What is the meaning of a; -|- 3 (— 2 5) ? 11. What is the meaning of a; - 3 (+ 2 J) ? 12. What is the meaning of a: — 3(— 2Z>)? 13. What is the value of x -\- ^ {-{• 2 h) ? Solution : 3(+2&)= + 6&; hence, cc + 3 (+ 2 &) = a; + (+ 6 ^') = a; + 6 & [P. 3]. 14. Since a; + 3 (+ 2 J) = a; + 6 ^ [Ex. 8], what is the value of + 3 (+ 2 Z*) ? Then a positive quantity multiplied by a positive quantity will give what kind of quantity ? ALGEBRAIC MULTIPLICATION, 27 15. What is the value of a; — 3 (- 2 J) ? Solution : 3 (- 2 6) = — 6 6 ; hence, iC_3(_2ft) = a;-(-6&) = a; + 66 [P. 4]. 16. Since a; — 3 (— 2^) =a; + 6 J [Ex. 15], what is the value of — 3 (— 2 Z>) ? Then a negative quantity multiplied by a negative quantity gives what kind of quantity ? 18. Since a positive quantity multiplied by a positive quantity gives a positive quantity [Ex. 14], and a negative quantity multiplied by a negative quantity gives a positive quantity [Ex. 16], we have, Prin. 5, — The product of two quantities with liTce signs is positive. 17. What is the value of a; + 3 (— 2 3) ? Solution : 3 (- 2 &) = — 6 & ; hence, x + d{-2b) = x + {-6b) = x-6b [P. 3J. 18. Since x-^3(-2h) =:x - 6lf [Ex. 17], what is the value of + 3 (— 2 J) ? Then a negative quantity multiplied by a positive quantity gives what kind of quantity ? 19. What is the value of a; — 3 (+ 2 5) ? Solution : 3(+2&)= + 66; hence, x-S{+2b)z=x-{+Gb) = x-6b [P. 4]. 20. Since x — 3{-^2b) = x — 6b [Ex. 19], what is the value of — 3 (-j- 2 3) ? Then a positive quantity multiplied by a negative quantity gives what kind of quantity ? 19. Since a negative quantity multiplied by a positive quantity gives a negative quantity [Ex. 18], and a positive quantity multiplied by a negative quantity gives a negative quantity [Ex. 20], we have, JPrin. 6. — TTie product of two quantities with unlike signs is negative. Note. — Principles 5 and 6 may be stated in one, as follows: In multiplication, like signs give plus^ and unlike signs minus. 28 ELEMENTARY ALGEBRA. SIGHT EXERC 1 SE. Name the products of the following quantities, reciting in each case the proper principle of signs : 1- (+3)x(+4) 7. (-3)x(+2/) 2. (-3)x(+4) 8. (-5)x(-7) 3. (-3)x(-4) 9. (-V,)x(-V3) 4. (+3)x(-4) 10. (_%)x(+%) 6.{-i-a)x{-x) 11. (+73)x(-%) 6. (- a) X (+ x) 12. (- x') X (- y^) Principle of Exponents, etc. EXERCISE 12. 1. Find the product of «* times aK Solution :a;^ = axaxaxa a^ = axaxaxaxa .'. a^xa^ = axaxaxaxaxaxaxaxa = aK Find the product of 2. aP Xa^ 4. x^ X3^ 6. a^ X a 8. r^ X r^ 3. x^ X x^ 5. TYv' X m^ 1. a} Xa^ 9, y^ X y^ 20. Since a'^xa^ — a^ = «*+^ [Ex. 1],- we have, Trin, 7. — The exponent of a factor in the product equals the sum of its exponents in the multiplicand and multi- plier, 4. Which is the greatest, axhc, {aXh)X c, oi h X (ax c) 1. When a = + 3, J = - 2, and c = - 3 ? 2. When « = + 2/3, !?=-%, and )*(a -3bf(a-3 bf 21. (- a:) (- a;2) (- x') 26. 3 a (a: - y)^ {x - yf {x - yf ALGEBRAIC MULTIPLICATION, 31 Problem 2. To multiply a polynomial by a monomial. EXERCISE 14. 1. Which is the greater, a{b-\-c — d) or ah-\-ac — ad 1. When a = 3, 5 = 4, c = 5, and 6? = 6 ? 2. When « = +4, 5 = + 5, c=-3, and ^= + 5? 3. When « = +%, * = +V4, c=-lV2, and e? = +6%? 26. Since «(J-|-c — (?) = aJ-f-ac — «rf for any values of a, h, c, and e? [Ex. 1], we have, Prin, 9. — Multiplying every term of a quantity multi- plies the quantity, SIGHT EXERCISE. Name the products in the following examples : 1. x(a-\-b-\-c) 9. + 4 (+ 3 a; — 2 ?/ + 3 2) 2. y{x-\-y-\-z) 10. — 2(-3« + 6& — 42;) Z, z{x — y — z) 11. —5(+2a; — 3^ + 42;) 4. m^a^-¥^(^) 12. -8(-a;2_2^2_322) 5. — c (« — 5 + c) I'i, ah {x -\- y — z) 6. +3^{x^-x^-\-l) 14. 2:3^(2; — y + ;2) 7. a:3(flfa;2-|_Ja;-|-c) 15. 2 a; (a;^ + a: - 1) 8. m^ (« w^ — 5 m — w) 16. — 4:X{x^ ^ x^ — x) WRITTEN EXERCISES. Illustration.— Multiply 2a^ — Sah + 6b^ hy —Sab. Solution : Since multiplying every term of a quantity multi- Form, plies the quantity [P. 9], we mul- 2fl^ — 3«5 + 6^ Sab tiply each term of the multi- plicand by —Sab, and obtain ~GaH + 9aH^- 18ab\ There- - Q aH -^ 9 aH^ - ISaP fore, Rule 2, — Multiply each term of the polynomial by the mo7wmial, bearing in mind the principles of signs. 32 ELEMENTARY ALGEBRA. EXERCISE la Multiply 1. 2a-35 + 4c by ha 2. aJ'-^ab + b^ by -3^5 3. 5a;2 + 3a;«/ — 2/ by bx^y^ 4. 7a^a:^ — 6aa;y + 9fl^^^ by '^axy 5. x^ — xy — y^ hj —2x^y^ 6. duH^x + 5aH^xy-'7ab^y^ by Sir^.y* 7. 6 m* + 5 ??i^ — 4 ?7i^ + 3 m — 5 by 5 m* 8- a5-fl^*^4-«'^'-«^^ + «^*-^' by aH^ 9. 6:r*^-5a;=^/ + 7aj2«/^-5a;2/'4-5/ by -6a^xy 10. a (a; + 2/) + ^ (^^ + ^) — ^ (i> + $') by aic IL (« + 5):r2_(^_^)^,^_|.^j^2 ]^^ 2a^y^ 12. 3«(z + 2/)-4^»(:i; + «/)2-2c(:r + ?/)3 by (iC + 2/)^ 13. 2x(a-^h)-3y(a + iy-\-4.z{a-\-I?y by2(a + ^f 14. ^2 (^ _ ^)3 _ ^ ^, (^ _ ^)3 + ^2 (^ _ ^)4 by a J (a; - yf 15. px(p + q)-qx{p + qf +i? §' (i? + ^)' by pqx(p + qY Algebraic Division. Principles of Signs. EXERCISE 16- 1. By what algebraic number must -f- 4 be multiplied to produce + 1^ ? Theu, + 1^ divided by + 4 will give what algebraic number ? Then, the quotient of two positive quantities is what kind of quantity ? 2. By what algebraic number must — 4 be multiplied to produce — 12 ? Then, — 12 divided by — 4 equals what algebraic number ? Then, the quotient of two negative quantities is what kind of quantity ? ALGEBRAIC DIVISION. 33 27. Since the quotient of two positive quantities is a positive quantity [Ex. 1], and the quotient of two nega- tive quantities is a positive quantity [Ex. 3], we have, Prin, 10. — The quotient of two quantities with like signs is positive. 3. By what algebraic number must + 4 be multiplied to produce — 12 ? Then, — 12 divided by -[- 4 equals what algebraic number? Then, the quotient of a negative quan- tity divided by a positive quantity is what kind of quantity ? 4. By what algebraic number must + 4 be multiplied to produce -|- 1^ ? Then, -f- 12 divided by — 4 equals what algebraic number ? Then, the quotient of a positive quan- tity divided by a negative quantity is what kind of quantity ? 28. Since a negative quantity divided by a positive quantity gives a negative quantity [Ex. 3], and a positive quantity divided by a negative quantity gives a negative quantity [Ex. 4], we have, Prin, 11, — The quotient of two quantities with unlike signs is negative. Note. — Principles 10 ancl 11 may be stated in one, as follows : In division, like signs give plus and vmlike signs minvts. SIGHT EXERCISE. Name the quotients in the following examples : 1. (+13)--(+3) 9. (+73)H-(-%) 2. ( - 18) ^ (+ 6) 10. (- 3 %) - (+ 'A) 3. (+ 24) - (- 4) 11. (4- 6 'A) ^ (+ 3 Vs) 4. (- 36) - (- 6) 12. (- 37 %) -r (- 6 'A) 6.(+xy)^(+x) 13. (+8 a:) -^(+4) ^■{-xy)-i-(-y) 14. (-8 a.) ^(-2) 7. (-xy) + (+a;) X6. (_9ar=).^(+3) 8. (+ xy) - (- y) 16. (- 6 x>) -f- (- 3) 34: ELEMENTARY ALGEBRA, Principles of Exponents. EXERCISE 17. 1. By what quantity must a^ be multiplied to produce a^ ? Then, a^ divided by w* equals what quantity ? 2. By what quantity must x^ be multiplied to produce x^^ ? Then, x^^ divided by a^ equals what quantity ? 29. Since a^-^a^ = a^ [Ex. 1], and x'^-i-a^ = a^ [Ex. 2], we have, PHn, 12, — The exponent of a factor in the quotient equals the difference of the exponents of the factor in the dividend and divisor. 30. a^-^a^^ a' [P. 12]. But a^-^a^ = 1, since lXa^ = a^, ftO = 1. Therefore, JPrin, 13, — Any quantity with an exponent of zero equals unity. SIGHT EXERCI SE. Name the quotients in the following examples : 9. (_j.a;i8)^(-a;i2) 10. {- x}') -^ {- x') 11. (+3^) --(-32) 12. (4-5^)-^(+52) EXERCISE 18. 1. What is the value of 54 -^ 3 ? What, then, is the value of (9 X 6) -V- 3 ? Is (9 X 6) -r- 3 = (9 ^ 3) X (6 -f- 3) ? Is (9 X 6) -^ 3 = (9 ~ 3) X 6 ? Is (9 X 6) -^ 3 = 9 X (6-^3)? 2. Which is the greatest, ai -r- c, {a -i- c) X b, or ax{b-^c) 1. If a = 8, 5 = 6, and c = 2 ? 2. If « = + 12, J = - 8, and c = - 4 ? 1. a^^ -^ a« 6. y^-^y 2. a^^ -^ d^ 6. z' -^ z^ 3. a^-i-a^ 7. (+a;^)^(+^) 4. m^' ~ mi« 8. (_a;io)^(^^5) ALGEBRAIC DIVISION, 36 SIGHT EXERCISE. 31. Since ah-^c = {a-^ c) xh =^ ax {1)-^ c) for any values of a, J, and c [Ex. 2], we have, Prin» 14, — Dividing one factor of a quantity divides the quantity, SIGHT EXERCISE. Name the quotients in the following examples : i.al)c-i-a 9. (^-6«)-^(^-3) 2. xyz-T-y 10. (-12:r2)^(_4) Z,pqr-^r ii. (+15 2)-^(-5) 4. a^aj-T-a 12. (-18 m) -^(+6) f^.xf.^y 13. (-^y)-^(r-^) 6. m'n^ -^n- 14. {-{- a^ z) -^ (- z) n, a^z-^x 15. (-:z^y')-^(+«/^) 5. as^-r-a^ 16. (+ r s^) -r- (— s) Definitions. 32. The process of finding how many times, and in what manner, one of two algebraic quantities must be taken to produce the other is Algebraic Division. 33. The general problem of division is: ^^ Given the product of two factors and one of them, to find the other, ^^ 34. The quantity to be produced, and corresponding to the product, is the Dividend. 35. The quantity taken to produce the dividend, and corresponding to the given factor, is the Divisor. 36. The quantity which shows how many times, and in what manner, the divisor must be taken to produce the dividend, and corresponding to the required factor, is the Quotient. 36 ELEMENTARY ALGEBRA. Problem 1. To divide a monomial by a monomiaL Ulustration.— Divide —Z^a'^hU by +8 a^ W. Solution : Since dividing one fac- tor of a quantity divides the quantity Form. [P. 14] - 32 X a« X *» X is divided ^ g „, J^^gg^ijs^ by + 8 X a^ X h\ if — 32 is divided '■ a zi3, by + 8, a^ by a\ ¥ by h\ and c by 1. — ^a b c - 32 divided by + 8 is - 4 [P. 11] ; «^ divided by a^ is a^, and h^ divided by h^ is h^ [P. 12] ; and c divided by 1 is c ; hence, the quotient is — 4 a^ &^ c. Therefore, Mule 1, — Divide the numerical coefficient of the divi- dend by that of the divisor ; annex to the quotient all the different literal factors that occur in the dividend, and give to each an exponent equal to its exponent in the divi- dend diminished hy its exponent in the divisor. Note, — Two equal literal factors, one in the dividend and one in the divisor, may be canceled, since their quotient is one. EXERCISE 19. Divide 1. 6«3by3«2 Z. -ISa^hHhY ^ahc 2. 12«2j by -4« 4. l^a^y^z by ^x^y'^z 5. — 21 m^ n^ y^ hy — '^ m"^ n^ y 6. maH^x^f by -^a^a^y^ 7. —mx^f^ by —12x^y^ 8. —^^ax^y^z by — 26 aa^yz 9. (« + by by {a + by 10. — (m — ny by {m — ny 11. 6a^a-{-xy by 2a(a-\-xy 12. 21xy{a-by by 7x{a-by 13. -26z^x-i-yy by -6z{x-\~yy 14. 36 0^ (x^ - y^y^ by - 9 rz;^ (^2 _ ^2)7 15. - 60 m' (a^ + ^>3)8 by 10 m^ (a^ -j- 53)6 ALGEBRAIC DIVISION. 87 Find the value of 16. {^3^y^zxQxy^z^)-^8a:^fz 17. (-9a:^z^-^d3^z) X 2xyz^ 18. 4^3^,2^ X (-SaH'c*-^2aH^c^) 19. (4:a^fz-i-23^y)-{12x^yH^-r-Sa^fz) 20. (-4:aH''cXSaH^(^d) + {lSaH^(^d-i-6aH^c^) 21. (-12a'P(^d-T-4:a:'b(fd) + (21a^I)^c^d-T-7a''Pcd) Problem 2. To divide a polynomial by a monomial. EXERCISE 20. 1. Which is the greater, (ab-{-bc — id) -i-b or a-{-c — d 1. When a = S, 5 = 6, c = 5, and 7« + 6&- -he 9x-5y-\-6z 4a — 85 —3a + 5&- -6c 9x — 7y-]-5z 4. 5. 2ab-\-dcd — 4:ad-{- e — 6a^-{-9y^ — 6xy 6ab-6cd-]-7ad + 2e Sa^-7f-5xy 6ab — 6cd—ead — 3e 9a^-{-6y^-6xy -8ab + *7cd-dad-{-6e 2x^-9y^-\-'7xy -4.ab-\-2cd-\-bad-4.e 6. Add hxy^2z\ 3xy-5z% -1xy-\-2z\ 4:xy-\-6z^, and dxy — 7z^ 7. Add 3m^ + 2mn-{-6n^ 1 m'' -3mn-^6n\ 6m^ — 6mn-}-'7n^, and —87n^ — 6mn-\-5n^ 8. Add 3ax^-5b^y^-\-Sabxy, 6b^y^ - ^ao^ - habxy, ^abxy-hax^-^Wy^, 3Wy^ -habxy -\-9ax^ SUBTRACTION OF POLYNOMIALS. 49 Add : 9. 3 (re + ?/) + 5 (m + w), - 2 (a; + i/) - 5 (m + n), 7 (a: + y) - 8 (m + 7^, 9 (r/i + w) - 5 (a; + y) 10. 7a(io + (7) + 6^(i?-(Z), 6«(i? + (7)-9^»(jo-^), 5^,(^_^) + 7«(j, + g.), Za{p^q)-hh{p-q), ei(p-q)-Sa{p + q) Subtraction of Polynomials. 50. Rule. — Arrange the terms of the subtrahend so that they stand under like terms of the minuend; then change the sign of each term of the subtrahend, or conceive it changed, and proceed as in addition of polynomials, Ulustrations. — Examples. Solutions. From 3ar-l-7a:y-2/ Zx^-\- Ixy-^y"^ take biii? — ^xy-\-hy^ —^a?-\- 9xy — 6y^ Difference = -2a^-j-Wxy - 7 y"" From 3 x — 4:y dx— 4i/ + take 7 x -{-6y — 4.z —7x— 6y-i-4:Z Difference = — 4:X — 10y -\-4:Z EXERCISE 28. 1. 2. From 3a:2_|_2^2 From 9x'-7xy + 3y^ take 4:X^ — 7 y^ take dx'-Sxy + Qy^ 3. 4. From 9 a + 6 * - -7c From lOm^n^ — 7 mn-\-Qn^ take 12a-7J + 9c take 4:m^n^-{-Smn-\-9n^ 6. 6. From Sa:^- 72:^2^- 9y^ From 9y* - 7^^ + 6y + 5 take ^T^'^-bx'y-ny^ take 8y^ + 5y' - 7y + 1^ 50 ELEMENTARY ALGEBRA. 7. From 9a:2 + 6a; + 5 take 8a^+7:r-10 8. From 25 a^ — 5 2/^ take ^a?-\-lxy-\-^y^ 9. From 10 m^ — 25 n^ take 8m^ -{-7 mn — dn^ 10. From :2;3 _|_ ^. _|. 9 take 5a^ - H x^ -{-2x - 6 11. From 8(2; + «/) + 5(:z; — «/) take 7(a;4-^) — 9(a; — ?/) 12. From the sum of 3x-\- 6y — 4:Z and 5a;— 7?/ + 52; take 9x — 7 y -\-5z X3. From 9^ — 3a;i/ + 7^^ take the sum of 3x^ — 5xy-\-l!y^ and 2a.-^ + ^^y-"5 2^^ 14. From the sum of 9 a^ + 7 a^ — 3 o^ + 5 and 6 «3 - 5 a^ + 7 « - 3 take the difference of 9a^-\-5a^ — '7a-\-6 and 3a^ + 5a^-6a-i-9 Symbols of Aggregation. Definitions. 51. The symbols of aggregation are the parenthesis ( ) ; the braces { } ; the brackets [ ] ; and the vinculum . They signify that the quantities inclosed by them shall be considered together as one quantity. 62. In a more general sense, the term parenthesis is made to include all symbols of aggregation. Principles. 63. The expression a-{-(b — c) denotes that the quan- tity Z* — c is to be added to a. If the addition be per- formed the result will be a-\-I} — c. Therefore, JPrin, 23, — If a number of terms are inclosed hy a paren- thesis preceded hy plus, the symbol and the sign before it may be removed without altering the value of the expression. SYMBOLS OF AGGREGATION. 51 54. The expression a — {h — c) denotes that the quan- tity J — c is to be subtracted from a. If the subtraction be performed, the result will be a — h-\-c. Therefore, Prin, 24, — If a number of terms are inclosed hy a paren- thesis preceded by minus, the symbol and the sign before it mxiy be removed, if the sign of every term inclosed be changed. SIGHT EXERCISE. Name at sight the equivalents of the following expres- sions, without parentheses : l-a^ + (3^-^) 5. 2 + (5 -3) 9. a -{-a) 2. 3x-(2y-\-z) 6. 4-(6-4) I0.3a — (2a — a) 3, a-\-(-b-c) 7. 3-(4-8) 11. 4:b + (Gb - 3b) ^a — {—b — c) 8. 6 — (—4) i2.2y — {—y — dy) 65. a-i-(b-c) = a-{-b-c [F. 23], .-. a-\-b — c = a-\-(b — c). Hence, Prin. 25. — Any number of terms may be inclosed by a parenthesis and preceded by plus, without changing the value of the expression. 66. a-{b-c) = a-b + c \V. 24], .*. a — b-\-c = a — {b^c). Hence, Prin, 26, — Any number of terms m^y be inclosed by a parenthesis and preceded by minus, if the sign of every term inclosed be changed. SIGHT EXE RCISE. Inclose the last two terms of the following trinomials by parentheses preceded by plus when the middle term is positive, and by minus when it is negative : 1. a-\-b — c b. m-\-n -{-p 9. m — 2n-{-3t 2. a — b-\-c 6. m — n-{-p 10. — x — 3y — 3z Z.a — b — c l.m — n—p li.y — 2x-\-3z 4. x-{-2y — z Q. 3x — 2y-\-2z 12. 3z — 2x-\-y 52 ELEMENTARY ALGEBRA. Problem 1. To simplify a parenthetical expression. niustrations. — 1. Simplify ^ x -\- {^x -\- % x — ^ x). Solution: 3a; +(4a; + 2a;-3a;) = 3a; + 4a; + 2a; — 3a; [P.23] = 6a;. 2. Simplify 5aJ-(3«5-2a^> + 7a5). Solution: 5a& — (3a& — 2a6+7a<5>) = 5a6 — 3a6 + 2a& — 7a6 [P. 24] = -3a J. 3. Simplify 2a-\Za + 2h-\- {4.a — dh -{^a- bb)\^ Suggestion. — Remove the inner symbols continuously until all the symbols are removed. Thus, 2a_[3a + 25 + |4a- 3& -(2a- 56)}] = 2a — [3a + 2& + i4a — 3& — 2a + 56[] = 2a — [3a + 2& + 4a — 3& — 2a + 5&] = 2a-3a-2&-4a + 3^> + 2a-5J = -3a-4& Note. — The operation may often be simplified by collecting the terms inclosed by a symbol at the time of removing it. EXERCISE 29. Simplify 1. ^x-\-{^x—^x-^^x) 4. Zx — '^y — {^x—'^y) 2. 4 6f — (3 «^ — 7 « + 6 «) 5. 5 m — (6 m + 2 m — m) *3. « + 2J + (6a~3&) 6.2x-\-{^x-(^x-^x)\ 7. 2a- {3a + (2a-^>)i 8. 'Hxy — {— ^xy-\-^xy — '^xy) 9. 'ix^^y—{4.x—{Zx-\-^y)\ 10. ^x^{x-\-y)-{2x-4.y) 11. ^xy-{^xy-{-(-'^xy-xy)] 12. 2 + [2-12 + (2)-2}+2] 13. (6-5)-{6-(5-6)-5} + {(5-6)~(5-6)-5( 1^. x-y-\x-{y-{z-x)-y\-z'\-\-{x-y-z) 15. \_a-\-{x-\-{a-\-x)-\-a}-\-x'\-\- [a -\- {x -\- a) -\- x\ 16. 1 _ [_ 1 + { _ 1 _ (_ 1 _ r+T - 1) - 1 } - 1] 17. x — [—x — {x-{-x) —x — x— {x-\- {x — x)}'] SYMBOLS OF AGGREGATION, 53 Problem 2. To inclose terms by symbols of aggregation. Illustrations. — 1. Express in binomial terms a — h — c-\-d. Solution : a-h-c-\-d = {a-h)-{c — d) [P. 25, 26]. Note. — It is customary to place before the symbol the sign of the first terra to be inclosed, and if this is negative the signs of the terms inclosed must be changed. 2. Express in trinomial terms a-\-h — c — d— e +/. Solution : a + h — c — d — e+f={a + h — c) — {d-\-e — f) [P. 25, 26J. 3. Express in trinomial terms having the last two terms of each inclosed by a vinculum, Solution : 3a — 26 + 5c — 6rf + 5e — 4/=(3a — 26^-5c) — (6(^_5e+4/)=(3a- 26-5c) -{(Sd- 5e-4f). EXERCISE 30. Express in binomial terms : 1. 2a-{-3b-{-6c — 2d 3. 7n-]-n —p + q 2. a — 2b-}-c — 2d 4. Sz/i — 2w — 4^ + 2^ 5. a — h-{-c — d — e-\-f 6. 2a-db-4.c-]-2d-6e-^(jf 7. x — y-[-2z — 3v — 6u-{-4:W 8. 5p — 3q-\-6z — 4:m-\-2n — 6r Express in trinomial terms : 9. Examples 5, 6, 7, and 8. 10. 27n — 3n-{-4:a — Gb-\-7c — 2d — 4:e-{-g — 2h 11. 4:a-2b-dc-4:d-{-5e-\-6f+'7g-2h-\-4:l 12. 2p-3q-\-4:r — 2s-\-6t-{-6u-7v-\-2w — 6t/ Express in trinomial terms, having the last two terms of each inclosed by parentheses : 13. Examples 10, 11 and 12. 14. X — y-\-z — m-{-n—p-\-q — r — 8 Form, aJ'-^-ah -\-W a^-aJ) -^h^ a^ + a'b-^aH^ -aH-aH^- aW -\-aH^-^ aP + b'' 54 ELEMENTARY ALGEBRA. Multiplication by Polynomials. niustration.— Multiply aJ" -\-al)-\-h'' hy aJ" - ah + lK Solution : a^ — ab + b"^ = a'' + (— a &) + (+&*) ; therefore a^ — ab + b"^ times a^ + ab + b^ equals the sum of 1. a^ times a^ + ab + b^ = 2. —ab times a^ + ab + b^ — 3. + &2 times a^ ^ab + ¥ = which = a^ -\-a^W -\-h^ MtUe 3,— Multiply the multiplicand hy each term of the multiplier and take the algebraic sum of the products. Note. — For convenience, arrange both multiplicand and multiplier according to the ascending or descending powers of some letter as- sumed as a leading letter. Thus, a better order of (a;2 _ 5 + 7a;) (3a: + 2a;2 - 5) is {x^ + 7a; - 5) (2a;2 + 3a; - 5). EXERCISE 31. Multiply 1. a-\-b by a-b 9. Sa^-\-2b by 4:a^-eb 2. a + b by a + b lo. 7a^-8b^ by da^ + Hb^ 3. a — b hj a — b li. ac — bdhj by — dx ^ 2a-{-db hj 2a-db 12. cc* - a:^ + 1 by o^ + 1 6. xy-{-12 hj xy — 6 13. a^ -{-ab -\-b^ hj a — b 6. x — a hj x — b 14. a^-\-2x-{-4o hj x — 2 1. b — X hy c -\- X 15. fl^^ — «* + 1 by «* + 1 8. m^-j-n^ by m2 + ^2 ^^ gl-Oc^ + c* by 9 + c^ 17. J*-4Z>2c24-l6c* by ¥ + 4:(^ 18. Sm^-}-2mn hj 2m^ — Sn^ 19. ex^y^-7y^z^ by 6 a^ y^ -+■ 7 y'^ z- 20. 9a^ + 36a-\-1Uhj 3a-12 21. cc^ — a^y-\-xy^ — y^ by x-\-y 22. a^ + aH + a^b^ + aP + b'^ by a-b DIVISION BY POLYNOMIALS, 65 Find the value of 23. (30:2^32.^^22^2) (3^__3^y_^2/) 24. (5fl2-f 7aJ + 4J2)(5a2-7aJ + 4Z>2) 25. (9a:2_|_42.^_^2)(9a;2_4^y + 2/') 26. (2a;2_4a.^6)(^_{.2a; + 3)(2a:*-4a:2 + 18) 27. (a* + 2a2 5 + 4a2^ + 8aJ=* + 16J^)(«-2J) 28. 81a;*-54a;2y + 3Ga:2y2_24a;i/3^16^)(3a;^2f/) 29. (2:*-a^2/2_^^)(,^_^^2^_|.2^2)(^_^^_|_^2) 30. (2a:5 + 3a.-*^-2a:32^2_j_4^^3_5^2^_^3y5) (3a;2^2a;y + 3r) Division by Polynomials. Ulnstration.— Divide a^ + W by a-^h Solution : « + & is con- tained in a' + ^3 as many Form, times as it can be taken out ^ i j ) ^3 _|_ j3 / ^2 _ ^ j _|_ ^ of it. a is contained in a* ^ , 9\ Or -T- d b a? times; taking a» times IL — _ (a + 6), or a3 + a» 6 out of —a^l)-\-l^ o» + 6' by subtracting it, — a^l — aW^ there remains — o' 6 + &». ^J2TI|r J3~ a is contained in — a' 6 xg i zs (— a 6) times ; taking (— o h) times (a + 6), or — a' 6 — a h^ out of — a^ 6 + 6' by subtracting it, there remains ah -\-h^. a is con- tained in aV^, (+ h) times; taking (+ &) times (rt + 5), or aW -k- J' out of db^ + &' by subtracting it, nothing remains. Therefore, (a + h) is contained in (a' + &*)> (a* — a 6 + &*) times. Suggestions. — For convenience, arrange the terms of the dividend, divisor, and the several remainders, according to the ascending or de- scending powers of some letter assumed as tlie leading letter. After each subtraction do not bring down any more terms than will ho. needed for the next operation. Always divide the first term of the dividend, or partial dividend, by the first term of the divisor to obtain the next term of the quotient. 56 ELEMENTARY ALGEBRA. EXERCISE 32. Divide 1. a2+9a + 18 by « + 3 6. x^ — y^ by x--y 2. o;^ — a; — 20 by ic — 5 l. a? -{-^1 y^ hj x-\-^y 3. a;2- 12a; + 35 by a; -7 8. a« + J« by aJ'-i-h^ 4. a;* + 4ic2-12 by a;^ - 2 9. «*-§* by a-J 5. x^-\-2xy-\-y^ by a; + 2/ 10. 8a;3 + 27y^ by 2rr + 3y 11. 4a:2_4^__24 by 2a; + 4 12. 4 a;^ — 4 a a; -— 3 6f^ by 2 ic + a 13. 8rc2-14aa?~15a2 by 4a; + 3a 14. 64 a;« - 125 / by 4a;3 - 5 / 15. d2x^-{-y^ by 2a; + y 16. 8m^-277i« by 2^3-3^2 17. x^ -\- x^ y^ -}- y^ by a:^ + a;^ + ^^ 18. x^-{-4:a^-\-lQ hj a^-2x-i-4t 19. a8 + a*J* + J« by a* + a2j2_^J* 20. 16 m* ^» + 36 m^ n^ + 81 m^ n!" by 4 m^ 7^* — 6 m^ n^ +9 m* w 21. a;* + 4a;3 + 6rr2-}-5iC + 2 by .T2 + 3a; + 2 22. 6a;* + 19a;3 + 10a;2^2a: + 5 by 2a; + 5 23. 5x* + 2a;3-20a;2-23a;-6 by 5a;2+7a: + 2 24. 8a«-16««-34a* + 32a2-6 by 2a*-7a^2 + 3 25. 2 a;^ — 3 a; 2f^ + 2/2 + rr ;2 — 2/ 2; by a; — ^ 26. x^ — ^xy-\-4.y'^ — ^z^ by a; — 2?/ + 3;2 27. 4a:2 — 9^/2— 6?/2 — ;22 by %x-{-Zy-{-z 28. 4a;2 + 12a;?/ + 9/ — 2;2 by %x-\-Zy — z 29. :x^ — x^y^ — %xy^ — 'f by a:^ + a; ^ + 2/^ 30. fl2 + 2«^ + Z'2-c2-2cJ-^2 by « + ^> + c + ^ ,4^2 NUMERICAL EQUATIONS, 67 Simultaneous Numerical Equations of Two Un- known Quantities. EXERCISE 33. 1. What is the value of x in the equation 2 a; + 3 ?/ = 24, if y = 4 ? Solution : Put 4 for y in the equation, 2a; + 12 = 24 (1) Transpose 12, 2 a; = 12 (2) Divide by 2, a; = 6 2. What is the yalue of x in the equation 4 a; — 3 y = 26 1. If y = 1 ? 3. If ?/ = 3 ? 5. If 1/ = 5 ? 2. If 2/ = 2 ? 4. If y = 4 ? 6. If 2/ = 6 ? 3. What is the value of x in 7 ic + 5 ?/ = 66 1. If 2^ = 3 ? 3. If 3^ = 5 ? 5. If ^ = 7 ? 2. If 2/ = 4 ? 4. If 1/ = 6 ? 6. If 3/ = 8 ? 57. A single equation containing two unknown quanti- ties can be satisfied with any number of pairs of values of the unknown quantities. 4. What values of x and y will satisfy- both 4:X-Sy = 26 (A) and 1lx + 6y = 66? (B) Solution : Multiply (A) by 5 and (B) by 3 to make the coefficients of y numerically equal [P. 18], 20 a:- 15 y = 130 (1) 21a; + 15y = 198 (2) Add (1) and (2) [P. 16], 41 a; = 328 (3) Divide by 41, a; = 8 Put 8 for X in (B), 56 + 5y= G6 Transpose 56, 5y= 10 Divide by 5, y z= 2 Verify by putting 8 for x and 2 for y in (A) and (B), 32 - 6 = 26, which is true. 56 + 10 = 66, which is true, a: = 8 and y = 2 are the only values of the unknown quantities that will satisfy both equations. 58 ELEMENTARY ALGEBRA, 5. If 2a; + 3?/ = 18 and 3 a; + 2?/ = 17 6 a; + 9 «/ = 54 Why ? 5?/ = 20 2ir+12 = 18 2a;= 6 i?;= 3 '' Verify: Are 3.+ 3, = 18).^^^ and 3a; + 2^ = 17j ^ 58. Two equations of two unknown quantities can be satisfied only by particular values of those quantities. Definitions. 59. When two equations express such relations between two or more unknown quantities that neither of them can be reduced to the form of the other, they are called Inde- pendent Equations. niustration.— 1. 3x-\-y=:l2 and 2x — dy = 6 are in- dependent equations. 2. x-\-y = 6 and 5 tr -|- ^ ^ = ^^ are not independent of each other, since the first multiplied by 5 will give the second. 60. If two or more independent equations are to be satisfied by the same values of the unknown quantities, they are called Simultaneous Equations. 61. To solve two simultaneous equations of two un- known quantities, we first deduce from them a single equa- tion containing only one of the unknown quantities. That is, we perform such operations upon the given equations as are necessary to get rid of one of the unknown quantities. This process is called Elimination. • ELIMINATION. 59 Elimination by Addition or Subtraction. niustration.— Solve 2x-\-3y=l% (A) and 3 a; + 5?/ = 19 (B) Solution : Multiply (A) by 3 and (B) by 2 to make the coefficients of X alike [P. 18], 6a: + 92/ = 36 (1) Qx + \Qy = m (2) Subtract (1) from (2) [P. 17], y= 2 Put 2 for y in (A), 2ic + 6 = 12 (3) Transpose 6, 2x= Q (4) Divide by 2, x= S Verify by putting 3 for x and 2 for y in (A) and (B), m — iq I ^^^^ ^^ which are true. Kote. — If the signs of the like terms in (1) and (2) were unlike, the equations would have to be added to eliminate x. EXERCISE 84. Solve : 1. 3a; + 4?/ = 29) 2x-\-Sy=^6 \ 8. 3x-4:y= 3) 5x-3y = lQ ) 2, 5x — 3y = 22\ 2x-{-9y = 19 ) 9. 5a;+7y = 0) 8x + 6y = 0\ 3. 4a;- 7y=-l ) 3a;+ll2^ = 48 j 10. 6x-\- 92^ = 45) Sx-i-16y='70\ 4.7a: + 8y = 2 ) 6x-2y=-U ) 11. Ux-{-19y = 25 ) 21a;~17?/=-190 j 6. 3x-{-7y = S3) 12. 16x-{-l()y = 105 10a;-15i/=-G5 6. 5a;-7i/ = 13 ) '7x-6y=-l\ *13. ^ + ^ = 36- 7. x-\-3y = 10 dx — 5y = 30 |. + |, = 25 Clear of fractions first. 60 ELEMENTARY ALGEBRA. Concrete Examples involving Simultaneous Equations. EXERCISE 38. 1. A and B together have $500 ; if A had three times and B four times as much as now, A would have 1800 more than B. How much has each ? Suggestion.— Let x = the number of dollars A has, and y = the number of dollars B has. Now, since they together have $500, a; + 2/ = 500 (A) Since 3 times A's sum exceeds 4 times B's by $800, 3a; -42/ = 800 (B) Solve (A) and (B) to obtain results. 2. Three times A's age added to twice B's equals 85 years, and twice A's added to three times B's equals 90 years. What is the age of each ? 3. If 3 apples and 4 peaches are together worth 10 cents, and 5 apples and 2 peaches 12 cents, what are they worth apiece ? 4. If 4 bushels of corn and 5 bushels of oats together weigh 374 pounds, and 3 bushels of corn weigh 48 pounds more than 4 bushels of oats, what are their respective weights per bushel ? 5. A house and barn together cost $3000, and three times the cost of the house exceeds five times the cost of the barn by IIOOO. What is the cost of each ? 6. If A's money were increased by $36, he would have three times as much as B ; and if B's money were dimin- ished by $5, A would have twice as much as B. Find the amount each has. 7. The sum of two numbers is 38, and twice the less is 18 times their difference. What are the numbers ? 8. Five coins of one kind and six of another are worth $4.25, but four of the first kind and seven of the second are worth $4.50. Required the value of each coin. CONCRETE EXAMPLES. 61 9. If 7 bushels of corn and 10 bushels of oats are worth $8.20, and 6 bushels of corn and 8 bushels of oats $6.80, what is the price of each per bushel ? 10. If 8 men and 12 boys earn $168 a week, and 9 men and 7 boys $150 in the same time, what are the daily wages of each man and boy ? 11. A drover sold 12 sheep and 8 cows for $392 ; had he sold 3 more cows and 5 less sheep, he would have re- ceived $482. What was the price of each sheep and cow ? 12. A merchant bought 40 grammars and 50 readers for $77 ; had he bought 50 grammars and 40 readers, they would have cost $1 less. What was the price of each book ? 13. If 8 men and 6 boys earn as much per day as 9 men and 4 boys, and the difference between the daily wages of a man and a boy is $1, how much does each re- ceive per day ? 14. A grocer has two kinds of coffee : if he mixes 12 pounds of the first kind with 1*8 pounds of the second, the mixture will be worth 20 cents a pound ; but if he mixes 24 pounds of the first kind with 6 pounds of the second, the mixture will be worth 16 cents a pound. What is the value per pound of each grade ? 15. If A buys 40 acres of land from B, B will have twice as much as A ; but if he buys 80 acres, they will have the same amount. How much has each ? 16. C has 60 acres of land : if A buys C's land, A will have as much as B ; but if B buys it, B will have three times as much as A. How many acres has each ? 17. The sum of two numbers exceeds twice their differ- ence by 30, and twice the first equals three times the sec- ond ; required the numbers. 18. If B were to give A $25, they would have equal sums of money ; if A were to give B $22, B would have twice as much as A. How much has each ? 4 62 ELEMENTARY ALGEBRA. Partial Treatment of Algebraic Involution. Definitions. 62. The result obtained by using a quantity two or more times as a factor is a Power of the quantity. 63. The number of times a quantity is used as a factor to produce a power is the Degree of the power. Illustration. — Thus, a^ is a power of the fourth degree when derived from a^, since a/^ X a^ X a^ X a^ = a\ 64. The degree of a power is expressed by a quantity called an Exponent, written on the right hand above the quantity. Illustration. — The 4th power of a^ is written {a^)\ 4 is the exponent, and denotes the degree of the power. 65. The process of raising an algebraic quantity to any power is Algebraic Involution, Principles. 66. {J^ay=(+a)x{-ha) = -ha' (-aY = (-a)x{-a) = -\-a^ (+«)* = (+«) X i+a) X (+«) X (+«) = +a* (- ay ={-a)x (- a) X (- a) X {- a) = -{- a'^ In the same way it may be shown that * ( ± «)^ = -j- a^ ; {±aY = -{-a^; (± «)i« = + «^« ; etc. Therefore, JPrin. 27 » — An even power of a positive or a negative quantity is positive, 67. (+«)3=(+«)x(+a)x(+«) = +a^ (- ay = {— a)x{— a)x(— a) = — a^ {+ciy={-i-a)x{+a).x{+a)x{+a)x{+a) = -\-a' (— ay = {— a)X{— a)x{— a)x{— a)X(— a) = — a^ * ± a is read plus or minus a. ALGEBRAIC INVOLUTION, 63 In a similar manner it may be shown that (± «)' = ± rt^ ; (± a)' = ± a* ; etc. Therefore, Brin. 28. — An odd power of a quantity has the same sign as the quantity. SIGHT EXERCISE. Give the true values of the following expressions : 1. (+2)« 6. (+3f 11. {±ay 2. (-2)2 7. (-3)3 12. {-xY'^ 3. (+2)» 8. (-a:)* 13. (+5)3 4. (-2)3 9.{^xf 14. (-5)3 6. (±2)* 10. (+rt)» 15. (±5)* 16. (4)' ■'■(4)' '-(^J 68. {a^Y = a^xa^Xa^Xa^ = a}'' = a^''\ Therefore, Brin, 29, — Multiplying the exponent of a factor by the exponent of a power raises the factor to that power. SIGHT EXERCISE. Give the true values of the following expressions : 1. {a^r 5. (-a')' 9. (±a')« 3.(x3r 6. (-a*)« 10. (-/)» 3. (a*)' 7. {^^Y 11. (- 2=f 4. (+aT 8. (±a»)* 12. (- Vf 13. + (DT -{-i 69. (a^l^cY^a^y'c X a^h^c X a^^'^c X a'l^c^ a^ X fl2 X «2 X a' X ^3 X J3 X J3 X ^,3 X c X c X c X c [P. 8] = (a')* X {h^Y X (c)^ Therefore, Prin. 30, — Raising every factor of a quantity to a given power raises the quantity to that power. 64 ELEMENTARY ALGEBRA, SI GHT EXERC ISE. Give the true values of the following expressions : 1. (2 a^ hf 5. (+ a^ h^ c'^f 9. ( ± 2 a^ h^f 2. (2«3J2)3 Q (_ ^^2^3)4 10^ [-.a}^h^c' 3. (3 a* h^f 7. ( ± m^ ^2^5)6 1 1. (+ m^^ n^ r^ 4. (- 2 a^ a;5)2 8. (+ 0.-* 2/^ ;25)5 12. ( ± 3 w^ 0,-2)3 klO 13. 14. i^^y^y 15. (|r*2/^)« i^o^f^ 16. (±|«^J^)* Problem 1. To raise a monomial to any power. Illustration. — Eaise —^a^¥ c* to the third power. Solution : Since raising every factor of a quantity Tora^, to a power raises the quan- (— 3 «^ Z>2 d^Y = — 27 «^ ¥ c^^ tity to that power [P. 30], (-' 3 a^ y^ &f = (- 3)3 X (a3)3 x (62)3 x (c4)3, (_ 3)3 = _ 27 [P. 28] ; (a3)3_a9, {b^f = h\ and (c4)3 = ci2 j-p, 29]; hence, the result is -27a»68c« Therefore, Mule, — Eaise the numerical coefficient to the required power, and multiply the exponent of each literal factor by the exponent of the power, EXERCISE 30. Find the value of 1. (aHc^y 7. (2aH^cdy 2. {2abHY 8. {-Sa^y^zy 3. {-dan^(^f 9. {2x'y^z^y 4. {-'2xyzY 10. {omn^z^Y 5. {^^x^y^zf 11. {(a-\-h) {c^d)]^ 6. (— 4m2^^^a;)3 12. [m ay^ {a -\- h) 3)4 ALGEBRAIC INVOLUTION. 65 13. J3(« + J)2 (x-yf]'' 15. {2aH^{x--\-ifY\^ 14. {a3^c(m + w)2}* 16. \rrv'{x-ijf{x-{-yf\'' 17. (Sa^Z^^c)^ X {-%ahc'Y X (|«'^'^)' 18. (9a:2^;2-3a;»2^^2)^-v-(6r^/2)2 19. {{^7?yH''y-{^x'f^f\X^x^y^z^ 20. j(5a:*y«;z^«)2+(a;2^^6)4|_^2a:*y^2^5 21. {a^ W (^Y"" + 2 («* J* c*)« - 3 (a» ^>8 c«)3 22. (4 7? y'' zf X{^xy^ z^f - 8 {x^ y^ zf 23. (2 a;* ^ ;z^)* ^ (- ^ y^ z^f X{-2xy''zY 24. {(3a:3 2^i2«)«X (- aa^/;?^}^ -^ (- 3^^'^)^ Problem 2. To square a binomial. Principles. 70. The square of the sum of a and b, or (a + J)" = {a-^ b) (a + b) = a^ + 2ab + b^ Therefore, JPrin, 31, — The square of the sum of two quantities equals the square of the first, plus twice their product, plus the square of the second. 71. The square of the difference of a and b, or (a — by = (a - b) (a - b) = a^ - 2ab + b^ Therefore, Prin. 32. — The square of the difference of two quan- tities equals the square of the first, minus twice their prod- uct, plus the square of the second. Note. — These two principles may be stated together as follows: ''The square of a bmamial equals the sum of the squares of its terms^ and twice their algebraic product.''^ ninstrationg.— Square 2a + 35 and da^ — by^. Solutions : (2 a + 3 &)« = (2 a)^ + 2 x 2 a x 3 6 + (3 &)« [P. 31] = 4a2 + 12a6 + 96». (3a;» - 52/T = (3 a;')' -2x3a;«x5y2 + {'iy^f [P. 32] = 9 a;* -30 a;* 2^* + 25 y. 66 ELEMENTARY ALGEBRA. EXERCISE 37. Find the value of i.{x^yy 11. (%a-\-dbf 21. {ah-cdf 2. (x — yf 12. {ha — ^hf 22. {xy-\-yzf 3. {m + nf 13. (2 a; + 8)^ 23. {2pq-3 rf ^ (m- nf 14. (3x- 5)2 24. (a^ J + a b^f 6. (iP + 4)2 15. (5 + 2a:)2 25. (2 a:^ ^ __ 3 ^ ^2^ 6. (x - 7)2 16. (6 - 3 ^)2 26. (^+T - 1)2 7. (2: + «)2 17. (xy-\-lf 27. (a - 6 + 1)^ 8. {x — af 18. (a; ^ + 5)2 28. {x-\-y-\- zf 9.{^x-\-yf 19. (1-C6?)2 29. {x-\-y- zf 10. (3 a; - 2()2 20. (1 + a; ^)2 30. (a - J + cf Problem 3. To cube a binomiaL Principles. 72. The cube of the sum of a and i, or (« + ^)^ = {a + l)){a + h){a-^J)) = a3 + 3«2 j 4. 3a^,2_^ 53^ There- fore, JPrin, 33, — The cube of the sum of two quantities equals the cube of the first, plus three times the square of the first into the second, plus three times the first into the square of the second, plus the cube of the second. 73. The cube of the difference of a and b, or {a — bf = (a-b){a-b){a-b) = a^-3aH^3ab^- b\ There- fore, JPrin. 34. — The cube of the difference of two quantities equals the cube of the first, minus three times the square of the first into the second, plus three times the first into the square of the second, minus the cube of the second. Illustration. — Cube 2x-{-dy. Solutions : {2x + Syf = {2xf + S X {2xf x (3y) + 3 x (2x) x (3y)2 + (3y)3 [P. 33] = 8x^ + dQx^y + 54:Xy^ + 27y\ COMPOSITION, (57 EXERCISE 38. Find the value of 1. (x -f- ijf 8. (2 + zf 15. {^^-^ If 2. {x - Iff 9. (2: + 2 zf 16. (a:^ - 2 y3)3 3. (y + 2)3 10. (a; - 2 yf 17. (2 a;^ - 3 /j^ 4.(?^-2)3 \i.{ax^l)yf \^.{x'-?>xyf 5. (l+a:f 12. (2a; + a;y)3 19. (a;^ y + a: ?/2)3 6. {x - \f 13. (a- - x^f 20. (3 a:2 - 4 2/^)3 7. (a; y + 1)3 14. (a^ + x yf 21. (a: + ^^ + ^)' Composition, I. Definitions and General Principles. 74. A quantity composed of two or more factors other than one is a Composite quantity. 75. A quantity composed of no other factors than itself and one is a Prime quantity. 76. The process of forming a composite quantity is Composition. Note. — Involution is a special kind of composition in which the factors used are all alike. 77. Since the product of two factors with like signs is positive [P. 5], and with unlike signs negative [P. 6], show that Brin, 35, — The product of any even number of factors with like signs is positive. Prin. 36, — The product of any odd ^lumber of factors with like signs has the same sign as the factors. Prin, 37, — If the signs of an even number of factors be chaiigedy the sign of their product will remain unchanged. Prin, 38, — If the signs of an odd number of factors be changed, the sign of their product will be changed. 68 ELEMENTARY ALGEBRA. SIGHT EXERCISE. Name the signs of the products of the following sets of factors : 1. (+«)x(+«)X(+«)X(+a) 2. (- a) X{-b)X (- c) X (- d) 3. (+2)x(+3)x(+4)x(+5)X(+5) 4. (- 2) X (- 2) X (- 2) X (- 2) X (- 2) 5. (- 3) X (- 4) X (- 5) X (- 6) X (- 7) X (- 8) 6. (- 2) X (- 4) X (- 5) X (- 6) X (- 7) X (- 8) X(-9) 7. (- a) X (- J) X (- c) X(-d)X (- e) X (-/) 8. (- x) X{-y)X (- z) X (- m) X{-n)x {- p) Tell which of the following expressions are true, which false, and why : 9,l{x-y)=-l{y-x) 11. {z -x) = -{x- z) 10. —l(z — y) = l{y — z) 12. 2 (7i — m) = 2 (7n — n) 13. xx{—y)x{-z) — xxyxz 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. — x)x{-y)Xz — xxyX{—z) y-x){z-y) = {x-y){y- z) ^-y){y-z){^-^) = (^- y) {y - ^) {^ - ^) n — m) (q —p) {s — r) = (w? — n) (p — q) {r — s) V — z){z — v){v-\- z) = {z — v) (z — v)(z-}- v) ^-y){y-z){^-'^) = {^- y) (^ -y){u- ^) ^-y){y-^)(y-^) = {y- ^f m — nY (n — my =z (m — nY ^ - yf {y - ^Y (^ - ^Y (^ - ^)' = (^ - yY (^ - ^Y COMPOSITION. 69 2. Special Principles and Applications. 78. The sum of a and h multiplied by their difference, or {a -b) {a-[-h) = a^- h^. Therefore, Prin. 39,— The product of the sum and difference of two quantities equals the square of the Urst minus the square of the second. Applications. Ulustration. — Find the product of ^x^-\-^f and 27?-^ if. Solution : (2a;« + 3y») (2x« - 3y») = (2a:«)« - (3y»)« [P. 39] = EXERCISE 80. Find the value of : 1. {x-\-y)(x-y) 4. (3m + 5/t)(3m-5w) 2. {a?-\-f)(^-y^) 5. (4aa: + 3^)(4aa;-3J) 3. {ax^\){ax-l) 6. (23^y-{-dyz)(2x'y-3yz) 7. {o^y^-{-4:xy){x^y^ — 4:xy) 8. {5x'y-7z^)(63^y-{-7z^) 9. (x'y^z-'7)(x'y^z-{-'7) 10. (12 a:* - 5 /) {12a^-\-5 y^) 11. \{a + l)-^l\{(a + h)-l} 12. {{:>?-\-f)-Vz^\{{x'^f)-z^ 13. (a: + y)(.^-^)(^^ + r) 14. (2a; + 4)(2a;-4)(4a:2+16) 15. (3a: + 52^)(3a;-5i/)(9a:2_^252^) 16. («2 r^ + ^2 ^2) (^2 ^.2 _ ^2 ^2) (^4 ^T* + J* «/*) 17. (a; + 2)(a;~2)(:r + 2)(x-2) 18. (2x-yY(2x^y){%x-^y) 19. {ax-\-byY{ax — b yY 70 ELEMENTARY ALGEBRA. 79. (1) (2) (3) X +4 a: +4 X -4 a; +3 X -3 a; -3 X^-\-A:X a;2 + 4a; a;2-4a; + 3a: + 12 -3a:- -12 -3a; + 12 x^^'lx + 12 a;2+ a;- -12 a?-tx-{-l% Notice. — 1. That, in each of the above examples, we have found the product of two binomials having a like term {x), and two unlike terms (4 and 3), the latter being both positive in (1), one positive and the other negative in (2), and both negative in (3). 2. That the first term of each product is the square of the like term ; the second term is the algebraic sum of the unlike terms times the like term ; and the third term is the algebraic product of the un- like terms. Therefore, Prin, 40, — The product of tiuo binomials having a common term equals the square of the common term, and the algebraic sum of the unlike terms times the common term, and the algebraic product of the unlike terms. Application. Illustration. — Find the product of a;^ + 8 and x^ — 3 Solution : {x" + 8) {x^ - 3) = {x^f + (8 - 3) a;^ + (8 x - 3) [P. 40] = EXERCISE 40. Find the value of : 1. (a: + 4)(a: + 5) 9. (2a; + 4) (2a; + 3) 2. (a;+5)(a: + 2) 10. (3a: + 3)(3.T+l) 3. (a: + 5)(a: + 6) 11. (5 a: + 2) (5a; + 3) 4. (a: + 7)(a: + l) i2,{x^2y){x-\-Zy) 6. (a: + 8)(a: + 3) 13. {x^Q){x-b) 6. (a;+2)) 19. (3a:2_2y)(4^_5^) 9. (2a;-3)(a;-5) 20. (3a; + 5) (5^ - 3) 10. (4 a; -1) (2 a; -7) 21. (5icy - 6) (4:?;^ + 5) 11. (:?;-2«/)(3a;-5y) 22. (3^^ + 5) (Sm^ - 7) Exact Division. Definitions. 81. Any quantity that divides a given quantity without a remainder is a divisor of the quantity. 82. All the different quantities that divide a given quan- tity without a remainder are the divisors of the quantity. Illustration. — The divisors of aJ^h are 1, a, a^, b, ah, and a^J). 83. The quantities that successively divide a given quan- tity and the resulting quotients, excepting unity, are the continued divisors of the quantity. They are the same as the factors of the quantity. 84. The process of finding one or more divisors of a quantity is Exact Division. EXACT DIVISION, 7a Special Principles. 85. If we let a and h represent any two quantities, then will a-\-h represent their sum, a — h their difference, and a^ — y^y a^ — d*, a^ — h^, etc., differences of equal even powers of them. Now, we may learn by actual division, 1. That a^—l^, a^—b\ and a^—b^, are divisible hy a — b. 2. That a'—b^, a^—¥, and a^—¥, are divisible by a + J. Therefore, Prin, 42. — The difference of the equal even powers of two quantities is divisible by both the sum and the difference of the quantities. 86. a*+^, a*+J*, and a^-\-b^, are not divisible by a-\-b. a*+^, a*+*S a^d a^-^-b^, are not divisible hy a—b. Therefore, Caution 1, — The sum of the equal even powers of two quantities is not divisible by either the sum or the difference of the quantities. SIGHT EXERCI SE. Tell at sight which of the following examples will give rise to entire quotients, and why : 1. (a^-b')^(a-b) il. (4:a^ -9b^) ^ (2a-3b) 2. (a* - b') ^(a-^b) 3. {a' - ¥) -r-(a-b) 4. (a« - b') ^(a-^b) 5. {a' + b')^{a-b) 6. {a'-\-b^)-^(a^b) 7. (a« - b^) 8. (a' - ¥) 9. (a^ + b") 10. {a^ - b'') -i- {a^ -\- b^) (a^ - ¥) («3 + b^) {a^ + ¥) 12. (4^2 _ 9 5,2)^ (2a + 3 ^>) 13. (4«2-|-9J2)^(2a_3^>) 14. (16a*-81J*)-^(2a + 3^>) 15. (16 a^ + 81 b^) -^ (2 a + 3 J) 16. (16a*-81J*)-^(2a--35) 17. (16a* + 81**)-^(2a^-3J) 18. (a*2 _ J12) ^ (^2 _ J2) 19. («12 _|_ J12) ^ (^2 _ J2) 20. (flj'^ a^ — Z>* y^) ^ (a a; — 6 y) 74 ELEMENTARY ALGEBRA. 87. If we let a and h represent any two quantities, then will a-^-l represent their sum, and a^ + h^y w" + V', a} + V, etc., sums of equal odd powers of them. Now, we may- learn by actual division : That a^ + l^, a^ + h\ and a? + IP are divisible by « + J. Therefore, Prifi. d3* — The sum of the equal odd powers of two quantities is divisible by the sum of the quantities, 88. a^-{-b% a^ + b^ and a'^-^b^ are not divisible by a — b. Therefore, Caution 2. — The sum of the equal odd powers of two quantities is not divisible by the difference of the quantities, ^ SIGHT EXERCISE. Tell at sight which of the following examples will give rise to entire quotients, and why : a' + b') ^ {a^ - ¥) a' + b') -V- {a' + *') a^-^b')-^{a^-b^) «3 _|_ J6) -^ (^ -|_ ^2) a^^2'7b')-^{a + dP) a^-\-27b')^ia-3P) a'^-{-b'')-^{a + b) ^12^J12)_j.(^2^^2) ^12 4_J12)^(«4_^J4) 1. (a' + x^)- -(a + 2^) 13. ( 2. (a'-^c^)- - (05 — :?r) 14. (< 3. (a' + f)- -(a + 2/) 15. ( 4. (x'-^f)- -(:?:-«/) 16. (( 5. {8a^-{-27b^)-^{2a + Sb) 17. (< 6. {Sa' + P)---{2a-b) 18. 7. {a'-\-d2b')-^{a + 2b) 19. (i 8. (a^^b')^(a'-\-b') 20. ( 9. (l + ^-^)-(l + ^) 21. (i 10. (2:^ + l)-(^+l) 22. 11. (8 + ^)^(2 + :^-) 23. 12. (8a:3 + 27)-T-(2:r + 3) 24. (< 25. («^«^-32^'^«)-^(«2_^2^>2) 26. (8«^ + 27b')^{2a^-{-3b') EXACT DIVISION. 75 89. If we let a and b represent any two quantities, then will a — b represent their difference, and a^ — P, a^ — b^, a' — i^ etc., differences of equal odd powers of them. Now, by actual division we learn : That a^ — b^, a^ — b^, and a? — b'^ are divisible by a — J. Therefore, Prin, 44, — The difference of the equal odd powers of two quantities is divisible by the difference of the quantities, 90. a^ — P, a^ — b^, and a"^ — ¥ are not divisible by a-\-b. Therefore, Caitti&n 3, — The differ ejice of the equal odd powers of two quantities is not divisible by the sum of the quantities, SIGHT EXERCISE. Tell at sight which of the following examples will give rise to entire quotients, and why : 1. {a^ - X') ^(a-x) 8. (a^' - ¥') ^ {a^ + ¥) 2. {a' - ¥) ^{a-\-b) 9. (a'' - ¥') ^{a-b) 3. (a« - J^) -f- («2 - ^2) 10. (8a3_275^)-4-(2a-3*) 4. (a* - ¥) -r {a^ + ¥) 11. (8 a^ - f) -- (2 « - y^) 6. (a« - ¥) -^ (a^ _ J3) 12. {7? - 27 /) ■^{x'-^ y^) 6. (a^« - 5^«) -4- (a^ - ¥) 13. (a:* - 27 ^«) ^ (rc^ + 3 y^) 7. (Sa:^ _ 1) ^ (2a; - 1) 14. (32 - x"^) -f- (2 - ar^) 91. By actual division we learn that : (16a* - 81 ^>*) H- (2« - 3 ^') = 8a^ + 12a2 J + 18«^ + 27 J' (16a* - 81 ^'*) -4- (2a + 3^) = 8a3 - 12a3 J + 18«^r^ -27J» (a5-32J^«) ^{a-'2¥) = a^ -\- 2 aH^ -^ 4: aH"^ + 8a^« + 16J« (a'^ + 32J^«) -^(a + 2J2) = «i _ 2a3d2 _|_4a2^* -8a^>« + 16i8 76 ELEMENTARY ALGEBRA, By careful inspection we may observe the following laws of the quotient : JPHn, 45. — 1. The number of terms equals the expo- nent of the power involved in the terms of the dividend, 2, The terms are all positive when the divisor is the difference of two quantities, and alternately positive and negative when it is the sum, 3, The first term is found by dividing the first term of the dividend by the first term of the divisor. U. Each succeeding term may be found by dividing the preceding term by the first term of the divisor, and multi- plying the quotient by the second term of the divisor, dis- regarding the signs, 5, The last term may also be found by dividing the last term of the dividend by the last term of the divisor. Note. — The fifth law may be used as a check upon the fourth to discover errors in work. 2. Applications. EXERCISE 42. Tell which of the following expressions will give rise to entire quotients, and according to what principle. Write the quotients according to the laws in [P. 45]. 1. {x^-y^)-^{x-y) 10. (81a^-16)-^(3a; + 2) 2. {x^ - y^) ^{x-y) 11. («^ + 32) -^ (a - 2) 3. {x^ - y') ^ {x - y) 12. {x^ - y') -^ (x - y) 4. {x' - y') ^{x- y) 13. {x^ - y') -ir{x-\-y) 5. {x^ - /) -^(x- /) 14. (x^ - y^) -^{x^- y^) 6. (82:^-27) --(22; -3) 15. {^ -^ f) ^ {^ -^ y") 7. (82:^-l)^(2a; + l) 16. {x^ ^- y^) -^ {^^ - y") 8. {a^ - 27 ¥) ---{a-db) 17. {x' -Sy')^ {a^ - 2/) 9. (1 + r^) -^(1+x) 18. (x' -Sy') -T- {3^-i-2y^) FACTORING. 77 19. (32 ar^» + 1) -T- (2 t> + 1) 24. {a}'' + Z»^°) -f- (a*^ + h') 20. (a^2 ^ J12) ^ (^3 _ j3) 25. (a^*^ + Z'^") ^ {a" + Z>^«) 21. (1 + 729 x^) -^ (1 + 9 o.-^) 26. («« - 729 /) -^{a-^y) 22. (64 - ««) -^ (4 + ^2) 27. (a« - 729 /') -^ (a + 3 ^) 23. (625 a* - 1) ^ (5 a + 1) 28. (a« + 729 y^) ^ (a + 3 y) 29. (16a*-81^*)-^(2« + 3^') 30. (x« + 64 1/'') -^ (a:^ _j_ 4 ^2) 31. (8a:« + 27/)-^(2a;2^3y3) 32. (a;^<> + 32 y"^) -^ (a:« + 2 /) 33. {7?y^ — 7^\f')-^{x^y — X %f) 34. (a3a;» + J3^«)-^(aa:5_^j^2) 35. (256 7^ + 10,000) -^ (4 a + 10) 36. (a« + 729 y«) -T- («« 4- 9 ^/2) 37. (512a^^^ + c3)-^(8a5 + c) 38. (8a;« + 273^*)-^(2 2:2_^3^3) 39. (a;i« + 32 /°) -^ (a:^ + 2 y*) Factoring. I. Definitions and Principles. 92. The quantities multiplied together to produce a given quantity are the Factors of the quantity. 93. The prime quantities multiplied together to produce a given quantity are the Prime Factors of the quantity. 94. A composite quantity may have two or more sets of factors, but it can have only one set of prime factors. Thus, a^b^ = ar X lr = arh x h = ab^ X a = ab X ab = axaxb" = a"xbxb = abxaxb = axaxbxb. The last is the only set of prime factors. 78 ELEMENTARY ALGEBRA. ' 95. The process of finding the factors of a quantity is Factoring. 96. ah -^ a = l) ', but, a and h are the factors oi ah-, therefore, JPrin, 46. — A divisor of a quantity is one of the two factors of the quantity/, and the quotient is the other. 97. Since a diyisor of every term of a quantity is a divisor of the quantity [P. 15], and a divisor of a quantity is a factor of the quantity [P. 46], it follows that, JPrin, 47. — A factor of every term of a quantity is a factor of the quantity. 2. Problems. 1. To factor a polynomial having a common factor in its terms. Illustration. — Factor a^c^ — ac^ -\-a^c. Fornii {a^(^ — ac^-\-a^c) = ac{ac — c-\-a) Solution : We see by inspection that a c is a factor of each term of the polynomial; it is therefore a factor of the polynomial [P. 47]. Dividing by a c, the quotient ac — c + a is the other factor [P. 46]. Therefore, (a^c^ — ac^ + a^c) = ac{ac — c + a). EXERCISE 43. Factor : 1. a^-^ab 7. Qx^y^ - 12a^y^ - 18xy 2. ah-hc 8. 10a^ + 152;3-20a;2 3. Qi?-\-axy 9. 7 r^ - 14 r^ + 21 r* 4. a;3 _|_ 3 ^ _ 2 a; jq. 2 « (a + &) + 3 J (« + 5) 5. 3 a^ — 6 a 5 + 9 a 5-' li. a{a — x) — l{a — x) 6. 2a^x-{-4:d^a^ — 6a''z'^ 12. c (m -\- n) -\- d {m -{■ n) 13. 12 aH^ c^ - Ua^bU'' -^dQa" I^ c FACTORING. 79 14. 10/ q^ + \bp^ f-'^Opqr 15. 24a:8«^«- 36 2:5^9^43^^10 16. ^c{a'^h^)-^d{ar-\-h^) 2. To factor the difference of two squares. Illustrations. — Factor a^ — W, 7^ — ^, and 7^ y^ — oc^y^. Solutions : 1. a^ - V^ = {a -\- h) {a - b) [P. 39 j. 2. a;*-y* = (a:« + y')(x^-y') [P. 30] = (a:» + y'){x + y) (a: -2/) [P. 30]. ^. 7fiy^-Qi^y^ = a^y'^{x^- y^) [P. 47 and 46] = x^y^ {x + y){x-y)[¥.m\, EXERCISE 44. Factor : 1. a^ - 4^>2 11. «* - ;2* 21. {a -\-hY-c^ 2. 4a2 - 25 ^^2 12. «« - 5* 22. (a - xf - if 3. 9a:2_49^2 13. IGa* - 81;2* 23. (m - 7z)2 - 1 4.a2«^2_4 14. 81 / - 256 ;z* 24.^-{x + ijf 5. 16 - ^8 15. a:*/ _ a;2^ 25. c^ - (a + ^>)2 6. ic^ - 64 16. a;« - / 26. c^ - (a - ^)2 l.^if- 100 17. 625 - 2^ 27. 25 a^-(x- yf 8. 81 - JZ^ 18. a^-f 28. 16 - (;z - a:)^ 9. a^W(?- 36 19. a;i2 _ ^^ 29. 1 - (a; - yf 10. x'y^-y^ z^ 20. m^ - ?i^^ 30. 49 - 4 (a; + yf 3. To factor the sum or difference of the equal odd powers of two quantities. niustration.— Factor c^ — l^, a^ + W, and a^ — W', Solutions : \. a^ -h^ = {a-h){a> ^ ah ^V^) [P. 44 and 45]. 2. a' + fe» = (a + &)(a« + a 6 + 6«) [P. 43 and 45]. 8. a« - 6» = (a» + &») (a* - 5») [P. 30] = (« + &) (a« - a6 + 6»)(a- 6)(a« + a& + 6') [P. 43, 44, 45J. 80 ELEMENTARY ALGEBRA, EXERCISE 43. Factor the following binomials. Three can not be factored : 1. 7? — y^ 11. a^-\-l 21. a?-\-if 2. o^-\-y^ 12. a^ — 1 22. x^ — y* 3. a? — \ 13. x^ + 8 23. a^-{-x 4. a^-\-l 14. a;« — 8 24. 2;5 — Sir* 5. a;^ — 8 16. x^ + 3/5 25. a;^ i/^ + 2^ 6. ar^ + 8 16. x^ — y^ 26. m^ + 7i' 7. 8 fl^=^ + 1^ 17. a;^ + ^^ 27. 64 m« + 125 n^ 8. 8a^ — ^^ 18. 32a;5 — 2/^* 28. 3^y^ — xy 9. 27 a^- 8 P 19. 16 x^ + 81 y^ 29. a;^« + a; ^/^ 10. 27a^ + 8^^ 20. 8a;«-27/ 30. (a^ + ^j^-^^^ 31. (x + i/)3 + z^ 33. a:^ + (^ + ;2)' 32. X^-{y + zy 34. (« + J)3_(c + c^)3 4. To factor a trinomial that is a perfect square. Ulustrations. — 1. a^ + 2ab-\-b^ = {a + h){a-]-b), since {a-]-by = a^-\-2ah-^h^ [P. 31]. 2. fl2 _ 2 ff J -f Z>2 _ (^ __ J) (^ _ j)^ since (a - If = a2_2fl^J + Z/2 [P. 32]. 3. 4a2 + 12a^5 + 9J2=(2« + 3^)(2a + 3^), since {2a-^^l)f = ^aJ'-\-12ab-\-^h^ [P. 31]. 4 4fl2-i2a$ + 9Z'^=(2«-3Z')(2a-3^), since (2«-3^f = 4a2_i2«J + 9^2 [P. 32]. 98. A trinomial is a perfect square when two of its terms are perfect squares, and the other term is ± twice the square root of their product. FACTORING. 81 EXERCISE 46. Factor the following trinomials. Three can not be factored : 1. a? + 'ilxy-\-y^ 13. IQx" -nx'y'' ^^ly^ 2. T^-'ixz-irz'^ 14. ar* + 22^2 + 4 3. x'^2x-{-l 15. x^''-\-%o^-\-l 4. a;2-4a; + 4 16. T^y"" ■\-4.x''y^ ^^.x" y^ 6. a:2_|_i8a; + 81 17. a:^ - 2 a:* 2^ + 3^ 6. 4a;8_i2a; + 9 18. 4ir* + 14a:2_|_49 7. 9a:2_^i2a;y + 4«^ 19. 4 a:« + 12 a:^ _^ 9 8. 25a:2_pioa; + l 20. 9:r« - 36ar*^2_|_ 35^^ 9. a;*-12a^ + 36 21. lOOa^ - 110 a; y + 121 y* 10. Q^-%j?^lQ 22. a;*^2_^2a:2y3^_^^^2 11. a:« + 2a:3^^yj 23. a;« + 42^6 - 4a:3^ 12. a2j2_2«Jcc? + c2j2 24. 16 2:8 + 256/ -128 a:*/ 6. To fjEUjtor a trinomial that io the product of two binomials having a like term. Illnstratioiis. — 1. a^ 4_ 3 05 4. 2 = (rt + 2) (ff + 1), since (a + 2) ( « + 1) = a2 + 3« + 2 [P. 40]. 2. a^ _ 5 flf _j_ 6 = (« - 2) (a - 3), since {a -2) {a- 3) = a''-6a + 6 [P. 40]. 3. a^ + 2 rt - 8 = (a + 4) (a - 2), since (« + 4) (a -- 2) = a^-\-2a-S [P. 40]. 4. (4a2-.4fl5_i5) = (2a + 3)(2a-5), since (2a + 3)(2a-5) = 4a2-4«-15 [P. 40]. 99. A trinomial is the product of two binomials having a like term, when the first term is a square, and the last term is the algebraic product of two factors whose sum. 82 ELEMENTARY ALGEBRA. multiplied by the square root of the first term, will give the middle term. The square root of the first term is the like term of the binomials, and the factors of the third term are the two unlike terms. EXERCISE 47. Factor the following trinomials. Three can not be factored : 1. ic- + 8a; + 15 13. 4.x^-\-ix^^ 2. x^-\-5x + 4c 14. 4a^-\-Ux-{-12 3. a^-\-6x-{-8 15. 9a^+9x-\-2 4. a^-7a-\-12 16. x^ -{- 4: a x -}- 3 a^ 5. a^-9a + U 17. x^-2ax-15a^ 6. «2_i3^4-40 18. 4:X^ — Sax — 3a^ 7. a;2_^2a;-15 19. 9 y^ -\-3y z - 2z^ 8. x^-^3x-28 20. dea^-\-24:bx-6i^ 9. a^-i-6x-16 21. 4:a^a^-4:ax- 16 10. x^-4:X-6 22. x^-i-Hx^-12 11. x^-4=x-21 23. a^-7aa^-\-12a^ 12. a^ — 2x — S0 24. 4:a^ + Sa^ — 3 6. To factor a trinomial that is the product of any two binomial factors. Illustrations. — 1. 2x^-{-6x-{-2 = {x-\-2){2x-]-l), since (x + 2)(2x + l) = 2x^-{-6x-{-2 [P. 41]. 2. 6a^ — 13x-{-6 = {2x-3){3x — 2), since {2x-3){3x-2) = Qa^-13x-]-6 [P. 41]. 3. 2x^-{-x — lb = {x-{-3){2x-5), since {x-]-3){2x-5) = 2a^-\-x-U [P. 41]. 4. 6 a;2 — 11 2: - 10 = (2 a: — 5) (3 a; + 2), since (2x-5){3x-\-2) = 6x^-nx-10 [P. 41]. FACTORING. 83 100. The first terms of the factors are the factors of the first term of the trinomial, the last terms of the factors are the factors of the last term of the trinomial, and the last terms of the factors are so arranged with the first terms that the algebraic sum of the products obtained by multi- plying the first term of each factor by the second term of the other will give the middle term of the trinomial. Note. — This and the following problem may be omitted until the class reaches page 192, if desirable. EXERCISE 48. Factor the following trinomials. Two can not be fac- tored. Why ? 1. 2a;2_j_5^_^3 jq. ISa^-f 20a-35 2. 2 2:2 _|_ 11 ^ _|_ 12 11. 2 ic2 _|_ 19 ^ _ 35 3. 6ar + 7a: + 2 12. 2a^-\-ab-h^ 4. Qa?-\-llx-{-^ 13. 2a2-|-5aJ_|-2J2 5. 2x'-'tx-\-Q 14. Qx^-llxy-^^y^ 6. 2 a,-2 + .T — 6 lb. Qu^-\-buv — Qv^ n. 2x'-x-15 16. 23^-7xy-dy^ 8. 12a:2^iia._i5 ^^^ 6a^-lla^-d6 9. 6a^-{-a-16 18. 2a^b^ + ab-Q 7. To factor a trinomial that is the product of two ttino- xnials of the form of x^ + xy + y^ and x^ — xy + y^. Solution : The product of x^ + xy + y^ and x^ — xy + y^ is a^ + x*y^ + y* ; therefore, a trinomial is the product of two trinomials of the form of x* + xy + y^ and x^ — xy + y^ when all its terms are positive squares and the middle term is the square root of the product of the other two. Rule. — The factors may be obtained by extracting the square root of each term and making the middle term of one factor positive and that of the other negative. 84 ELEMENTARY ALGEBRA. Illustrations. — 1. a*4-«2 4-l = («2-|-^_|-l)(a2_«-|-l). EXERCISE 49. Factor the following trinomials. One can not be fac- tored : 1. a;* + a:2_|.i 10, %lx^ + mQ?y^^Uf 2. a;* + 4a;2 + 16 ii. a^ -\- a^ y^ -\- y"^ ,16 3. a* + a2^>2_|_^4 12^ ttH-^ + a^ic^^g.^^ 4. a^-^aW-\-(^ 13. a'^y^-^a^y^-\-a^f 5. 16a* + 4a2 + l 14. a8&8_^4a*5* + 16 6. a8 + 4«4j2 + i654 15. 81 + 9^2 _j_j4 7. a^ + a;*/ + / 16. 625 + 25 a.-^ / + a;* / 8. a:8 + iC*«/^ + «/^3 17, 256 + 16;2* + ^« 9. ic^ — a^^ + / 18. a;*/ + a:2^2;2_^/i2* Ulnstrations.- 8. To factor polynomials. 1. Factor ax '-{-hy-hx-ay. SoltLtion : ax + hy — hx — ay — ax — hx — ay + &2/ = x{a — h) — y {a — h) = {a-b){x-y) [P. 46, 47]. 2. Factor a? -2a; 2/ + /-;^^ SoltLtion : ic^ — 2a:y + 2/2 — £2 = (x^-2xy + y^)-'Z^ = (x-yf-z^ = {x-y-z){x-y + z) [P. 39]. 3. Factor x^ -2/'-2«^2;-;z2, Solution : x^ — y^ — 2yz — z^ = x^-(y^ + 2yz + z^) = x'-^(y + zf = {x + y + z){x-y + z) [P. 39] = (x + y + z){x-y-z) [P. 23, 24]. FACTORING, 85 EXERCISE 80. Factor : 1. ax-\-ay + hx-\-ly ii. or -\-^ah-\-lr - c^ 2.hx — hy-\-cx — cy 12. a? — ^x-\-l— y^ 3. ax — az — bx-{-bz 13. a^ — y^ — 2y — 1 4. ab + 2i-\-3a-\-Q 14. 4:0^ + 4:xy -\-y^ - z^ 5. 9-^3x — dy — xy 15. 4:Z^ — 4:0^ — 4:X — 1 6. 2ax-i-3ay-{-4:bx-^6by 16. a^-\-2ab-{-b^ -16 7. 6ax-\-4:ay—dbx — eby 17. 25— a^ — 2«a; — a^ 8. aba^'j-2ax-\-3bX'i-6 18. a^ — a? -\-2x — I 9. «a;y4-6a — Ja;y — 6J 19. a:* + 2a:2^2_^^ _ ^2 10. a^ 7? ^ a^ y^ -V^ Qi?-W y^ 20. m* -j9* -2fq- q^ Miscellaneous Examples. EXERCISE 81. (Take out monomial factors first.) Factor : 1. a^b — ab^ 12. ^a7^ — 6a3^f-{-3axy^ 2. 3fl3 — 12a 13. 4ty'^-i-Sy* + 4:y 3. 2a^-2ab^ 14. 2 a:^ _^ iq ic^ + 12 a; 4. 3a35-f3J4 16, a:3^_92^y_^20rz;«^ 6.x^y — xy^ 16. 4: a^b-{- 4: a^b — lGSab 7. d}b-aV 18. a» + a2(& + c)^ 8. 2a»J2_^2a2J8 19, a^^ - a^c^ - 2ac3 9. 6a« + 10ay + 5i/2 20. 4a5 + 8fl + 12 J + 24 10. 2a3c + 12a2c + 18ac 21. axy — bxy—ay^-{-by^ 11. 7? y^z — 2oc^yz-\-xz 22. a^ — y^ — x-{- y 86 ELEMENTARY ALGEBRA. 23. (xJryf-x-y 36. «2 + 2 a + 1 - Z»2 24. {a^hfx^-c^x^ 37. aH^-a^-2ah- 25. ^-%x''y-^xy'-x%'' 38. \-x^-%xy-y^ 26. a^-^x^y-]-^xy^-y^ 39. a^ — m^ -\-2mn — 27. a:* + 6 a;3 + 12 2;2 + 8 ic 40. 1 - (a + ^')^ 28. a*/ - 1 41. a^ -\-aH^-^ h^ 29. a? y^ — z^ 42. {x + «/)^ — (ic — yY 30. a;i2^y2 43^ a» + «8 2/7 31. x"^ - y^^ 44. (a + 2')2 - 2 (« + J) + 1 32. TTv'n^ — m 45. a;^ — (a; + i/ + ;2;)^ 33. 121 «^ + 144 ¥ + 264 a^ J2 45, a;* - (2/ + ;^)* 34. 16 a^ -f 8 tt ^ - 3 ^2 4n. 7?-\-^x^y-\-^xy^^y^ 35. 3 m 7^ — a m ?^ + 2 a — 6 48. 1 — (« — ^)^ 49. aHc-^ahH^aHd-^-aTy'd 50. 3a^ — 15a:?/ — 2^a;+10J^ Highest Common Divisor. I. Definitions and Principles. 101. A divisor of each of two or more quantities is a Common Divisor of the quantities. 102. The common divisor that contains the greatest number of prime factors is the Highest Common Divisor. 103. Quantities that have no common divisor except one are prime to each other. 104. Since a quantity equals the product of its prime factors, it is divisible by the product of any two or more of them ; hence, too, each of two or more quantities is HIGHEST COMMON DIVISOR. 87 divisible by the product of any two or more of their com- mon prime factors ; and therefore, Prin, 48. — The highest common divisor is the product of all the common prime factors, 106. The abbreviation H. C. D. stands for highest com- mon divisor. 2. Problems. 1. To find the highest common divisor of monomials. niustration.— Find the H. C. D. of Ua^l^d", Ua^l^(P, and ^^Qa^'Pd'K Solution: Ua'^Pd' =2 X 2 X 2 X 2 X a^ X h^ X C*' 24a3J2c2 =2x2x2xSXa^Xb^X(^ ^QaH^d^ = 2 X2x3x3x«*XJ'Xrf2 .-. H;0. D. =2x2Xa^X*^ = 4«2 52[p. 48J. Another Solution : The H. C. D. of the coefficients is 4 ; of the o's is a' ; of the 6's is 6* ; and c and d are not common to the three quantities. Therefore, the H. C. D. is 4 a' 6*. Mtde. — Find the highest common divisor of the numeri- cal coefficients, annex to it the different common literal fac- tors, giving each the lowest exponent it has in any one of the quantities, EXERCISE 82. Find the H. 0. D. of : 1. ^x^y, 12xy^, and 24a;2^2 2. 15aH^c^, 25 a- b^ and 30^2^2 3. 20xy^z, SOs^yz, and AOxyz^ 4. 20aH\ 26an^c, and S5aH^(^ 6. IS m^n\ 24:am,^n^ and d6bm^n^ 6. {a-\-by, (a-i-by, and {a -{-by 7. 3{x-{-yy, 6{x-\-yy, and 9(x-{-yy 8. m{m-\- ny, m^ (m + ny, and m^ {m -j- n) 88 ELEMENTARY ALGEBRA, 9. 2ax{x^-\-y% 4.ax^x'^y^)\ and ^ a^ a? {a? -\- y-f 10. 4 (m — ny, 6{n — rnf, and 8 (m — vif 11. xy{a — if, o?y(a — lY, and if{a — bf 12. (fl^ + ^') (a - ^), {a + 5)2 (^ - hf, and (a + 5)2 {a - hf 13. 2 (rr - yf, 4 (a; - 2/)^ 6 (a; - ^)S and ^{x- yf 14. 3a(m — 7e)^ Qah{m — nf, 9a^(m — n)\ and 12 aW {m — nf 15. (m + ?i) (m — 7i), (m -\-n){n — m), and (m + nf (m - nf 16. {p -q){a- h), {q - p) {a - I?), and {p - q) {i - a) 17. {a - J))\ (b - of, and (a -l)){h- a) 18. ah{a — V), —l){h — a), and a {a — hf 2. To find the highest common divisor of polynomials. ninstration.— Find the H. C. J), ot xf^ — f, x^y — xf, and Q? — 7? y — X y^ -\- y^. Form. ^-t^i.x^-f)i.^^^t) = {x-y){x^y){x^-^f) 7^ y - xy^ =z xy {x^ - y^) = xy {x - y) {x -\- y) a? — x^y — xy^-\-y'^ =:a?{x — y) — y^{x — y)=^ (^ - y^) i^-y) = (^ + y) (^ -y)(^- y) H. Q.T>. = {x^y){x-y) = x'- / [P. 48]. Solution : We resolve the quantities into their prime factors, and observe that a + 5 and a — h are the only common factors ; therefore, {X + y){x- y\ or x^ - y\ is the H. C. D. [P. 48]. EXERCISE 63. Find the H. C. D. of : \. a-{-h and a/^ — h^ 3. {a — hf and a^ — W 2. (a + hf and a* - 5* 4. (a + hf and a^ + 5^ 5. x^ — xy, x^ — y^, and x^ — 2xy -\-y^ HIGHEST COMMON DIVISOR, 89 6. {x -{- yY, 7? — y^, and o? -{-xy 7. or — y^, 7? — 1xy-\-y'^, and x^y ^x^ 8. {a; + y)^ a?-\-7?y, and 2:^4-^^ 9. a;2_42:, x^-^x + lQ, and a;2_2a;-8 10. «* + 4a3 + 4a2, a35-4«^, m^a^b-^ba^h-^Qa^l) 11. a,*^ — 8 a:^ + 15 a;, ic^y — 8a;^ + 15?/, and x^z — ^xz-\-lbz 12. 3a-2-3/, cc*- 2 2:2^2^^^ a^jj x'y-f 13. a:^ + a: y 4- ^ ^> ^ V -{- V^ -\- V ^i ^^^ ^ — (^ + ^)^ 14. a:2 + a;_6, ar + 7a;4-12, and x'-^x-lb 15. a:^ + 27, a^ + 5 a; + 6, and a:^ _|_ g ^ _^ 9 16. 3? -\-ax-\-hx-\-al) and 3? -{-ax-\- cx-{-ac 17. vf? — v^, am-{-an-\-lm-\-ln, and w^ + 2 ?/i 7i + w* 18. 7^-\-xy^, x^ -^%7?y^ -\-y^, and aa:^ + «y^ 19. a:* + 4 a:^ y + 4 3/2, a^ — 4^^, and a:^y + 2a;^* 20. 7? —{y-\- zY, y^ — (x-{- zf, and z^ ^{x-{- yf 21. a;* - (2/ + 2)«, y^-{x^ 2)^ and 4 - (a; + 3/)^ 22. a;^^-y^ ^y + 2«*y* + a;y^ and 23. 3^y-\'12x^y-}-Z5xy, ic^ -f- 3 a:* - 28 a:^^ and a:^;2 — a:^;2! — 56a?jj; 24. 6a:2 _|_i0a;-24, 2a;2_2a;_24, and 8a;« + 22a;-6 25. x^-y^ a^ + a:2«/^ + /, and a:^ y + 3^5 ^^2 _p 3. ^ 26. a^ + 5x^-]-6x, a^y-^z^y-exy, x^-x^-12a^ 27. a:*-16, ai^-^Sx^ + 16, a:*4-2a:2_8 28. a^'-a', x^'-2a^o?-\-a?', x' - a7? - a'x-^-a^ 29. x^ — f, ^-\-^y + y% x*-\-3^y^-\-y* For highest common divisor by successive division, see Appendix. 90 ELEMENTARY ALGEBRA. The Lowest Common Multiple. I. Definitions. 106. A quantity that exactly contains a given quantity is a Multiple of the quantity. 107. A quantity that exactly contains each of two or more given quantities is a Common Multiple of those quantities. 108. The common multiple that contains the least num- ber of prime factors is the Lowest Common. Multiple, 109. The abbreviation L. C. M. stands for lowest com- mon multiple. 110. The L. 0. M. of aH^c, aW(?, and aHH^ must contain each of these quantities [Art. 107] : ah^c^ =aXhXhxhXcXc a^h^c^=zaXaXhXhXcXcXc To contain a^ W c, the L. 0. M. must contain the prime factors a, a, a, h, h, c. To contain a ¥ c^, it must contain the additional prime factors l and c. To contain a^l)^(^, it must contain the still additional prime factor c. Since these are the only factors required to contain each of the quantities, and all are necessary, the lj.G.lA. = aXaXaXhXiXhXcXcXc = a^b^c^, Therefore, Trin. 49. — The loioest common multiple of two or more quantities equals the product of all their different prime factors, each taken the greatest number of times it occurs in any one of them. LOWEST COMMON MULTIPLE. 91 2. Problems. I. To find the lowest common multiple of monomials. niustration.— Find the L. C. M. of UaHH, 3Ga^JV, and bQaH^c^, Solution '. %4:a^h^c =2x2x2xSXa^Xb^Xc 3eaH^(^ = 2x 2 X 3 X3 X«^ X &2><^3 56a2^V = 3x2x2x 7 X a^ X ^'* X c^ ,-. L. CM. =2X2X2X3 X3X7X«'^X^'XC3 = 504:a'b^(^ [P. 49J. Another Solution : The L. C. M. of the coefficients is 504 ; of the a's is a* ; of the 6's is b* ; and of the c's is c\ Therefore, the L. C. M. of the quantities is 504 a' 6^ c^. Utile, — Find the least common multiple of the numeri- cal coefficients ; annex to it all the different literal factors found in the quantities, giving each the highest exponent it contains in the quantities. EXERCISE 84. Find the L. C. M. of : 1. lOx^y, 15xy\ and 20a^y^ 2. na^b\ ISai^c, and 24:a'^c^ 3. 24:aa^, d2bxy, and 48c/ 4. 22a:*/, Sdx'z^, and Uy^z"" 5. 2Aab^a^, 36a^x^z, and ^Sb^z^ 6. 48m^w^ bem^nx, and 63^^0:3 7. (a-\-by, (a + J)^ and {a-\-by 8. 25{x + yY, 60{x-i-y)^ and 100(x-\-yy 9. a^ (x — y), a^ {x — yf, and a* {x — yY 10. 8 a* (a; + ;?)*, 12 ah {x-^-zf, and 24.b^x + zy II. Qa^^^iT'-i-y^y, lSx^{a^-{-y^y, and 36a^z(a^-}-y^y 12. {a-{-b){a-b), {a^ + b^){a-{-b), and (a"" -\- b') (a - b) 92 ELEMENTARY ALGEBRA. 2. To find the lowest common multiple of polynomials, niustration.— Find the L. C. M. of a^ - V^, a" - ^>^ and Form. a^-i^ =(a-^b)(a-b) a^ -\- at = a {a -{- h) L. C. M. = a {a-\-h) {a - h) {a^ -{.ah-{-¥) Solution : To contain a? — b^, the L. C. M. must contain the prime factors a + b and a — b, and to contain a^ — b^ it must contain the additional factor a^ + ab + b^, and to contain a^ + ab it must con- tain the additional factor a. Therefore, the L. C. M. is a{a + b) (a-b){a^ + ab + b^). EXERCISE SS. Find the L. C. M. of : 1. {a + by and a^ - W 3. o? - / and ar* - ^ 2. {a — by and a* — 5* 4. x^ — y^ and x^ — y^ 5. x-\-y, 7? — y^, and o? -^-^xy -\-y^ 6. x — y, x^ — y^, and a? — 2xy -\-y^ 7. x^ — y^, x^ — y^, and x^ — 2xy-{-y^ 8. {x -\-a){x-{- b), {x-{-a){x-\- c), and (x -\-b){x-\- c) 9. a (a — b), b{b — a), and — c(a — b) 10. {a — b){b — c), {b — a){b — c)y and {b — a){c — b) 11. {x-{-yyy ^'^ + ^^ and x^ — y^ 12. a;2+5a:+6, a;2 — 2a; — 8, and a;2 — a;-12 • 13. a^ + 3a;-4, x^-Qx-\-b, and a:^ _ ^ __ 20 14. « m 4- ^ ^^ + ^ ^^ + ^ ^ and ap-\-aq-\-bp-\-bq 15. ax — bx — ay-\-by and ax — ay -\-bx — by 16. «2 _ (J ^ ^)2^ ^' - (« + c)^ and c2 - (« + ^)' 17. a3 + 3«2^' + 3«Z»2_^J3 and a^ _ ^ J2_^«2 j _ 53 CANCELLATION. 93 18. x^-^t/y a:* + ari/2_j-y4, and 7? — y^ 19. 2a:« + lla; + 15, ^7?-\-x-10, and x'-\-x-Q 20. 6ar + 13x + 6, 6ar — 5a;-6, and 4:7? — ^ 21. ax^ — ay^f 7^ — xy^, and a.-^y + ^ 22. a;^ + 5 ^ + 6 ^^ 7?y — 7? y ^^xy, and 7?y — ^y 23. 2:^ + 2;^ + ?/^, 7? — xy-\-y^, and re* + ^2/^ + ^ 24. 2:^ + /, a:* — ^^3/^ + /, and a;^ + / Cancellation. I. Definitions and Principles. 111. Multiplying a quantity by a factor is called insert- ing a factor. 112. Dividing a quantity by a factor is called eliminat- ing a factor. 113. Crossing out a quantity and writing in its stead the result obtained by inserting or eliminating a factor is Cancellation. 114. ah Xac=^a^hc. If we now eliminate a from a h and insert it in a c, we have h X oj^Cy which also equals a* h c. Therefore, Brin, 50. — Dividing one quantity and multiplying another by the same factor does not alter their product. 115. ahcd -^ ah = cd. But if we insert l in ahcd we have a¥cd-i-ab, which equals bed. Also, if we eliminate b from a b, we have abcd-r-a, which equals bed. Therefore, Prin, SI. — Multiplying the dividend or dividing the divisor multiplies the quotient. 94 ELEMENTARY ALGEBRA. 116. ahcd-ir ah = cd. But if we eliminate c from ahcdy we have ahd-^ ah, which is d. Also, if we insert c m ah, we have ahcd-^ ahc, which is d. Therefore, Prin. 52. — Dividing the dividend or multiplying the divisor divides the quotient. 117. ahcd-T- ah = cd. If we now insert h in both abed and a h, we have ah^ cd-^ aW, which is c d. Also, if we eliminate h from both ahcd and ah, we have acd-^ a, which is c d. Therefore, JPrin, 53, — Multiplying or dividing hoth dividend and divisor hy the same quantity does not alter the quotient. 2. Problem. To multiply or divide by cancellation. niustratioiis.— 1. Multiply 36 by 25. Solution : Since dividing one quan- tity and multiplying another by the Form, same factor does not alter their product 9 ]^ QO [P. 50], we divide 36 by 4 and multiply *0 w ^rt __ qqq 25 by 4, and obtain 9 x 100, which is 900. 2. Multiply {a + hf by {a - h). Form, a-^h a^-h^ (ar-\^f X («^--^) = a^-a¥-\-aH-h^ Solution : Dividing {a + hf by {a + b), and multiplying {a — h) by (a + I) [P. 50], we have {a + b) x (a" - ¥\ which \s a^-ab'^ + a'^b- b\ 3. Divide {a^ -f ¥) {a' - h^) by {a + hy. Form. (a^-\-h^) {a^-h^ ) _ (^H^) (a^-ah + h^) (^H-J) (^ - ^) ^ a^-2a^h-^2ah^-h\ Solution : Dividing both dividend and divisor by (a + b){a + b) [P. 53], we have {a^ — ab + b^) {a — b), which is a^ — 2a^b + 2ab^ — bK CANCELLATION. 95 EXERCISE 86. Solve by cancellation : 1. 44 X 25 4. 42 X 16% 7. 26% -r- 6% 2. 36 X 15 5. 56 X 12% 8. 35% -r- 7% 3. 27 X 33% 6. 48 X 36 9. 144 -4- 36 16 x — y^^' 25X36 (,; + y)2(^_y). 6%X7% '^' :^-y^ '^ 2%, "• i^^T (a:»-.4)(a:^ + 6a: + 8) (a^H-2a:-8) (g»+'7a; + 12)(a:^ + l l a; + 30) "• (a;4-6)(a:2_^3^_^15J ((x'-\-ac-^al-\-lc){W^lc-\-ld-\-cd) {al-\-ac-\-l^-\-ho){ah-\-ad'\'l)c-\-cd) (a + * + c) (a + * - c) Find the value of the following expressions, when a = 10, * = 8, c = 6, and tZ = 4. (a^-2gZ' + ^^)(a^ + 2g^ + Z>g) a^-(bJrc)\a + h-\-c ^'- (^^{a-\-hf^ a-b-c ^ {h-^cf-d^^ a-h-^c 23 (^^-^^H^M:^) ^ c^d (a-b){c-\-d) ^a'-^ab^l^ \{a + bY^(c^dY] \{a^bY-(c^dy\ (a-{-b-\-c-\-d){a-b-\-c-d) 96 ELEMENTARY ALGEBRA. Simultaneous Numerical Equations of Three Unknown Quantities. Elimination by Addition and Subtraction. Direction to Pupil. — To solve three equations of three unknown quantities, select one of the unknown quantities to be eliminated. Combine any two of the equations so as to eliminate this quantity. Then combine either one of these two with the third in like manner. You will then have two equations having only two unknown quanti- ties, which you already know how to solve. Illustration. — Solve : Solution 6x-4:y-^2z=Q (A))- 2x-{-Sy-4:Z = ll (B)y Sx + 2y + 6z = d7 (C)) : Multiply (A) by 2 and bring down (B), 10x-8y + 4z = 12 (1) 2x + Sy-4z = n (B) Add (1) and (B), 12x-5y = 2S (2) Multiply (A) by 5 and (C) by 2, 25x-20y + 10z = d0 (3) Qx + 4:y + 10z = 74: (4) Subtract (4) from (3), 19 a; -24 2/ = -44 (5) Multiply (2) by 24 and (5) by 5, 288 a; - 120 y = 552 (6) 95a:-120y=-220 (7) Subtract (7) from (6), 193 a; = 772 (8) a; = 4 * Substitute the value of x in (2) and y = 5 Substitute the values of x and y in reduce, (B) and reduce, z = S Verification : Put 4, 5, and 3 for x, y, and z, in (A), (B), and (C), 20 - 20 + 6 = 6 ; which is true. 8 + 15-12 = 11; 12 + 10 + 15 = 37; * Substitute means put in place of. CONCRETE EXAMPLES, 97 EXERCISE 87. Solve : ^-yj^z= 5V 3a;-4t/-f22= 4V x-\-y-z= 3) 5:r + 3i/-7. = -16) 2:^ + 3^-. = 12) 8.5a;-62/ + 2.= 7| 3^ + 3^ + . = 16) 2a:-3y + 4.= 10) 9. 2x-6y-3z = -14:) 5x-7y + Gz= 4| 3^_4_5, = ^4 2^ + 4^-5.= 6V 7^^6y+8.= 27 a; + 3i/ — 2;2 = 10 ) -^ * 10. a: + ^ = 9 ) 2:,_3y + 4^ = 10) x + z = 10} 3^ + 2;/-2. = 17V y + z = ll) x + 5y- z = 22) ^ ^ * II. x + y-z= 3) — dx-\-2y-6z = —26\ x — y-\-z= 9V 3x-2y-\-U= 21V - x-{-y + z=ll) 4:X — 2ti—^z——2\) ^ ^ . , ^ -^ *12. a;H-y + ;2; = 6 x + 2y — 3;z = — 1) x-\-y^u=l 4a; — 4?/— z= 8/- a;+^ + w=8 3x-\-^y-\-2z = -b) y-\-z-\-u=9 ) Concrete Examples involving Simultaneous Equa- tions of Three Unknown Quantities. EXERCISE 88. 1. The suDi of three numbers is 90 ; twice the first, minus three times the second, plus four times the third, is 200 ; and three times the first, plus twice the second, minus the third, is 10. What are the numbers ? Suggestion. — Let x = the first, y = the second, and z = the third. * The 10th, 11th, and 12th are most readily solved by comparing each equation with the sum of the three. 98 ELEMENTARY ALGEBRA. 2. The sum of three numbers is 90 ; twice the first, plus three times the second, is 30 less than four times the third ; and the third is 10 less than the sum of the other two. Required the numbers. 3. A, B, and together have $3500. If A had twice as much, B three times as much, and C four times as much as now, they together would have $9900 ; and twice C's amount exceeds the sum of A's and B's amounts by $400. How much has each ? 4. A, B, and together have 500 acres of land ; if A buys 25 acres from each of the others, he will have 50 acres more than B and 25 acres less than C. How many acres has each ? 5. The cost of two bushels of corn, three bushels of oats, and four bushels of rye is $5. 60 ; of three bushels of corn, two bushels of oats, and one bushel of rye, $3. 40 ; of four bushels of corn, one bushel of oats, and five bushels of rye, $6.80. Required the price per bushel of each kind of grain. 6. If a horse and cow together are worth $160, a horse and sheep $108, and a cow and sheep $68, what is the value of each ? 7. A has as many horses as cows and sheep together ; twice the number of cows is 12 less than the number of horses and sheep together, and the number of horses and sheep together equals four times the number of cows. How many of each has he ? 8. The sum of A's, B's, and C's ages is 60 years ; the sum of A's and C's is twice B's ; and C's alone equals the sum of A's and B's. Required the age of each. 9. A man has two horses and a saddle, together worth $180. The first horse saddled is worth twice as much as the second horse ; the second horse saddled is worth $20 less than the first horse. What is the value of each horse and the saddle ? CHAPTER l(. ALGEBRAIC FBACTIOJ^S. Preliminary Definitions. 118. An algebraic fraction, in the most general sense, is an expression denoting that one algebraic quantity is to be divided by another. The dividend written above a horizontal line is called the Numerator, and the divisor written below the line is called the Denominator. Illustration. — Thns, ^ , read a-\-h divided by c — J, C Uf in which a, h, c, and d may have any values, positive or negative, integral or fractional, is an algebraic fraction. 119. The numerator and denominator are called the Term.s of the fraction. 120. In a very limited sense, in which the terms are restricted to arithmetical integers, an algebraic fraction may be defined as **a number of equal parts of a unit." niustration. — Thus, -r, read a — &th, denotes a of the h equal parts of a unit. 121. Since a of the h equal parts of a unit is equivalent to one of the h equal parts of a units, or -^ of 1 = ^ of a, -T in the more restricted sense may still be regarded as an expression of division. 100 ELEMENTARY ALGEBRA. 122. An algebraic fraction is usually preceded by the positive or negative sign to indicate whether it is to be used additively or suhtr actively, 123. The value of a fraction is the result obtained by performing all the operations indicated. Illustration. — Thus, the value of — -^ when a = ~ 6 — r> and J = + 2 is -—-=-(- 3) = + 3. -r <> 124. The apparent sign of a fraction is the sign pre- ceding it, the real sign the sign of its value. 125. An Integral Quantity in the literal notation is a quantity that is not fractional in form. It may be in- 2 tegral or fractional in value ; as, a = 8, or « = — . o 126. A Mixed Quantity in the literal notation is one that is partly integral and partly fractional in form ; as, c 127. A Proper Fraction in the literal notation is one that can not be reduced to the integral or the mixed form ; a + h as, — ' — . c 128. An Improper Fraction in the literal notation is one that can be reduced to either the integral or mixed form ; as, — -^ = a — o, or — ' — = a-\ — . a-\-h a a 129. A Compound Fraction is a fractional part of an integral or fractional quantity ; as, -^ of c, read the a — hth. ft Q part of c ; or -^ of -, , read the a — hih. part of c divided by^. 130. A Complex Fraction is a fraction one or both of whose terms are fractional in form. REDUCTION OF FRACTIONS. 101 131. The inverse of a fraction is the fraction resulting from an interchange of its terms. Illustration. — Thus, - is the inverse of -7. a 132. The reciprocal of a quantity is unity divided by the quantity. Reduction of Fractions. Definition. 133. Reduction is the process of changing the form of a quantity without altering its value. Principles. 134, Since multiplying the dividend or dividing the divisor multiplies the quotient [P. 51], it follows that, JPrin, 54. — Multiplying the numerator or dividing the denominator multiplies the value of a fraction, a Bemark. — If — be regarded as a of the b equal parts of a unit, b it is evident that multiplying a by n and leaving b unchanged will multiply the number of equal parts taken by n without altering their size, and therefore will multiply the value of the fraction by n. And, dividing 6 by w and leaving a unchanged will divide the number of equal parts into which the unit is divided by n; and, hence, make each part n times as great without changing the number of parts taken, which will also multiply the value of the fraction by n. SIGHT EXERCISE. Name at sight the products in the following examples : 1. ^ X c 4. — , X c 7. 2 ,8 X (a + b) b cd a^ — 0^ ^ ' 3.^x. e.«^>(a-*) ..,-Sx. 102 ELEMENTARY ALGEBRA, (a-\-'bY , ,x x — a / , X 10- 5^. X (« - i) 12. ^-^^ X (« + «.) 136. Since dividing the dividend or multiplying the divisor divides the quotient [P. 52], it follows that, JPrin, 55, — Dividing the numerator or multiplying the denominator divides the value of a fraction, Remark. — If — be regarded as a of the b equal parts of a unit, h dividing ahj n and leaving h unchanged divides the number of equal parts taken by n without changing their size, which evidently divides the value of the fraction by n. And, multiplying 6 by n and leaving a unchanged makes the number of equal parts into which the unit is divided n times as great, and, therefore, each part only — as great, without altering the num- ber of parts taken, which divides the value of the fraction by n. SIGHT EXERCISE. Name at sight the quotients in the following examples : a^ aH^ ^ m 1. T^ -r- a 2. '- ao 3. — '-p ¥ c n ^ 4. - -^ V 10. — -^-^ -^(xA-y) y xy ^ ^ ^' m^ -\-n^ , . m^ — n'^ , ^ 7. , r- (m — n) 13. 7 '- {ni — n) m-\-n ^ ' ao ^ y ^{x^-f) 14. -—--^{x^Z) rg + 2 x-^Z ' — ^ -^ (« 4- ^) 15. -~ -^ {x + 1) REDUCTION OF FRACTIONS. 103 « 136. Multiplying the numerator multiplies the value of a fraction [P. 54]. Multiplying the denominator divides the value of a fraction [P. 55]. Therefore, Prin, 56, — Multiplying hoth terms of a fraction by the same quantity does not alter its value. 137. Dividing the numerator divides the value of a fraction [P. 55]. Dividing the denominator multiples the value of a fraction [P. 54]. Therefore, JPrin, 57. — Dividing both terms of a fraction by the same quantity does not alter its value. SJGHT EXERCISE. Tell at and why : ; sight which of the following equations are true. -1= a^ ah « «5 2. — = ac 5 c 3 ^^- a "h X 4. - = y x(x^y) y{^-\-y) 7. m n mn-\-m n^ -\-n X — a _ a^ — (^ c? — '3? _ a?' — 7^ ®' x^a ~ {x + af ^' oH^ "" («M^ x-\-y ^ rs p'TS Complete at sight the following equations : 10. ^(^ + ^U - 13. ^'^*'- '- a{(i-\-x) a a^-h^ a"" -\- ah -\-l^ ^^ ^^-y^ _ y^-t * ^^ {m-\-n)a ^m^n ' 7? — y^ * a{a — b) m m^n^ r^s^a^ 12. = 15. — ^ = —5 n • rs^q^ 8^ 104 ELEMENTARY ALGEBRA. 138. Multiplying both terms of a fraction by — 1 will change the signs of both terms. And multiplying both terms of a fraction by the same quantity does not alter its value [P. 56]. Therefore, Prin. 58. — Changing the signs of both terms of a frac- tion does not alter its value. a 139. Let + -^ = + ^ -a a - a then -y- or —7 = — q {F, 54] -, — a a / x . .a and ---^or - ~- = ^ {- q) = + q = J^ Therefore, Prin, 59. — Changing the apparent sign and the sign of either term of a fraction does not cha^ige the value of the fraction. Remarks. — 1. Changing the sign of one factor of either term of a fraction changes the sign of that term. Why ? 2. Changing the sign of every term of either numerator or denom- inator changes the sign of that term of the fraction. Why ? SJGHT EXERCISE. Change at sight the following fractions to equivalent ones having apparently positive terms : — a a — h X — y 1. — ^ 6. 11. ^ — — c —z — a _ —a? —x — y — i 2. ^— 7. — — ^ 12. ^ o y^ xyz c ' y{a — b) ' — (m^n) 4 ^ 9 -(a' + b^) P + q -b c^ + d^ -ip^ + q') — a-\-b xA-y — 7n — n 5. ~-^ 10. =J-^- 15. — c-i-d —m — n —x — y REDUCTION OF FRACTIONS. 105 Change at sight the following fractions to equivalent ones having positive apparent signs : a(x-\-y) 16. a — x a-\-x 17. d — c io (a-h){c- -d) 20. y -X ^^ + / (m -\-n)(m — n) {a^J)){c-^d) ^^' m^-\-n^ Problem 1. To reduce a fraction to its lowest terms. 140. A fraction is in its lowest terms when its terms are prime to each other. c? — b^ Illustration.— Reduce .. , ,o to its lowest terms. fl^ + ^"* Form. a^-b^ _ {^^^) {a - b) ^ a- h a^ + b^" (cC^^)(a^-ab^¥) " a^-ab-\-b^ Solution: Since dividing both t^rms of a fraction by the same quantity does not alter its value [P. 57], we divide both terms by their H. C. D. (a + h). The resulting fraction is in its lowest terms, since the terms are prime to each other. Therefore, IRvle* — Divide loth terms by their highest common di- visor. EXERCISE 89. Reduce to lowest terms : 4:0,^1^ %x''y^z^ a + h ^' 6aH ^' lOx'y z^ '^' a'-W ^aH'(^ Hx±yY {a + xf 12aHV ^{x^-yf a^-^ Ibaf'y^^ ^' aH{a?-y^) (x-^%f ^°- {x-\-%yy "• (2a:-3y)« 106 ELEMENTARY ALGEBRA. ' 2;* + 2a;2«/2 + / ^"- a;2 + a;-20 13 ^ + y^ _- 0: ^-12 a; + 35 4:3^- 26 y^ Sa^-27P ^^ {2x + 5yf ^^' (2a-3*f ' a?-{-xy-\-xz {f-a^f {a-\-Vf-(? a;« + ^/« ac-\-hc — G^ ' x^ — y^ a^^ + 5^ + 6 ^j-2^V±/ ir2+7:i; + 12 " a;8-/ • {a -i){a + bf Problem 2. To reduce a mixed quantity to an improper fraction. lUustration. — ^2 _|_ ^ Reduce a-^-x to an improper fraction. a-\-x — a — X Form and Solution, a^-ira? a + x a^-\-x^ a^ - x^ a^-\-a^ a — X 1 a — X a — X a — x [P. 56] = -A_ (^F^^ _ ^f+^) [P. 47] = _!_(_ 3^) -^^ ^^[P.59]. a — x a — a; EXERCISE 60. Reduce to improper fractions : , a , x-\-a , , a^ 1. a + — 3. ^H ' — • 5. a-\-x-\- X ' X '11 ^_|_^ c mx4-n ^ , x 4- a 2. c 4. m ' — 6. 1 H ' — y X X — a REDUCTIOIi OF FRACTIONS. 107 .t2 „ , ^ 3a-5 7. a — x i — 10. 2 a + 7 — -. r-x fl^ — rr^ 8. a; + « — ^^ ^ 11. a^ + a;?/4-y^-} 12. a; + 7 x — y 3a; + 5 Problem 3. To reduce improper fractions to integral or mixed quantities, ninstrations. — 1. Reduce 7^ to a mixed quantity. ^-^+i_/„, Forait (a;2 - a; + 1) ~ (2: + 1) = a: - 2 + Solution : Since a fraction indicates division, and the numerator is partly divisible by the denominator, we perform the division and obtain a quotient of a; — 2 and a remainder of 3. As 3 is not divis- ible by a; + 1, we simply indicate the division and add the result to Q x — % which produces the mixed quantity a; — 2 + ~. ^ A-x 4 2. Reduce — — — z — to a mixed quantity. X — i Method. ^'^^~'^ = {x^-^x-^)---{x-l) = x-lr2-{- ~^ x — 1 ^' ^ ' ^ ' x — 1 x + 2-^[P. 59]. EXERCISE 61. Reduce to whole or mixed quantities : ^ ac + b ^ 7? ^ Q^-Yxy + y* c x — 1 ' x-\-y ^ ax-a ^-\-y^ Zx^-\-%x-Z 2. 6. ; 8. -. — :; X ^-\-y ^ + 1 3. ^^ + a^ + l g a^-y"" ^ a:* + ar/4-y* X ' x-\-y ' a^-\-xy-\-y' 108 ELEMENTARY ALGEBRA. 10. '-^- 11. 12. x — y %x — l 5a;— -6 13. — — ^ ^^—- 14. ^ — ~- 2x 2 a; + 1 Problem 4. To reduce fractions to similar forms. I. Definitions and Principles. 141. Fractions having a common denominator are similar, 142. Dissimilar fractions in their lowest terms must be reduced to higher terms to have a common denomina- tor. This is done by mutiplying both terms by the same quantity [P. 56]. Therefore, the common denominator must contain each of the given denominators. Hence, JPrin, 60, — Any common multiple of the denominators of two or more fractions is a common denominator of the fractions. Frin, 61. — The lowest common multiple of the denomi- nators of two or more fractions in their lowest terms is the lowest common denominator. Note. — L. C. D. stands for lowest common denominator. 2. Examples. niustration. — Reduce ^- , — , and — r to similar fractions. oc ac ao Solution : The L. C. M. of the denominators is a be, which is, therefore, the L. C. D. [P. 60] : a _ a X a rp f---. _ a* h c~ b c y. a ' ^~ abc ac ac X b ^ -' abc c c X c ^-^ ^^, c ab ab X c *- -' abc Note. — To determine the factor to be inserted in both terms of any fraction, divide the L. C. D. by the denominator of that fraction. REDUCTION OF FRACTIONS. 109 EXERCISE 62. Reduce to similar fractions having the L. C. D. : X y ^ z ^ ax hx , ex 1. —,. -^, and T- 3. 7—, — , and -^ ah ac he hy ay^ ah a-\-h a— h J h a h j<^ 2. — - — , , and — 4. — r^, 7, and -o — to xy ' xz ' yz a-\-h' a — h' a^ — W a — x a -\- X a^ — TT a h ^ c 3 , 5 1 1 and ^' {x^af-W {x^hf-a^ 2 3 4 ^* (a:-l)(2:-2)' {x-2){x^dy^^^ {x-l)(x-3) X 2x — l - 2a; + l 10. 7—2 7, n rT» ^"^ ^i 1* 4:X^ — 1 2ic + l 2a; — 1 a h , c "• (a-c)(h-cy (a-c)(c-hy ^"""^ (c - a) (c - h) Solation : ^a-c){e-b) ^ (a-c){b-c) ^^' ^^^ ^ ~ {a-c){b-c) ^^' ^^^ c c c (c — a){c — b) ~ — {a — c) x —{b — c) ~ {a — c){b — c) a a {a — c){b — c) ^ (a — c){b — c) ah c ^34 5 12. , -, ;; T 13. l-.x'x-Vl-x' *"• 2-a;' a;-2' (x-2)« 1 2,3 1^ 7 TV7Z \» 7 SVTo \' ^^^ (X -l)(^^xy {x- 2) (3 - xy ""^ (1 -x)(x- 3) 15. 7 w r, 7 r-7T r, and {a-x)(x-cy (x-a){h-xy (c-x)(x-h) 6 110 ELEMENTARY ALGEBRA. Addition and Subtraction of Fractions. 1. Principles. ^^^•1 + 2+2 = 1^'' + ^ + '^ [P. 47]-^Li.. Therefore, JPrin, 62, — The sum of two or more similar fractions equals the sum of their numerators divided hy their com- mon denominator, 144. ^ _ 1 = 1 (^ _ J) [P. 47] == ^Jzi. Therefore, c c c ^ ^ ^ ^ c Prin, 63. — The difference of two similar fractions equals the difference of their numerators divided hy their common denominator, 2. Problem. To add or subtract fractions. Dlustrations.— 1. Find the sum of and — ; — . a — x a-{-x Solution : The L. C. D. = a^ - x^ a + X _ {a ■{■ x) (a + X) _ a;^ + 2 ax ■\- x^ a — x~ {a — x) {a + X) ~ o* — a;* a — x (a — x) (a — x) a^ — 2ax + x^ a + X {a + x)(a — x) a^ — x^ a'' + 2ax + x^ a^-2ax + x^ _ 2a^ + 2x^ a h a^^-h^ [P. 62] 2. Find the value of Solntion a — I) h — a a^ — W a(a + b) a^ + ab a-b {a — b){a + h) a^ - b'^ h _ b b{a + b) _ab + b* b — a a — b {a — b)(a + b) a^ — b* _aM^_ - CT^ - 6» a2 _ 53 - + a2 - 62 a^ + ab ab + b^ - a^ - b^ '2ab + -5 T^ + J2 -r a«-62 ^ a^-b^ ~a^-b^ ADDITION AND SUBTRACTION OF FRACTIONS. \\\ Note.— Sometimes it is better not to combine all the fractions at one time. 3. Find the value of ; l+a;"^l 1 — X 1 l+a;^- 8oliiUon:^^^ + ^_^ = \-x 1 + x 2 2 1 2 4-2a;« 1- X* l + 3a;« 1-x^ 1 + x^ ~ l-o;* 1- a;*" 1-a:* 4. Find the sum of 2 Method. 2«- — and 3 a ; + 2J 3«-5 c 2a 36 + ^''" l^ = 2a- 36 + 2a-6 c 8a + 2& ^"^ c l^ = 3a + 26 + -3a + 6 c Sum = 5a- b- EXERCISE 63. Find the value of : 1 + a 1 — a a a a-\-x a — x ah , ac , he c ' h ^ a 1,1 1 ao ac be ^a±z_a±y ^^ X y p-1 q—p -(¥+'4*)-(t-'-I*) x'^f ^^-f m n m — n n — m P 1 p> p-q^ f-f " 1 a c? a-x 1 a-\-x x^~ a« o + i a-h 112 ELEMENTARY ALGEBRA. ic / ^ + ^ , c^d\ (a-h , c-d\ \x y ^ zj^\x y z) '• V3cB"^4y 3«y"'"l^3a: 3y+4«^ \3a; 5y'^6z) \5x^ S.y 4.z) a + a; a — a; {a + xf ' (a — :z;)^ 20. ^ I ^y I ^-y X xy 7? 21. '^ 22. ^^ + I + I— x^ — dx-\-^^x^-4.x-\-Z^x'' — bx-\-Q 2 4 2 24. a?-^x-\-\% a?-^x-\-^^ T^'-hx-^^ -1 + 1 2 3 2a; — 3 25. 26. 27. ic 2a;-l 4a;2-l 3 + 2a; 2-3a: 16:r — a^ 2 -^ aj 2 + a; "^ a;^ _ 4 3 y 4 -20a; l-2a; l + 2a; 4a;2-l MULTIPLICATION AND DIVISION OF FRACTIONS. 113 Multiplication and Division of Fractions. Problems. 1. To multiply or divide a fraction. niustrations. — 1. Multiply . by a — h. » ^ .- a + h . ,, ( a + h){a-h) a^-b^ Solution: -^ x (a-h) = ^ [P. 54] = -^^ 2. Multiply ^2^_^ by a + 5. solution: ^^ x (a + 5) = ^-^_^,g^^^ [P. 54] = ^ 3. Multiply |±| by a^ - h\ Solution : ^ X (a* - &») = ^^ "^ ^^^'^ [P- 57] = (a + 6)« 4. Divide 7 — by a-\-'b, ah ^ ' Solution : j— -s- (a + 6) = ^ '-^ [P. 55] = — 1— ab ^ ' ao ^ ^ ao 5. Divide — tt: by a — h, 6. Divide ^'7^^ by a^-^. Solution: t^— r h- (a« — 6«) = tts— — rrr-i — j:5^ = o( — Canceling a factor common to the numer- ators or the denominators of two fractions does not alter their quotient, SIGHT EXERCI SE. Name at sight the quotients of the following fractions : a^ a x^ 7? x^y^ ' xy fl2 _ ^ a - a: T^ia-^xY . (a + xY ^' s'ia-xY ' {a-xY xy X (x-^-'^Y a;H-2 m^ -\-n^ m-\-n mr — n^ m — n a; + 3 * x-\-Z a{p + q) , p-^q (x + %)(x-Z) .x-Z b(p — q) p — q {x—2){x-{-d)'X'-'2 WRITTEN EXAMPLES. Ulnstrations.— 1. Multiply -TZTRt ^^ ~T~~* 1 Solution : -, — p, x — — = ^^ . , x ^!!^ [P. 65] = —. ^r . a — b 2. Divide 7— by ^ . cd -^ de a-\- b solution: -^ + ^_ = _^^ J^[P. 67] = i±*xl[P.66]=^, 118 ELEMENTARY ALGEBRA, 3. DiYide 1+- by 1-- y ^ y Solution 1 + — (l + — ) X y ^ V y) ' \ y) i_£ (i-^)x2/ 2^-^' Or, y \ y) ^ \ y) \ y) y ' y y y — ^ y—x' EXERCISE 68. Find the yalue of : ax ex a-\-x a^ — x^ hy dy cd de 2^3 2. — ^-^ X ^ «* «*a;* xy^z^^ a^h^i? a^^o? ' a^ ^ x^ ax hx ^^ a^-\-x^ ^ a^ — x^ ^' '^ a-\-x ' a + y ' a^'-y^ a a" ax X 11. X p-q p-\-q ^V^V^ 12 (^ + ^)' x ^'-y' by^dx^ad ^-y aJ' + ab Ji/^ * dy^ k-M^-S} 16. a;2 + 7a; + 10 * a:2-a:-30 ac — ad — bc-\-bd c-\-d 18. a:3-f27f/^ a;^ - 3a;y + 9.y^ COMPLEX FRACTIONS, 119 Complex Fractions. To reduce a complex fraction to a simple one. a Illustration. — 1. Reduce — to a simple fraction. a a . , a «... ^ *■ ad ^ h a c a d ad Solution : — = = -r— ; Or, -. = ^^ — = — x— = -j— . c c ^ , be c d b c be d-d. d 2. Reduce to a simple fraction. 3^ — — Solation : ic' _ \ a:' / _x^ — X ^ V ^V a;(a:« + l) Or, a;* + a:« + 1 "**** »_1 ^-^ a: ^ £__fL-?llll ^•g^. a;(a:«+l)(a;4-l)(a;~l) _ 1 ~ a;«-l ~ ^« ^a^-l~(a;+l)(a;«-a;+l)(a;-l)(a;«+a;+l)~ EXERCISE 66. a;(a;» + l)((^H4)(^^) Simplify : a ^ + 1 X ''1 y x+i ax hy *2. -i xy ab a c 5. a; + 2 6. a:-3 a:-2 .T + 3 a-f ■x 7. a: a — a 8. 1 "l X 1 + 120 ELEMENTARY ALGEBRA. 1 X 1 9. 11. a: 4 x-\-- x-\-- X ^ X , ^ , 1 .2a x—% ' «— 3 10. 12. „ , 1 2a x-{-Q a — 3 Involution of Fractions. 146. Involution of fractions is the process of raising a fraction to any power. Problem. To raise a fraction to any power. Illustration. — 1. Raise -r to the fifth power. Solution '(j)=j^j^jXj^j = Y.- Therefore, Prin, 68. — Raising both terms of a fraction to any power raises the fraction to that power. SIGHT EXERCI SE. Name at sight the indicated powers of the following fractions : '■©■ -(-f;)" HffefF \by/ \ Pq) \ ^^ P ) ( ^2X4 (a-bV I r^s^'V *■ \3¥) ^' \x^y) ^^- \ mH^j INVOLUTION OF FRACTIONS, 121 WRITTEN EXERCISES. EXERCISE 67. Find the value of : ■•(14)" A'§m Miscellaneous Examples. EXERCISE 68. /p2 a; 30 1. Reduce « — — r^r to its lowest terms. 7^ — 1^ 2. Reduce -» — ^ to its lowest terms. 7^ — y^ 3. Reduce r^^ ^r-^r — rTTTZJ ^ i^ lowest terms. ac-\-Sad-\-5bc-{- 16 bd 4. Reduce 1 — \o. 2 ^^ ^^ improper fraction. (x — yV 6. Reduce 2x — y-{- ^^ to an improper fraction. 1 -_ 2ar 6. Reduce — — ; to a mixed quantity. 1-^x 122 ELEMENTARY ALGEBRA. 7. Simplify 3 ^ 4-20« 8. Simplify l-2a l + 2a 4a2_i fl^^ + ^' _ _«" ^__ ah al^W~ a^^ah 2 9. Simplify ^±i.:+ii^,i^i^ 10. Simplify -^ X L_^ — «{'+(^)M-(sl)l .,.M.„iplj| + £b,J + | ^ Operation. a c J + d c d — 4- h a c c^ J + ad c. + b^ + h 2e c^ ad T + ^ + -¥- 14. Multiply I- I by ^ + ^ 15. Multiply ^ + f by ^^ + g 16. Square - + 1 ; also - - ^ [See P. 31, 32.1 a ^ h^ y a ^ -■ 17. Cube 1 + - ; also --b [See P. 33, 34.1 18. Find the value of (^ + ^ W^ - ^^ [See P. 39.] 19. 20. 21. MISCELLANEOUS EXAMPLES. 123 Find the value of (^ + ^) [^ — ^) Find the value of f a; + 1 + - j ; ix — l-A Find the value of (^±^ + ^) (^±^ _ ^) 22. Divide -§ — ^ by y [See P. 44.] 23. Divide T5 + -3 by y + - 24. Divide ^--, by ---; also by - + - 26. Factor a^ - ^ ; a^f--^; h^~^^^ 26. Factor 1-^; «*--,; -,-^ 27. Factor 2;' + ^; ^-^5 ^"7 28. Factor ^ + '^; — « - ^ ; l — lr-Ml y^ ^ x^^ m^ n® S^ — yJ 29. Divide a:'^ + -5 by a: + - 30. Find the value of : a; + y a;- + / ^ X — y ' __x^ — y^ x-\-y x^ + y^ -,,,., a: a , y h . x a y , b 31. Multiply Kt by t + - as m ^•^a x^ b y a X b ^ y multiplication of entire polynomials. 32. Find the value of ^ when z = b — z a-\-b CHAPTER 111. GEJ^ERAL TBEATMEJ^T OF SIMPLE EQUATIONS, General Definitions. 147. An equation in which the known quantities are numerical is a Numerical equation ; as, dx-{'2 = 6x-4:. 148. An equation in which some or all of the known quantities are literal is a Literal equation ; as, 2ax — 4:bx = c. 149. The degree of a term of an equation is determined by the number of unknown prime factors it contains. Thus : In ax^-\-bxy-\-cy^-\-dx-\-ei/ -\-f =0, a oi^, hxy, and cy^ are of the second degree; dx and ey oi the first degree ; and / of no degree. 150. A term in an equation that does not contain an unknown quantity is an Absolute term. 151. The degree of an equation is the same as the degree of its highest term. 152. An equation of the first degree is a Simple equa- tion ; as, 2x — Sx-{-5x=zl2 ; or, ax -\- by = c, 153. An equation of the second degree is a Quadratic equation ; as, a^-{-4cX = 6 ; or, x^ -{- xy = 12, 154. An equation of the third degree is a Cubic equa- tion ; SLS,x^ + dx^-{-2x + 6 = 0', or, a^ + x^y -\- x -{■ y = 12. TRANSFORMATION OF EQUATIONS, 125 Transformation of Equations. Definition and Principles. 155. The process of changing the form of an equation without destroying the equality of its members is trans- formation of equations. 166. An equation may be transformed : Prin, 69. — 1. By adding the same or equal quantities to both members. 2. By subtracting the same or equal quantities from both members, 3. By multiplying both members by the same or equal quantities. 4. By dividing both members by the same or equal quantities, 6, By raising both members to the same power, 6, By taking the same root of both members. 157. If we take the equation ax — b = cx-\-d, (1) and add b to both members [P. 69, 1], we obtain ax = cx-{- d-\-b, (2) K we now subtract c x from both members, we obtain ax — cx=.d-\-b. (3) If we now compare (3) with (1), we observe that : Prin, 70, — Any term of an equation may be transposed from one member to the other if its sign be changed. 158. If we take the equation | + ¥ = T-^' (1) and reduce all the terms to a common denominator, we , ^ . 6 a; , 4 a: 15 a; 36 ,„, 126 ELEMENTARY ALGEBRA. If we now multiply both members by 18 [P. 69, 3], we obtain 6a; + 4a; = 15ic — 36, an equation without frac- tional terms. But 18, the common denominator of the fractional terms, is a common multiple of the denomina- tors of the fractions. Therefore, Trin, 71* — An equation with fractional terms may be cleared of fractions hy multiplying loth members by a com- mon multiple of the denominators of the fractions. Simple Equations of One Unknown Quantity. I. Solution of Numerical Equations. Illustrations. — 1. Given hx-\-^ = Zx — h-^^x to find the value of x. Solution: Src + 7 = 3ic — 5 + 6a; (A) Transposing 7, 3a^, and 6 a; [P. 70], 5a;-3a;-6a;=-5-7 (1) Uniting terms, — 4 a; = — 12 (2) Dividing by - 4, a; = 3 [P. 69, 4]. Proof : Substituting a; = 3 in equation (A) 15 + 7 = 9 — 5 + 18; whence 22 = 22. Q /v ^ T ^ '7' 7 2. Given ~ 4 + -^^ = —- + - to find the value of x. ^ x = an — bm a , h ^ y (A)' Solve - a c ^ U-y = ' (B) Solution: Subtract (B) from (A), b + c y -"-' (1) b + c=z(c — d) y (2) b + c y = ._^ SOLUTION OF LITERAL EQUATIONS. 147 EXERCISE 77. Solve 1 iax-\-hy = c) ax — by = d\ {x-jry = m ) \ ax-\-hy = n ) ^ , y a 1=^' j mx-\-ny=p \ rx-\-sy = q a , b X y X y j ax \bx + my -my '•! X — y = a mx-{-ny 10. X + ^.= 1 a a- a X y - 1 b w~ b {cx — dy = b ) ( mx — ny=- b \ a mx + b __ ny~ c a nx + b _ my d Examples involving Simple Equations of Two Unknown Quantities. EXERCISE 78. 1. A man bought two farms, one of 80 acres and one of 50 acres, for $22190. Had the first contained 70 acres and the second 60 acres, the second would have cost ^%8 as much as the first. What was the price of each farm per acre ? 2. Two thirds of A's fortune added to three fourths of B's is $700, and B's increased by $100 is five sixths of A's. What is the fortune of each ? 3. The sum of two numbers is 7, and if the larger be added to the numerator and the smaller to the denominator of the fraction Y^, the result wiU equal Y*. What are the numbers ? 148 ELEMENTARY ALGEBRA, 4. The sum of two fractions is ^Yig, and their difference is ^Yie . What are the fractions ? 5. A man has a certain quantity of oats and corn. If he mixes two thirds of his oats with one half of his corn, he will have a mixture of 60 bushels ; bui) if he mixes all his corn with four fifths of his oats, the oats in the mixture will exceed the corn by 8 bushels. How many bushels of each kind has he ? 6. A man has two watches and a chain. The first watch is worth $60. If the chain be put on the first watch they together will be worth % as much as the second ; but if the chain be put on the second watch they together will be worth twice as much as the first. Required the value of the second watch and chain respectively. 7. If 2 be added to the numerator of a fraction it will be V2, but if 3 be added to the denominator it will be Vs. What is the fraction ? Let — = the fraction ; then, = -^ , and :^ = 77 . y ' ' 2/ 2' y + S 3 8. If 3 be added to both terms of a fraction it will be %, but if 3 be subtracted from both terms it will be %. What is the fraction ? 9. The difference of two numbers is 5, and if the greater be subtracted from the numerator, and the less from the denominator of ^y^, the result will be 7?. What are the numbers ? 10. There is a number consisting of two figures, such that if 9 be added to the number the figures will change places, and the sum of the figures is 7. Required the number. Suggestion.— Let x = the ten's figure and y the unit's figure. Then, lOx + y = the number, and lOy + x, the number with the fig- ures interchanged ; whence, lOx + y + 9 = 10y + x (A) x + y = 7 (B) EQUATIONS OF TWO UNKNOWN QUANTITIES. 149 11. The sum of the two digits of a number is 12, and if 54 be added to the number the digits will change places. What is the number ? 12. A certain number is four times the sum of its two digits, and if 9 be added to the number its digits will change places. Required the number. 13. The difference of the two digits of a number is Yig of the number, and if 6 be added to the number its value will be five times the sum of the digits of the original number. Required the original number. 14. A and B together can do a piece of work in 8 days, and A can do as much in 3 days as B can do in 5 days. In how many days can each alone do it ? Suggestion. — Let x = the time in which A can do it, and y = the time in which B can do it. — = the part A can do in 1 day. — = the part B can do in 1 day. ■^ = the part both can do in 1 day. Ill' Since A can do as much in 3 days as B can do in 5 days, '^='- (B) X y ^ ^ 15. If A works 3 days and B 5 days, they can accom- plish a piece of work ; but if A works 2 days and B 3 days, they will accomplish only % ot it. In what time can each alone do it ? 16. Two thirds of what A can do in a day equals three fourths of what B can do, and they together can do a job in 8 days. How long would it take each alone to do it ? 17. Five men and 3 boys can do a piece of work in 6 days, and 4 men can do as much as 6 boys. Ilow long would it take 1 man and 1 boy each to do it ? 150 ELE3IENTARY ALGEBRA. 18. A field may be divided into 8 lots of one size and 9 lots of another size ; but 4 lots of the first size and 10 of the second size together will occupy only ^1^ of the field. Into how many lots of each size may the field be divided ? 19. The distance around a room is 52 feet, and if 4 feet be added to the length it will be twice the width. Ee- quired the length and width respectively. Suggestion. — Let x = the length and y the width, then 2x + 2y = 52f the number of feet around the room ; also, X + 4: = 2y, twice the width. 20. A man has two square fields, one of which is 6 rods longer than the other, and the sum of the distances around them is 96 rods. What is the length of each field ? 21. A man has a rectangular lot, such that twice the length increased by 6 yards equals four times the width diminished by 4 yards, and the distance around it is 50 yards. Eequired the length and width respectively. 22. A rectangular field has a perimeter of 52 rods, and if its width be increased by 6 rods and its length by 8 rods, the width will be % of the length. Required the dimen- sions of the rectangle. 23. A certain fishing-rod consists of two parts : the length of the upper part is y^ of the length of the lower part, and 9 times the upper part together with 13 times the lower part exceed 11 times the whole rod by 36 inches. Find the length of the two parts. 24. A and B ran a race which lasted 5 minutes : B had a start of 20 yards, but A ran 3 yards while B ran 2, and won by 30 yards. Find the length of the course and the speed of each. 25. A man having worked 20 days and been idle 8 days, saved $50. Had he worked 24 days and been idle 12 days, he would have saved $57. What were his daily wages, provided he maintained himself ? EQUATIONS OF TWO UNKNOWN QUANTITIES. 151 26. If the length of a rectangle be increased by 2 feet and the width by 3 feet, the area will be increased by 42 square feet ; but, if the length be diminished by 2 feet and the width be increased by 4 feet, the area will be increased by 12 feet. Kequired the length and width of the rect- angle. Suggestion. — Let x equal the length and y equal the width. 27. If a farmer had planted 5 more hills of com in one row, and had planted 5 more rows, he would have had 700 hills of corn more ; but, had he planted 5 hills less in one row, and 4 rows less, he would have had 620 hills less. How many hills did he plant ? 28. If there had been two more persons at a dinner- party, and each person had paid one shilling less, the entire bill would have been 4 shillings more ; but if there had been two persons less, and each person would have paid two shillings more, the bill would have been 2 shillings less. Kequired the bill and number of persons. 29. If the length of a rectangle were diminished by 5 feet and the width increased by 4 feet, the area would remain unchanged ; but, if the length were to remain un- changed and the width increased by 7 feet, the area would be increased by 224 square feet. Required the dimensions and area of the rectangle. 30. A certain sum of money, put out at simple interest, amounts in 6 years to 1780, and in 10 years to $900. Re- quired the sum and rate per cent. 31. A certain principal in a given time at 3 per cent amounts to 1920, and at 5 per cent for the same time to $1000. Required the principal and time. 32. If two trains start from two stations 40 miles apart at the same time, and approach each other, they will meet in one hour ; but if they run in the same direction it will require the faster train 4 hours to overtake the slower. What are their respective rates of running ? 152 ELEMENTARY ALGEBRA. 33. A passenger-train 200 feet long will pass a freight- train 680 feet long in 30 seconds, if they run in opposite directions ; hut if they run in the same direction it will require 1 minute to pass it. How many miles per hour does each train run ? 34. A and B jointly loan C a sum of money which in five years at 6 per cent amounts to 11170 ; 60 per cent of A's share of the principal equals 75 per cent of B's share. How much of the amount belongs to each ? Simple Equations of Tiiree or more Unknown Quantities. Illustrative Examples. — 1. Solve 2 + 3 + 4-^^ - 4- ^ 4- - = 47 3^4^5 4"^5"^6 38 Solution: Clear equations A, B, and C of fractions, Qx+ 4y+ Sz= 744 (1) 20rc + 15y + 132 = 2820 (2) 15a; + 12y + 102 = 2280 (3) Multiply (1) by 4 and bring down (2), 24 a; + 162/ + 122 = 2976 (4) 20a: + 15?/ + 122 = 2820 (2) Subtract (2) from (4), 4:X + y= 156 (5) Multiply (1) by 10 and (8) by 3, 60a; + 40y + 302 = 7440 (6) 45a; + 36y + 302 = 6840 (7) Subtract (7) from (6), 15 a; + 4^=600 (8) Multiply (5) by 4, 16 a; + 4 2/= 624 (9) Subtract (8) from (9), x= 24 Substitute value of x in (8), and reduce, y= 60 Substitute values of x and y in (1), and reduce, z= 120 EQUATIONS OF THREE UNKNOWN QUANTITIES. 153 [^ + ^ + ^ = 29 (A)' 2. Solve - ^-i+l= 9 (B) \x ' y z (C) Solution : Add (A) and (B), M=- (1) Multiply (A) by 4 and (C) by 3, (2) X y z (3) Subtract (3) from (2), - 1 + ^ = 123 X z . (4) Multiply (4) by 7 and bring down (1), X z (5) 1+1 = 3« X z (1) Add (5) and (1), ?^ = 893 (6) Divide by 223, i= 4; whence z- 1 ~ 4 Substitute the value of z in (1), and reduce, 1 Substitute the values of x and z in (A), and reduce, 1 2^=0 (rr + y = 14 (A) 3. Solve \x^z = \^ (B) (y + 2:=18 (C)^ Solution : Take the sum of (A), (B), and (C), 2x + 2y + 2z = 48 (1) Divide (1) by 2, x + y + z = 24: (2) Subtract (A) from (2), z = 10 Subtract (B) from (2), y= S Subtract (C) from (2), x= Q 154: ELEMENTARY ALGEBRA. Solve : EXERCISE 79. 3x-27/ + 4:Z= - 4,x-\-2i/~5z= - 2x-\-4:y--5z= 6 I x-{-Sy — 2z = 10) 3. 1/ 2x-^dy-z=l2 \ 4:X — 2y-{-z= 3 [• i^+iy + i. = 23 3^+2^+4^ 25 ia; + iy+l. = 27 = m = ?^ = r 15 5 3 20 34 44 = 12 X-}- z = ^ + '=2 X y X z 11. 12. 13. 1 1,1 2^-4^+8^ j a;-f-2?/ — 3^ = 4a: — 4?/— z = [ 3x + 8y-i-2z = fx-i-y-{-z = 12 x-{-y-J^u = ll y-^z-\-tt= 9 J ^ + ^-^=. 8 X y z a: y ' z 14. x-{-2y = 5Z'-10x — y-{-z = 60 15. r «iC+^?/-C^=z:«2_|_^2_^2 i ax — h y -\- c z = a- — b'^ -{- c^ ( — ax -\-l y -\- c z = h^ -\- c^ — a^ y z EQUATIONS OF THREE UNKNOWN QUANTITIES, 155 Examples involving Simple Equations of Three or More Unknown Quantities. EXERCISE so. 1. If 5 bushels of corn, 6 bushels of oats, and 8 bushels of rye together are worth $10.30 ; 3 bushels of corn, 5 bushels of oats, and 8 bushels of rye, $8. 75 ; and 1 bushel of oats mixed with 1 bushel of rye is worth as much as 1% bushel of corn — what is the value of each per bushel ? 2. A's farm, plus Y3 of B's and C's, equals 230 acres ; B's, plus Y4 of A's and C's, equals A's ; and C's, plus Vg of A's and B's, equals 170 acres. How many acres are there in each farm ? 3. If A should give B one half of his money, and then B give C one half of his, C would have $550 ; if B should give C one half of his money, and then C give A one half of his, A would have $800 ; if C should give A one half of his money, and then A give B one half of his, B would have $750. How much has each ? 4. A, B, and C together can do a piece of work in 5 7ii days ; A can do twice as much as B or three times as much as C in a day. How long will it take each alone to do it ? 5. The sum of A's and B's ages is 55 years ; the sum of A's and C's is 62 years ; and the sum of B's and C's is 77 years. Required the age of each. 6. A and B can do a piece of work in 4 days, A and C in 5 days, and B and C in 6 days. In what time can each alone do it ? 7. Two supply-pipes, A and B, and one discharge-pipe, C, are connected with a cistern. If the three pipes run together for 2 hours, the cistern will be Veo full ; if A runs 3 hours, B, 4 hours, and C, 2 hours, it will be V30 full ; and if A runs 5 hours, B, 3 hours, and C, 2 hours, it will be Yio full. How long will it take A and B each to fill it, and C to empty it ? 156 ELEMENTARY ALGEBRA. 8. A man bought a horse, carriage, and harness for 1500. The horse cost 15 more than the carriage and har- ness, and the carriage cost Ys as much as the horse and harness. Eequired the cost of each. 9. There is a number consisting of three digits : the sum of the digits is 13 ; the middle digit is Yg of the other two ; and if 297 be added to the number the unit's and hundred's digits will change places. Eequired the number. 10. A's money in 9 years at 6^ will produce as much interest as the sum of B's and O's in 473 years at 4^ ; B's in 8 years at 5 ^ as much as A's and C's in 3 Y3 years at 6fo ; and C's in 7 years at 3^, $42 more than A's and B's in 3 years at 4^. Eequired the principal of each. 11. A, B, C, and D received $1000. B got half as much as A. The excess of C's share oyer D's was Y3 of A's share, and B's share, increased by $100, was equal to the sum of C's and D's shares. How much did each receive ? 12. If 40 peaches are worth as much as 20 quinces and 4 oranges ; and 40 quinces are worth as much as 30 peaches and 12 oranges ; and 40 oranges, 70 peaches, and 20 quinces are worth $4 — what is the price of each apiece ? 13. A man has $180 in three parcels. If he takes $20 from the first parcel and puts it with the second, and then takes one half of the second and puts it with the third, the third will be worth twice as much as the other two ; but if he takes $20 from the third parcel and puts it with the second, and then takes one half of the second and puts it with the first, the value of the first will be % of the value of the third. Eequired the value of each parcel. 14. If 5 casks, 3 cans, and 2 jugs of oil be drawn from a barrel containing 60 gallons, it will remain ^Yso ^^11 ? i^ 4 casks, 5 cans, and 8 jugs be drawn, it will remain Y20 full ; and if 3 casks, 5 cans, and 10 jugs be drawn, it will remain Ye f^li- What is the capacity of a cask, a can, and a jug respectively ? GENERALIZATION AND SPECIALIZATION. 157 Generalization and Specialization. 1. Definitions. 166. Any question proposed for solution is a Problem. 167. A problem whose given quantities are literal, or general, is a general problem. 168. A problem whose given quantities are numerical, or special, is a special problem, or an example, 169. A number of examples with different given quanti- ties but like conditions and requirements constitute a class. 170. A general problem involves a whole class of ex- amples. It is the type of a class, and its solution the solu- tion of a class. 171. The solution of a general problem gives rise to a formula, which, interpreted, gives a rule for the solution of every example of a class. 172. The process of converting a special problem into a general one, by substituting literal for numerical quanti- ties, is Generalization. 173. The process of converting a general problem into a special one, by substituting numerical for literal quanti- ties, is Specialization. 2. Examples. Ulustrationg. — 1. If A can do a piece of work in 4 days and B can do it in 5 days, in what time can they do it working together ? Generalize this question and solve it. Solution: Put a for 4 and b for 5. Let x equal the time re- quired for both to do it. Then i + ^ = i (A) , ab 4x5 20„2, whence, x — r = t — ~ = tt = 2 tt days. 8 158 ELEMENTARY ALGEBRA. 2. A and B can do a piece of work in a days, A and C in h days, and B and C in c days ; in what time can each alone do it ? Solve this problem and specialize for a = 10, J = 8, and c = 6. Solution : Let x equal the time required hy A,y the time required by B, and z the time required by C ; then X y a (A) X z h (B) 1+1=1 y z c (C) Adding (A), (B), and L (C), and subtracting from the sum twice (A), twice (B), and twice (C) respectively, we have 2_ z ~ ac + ah — he ahc (1) 2 _ y~ ah + he — ac ahc (3) 2_ X be + ac — ah ahc (3) whence x = 2ahc (a) ' be + ac — ab y = 2abc (*) ' ab + be- ae Z — 2abc (c) 'ac + ab — ac Put a = 10, 1 = : 8, and c = 6, X- 3 X 10 X 8 X 6 = 34| 48 + 60-80 " y = 2 X 10 X 8 X 6 80 + 48-60 -4t z = 2 X 10 X 8 X 6 • 60 + 80-48 = -i EXERCISE 81. 1. The sum of two numbers is 20, and their difference is 8. Find the numbers. Snggestion. — Generalize by putting a for 20 and h for 8 in the problem, then 20 for a and 8 for b in the result. GENERALIZATION AND SPECIALIZATION. 159 2. A's age is three times B's, but in 12 years it will be only twice B's. Kequired the age of each. Suggestion. — Put m for 3, n for 2, and t for 12 in the problem, and 3 for m, 2 for n, and 12 for < in the result. 3. A and B have $170, and % of A's share, equals Y4 of B's. How much has each ? Suggestion. — Put m for ^/g, n for '/4, and c for 170, etc. 4. A can do a piece of work in 6 days and B can do it in 8 days. In what time can they do it working together ? Generalize and solve. 5. A has $400 more than B, and B has $500 less than C, and they together have $1800. How much has each ? Generalize and solve. 6. If a certain number be increased by 20, the result will be twice as great as when the number is diminished by 10. Required the number. Generalize and solve. 7. What number added to both terms of the fraction Y7 will give the fraction ^4 ? Generalize and solve. 8uggestion.^Put — for ^ and — for -7-. 8. B has 40 acres more land than A, but if A buys 60 acres from B, A will have 175 times as much as B. How many acres has each ? Generalize and solve. 9. If a man work 5 days and a boy 3 days, they to- gether earn $23, but if the man and boy each work 4 days they together earn $20. Required the daily wages of each. Generalize and solve. 10. The sum of A's and B's ages is c years, and A is d years older than B. Required the age of each. Special- ize by making c = 3G and c? = 8 in the result. 11. Mr. Jones has a coins worth a dollar; some of them are c-cent pieces, and the rest are d-cent pieces. How many of each are there ? Specialize by making a = 14, c = 10, and c? = 5. 160 ELEMENTARY ALGEBRA. 12. The sum of three consecutive numbers is 18. Re- quired the numbers. Generalize and solve. 13. James is a years younger than William ; but if m times James's age be subtracted from n times William's, the remainder will be d years. How old is each ? Spe- cialize by making « = 4, m = 2, n = ^y and d = 22. 14. If a cows and b oxen are worth m dollars, and c cows and d oxen, n dollars, required the value of a cow and of an ox. Specialize by putting 5 for a, 7 for b, 10 for c, 3 for d, 370 for m, and 355 for n, 15. A and B can do a piece of work in d days. After working together c days, B leaves, and A does the balance in a days. In what time could each do it alone ? Special- ize by putting 30 for d, 18 for c, and 20 for a. 16. If a certain rectangle had been a feet broader and b feet longer, it would have been c square feet larger. But, if it had been b feet wider and a feet longer, it would have been d square feet larger. Required its dimensions. Specialize by making a = 2, b = 3, c = 64, and d = 68. 17. There is a number consisting of two digits whose sum is a, and if b be subtracted from the number, the digits will change places. Required the number. Special- ize by putting 13 for a and 27 for 5. 18. The wages of a men and b women in one week amount to c dollars, and b men receive d dollars more than e women. What does each receive per week ? Put 5 for a, 7 for b, 170 for c, 80 for d, and 6 for e. 19. Three children, taken two at a time, weighed a pounds, b pounds, and c pounds. What was the weight of each ? Fut a = 94, b = 76, and c = 90. 20. A purse holds a crowns and b guineas ; c crowns and d guineas fill ^Yes of it. How many will it hold of each ? Put 19 for a, 6 for b, 4 for c, and 5 for d. Enun- ciate the special problem thus formed. CHAPTER IV. POWERS AJ^B ROOTS. Involution of Binomials. I. Principle 174. We may learn by actual multiplication that : (a + Z»)3 = «3 _^ 3 «2 J + 3 « 52 _|. J3 (a-\-bY = a"" + 4:aH -\- Qa^h^ ^ 4.aW -\-¥ {a -hY=ia''-4.aH^QaH^-^a¥-\- 5* By a careful inspection of the above results the follow- ing laws will appear : The Binomial Theorem. Prin, 72, — 1. The number of terms in each result is one greater than the exponent of the binomial, 2. When the binomial is the sum of two quantities, all the terms of the power are positive ; when the difference of two quantities, the terms are alternately positive and nega- tive. 3. The first letter occurs in all the terms but the last, and the second letter in all the terms but the first, Jf, The exponent of the leading letter in the first term is the same as the exponent of the binomial, and decreases by 162 ELEMENTARY ALGEBRA. unity in each succeeding term. The exponent of the second letter is one in the second ter?n, and increases hy unity in each succeeding term. 5. The coefficient of the first term is one ; of the second term, the exponent of the Mnomial ; and that of each suc- ceeding term may be found hy multiplying the coefficient of the preceding term hy the exponent of the leading letter in that term^ and dividing the product hy the number of that term from the Note. — The coefficients after the middle term are the same, in an inverse order, as those before it. When the exponent of the binomial is odd, there are two middle terms with like coefficients. 2. Applications. EXERCISE 82. Expand : 1. {c-\-dY 5. {m-\-nf 9. (x — yf 2. (a - df 6. (m - nf lo. {c + zf 3. (x-\-yy l.{c — xy \i.{y — xf'' 4. (x — zf 8. {x-\-zf 12. {z-^yf^ 13. The fourth power of the sum of two quantities equals what ? Suggestion.— Since {a -{-})f = a* + Aa^h ■\- Qa^ b"^ + 4:ah^ + h\ the fourth power of the sum of two quantities equals the fourth power of the first + 4 times the cube of the first into the second + 6 times the square of the first into the square of the second, etc. 14. The 4th power of the difference of two quantities equals what ? 15. The 5th power of the sum of two quantities equals what? 16. The 5th power of the difference of two quantities equals what ? 17. (:r+l)*=? 19. {x^iy=? 21. (1+;^)^=? 18. {x - 1)* = ? 20. {x - 1)^ = ? 22. (1 -zf^'^ INVOLUTION OF BINOMIALS. 163 Expand (^x'-2fy. Solution : Let m = 3 a:« and n = 2 y\ then (3 a:« - 2 y^* = (m — nf = m* — 4: w' n + G m^ n^ — Amn^ + n* = (3 a:*)* — 4 X (3a;«)« x 2y» + 6 x (3a:»)« x (23/3)2 _ 4 x 3a:2 x (22/')» + (2yY = 81a;«-216a:«y« + 216a:*y«-9(>a;2 3/9 + 16 y«. Expand : EXERCISE 83. 1. (a -{-2 by 7. (l-2a:2)5 13. (2a« + 3a;3)« 2. (3a -25)* 8. (T'^fY 14. {-(x^-f)}^ 3. (3 a: +1)5 9. {a' - b'Y 15. (-2a:-3y)5 ^(-p)' ■^ (-I)' ...(i-i,)- ••(^5)' /2 a 3JV ''•[3 6- raj ■'■(»''-• ..y <-i)' -d-i")- ...(..--■)• Involution of Polynomials. I. Principles. 175. By actual multiplication : 1. {a-{-b -\- cY = {a-{-b-{- c) (a-i-b-^ c) = a^-^b^-^(^-]-2ab-\-2ac-^2bc. 2. {a-b-^cy = (a-b-\-c){a-b-j-c) = a^-]-b^-i-(r-2ab-j-2ac-2bc, 3. (a + 5 + c4-^)* = a' + ^ + c2 + <^3 + 2a5 + 2ac-\-2ad-\-2bc-\-2bd-i-2cd, 4. (a - 5 + c - J)2 = a^ -j_ J2 _^ c2 + ^2 _ 2 « ^, _|- 2ac — 2a^ — 2Jc + 2Z'^ — 2c;8 4. Va^«^2o 8. Va}H^z^ 12. V«^«(a + ^>)25 184. Any even power of a positive or a negative quantity is positive [P. 27]. Therefore, Prin. 77 » — Any even root of a positive quantity may ie either positive or negative. Illustration : a/+ 64 = ± 8, since (±8)^ = 4- 64. 185. Any odd power of a quantity has the same sign as the quantity [P. 28]. Therefore, Prin, 78, — Any odd root of a quantity has the same sign as the quantity. lUustration : V+ 27 = + 3, since (+ 3)^ = + 27. V-27 = - 3, since (- 3)^ = - 27. 186. Since no even power is negative [P. 27], Prin, 79, — An even root of a negative quantity is im- Illustration : v — 16 is neither ± 4, since (± 4)^ = 16. Note. — The indicated even root of a negative quantity, as V— 16, is called an imaginary quantity. SIGHTEXERCISE. Name at sight, giving the proper signs : 1. Vl6 4. V81 5. V-X^ 6. V-32 7. V«* 10. VaH'^ 2. V8 3. V-27 8. V-x^y^ 9. Vx^^ 11. V-4. 12. V-16 ALGEBRAIC EVOLUTION, 167 187. Since raising both terms of a fraction to any power raises the fraction to that power [P. 68], rrin. 80, — Extracting any root of both terms of a frac- tion extracts that root of the fraction. niustration. — ±-^=±^, since \ 3/ ^ (3f ^ 9 SIGHT EXERCI SE. 3 / X 3a 10. 11. •y-3^ 3 /T T V,27^64 y 8 ^ 27 Problem 1. To find a root of a numerical quantity by factoring. niustration. — Find the cube root of 1728. Solution: Since the cube root of a number is one of the three equal factors of the number, we resolve 1728 into its prime factors, and take one of every three equal ones, and find their product. Note. — To find the square root, take one of every two equal factors ; to find the fourth root, one of every four equal factors, etc. Form. 2 1728 2 864 *2 432 2 216 2 108 *2 54 3 1 27 3 9 *3 Vl728 = 2X2X3 = 12 168 ELEMENTARY ALGEBRA, EXERCISE 8B. Find the value of : 1. V324 4. V512 2. Vl296 5. V3375 3. ^2304 6. V5832 7. V4096 8. V20736 9. V248832 Problem 2. To find a root of a xnonoxnial. Ulustration. — Find the 5th root of ^o™* Solution: Since taking a root of every factor of a quantity takes the root of the quantity [P. 76], ^ X V^ X V^ X v^. y^^33 = -2 [P. 78]; V^=ia^, V^=Z», and V^=6'* [P. 75]; hence the result is — 2 «^ ^ C*. Therefore, Mule. — Tahe the required root of the numerical co- efficient and divide tJie exponent of each literal factor hy the index of the root. EXERCISE 86. Find the value of 1. Vc^YJ 8. VlMo^^V^ 2. V4.anu^^ 9. V- 729 (6^ + 0:)^ 3. Vx^y'z'^ 4k. V8 10. V256(«-:r)« twn'^ 5. V-27a:«y2 6. wmFvFp^ 7. V-^^a^H^'^c^ 11. V-(« + Zi)5ci« 12. VlOOOOa:»(a; + ^)i» 13. V- 243 {m + w)^« 14. V64 (:^ - «/2)i2 15. V^^y^x^^~yY, when a; = 6 and ?/ = 3 16. Vie flS ^,12 (^2 _ j,2Y^ ^hen a = 5 and J = 3 ALGEBRAIC EVOLUTION. 169 17. V(ar - b^f -f- V{a + bf when a = 3 and J = 2 18. l/(a + a:)« + V(a -- a:)^ -|- VC^^ - 3^)\ when a = 4 and a; = 2 19. /Va«1^+ f'Va^^- Va^, when a = - 3 and J = 5 Find the value of : y 16' r 9' T 16' y 25' y si ^/Z 3 /_i V ^ 3/ 8a:«y»" 3 / a;« (<^ -f a;) 3' y27' y 8^^^' y 27«=^i^2' y ^9^18 ^'*- y («_^)4' y (a;+i/)»' y (a-^)«' y z%a>^ y 27' y 3 ^ 9' y 128' y («-a:r' y m 20. 21. 22. Problem 3. To extract the square root of a polynomiaL I. Method. 188. Since the square of a polynomial is the sum of the squares of its terms and twice the product of each term into all the following terms [P. 73], the square root of a polynomial that is a perfect square may be obtained by inspection, if no terms of the power have disappeared by collection. Illustration. — V(a»-h2a^4-^ + ^gg + ^^g + g^) = 'v/(a* + J* + c24-2ad + 2flrc-f~2T(c) = a + J + c, since {a-{-h-\-cY = a--\-¥-\-c^^2ab-\-'ilac-\-%bc, { UNIVEP 170 ELEMENTARY ALGEBRA, EXERCISE 87. Extract the square root of : \. a^-\-¥^^al) 3. a;2 + 16 + 8:2; 2. x^-{-y^ — 2xy 4. :r2 — 6 :r + 9 5. x^-\-y^-\-z^-\-%xy-\-2xz-\-2yz 6. x^-\-y^-^z^ — %xy — 'ilxz-\-%yz 7. a^-\-^h^-\-^c^-\-^al)^^ac-\-l'^hc 8. 4.x^-^y^-{-^ — 4.xy^l%x-'^y 9. x^-^y^-i-2x^y-^2x^ + 2y-}-l 10. 9x^-24:xy + 16y^ 12. 4:X^-{-^y^-i-9-^2x^y-{-12x^-^3y 189. When the law of development does not appear by inspection, the following method must be resorted to. Illustration. — Extract the square root of x^-\-f-\-3x^y^-2x^y-2xy\ 7n-\-n {m + nY = m^ -\- {2 m -\- n) n m-^ 71 2a^y-\-3x^y^ — 2xy^-\-y^\ x^ — xy -\-y^ x^ %x^ — xy 2x?y -\-Zx? y^ 2x^y -\- x^y^ 2 o;^ — 2 :c y + y^ 2xry^ 2a^y^ 2xy^-^f 2xy^-\-y^ Solution : Having arranged the terms according to the descending powers of x assumed as the leading letter, we will proceed to take out of the polynomial the square of the first two terms of the root. For this purpose we let m + w represent the first two terms of the root. Now {m + nf = m^ + 2mn + n^, or m"^ + {2m + n)n. m' ob- viously equals a:*, or w = x^. Subtracting ic* from the polynomial ALGEBRAIC EVOLUTION. 171 and bringing' down the next two terms, we have — 2.r'y + 3a;'y'. This remainder consists mainly of (2 w + n) n ; hence if we use 3 w, or 2x', as a trial divisor^ we will obtain the value of w, which is — - 2 x* y -*- 2 a;', or —xy; adding this value of n to that of 2 m, we have 2x^—xy, the complete divisor ; multiplying the value of 2m + n, or x^ — X y, by the value of n, or —xy, we have — 2 a:* y + a;* y*. Subtracting this product from —2a^y + ^x^y^ and bringing down the remaming terms, we have 2x^y^ — 2xy^ + y*. We now let m represent x^ — xy and n the next term of the root, and proceed as before to take out of the polynomial the square of m + n, or w* + (2 m + n) n. m^ or (x^ — x yf has already been re- moved, hence the remainder 2x^y — 2xy'^ -v i^ is composeti of (2 m + n) n. Using 2 w, or 2x^ — 2xy, as a trial divisor, we obtain y' for n ; adding this to the value of 2 m, or 2x^ — xy, we have 2x'^ — 2xy-\-y^ for the complete divisor. Multiplying the complete divisor by y' and subtracting, nothing remains. Therefore the given polynomial is the square of x? — xy •\-y^, or x^ — xy -^ y^ is the square root required. From an inspection of the above solution the following rule will appear : 1. Arrange the terms of the polynomial according to the ascending or descending powers of some letter assumed as a leading letter, 2. Take the square root of the first term of the poly- nomial for the first term of the root. Subtract the square of this term of the root from the polynomial. 3. Double the root found for a trial divisor. Divide the first term of the remainder by the trial divisor for the next term of the root. 4. Add the last term of the root found to the trial divisor for the complete divisor. Multiply the complete divisor by the last term of the root found, and subtract the product from the remainder, and bring down such terms as are needed. 5. If the root has more than two terms, double the root already found for a new trial divisor, and proceed as be- fore to obtain the next term of the root and the complete divisor. Continue this process until all the terms of tlie polynomial have been used. 172 ELEMENTARY ALGEBRA. ^ote. — In the formula m^ + (3 m + w) n, m represents the root as far as found, 2 m the trial divisor, n the next term of the root and also the correction, and 2 m + n the complete divisor. EXERCISE 88. Extract the square root of : 1. x^-{-2a^-{-3x^-]-2x-^l 2. x^-4:X^-^ex^-4:X-^l 3. x^-^4.x^ + 10x^-\-12x-}-9 4. x^-6x^y-\-lda^y^-12xy^-\-4.f 5. x^-4:X^-\-10x^-12x^-i-9x^ 6. 4:X^ — 20x^y + S7x^y^-dOxy''^df •'■^' + ^ + |^^ + i^ + ^ 8.^^ + 4:^:^ + 6 + ^ + 1 9. x'^ -]- 2x^ — x^ -]- 3 x^ — 2 X -\-l Problem 4. To extract tlie square root of numerical quantities. I. Definition and Principle. 190. Every numerical quantity of two or more figures may be considered a polynomial. Thus, 123456 = 12 ten-thousands + 34 hundreds + 56 units. 191. The square of a unit is a unit, the square of a ten is a hundred, the square of a hundred is a ten-thousand, the square of a thousand is a million, etc. ; hence, the square denominations" in order are the unit, the hundred, the ten-thousand, the million, etc. Therefore, Prin, 81. — i/" a number he pointed off into terms of two figures each, beginning at the units, the unit of each term will J)e a perfect square. ALGEBRAIC EVOLUTION. 173 2. Examples. UlustratioiL — Extract the square root of 105625. Fornit 1 0'5 6'2 5 9 325 62 156 124 645 3225 3225 Solution: We point the number off into terras of two figures each to keep the unit of each terra a perfect square [P. 81]. 105635 equals 10 ten- thousands + 56 hundreds + 25 units. The square root of 10 ten-thou- sands is 3 hundreds, the first term of the root. Squaring 3 hundreds, we have 9 ten-thousands; subtracting 9 ten-thousands from 10 ten-thousands and bringing down the next term, we have 156 hundreds. Doubling the root already found for a trial divisor, we have 6 hundreds ; dividing 15 thousands by 6 hundreds, we have 2 tens for the next terra of the root ; adding 2 tens to the trial divisor, we have 62 tens for the complete divisor; multiplying 62 tens by 2 tens, we have 124 hundreds; subtracting 124 hundreds from 156 hundreds and bringing down the next term, we have 3225 units. Doubling 32 tens, we have 64 tens for a new trial divisor; dividing 322 tens by 64 tens, we have 5 units for the next term of the root ; adding 5 units to 64 tens, we have 645 units for the cora- plete divisor; multiplying 645 units by 5, we have 3225 units; sub- tracting this product from 3225, nothing remains. Therefore the square root of 105625 is 325. Note. — The square root may also be obtained by means of the formula {m + nf = m' + (2 m + n) ;i, as in example (Art. 189). EXERCISE 89. Extract the square root of : 1. 289 6. 2704 2. 676 6. 4761 3. 1225 7. 5041 4. 1849 8. 7056 13. What is the value of : (•1)2? (-01)2? (-001)*? V-01? V-0001 ? V -000001 ? V -00000001 ? 9. 16129 10. 60025 11. 104976 12. 166464 (-0001) 2? 174 ELEMENTARY ALGEBRA. 192. The square decimal units below one are the hun- dredth, the ten-thousandth, the millionth, the hundred- millionth, etc. Therefore, 193. If a decimal contain, or be made to contain by annexing ciphers, an even number of figures, its unit will be a perfect square. Illustration. — •24 = 24 X -01 -536420 = 536420 X '000001 •3645 = 3645 X '0001 5-000000 = 5000000 X '000001 14. Extract the square root of 5 to thousandths. Solution : Since the square P root is to be expressed in > i , thousandths, the number " 5*000000 |2-2364- must be reduced to mill- 4 ionths. 5 = 5,000,000 mill- ionths. The square root of 5,000,000 is 2236 +, and the square root of a millionth is a thousandth. Therefore the square root of 5,000,000 mill- ionths is 2236 + thousandths, or 2-236 + . Extract the square root of : 15. -0049 19. -00000016 23. 108-5764 16. -0625 20. -104976 24. 1024-6401 17. -000144 21. 33-8724 25. 99-980001 18. 882-09 22. 11-4244 26. 8010-25 27. Find the square root of 10, 11, 12, and 13 to within one ten-thousandth. 7 . /T to 4 28. Find the value of V2, decimal places. 29. Find the value of a/40, V4i, a/42, and a/43 to within one thousandth. 42 100 84 44: 5 1600 1329 446( 3 27100 26796 ALGEBRAIC EVOLUTION. 175 Problem 5. To extract the cube root of a polynomiaL Illustration. — Extract the cube root of Fornu ^'^ + ^i (m + n)' = m» + (3m»+3mn + n«)/» w+n x^->fxy-\-y^ T. D. =3x4 1st Cor. = Z7?y 2d Cor. = xhf CD. =3x4 + 3a;»i^+a;V 3a;»y + 6x^^2 + 7u;V T.D. =3x* + 6x»i/ + 3xV 1st Cor. = 3a;V- 2d Cor. = f3xy» 3x*2/* + «-^/ + 6a:V+3a:3/*+3^ C. D. = 3a;4+6a:»y+6xV + 3a:!/3 + y* 3 a;4y' + 6 a: V + 6 ;'^y + 3 X2/6 + 3/« Solation : Having arranged the terms according to the descending powers of x, assumed as the leading letter, we proceed to take out of the polynomial the cube of the first two terms of the root. For this purpose we let wi + w represent these terms. Now, (m + ixf = m^ + 3 m' 7t + 3 w n^ + w', or m^ + (3 m^ + 3 m w + w') n. m^ obviously equals JK*, or m equals a;'. Subtracting x^ from the polynomial and bringing down three terms, we have for the first remainder, 3a:^y + 6a:*2/* + 7 x* y*. This remainder consists mainly of (3 w* + 3 m n + w*) n ; hence, if we use 3m^ or 3 a:*, for a trial divisor, we will obtain the value of n, which is 3 aH* y -s- 3 a:*, or xy. Substituting the values of m and n in Zmn and w*, we have 3 a;* y for the first and x^ y^ for the second correction ; adding the two corrections to the trial divisor, we have 3a:* + 3a:^y + a:*2/' for 3m* + 3m7i + n*, the complete divisor ; mul- tiplying the complete divisor by xy, the value of n, we have 3a:'y + 3a:*y' + a:^2/' ; subtracting this from the first remainder, and bringing down the remaining terms, we have 3a:*y^ + 6a:'i/' + 6a:*y* + 3a:2/'+y*. We now let m stand for x* + xy, the root already found, and n for the next term of the root, and proceed as before to take from the polynomial the cube of m + n, or m* + (3 m' + 3 m n + w*) w. m^, or (a:* + X y)\ has already been subtracted ; hence, the remainder consists of (3w^ + 3mw + n'^)n. Using as before 3 w^ or 3a:* + Qx^y + 3x*y', as a trial divisor, wo obtain y^ for ?» ; substituting the values of m and n in 3 win and w', we have 3 a;' y' + 3 a; y* for the first and y* for the second correctim, and 3a:* + 6a:*y + 6a:*y* + 3a:y* + 2/* for 176 ELEMENTARY ALGEBRA. the complete divisor. Multiplying the complete divisor by 2/^ and subtracting the product from the last remainder, nothing remains. Therefore, x^ + xy + y^ is the cube root required. Note. — In the formula m^ + (Sm^ + Smn + n*)n, m staTids for the root already found, 3 m^ for the trial divisor, Smn for the first cor- rection, n^ for the second correction, d m^ ■{■ 3 m n + n^ for the com- plete divisor, and n for the next term of the root. From an inspection of the above solution the following rule will appear : 1. Arrange the terms of the polynomial according to the ascending or descending powers of some letter assumed as a leading letter, 2. Take the cube root of the first term of the polynomial for the first term of the root. Subtract the cube of this term of the root from the polynomial. S. Take three times the square of the root already found for a trial divisor. Divide the first term of the remainder by the trial divisor for the next term of the root. Jj.. Add to the trial divisor three times the last term of the root found multiplied by the preceding part of the root, and the square of the last term found, for a complete di- visor. Multiply the complete divisor by the last term, of the root found, and subtract the result from the remainder, bringing down only such terms as are needed. 5. If the root has more than two terms, take three times the square of the root already found for a new trial divisor, and proceed as before to obtain the next term of the root, the new corrections, and the new complete divisor. Con- tinue the process until all the terms have been used. EXERCISE 90- Extract the cube root of : 1. a:^ + 3a^ + 3a; + l 4. 8a;« + 36ir* + 54.^'2 + 27 2. a^-^a''b^ZaJy'-¥ b. x^ -\-^x^ - bx^ -\-Zx-l 3. ic3-j-12a:2 4-48a;-f64 6. i/^ - 3^ + 5 «/3 - 3y - 1 7. a^a^ — ^a'^bx^-^-daWx-b^ 8. Sa^x^-dQaHx^y + h4.al^xy^-21b^f ALGEBRAIC EVOLUTION. 177 9. a:^ - 6 ar^ + 21 r^:* - 44 2^3 _^ 53 ^ _ 54 2. _j_ 27 12 10. Q^ + ^X^--\-\ 11. a^o^ — Qax-{- 8 ax d^7? 12. ^« + 3a:* + 6a:^+7 + | + | + ^ 13. a^-\-¥^(?^Zd^l^Za^c-\-ZaW-^Z}rc^ Problem 6. To extract the cube root of niunerical quantities. 1. Principle. 194. The cube of a unit is a unit, the cube of a ten is a thousand, the cube of a hundred is a million, the cube of a thousand is a trillion, etc. ; hence, the cubic denom- inations in order are the unit, the thousand, the million, the trillion, etc. Therefore, Prin, 82, — 7/" a number he pointed off into terms of three figures each, beginning at the units, the unit of each term will be a perfect cube. 2. Examples, niustration.— Extract the cube root of 16387064. ForiQf (m -\-nf = m^-\-{Zm^ + ^mn-\- n^) n 16'387'064 ^3= _8 3^2 =3x(2..)2 =12. . . . 8387 Zmn =3X2..X5.= 30 . . n- = (5.)2= 25. in-¥n 2 5 4 3 7^2-1-3 w. 71-1- ;r = 1525 7625 37/12 =3X(25.)2 = 1875 37/i7i =3x25.X4 = 300. w* = 42= 16 762064 37^^4-3 m n-\-n^ = 100516 762064 178 ELEMENTARY ALGEBRA. Solution : We point off the number into terras of three figures each to make the unit of each term a perfect cube [P. 82J. We find thus that 16,387,064 = 16 million, + 387 thousand, + 64 units. The cube root of 16 million is 2 hundred + . Cubing 2 hundred, we have 8 million ; subtracting 8 million from 16 million and bringing down the next term, we have 8387 thousand. Taking 3 times the square of the root already found (3 m^) for a trial divisor, we have 12 ten-thousands ; dividing 83 hundred-thousands by 12 ten-thousands, we have 5 tens (7?), the next term of the root ; taking 3 times the root previously found (3 m) and multiplying it by the last term found (n), we have 30 thou- sand for the first correction, and squaring the last term of the root {n% we have 25 hundred for the second correction ; adding the trial divisor and the two corrections, we have 1525 hundred for the complete divisor (3 m^ + 3 m w + 71^) ; multiplying the complete divisor by the last term of the root {n\ we have 7625 hundred ; subtracting 7625 hundred from 8387 hundred, and bringing down the next term, we have 762064 units. Taking 3 times the square of 25 tens (the new value of m), we have 1875 hundred (the new trial divisor) ; dividing 7620 hundred by 1875 hundred, we have 4, the next term of the root {n) ; finding as before the values of 3 m w and w^, we have for the two corrections 300 tens and 16 units, and for the complete divisor (3 m^ + 3 m /i + n^) 190516 ; multiplying by 4, or n, we have 762064, which subtracted from 762064 leaves nothing. Therefore the cube root of 16,387,064 is 254. Abbreviated Rule. 1. Point off the nwnber into terms of three figures each, 2, The cube root of the first term gives the first figure of the root. S. Three times the square of the root already found always gives the trial divisor. J/.. The remainder, exclusive of the two right-hand fig- ures, divided hy the trial divisor, gives the next figure of the root. 5. Three times the root previously found multiplied hy the last figure found gives the first correction, and the square of the last figure found, the second correction. 6. The right-hand figure of the first correction is placed one order to the right of the trial divisor, and that of the second correction one order to the right of the first correction. 7. The sum of the trial divisor and the two corrections gives the complete divisor. ALGEBRAIC EVOLUTION. 179 EXERCISE 91. Extract the cube root of : 1. 2744 5. 373248 9. 1815848 2. 19683 6. 592704 10. 10941048 3. 42875 7. 681472 11. 28372625 4. 300763 8. 941192 12. 74088000 13. What is the value of : (•1)3? (-01)3? ' (-001)3? V-001 ? V -000001 ? V -000000001? 195. The cubic decimal units below one are the thou- sandth, the millionth, the billionth, the trillionth, etc. Therefore, 196. If a decimal contain, or be made to contain by annexing ciphers, a whole number of times three figures, its unit will be a perfect cube. niustration : -325 = 325 X '001 ; 4-25 = 4-250000 = 4250000 X '000001 14. Extract the cube root of 3*25 to hundredths. Form. 3-'2 5 0'0 0[lj47 1 3 X P = 3 3X1X4 =12 42= 16 436 2250 1744 3 X 142 = 5 88 3X14X7 = 294 72= 49 61789 506000 432523 Solution : Since the cube root is to be expressed in hundredths, the number must be reduced to millionths. 3*25 = 3,250,000 millionths. The cube root of 3,250,000 is 147 + , and the cube root of 1 millionth is 1 hundredth. Therefore the cube root of 3,250,000 millionths is 147+ hundredths, or 1-47+. 180 ELEMENTARY ALGEBRA, Extract the cube root of : 15. -008 18. '-015625 21. 39-304 16. -001728 19. -029791 22. 13-824 17. -000027 20. -125000 23. 8I-37V27 24. Find the value to thousandths of V^, A/ — , V-27 Problem 7. To extract higher roots of quantities. 197. Since {ay=a\ Va^=Wa^; also, smce (aY=a\ Va^=VV^; again, since (a^)'^ = a^, X/c^=Vl/c^, Therefore, To extract the fourth root of a quantity, extract the square root of the square root ; to extract the sixth root, extract the cube root of the square root; and to extract the ninth root, extract the cuhe root of the cube root, EXERCISE 92. 1. Extract the fourth root of 2^ + 4ic«/ + 6 a:V + 4^^2/^ + 2/^ 2. Extract the sixth root of a;6 -f 6 ^5y + 15 2;* 1/2 _|_ 20 a:3^3 _|_ 15 ^2^ _|. g ^^5 _}_ ^e 3. Find the value of V256, V729, Vl953125, -V-0016 Factoring with the aid of Evolution. 198. If a polynomial is the difference of the squares, the difference of the cubes, or the sum of the cubes of two quantities, it may be factored by P. 39, 44, 43. niustrations.— 1. Factor x'^ -\-%x^ ^ha? -{-^x-\-^. Solution : By extracting the square root of the given polynomial we find that it lacks 1 of being the square of a:;^ + a; + 2, .-. a;* + 2a;3 + 52;2 + 4a: + 3 = (a;2+a; + 2)2-l = (a;2 + cc + 2 + 1) (a;2 + a; + 2 - 1) [P. 39] = {x^ + x + ^){x^ + x + 1). FACTORING WITH THE AID OF EVOLUTION. 181 2. Factor oi? -[-Qx^ +l^x-\-l. Solution : By extracting the cube root of a:^ + 6 a;* + 12 a; + 7, we find that it lacks 1 of being the cube of a; + 2, .-. a? + Qx^ + \2x-k-l = ix + 2f-l; ,'. a:^ + 6a;« + 12a; + 7 is divisible by a; + 2 — 1 or a; + 1 [P. 44] ; .-. a;3 + 6a;« + 12a; + 7 = (a; + l)(a;» + 5a; + 7). 3. Factor x''-{-3ax^-{-3a^a^-\-9a\ Solution : By extracting the cube root, or by inspection, we see that a;* + 3 a a;* + 3 a* a;* + 9 a' is 8 a' more than the cube of a;' + a, .-. a:« + 3aa:* + 3a*a:» + 9a» = (a;« + a)» + 8a«; . • . a;* + 3 a a;* + 3 a* a;* + 9 a' is divisible by a;* + a + 2 a, or a;« + 3a [P. 43]; .-. a:« + 3 aa;* + 3 a^a;^ + 9a« = (a;2 + 3 a) (a;4 + 3 a»). EXERCISE 93. Factor : 1. 3^-j-a'^a:^-{-a^ 4. 25 a^ - 9 r^ y^ -{- 16 y* 2. a^^ex-}-6 5. 4jo*-37yg2 + 9^ 3. 4.a^ + 12xy-^6y^ 6. 64a* + 128^2^ + 81^* 7. a^-^2:x^-\-33^-{-2X'-3 8. 4:a^-{-20x^y-{-29x^y^ + 10xf-3y* 9. 8a:3_j_60a:2 4-l50a; + 61 10. 27a^-54:a^y-}-36xy^-7y^ 11. a;« + 6a^«/8 + 12a:2^_19^6 12. 8a^-12a'b^-\-6aH'-9b^ 13. a^-]-303^-\-300x^S75 14. 8 a» - 12 a« + 6 a^ + 7 15. fl^ + 3a«^>3_|_3^3j6_|_9j9 16. a^2_3a8J4_p3^4 58_28J12 17. 4a2+12aJ + 8^>2_^16ac + 225c + 15c3 18. 27a3-135a2& + 225a^-61J^ 19. a»a;3_^3^j2^y^3^2j4a.^2_7j6^ CHAPTER V. QUADRATIC EQUATIOJTS. Quadratic Equations of One Unknown Quantity. Pure Quadratics. I. Definitions and Principles. 199. A quadratic equation containing only the second power of an unknown quantity is a pure or incomplete quadratic equation ; as, 3 a:^ = 8, or — + — - = - . ODD 200. Every equation containing fractional terms may be cleared of fractions [P. 71]. All the unknown terms in the second member may be transposed to the first, and all the known terms in the first member to the second [P. 70]. All the unknown terms in the first member may be united into one and the coeflBcient represented by a, and all the known terms in the second member may be represented by i. If a is negative, the equation may be divided by — 1. Therefore, Prin. 83. — Bvery pure quadratic equation of one un- known quantity/ may be reduced to the form of ax^'=^l, in which a and b are integral and a positive. 201. Any value of the unknown quantity that will satisfy an equation — that is, will make the two members equal — is a root of the equation. PURE QUADRATICS. 183 202. Take the equation au? = 1. Divide by a, a;^ = — . Take the square root of both members [P. 69, 6], --/!■ Therefore, JPrin. 84:, — Every pure quadratic equation of one un- known quantity has two roots, numerically equal, hut op- posed in sign. 2. Solution of Pure Quadratics. lUustratioiis.— 1. Solve V + 5 = 2 a;^ _ -3. Solution : Given ?|^+ 5 = 2a;»-3 (A) Clearing of fractions, 3a;» + 10 = 4a;« — 6 Transposing, — a;» — — 16 Dividing by — 1, a:^ = 16 Taking V, x- ±4. (1) (3) (3) 203. Sometimes equations have the pure quadratic form, and may be solved as such when they are not really such. Their roots may not be numerically equal. 2. Solve (£±^ + 3a» = ii^'-2a». (A) Suggestion. — Clear of fractions, {x + a)« + 12 a« = 6 (a; + a)« - 8 a« (1) Transpose terms, — 5 (f + a)* = — 20 a« (2) Divide by - 5, (a; + a)« = 4 a« (3) Take V, x + a=±2a (4) Transpose, a;= + 2o — a;or — 2a — a (5) Collect terms, x=z a or — 3 a. EXERCISE 94. Solve : 1. -^ + -3- = lU 3.0; + - = - ^--4 3-2a:2^5 3a:«-5_, ^•""2 r~~-4 ^—^--^ 184 ELEMENTARY ALGEBRA. X _x — 2 X a-\-x x-\-2 2x ' a — x H) 9 6. l^ + ^l =4ic2 10. (Sx^-9y:=-a^ , a X , ' ^+5 a;_3a: 17 8. --^ = - 12. [ax I =-a^x^ X c \ x] ^ 13. i(^ + 4)^ = |(:.+ 4)2-6 14- 2-5 (5^ + 36)^ = jg (8 a^- 4)= 15. If the side of a square be doubled, its area will be increased 75 square rods. What is the side of the square ? 16. Four times one number equals five times another, and the difference of their squares is 81. Find the numbers. 17. A man bought a tract of land for $5000, paying twice as many dollars per acre as there were acres in the tract. How many acres were there ? 18. I sold a horse at a gain of $81, and thereby gained as many per cent as there were dollars in the cost. What was the cost ? 19. Three times the sum of two numbers equals 10 times the smaller number, and if the sum be multiplied by the greater, the product will be 630. Required the numbers. 20. If the dimensions of a certain cube be quadrupled, the entire surface will be increased by 7290 square inches ; what are its dimensions ? 21. A man received % as many dollars per day as he worked days ; had he worked only ^5 as many days and received Yg as many dollars per day, he would have re- ceived $26 less. How many days did he work ? AFFECTED QUADRATICS. 185 Affected Quadratics. I. Definition and Principles. 204. A quadratic equation which contains both the first and second powers of an unknown quantity is an affected or complete quadratic equation ; as, 3ar + 72: = 15, or— -5 = — +6. 206. It may be shown, as in Art. 200, that : Prin, 85. — Every complete quadratic equation of 07ie unknoiun quantity may he reduced to the form of aoi? -\- J)x = Cy in which a, h, and c are integral, and a positive, 206. Take the equation aa^ -\-l)x-= c. Divide by a, a^ H — x= - , •^ ' ^ a a h c Put p for — and a for — , x^ -\-px=z n. Therefore, ^ a ^ a ^ ^ Prin, 86, — Every complete quadratic equation of one unknown quantity may he reduced to the form of x^ -\-px = q, in which p and q may he integral or fractional, positive or negative. \ Since p and q may be either positive or negatiye in the equation x^ -^ p x = q^ it follows that : Every complete quadratic equation may he reduced to one of the four following special forms : 1. or -\-px= -\- q 3. x^-\-px=— q 2. x^—px=-\-q 4:. 3r — px= — q 2. Solution of Numerical Affected Quadratics. 208. The first member of an affected quadratic equation can always be made the square of a binomial by a process 186 ELEMENTARY ALGEBRA. called '^ completing the square,^'' then the square root of both members may be taken, and the resulting simple equation solved. Illustration.— Solve 8 a;^ — 3 ic = 26. (A) Solntion : Multiply by 2 to make the first term a perfect square, 16a:2-6a; = 53 (1) Regard \Qx^ — Qx as the first two terms of the square of a bino- mial, then Vl6a;^ or 4ic, is the first term of the binomial, and —Qx is twice the product of the two terms; therefore {—Qx)-^{2 x 4a;), Q or (— 6) -4- (3 X 4), which is —-j-, is the second term of the binomial, and ( — -J ) ' ^^ i7> ' ^^ ^^^ third term of the square of the binomial. Add this to both members, 16.^-6x-H^ = 53 + ^ = § (2) Q OQ Extract V» 4 a; - -^ = ± ^ (3) 13 Transpose, 4 a; = 8 or — ^ (4) K Divide, a; = 3 or — 1 -5- o 209. Hence we have the following rule : 1. Reduce the equation to one of the typical forms. 2. Multiply or divide loth memhers of the equation ly any quantity that will render the first term a perfect square. 3. Add to loth members the square of the quotient ob- tained hy dividing the coefficient of x ly twice the square root of the coefficient of cc^, to complete the square. J/,. Extract the square root of both members and solve the resulting simple equation. Illustrations.— 1. Solve %a? — l%x = %. (A) Solution : Divide by 3, 4 a;^ — 6 a: = 4 (1) -G-7f)'-l' --e..| = 44.- (3) Extract the V, 2 a; - |- = ± I (3) Transpose, 2 a; = 4 or — 1 (4) Divide, a; = 2 or — -^ NUMERICAL AFFECTED QUADRATICS. 187 2. Solve 3 2;2 + 2a: = :33. .» + |.= (A) Solntion : Divide by 3, = 11 (1) Add (| + 2vT)', or ( a;« 4-1= = "4 _100 9 (3) Extract V> 1 *+3 = -V" (3) Transpose, a; = r3 or — 4' Therefore, Scholium, 1. — When the coefficient of 7? is 1, the quan- tity to be added to both members to complete the square is the square of half the coefficient of x. 3. Solve 3 3:^-5 a; = 28. (A) Solution : Multiply by 3, 9a;»-15a: = 84 (1) -(;;.)'-(! )• 9a;» 25 ^, 25 361 -15x + — =:84 + -p = -p 4 4 4 (2) Extract the -y/. --|=±V' (3) Transpose, 3a; = 12 or —7 (4) Divide, a; = 4 or -2^. o Therefore, Scholiuni 2. — When the equation is multiplied through by the coefficient of a^, the quantity to be added to both members is the square of half the coefficient of x in the typical equation. This method is generally the best, as it avoids all fractions above and below fourths. 4. Solve 2a:« + 3a; = 14. (A) Solution : xMultiply by 4 x 2, 16a;« + 24 a; = 112 • (1) / 24 \« Add (^-7--) , or 3«, lex* + 24 a; + 9 = 121 (2) Extract the V, 4a; + 3 = ± 11 (3) Transpose, 4 a; = 8 or —14 (4) Divide, a; = 2 or —^2' Therefore, 188 ELEMENTARY ALGEBRA. Scholium 3, — When the equation is multiplied through by four times the coefficient of a:^, the quantity to he added to loth members is the square of the coefficient of x in the typical equation. This is called the Hindoo method of completing the square. It avoids all fractions, hut often gives rise to very large whole numbers, EXERCISE 9S. Solve : 1. a;2 -f- 2 a; = 8 20. 16 rr^ — 16 o^ = 45 2 . r..,_ a 3 9 2. x^-'Zx = 24. 2,7 10 21. X^ -\-—X=z Z. x^-\-bx=-Q 4:. x^-dx= — 20 5. ^2 _ 3 ^ = 18 6. 2;2 — a: = 20 7. :r2 - 11 a; = - 28 8. x^-^4:xz=60 9. x^-^x = 66 10. x^ — x = 110 11. 4:X--j-10x=-6 12. Sx^-'i!x = 6 13. 2a;2-7a; = 30 14. 6x^-\-dx = Sl 15. 6x^-\-3dx=-28 29. x^-2x=-~ 16. 6x^-4.7x = (j3 17. 6^^ + 190:= -15 30. -^ + ^^ = 2^ 18. 2^:^-9^ = 35 ^ ^ ^ 31. 22. a;2 + a; = 8| 23. ^2-f 1 a; + 2 =4 5 ' 3 24. 2 2^2_3 ^_g 7 14 =4 25. 2:2-4a;=-l 26. 2;2-6a; = ll 27. X 4 28. ir+1 a; 13 u; x-\-l 42 19. 9:^2 + 27 a:= -14 a^'+l x-1 3^-1 x^-{-x^l a;^ - a; + 1 _ 23 ^^' ^ 6 5 - 30 LITERAL AFFECTED QUADBATICS, 189 3. Solution of Literal Affected Quadratics. ninstrations. — 1. Solve oc^ -\-ax = b. Solution : C'omplete the square, Extract the V» x-\-^ = ±-^ V4& + a« Transpose, a: = -|^ ± ^V4& + a2= -^(a T V46 + a2) 2. Solve a7?-\-lx = c. (A) Solution : Multiply by a, a'a;' + a&a: = ac (1) Complete the square, 6* &« 4ac + 6« ^, a«a:« + a&a; + -j=ac + -j= ^ <^) Extract the V, aa; + |=±|V4ac + 62 (3) Transpose, a a- = — -s-±:s-V4¥c + &« ==—-^(&TV4ac + &2) (4) 3 -" 3 1 Divide by a, a; = — 5— (J T V4 a c + 6*) EXERCISE 96. Solve : 1. x^^'iax — 3a^ 4. a^oi^ — acx = 'i(^ 2. a:2_3Ja:=:_2J2 5. x^-^(al)^y)x^ -al^ 3. a:^ + m a: = 6 m^ 6. a:2_|_fl2_^^ 7.2-^-^ = 3 a a; 10.^ + ^=^ + ^ 3,« + x a _^ a a — X x a — x x--b h ^' a "x-h 12. ^ + l = a + i 13. (^ + i)(x4-^) = — 4. 190 ELEMENTARY ALGEBRA, (x -\- m) (x — n) imx — n^ 14. _ = _ 15. acx^—'bcx-\~adx--'bd=0 16. aJ)x^-{a^-¥)x-al = Q Equations in the Quadratic Form. Definitions. 210. When an equation contains two and only two ex- ponents of the unknown terms, and one of them is twice the other, it is said to have the quadratic form ; as, ic*+6a;2 = 16, ax^^hx^^zc, or {a-\-'b xy -[-p {a-{-d xf = c, 211. Any equation haying the quadratic form, whatever its degree, may be solved by any of the methods employed to solve an affected quadratic. lUustrations.— 1. SoIyo x^ -\- 6 a^ = 16, (A) Solution : Complete the square, ^4 + 6^2 + 9 -25 (1) Extract V, a;^ + 3 = ± 5 (2) Transpose, x^ = 2 or — 8 (3) Extract ^/, a: = ± V 2, or ± 2 V^' 2. (x + 4.Y-j-{x + 4.Y = S^. (A) Solution : Complete the square, {x + Af + i ) + -i- = 4 (1) Extract V, (^^^ + 4)* + i = ± 2 (2) Transpose, {x + 4)^=1-^ or — 2 -^ (3) 4 . • 4 , Extract ^, a; + 4 = ± 4/1 J_ or ± 4/ _ 2^ (4) Transpose, re = 4 ± y^ JL or 4 ± 4/_2 — EQUATIONS IN THE QUADRATIC FORM. 191 EXERCISE 97. Solve : 1.^ + 2x^ = 24 ,.81.+ !=^-^ 2. x«-9x'=-8 ^^ 3. a;» + a:* = 6 8. a:^ + ^^^^ = 74 :c 6. ^- + 31.^ = 32 10..^ + ^ = ^ 11. (a; + 2)* + 4(2: + 2)2 = 21 12. (2a; + l)2 + 3(2a;+l) = 70 13.(. + iy-(. + l) = 6 14. (x2^a; + 2)2 + a:2^^^2 = 6 16. {a? -\-2xf -2^(7^ -\-%x) -\-VZO = 17. (a:2_5^)2_8a;2 + 40a; = 84 15 19. 2^|a:2_l\ _ 11 (3^:2 _ 2) = -10 21. a;'* + 4j:'' = 12 22. ar^4-a: + -^-=3 ar ' 'a: 4 192 ELEMENTARY ALGEBRA. Solution of Equations by Factoring. Illustrations. — 1. Solve 4 a:^ = 1. (A) Solution: Transpose 1, 4a;2 — 1 = (1) Factor, {2x + \){2x-l) = [P. 39] (3) Divide by (2a; + 1), 2x — l = Transpose and divide, ^ = o" Divide (3) by (2x - 1), 2a; + 1 = Transpose and divide, x=: — — 2. Solve a;2 + 5 a.- + 6 = 0. (A) Solution : Factor, (a; + 2) (a; + 3) = [P. 40] (1) Divide bya; + 2, a; + 3 = Transpose, a; = 3 Divide (1) by (a; + 3), a; - 2 = Transpose, a; = 2 3. Solve Qx^^llx-10 = 0, (A) Solution : Factor, (3 a; - 2) (2 a; + 5) = [P. 41] (1) (3) (3) Divide by (3 a; — 2), 2a; 4-5 = Transpose and divide. a;=- 2- 2 Divide (1) by (2 a; + 5), 3a;-2 = Transpose and divide, 2 ^=3 4. Solve a?-l = (}. (A) Solution : Factor, {x - 1) (a;^ + a; + 1) = [P. 44] (1) Divide by (a;^ + a; + 1), a; — 1 = Transpose, a; = 1 Divide (1) by (a; - 1), a;^ + a; + 1 = (2) Solve (2), a; = ^±iv^r3 5. Solve re* + a.-^ + 1 = 0. (A) Solution : Factor, (a;^ + a; + 1) (a;^ - a; + 1) = [page 84] (1) Divide by (a;^ + a; + 1), a;^ - a; + 1 = (2) Solve (2), a; = 4" ± T ^ 2-^2' Divide (1) by {x^-x + 1), (a;^ + a; + 1) = (3) Solve (3), a; = - i ± i V^^ SOLUTION OF EQUATIONS BY FACTORING, 193 6. Solve 3^-a7r-a^x-\-a^ = ^. (A) Solution : Factor (A), x^ {x- a)- a^{x- a) = (1) Factor (1), ' (x* - a«) {x-a) = (2) Factor (2X {x - a){x + a)ix- a)=zO (3) Divide (3) by {x - a) (.c + a), x — a = (4) Transpose, a: = o Divide (3) by (a; — a) (a; — a), a: + a = (5) Transpose, x= — a Divide (3) by (a; + a) (a; - a), a: — a = (6) Transpose, x = a 7. Solve 4a^ + 12a;3 + 29x2_|_30a;_j_2l. (A) Solution : By extracting the square root of the first member we find that it lacks 4 of being (2x^ + Sx + 5)«; .-. (2a;^ + 3a; + 5)«-4 = (1) Factor, (2^+3^+5"+ 2) (2a;«+3a: + 5- 2) = [P. 39] (2) Collect terras, (2a:*+3a: + ' 7){2x*+Sx + 3) = rO (3) Divide (3) by 2 ia;«+3a; + 7, 2a;2+3a; + 3 = = (4) Solve (4), X = = - i(3±/^ -15) Divide (3) by 2 la;»+3a: + 3, 2a;2+3a; + 7 = = (5) Solve (5X x = --- ^(3±V- -47) Kote.— This equation may also be solved by adding 4 to both members and extracting V« EXERCISE 98. Solve by factoring : 1. ar~4 = 10. 9a;2 - 24a; + 16 = 2. 4a:2-9 = 11. 2x'-'7x-15 = 3. 9a:2_4^2_o 12. 6a;2 _ 13^ _|_ g _. q 4. 2:2 _ 1/^^ = 13. S3^+Ux-15 = 6. a:2 4-7a;_|_io = 14. 3x^ -{-Sx- 35 = 6. x^-6x-\-S = 15. ar^-8 = 7. ar + 3a:-18 = 16. 2:^ + 1 = 8. 42:2_2a;_i2 = 17. a,'^ + 8 = 9. 42r -I- 12 a; -1-9 = 18. a:^ - 27 = 194 ELEMENTARY ALGEBRA. 19. a;^ + 27 = 22. a;« — 1 = 20. ic* — tt* = . 23. x^ — a^ = 21. x^ — 81 = 24. 82^ - 27 a^ = 25. :?:3 + 6 a;2 — 4a; - 24 = 26. ic^ — a;2 — a; + l = 28. a;-^ + 3:z;2_^3a; + l = 27. a;* 4- a;2 + 1 = 29. a;* — 13 a;2 + 36 = 30. ic^-2a;3 + 3a;2-2a; + l = 31. 0:3 + 60:2 + 12 a; + 8 = 32. 0:* — 8o:3 + 24o:2 — 32o: + 7 = 33. {x-a){x-hy^x''-{a^h)x-\-ab = Formation of Quadratic Equations. I. Principles. 212. If we solve the general equation x^-\-px=.q, we will find the roots to be : W\/\ ^P^ + q and The sum of these roots is — ^ ; Their product is — -rp^ — j -jp^ + 2' ) = — S'* Therefore, rrin, 87^ — The sum of the two roots of an equation of the form of x^-\-px = q equals the coefficient of a?, with the sign changed. Frin, 88, — The product of the tivo roots of an equation of the form of x^-\-px = q equals the absolute term with the sign changed. FORMATION OF QUADRATIC EQUATIONS. 195 2. Examples. ninstrations. — 1. Find the equation whose roots are + 4 and — 6. Solution : The coefficient of a: = - (4 — 6) [P. 87] = + 2 ; The absolute term = - (+ 4 x - 6) [P. 88] = + 24 ; Therefore the equation is a;'* + 2 a; = 24. 2. Find the equation whose roots are 2+V^ and 2— Vsl Solution : The coefficient of a; = - {(2 + V3) + (2 - ^/3)} [P. 87] = - 4; The absolute term = - (2 + V3 ) (2 - VS") [P. 88] = 1 ; Therefore the equation is a;* — 4 a; = 1. EXERCISE 99. Form the equations whose roots are : 1. + 2 and + 4 lO. 1 + a/2 and 1 - a/2 2. - 3 and + 5 ii. 3 + V2 and 3 - a/2 3. _8and +3 ^^ 2a-dand2« + d 4.-5 and — 4 6. 2« and a 6. 3 jt? and - 2^ 14. -^ and - 7. a and — 8 a 4 3 . T 1 -L 15. TT and -r 8. a + c* and a — o 3 4 9. a^ + ^ aiid c? — W 16. a -f 2 m and a — 2 m 13. a 4- VJ and a—^/h Formation of Equations by Composition. Illustrations. — 1. Form the equation whose roots are + 2 and — 2. Solution : If a; = + 2, a: - 2 = (1) Ifa:=-2, a: + 2 = (2) Multiplying together (1) and (2), (a;-2)(.c + 2) = (3) Expanding, x' — 4 = 196 ELEMENTARY ALGEBRA, 2. Form the equation whose roots are — 4 and -)- 7. Solution ; If a; = — 4, a; + 4 = (1) Ifa;= + 7, iB-7 = (2) (a: + 4)(a:-7) = (3) Expanding, a:^ — 3 a; — 28 = 3 2 3. Form the equation whose roots are — -r and -. 4 3 Q Solution: lix = — -, 4a;= — 3, and 4ic + 3 = (1) 2 Ifa; = -g, Zx = 2, and 3^-2 = (2) (4a: + 3)(3a;-2) = (3) or, 12a;2 + a;— 6 = 4. Form the equation whose roots are -{-'^y —2, and -1-3. Solution ; It x= + 2, {x — 2) = (1) It x = -2, (a; + 2) = (2) If a; = + 3, (a^ - 3) = (3) ,-. (a:-2)(a; + 2)(a:-3) = (4) Expanding, a:^ - 3 a;^ - 4 a; + 12 = (5) Observe that an equation always has as many roots as there are units in its degree. EXERCISE lOO. Form the equations whose roots are : 1. -}-3 and — 3 2. + 5 and — 7 3.-5 and -\- 7 4.-5 and — 7 5. -|-5 and +7 6. 2 ,3 ^and^ ' 7. 5 and — ^ 8. 3,2 -^and-- 9. + 1 and -3 10. a and — 2 a 11. 2, 3, and —3 12. 3, — 3, and 4 13. 1, 2, and 2 14. 3, 0, and 1 15. 11 ^ 1 16. -, -, and 1 17. 0, -, and -^ 2 3 ,4 18. 3, 4, and g MISCELLANEOUS EXAMPLES, 197 Miscellaneous Examples. EXERCISE lOl. Solve : 1, 7^-a^ = a^- {x - af 2ar _ 14 3 , 6 3 6. 3a;+180 9a;-15 2a; + 12^ic 2r2: + 4 10 10_ _ 16 12 + 3a; 3a;-12"~ 9 6 3:^4-10 2^:^ + 58 _234 2a;+14 2x-U _ 14 8a Zx'-^ix 3 a^ -^Wx 3 a;2 _ 219 10. ic(a:-l) = ^(a;2_|.22;) 2 2a:-10 , 2a:+6 , 1 11. — * ' — = 1 — 3 8-a; ^6-32: 3 x-\-3 _ 3 — 2x x — S ^^' x + 'Z" 1-x ^2-x X , m X , n o^o...! 13. — + — = -4-- 16. aar^ — 2a^x-]-a^=- m X n X a acT? lex , 1 11.1 14. ax ^— =^ h 16. :r~. — = T + - d a a — o-\-x a ox Find the approximate values of x in the following equations : l7.^-12a: = 21 19. 1 - ^ = - i 2 X of 5 a; + l 198 ELEMENTARY ALGEBRA. ^ 39 a:^ '^Qx^ 13 21 25 x 28 42 14 10 - 2 a; ~ 14 14 (5-^H^+5)_ll 5 3^+l_l Solve : 25. ^^ + A = «' + ^ 26. 0:^ + 1 = 27. 4i?;* + 16.T3 + 16a; + 4 = 57a;2 28. ^2_2:x;_2 + --i^ = 29. Find the three cube roots of 1. Suggestion. — Let a; = V 1 , then a;^ = 1, or a;* — 1 = 0. 30. Find the three cube roots of — 1. 31. Find the four fourth roots of 1. 32. Form the equation whose roots are a, —a, h, — hy c, — c. Examples involving Quadratic Equations of One Unknown Quantity. EXERCISE 102. 1. The product of two consecutive numbers is 156. "What are the numbers ? 2. The sum of two numbers is 17, and their product is 42. Required the numbers. 3. The difference of two numbers is 8, and their prod- uct is 105. What are the numbers ? 4. There are 900 trees in an orchard, and the number in one row exceeds twice the number of rows by 5. How many rows are there ? 5. A man bought some cloth for $90 ; had he bought 15. yards more for the same money, he would have paid $1 a yard less. How many yards did he buy ? EQUATIONS OF ONE UNKNOWN QUANTITY. I99 6. The sum of two numbers is 30, and their quotient is the less number. Required the numbers. 7. A lot that is 2 rods longer than wide contains 48 square rods. What are its dimensions ? 8. If a train would increase its speed 5 miles an hour, it would go 360 miles one hour sooner. What is the rate of the train ? 9. A can do a piece of work in 2 days less than B, and they together can do it in 2^5 days. In what time can each alone do it ? 10. One pipe can fill a cistern 3 hours sooner than an- other can empty it, and if they run together the cistern will be filled in 13 Ya hours. In what time could the first fill it ? 11. A certain number exceeds its square root by 30. Required the number. Suggestion. — Let «' equal the number. 12. If the circumference of a wheel were increased by 4 feet, the wheel would make 110 revolutions less in going a mile. What is the circumference of the wheel ? 13. A man sold a horse for 175, and thereby gained as many per cent as there were dollars in the cost. Required the cost. 14. A sold his farm at $48 an acre, and thereby lost one half as many per cent as there were dollars in the cost. Required the cost per acre. 15. A man increased his capital stock by $500 without increasing his gain, which was $500, in consequence of which his rate of gain was lowered 5j^. What was his original stock ? 16. How much must be added to both the length and the width of a rectangle, 18 by 20 inches, to make it con- tain 483 square inches ? 200 ELEMENTARY ALGEBRA. 17. Eggs rose 5 cents a dozen, in consequence of which 3 eggs less could be purchased for 25 cents. What was the price per dozen before the rise ? 18. A and B together earned $432. B earned $6 a month more than A, and the number of months they worked was one third of the number of dollars A earned in a month. How much did each earn per month, and how many months did they labor ? 19. A boy rowed 3 miles down a river and back again in 1 Ya hour. The rate of the current was 2 miles an hour. Determine his rate of rowing in still water. 20. A and B are 320 miles apart. If A travels 8 miles a day more than B, they will meet in one half as many days as B travels miles per day. How far does each travel per day ? 21. Twice the length of a rectangle equals three times the width, and if 2 feet be added to the length and 3 feet to the width, the area will be 56 square feet. What are the dimensions ? 22. A and B have each a debt of 1150 to pay. A pays 13 a week more than B, and pays his debt 8V3 weeks sooner. How much does A pay per week ? 23. A and B undertook to earn 1640. A earned $8 a week more than B, and the number of weeks required was one fourth of the number of dollars that B earned in a week. What were the weekly wages of each ? 24. Around a flower-bed, 18 feet by 12; is a gravel-walk whose area equals that of the flower-bed. What is the width of the walk ? 25. A farmer sold 7 pigs and 12 lambs for $50, and found that he had sold 3 more pigs for $10 than he sold lambs for $6. Required the price of each. 26. One person husked 48 shocks of corn in a day ; another husked the same number two hours sooner, and EQUATIONS OF TWO UNKNOWN QUANTITIES, 201 husked 2 shocks per hour more than the first. How many shocks per hour does each husk ? 27. A and B were engaged at different rates of wages. A worked a certain number of days and received $24, and B, who worked 6 days fewer, received $13 Vg. If A had worked 6 days fewer and B 6 days more, they would have received the same sum. How many days did each work ? Quadratic Equations of Two Unknown Quantities. Definitions. 213. A quadratic equation of two unknown quantities is complete when it contains all the second degree and all the first degree terms possible ; as, ax^-{-l)xy-\-cy^-{-dx-\-ey +/ = 0. 214. A quadratic equation of two unknown quantities is pure, or homogeneous, if all the terms containing un- known quantities are of the second degree ; as, aa? -{-hxy -\-cy'^ =^ d. 215. Any equation containing two unknown quantities is symmetrical if the unknown quantities may change places without destroying the equation ; as, 2a;« + 3a:y + 2^2^12 and 2y^ -{-Zyx-^23? = \%. 216. The solution of two simultaneous quadratic equa- tions of two unknown quantities often involves the solu- tion of a bi-quadratic equation. Ulnstration. — Given I ^' ~ y^ "" I (,t! \ to find x and y. U - / = 3 (B) j Solution : Transpose (A), y = re' - 2 (1) Square (1), y» = a:* - 4 j:« + 4 (2) Substitute (2) in (B), x-* — 4a;» — a; = — 7, a bi-quadratic. 202 ELEMENTARY ALGEBRA. Solvable Classes. I. When one equation is of the first degree and the other of the second degree, they are solvable as quadratics. niustrations.— 1. Solve P ^ ^ ^ ~ f Solution: Transpose (A), 2x = S — y (A)) (B)f (1) Divide (1) by 2, ^ = ^i^ Substitute (2) in (B), ^^~^'-2y^ = 2 (3) (3) Clear of fractions, Sy — 2/^ — 22/^^ = 4 Rearrange terras, 32/2_8y = — 4 Complete the square, 92/^ — 24?/ + 16 = — 12 + 16 Extract the V, 3 2/ — 4 = ± 2 = 4 (4) (5) (6) (7) o Transpose and divide, y = 2ov-^ (8) Substitute (8) in (2), x = i(8-2) or K- -1) <«) Reduce (9), a; = 3or3|- o 217. Many equations of this class may be solved by more elegant methods. ^- ^°'^« 1 xy = n (B)f Square (A), x"^ + 2xy + y^ = A9 (1) Multiply (B) by 4, 4xy = 48 (2) Subtract (2) from (1), x^ — 2xy + y^=l (3) Extract V> x — y=±l (4) Add (4) to (A), 2a; = 8or6 (5) Divide by 2, a; = 4 or 3 Subtract (4) from (A), 22/ = 6or8 (6) Divide by 2, ?/ = 3 or 4 Therefore y = S when a; = 4, and y = 4 when x = d. In a similar manner may be solved equations of the form of : ix~y = a) {x^ + f = a) ix^-{-i/ = a) \ xy=h)' \ xy=b)' \ x ± y = h ) EQUATIONS OF TWO UNKNOWN QUANTITIES. 203 EXERCISE 103. 1. \x-^y= 9) \ xy = %0) 2. \x-y= 6) 4. (a:« + / = 73) \x^y =^1%\ \x^^4.f 25 6 a:2 + r = 52 cc?/ = — 24 1 11. ( a;2-9«/2 = 16 \ \x -Zy = 2) 12. ( 2 a; + 3 3/ = 40 ) ( a;?/ = 50! 13. {4:a^-9y {3^-\-y^=50) \x —y = 10 ) X -\-y =7 f ^ +y =a -\-l 14. 15. - 9 / = 108 ) -Sy =-6f (a: + 2y = 17) |a;« + / = 61 j (30:2 = 4/ + 48 I a:^ - / = «2 _ j2 I (^ +?/ =« +5 j ( a; — ?/ = 4 i -I 17. a; + 3«/= 2 ) x'-{-xy = ^8 \ i 3x-7y=-5 \ xy — y^ = 25 7 - + ^ 1 1 __ 25 9. r_l _ j^_ 5 ] 18. 1 _!__ J^ a; y "" 6 218. Some equations of a higher degree may be reduced to this class by division. niustratioiL-Solve K " 2/' = 56 (A) ) \x -y = 2 (B) j Solution : Divide (A) by (B), x^+xy + y^ = 28 (1) Square (B), x^ — 2xy + y^ = 4: (2) Subtract (2) from (1), 3xy = 24: (3) Divide, xy = S (4) Add (4) to (1), x^ + 2xy + y' = 3Q (5) Extract the V» x + y=±G (6) Add (B) and (6), and divide, a; = 4 or - 2 (7) Substitute (7) in (4), and divide, y = 2 or — 4 (8) Therefore, when x = 4, y = 2, and when x = — 2, y= —4, 204 ELE3IENTARY ALGEBRA. In a similar manner may be solved equations of the form of : io^-{-if=:a) {a^-f = a\ { x^ -{- a:^ y"^ -{- f = a ) \x-^y=b\' \x'±y^ = b\' \x'±x y -{-y'=b\ EXERCISE 104. Solve : {x'-^y^ = 35) 7. ( c^-^ + 27y3 = 243 ) \x -\-y = 6\ \x + Sy = 9\ ix^-y^ = 61) e. iSa^-y^ = 9S) \x -y = l\ \2x -y = 2\ {a^-f = 80) 9. ( 8a;3 + 27^3 = 35) \a^-y^= 8f |2^ + 3^ = 5S j ic* - / = - 65 ) 10. ( :c* - 16/ = 80 ) \x^ + y^= 13 \ \x^- 4.y^= s] {x'-\-a^y^-\-f=:4:Sl) ii. {Sla^-16f=176) \o^-x y -^y^= 13) ( 9a;3+ 4/= 25 f ''^ + ^^/ + «/* = 21 I 12. j 16ic4+4a.V+/=91 ) ^ + ^ y +/= 3 ) ( ^.x'-^xy -]-y^= 7 j 219. Sometimes there is a common factor in the first smbers of two equations that may be removed by ' division. lUustration.— Solve \ ^^ ~ \xy- ■y^ = 2 (A)) (B)i Solution : Factor (A) and (B), {x -^ y){x — y) = ^ (1) y{x-y) = 2 (3) Divide (1) by (2), x + y 5 y ~ 2 (3) Clear of fractions, 2x + 2y = 5y (4) Transpose, ,2x = 3y (5) Divide, 3 (6) Substitute (6) in (2), jy' = ^ (7) Reduce, y=±2 (8) Substitute (8) in (6), x=±3 (9) EQUATIONS OF TWO UNKNOWN QUANTITIES. 205 EXERCISE lOS. Solve : 1. {x^ -y^=12) 4. {4:0^ -9/= -108) \xi/-\-7/ = 12^ \2xi/-Sf=- 24S 2. {x^-\-2xy =27) 5. { 4ta^-{-8xy-\-3y^=96 [ \x'-{-3xj/-\-2f=D4:\ \ 2a^-{-xy=^S') 3. {x^-\-3xy-{-2f=12) 6. (16a;2-9/ =319) \3^-^4.xy-}-dy'-=16] \ Sa^ -^6xy = 2do\ 7. i 4ar + 4a;y + «/2 = 169 ) t 2a:y + 2/'= 39 f 8. {a^-3xy-4y^ =-150) ( 2a^-Sxy=-160) 1 9a:y-6r = 30) 10. ( Ga.'2 + 19a;y + 15/ = 40) \6x^-xy-15y^=-lo\ 220. Sometimes one or both equations have the quad- ratic form, or may be reduced to the quadratic form, ninstrations. — • ^ ^ I x-y = i (B)f Solution : Complete the square in (A), 1 1 81 (X + yy + (X + y) + j = 20 + j = ^ (1) Extract the 7, x + 2^+2 = ^2 (2) Transpose, Add (B) to (3), a; + y = 4or — 5 2a; = 8 or —1 (3) (4) Divide, a; = 4or -^ (5) Subtract (B) from (3), 2y = or — 9 (6) Divide, y = or -4-^ Therefore, when a; = 4, y = 0, and when x= — ^, y= — 4 10 206 ELEMENTARY ALGEBRA. 2. Solve 1^^ + ^^ + ' + ^ = '' (^H Solution: Add (B) to (A), x^ + 2xy + y^ + 2x + 2y = m (1) Factor, {x + yf -\- 2{x + y) = 63 (2) Complete the square, {X ■\- yf + 2{x + y) + \ = U (3) Extract the V» a; + y + 1 = ± 8 (4) Transpose, a; + y = 7 or — 9 (5) Substitute (5) in (A) and (B), and transpose, a;2 + 2/' = 25 or 41 - (6) 2xy = M or 40 (7) Subtract (7) from (6), x^—2xy + y^=\ovl (8) Extract the V» x — y=±\,oY±\ (9) Add (9) and (5), 2 a; = 8 or 6, — 8 or — 10 (10) Divide, a; = 4, 3, -4, or -5 (11) Subtract (9) from (5), 2 3/ = 6 or 8, — 10 or — 8 (12) I>ivide, y = 3, 4, - 5, or - 4 EXERCISE 106. Solve : 1. i a;2/ + a:i/ = 42) 8. ( a;2 + /=100| I x-^y= 5) \2xy-{-x^y=llo\ 2. i X^ , 2X . 4 ) 9. [ X^ 7/2 ^^r^^_^y^27 "^ y ^9^ \y''^x^~^l} + '^-^ [ x-y = l ) ( ^.^^2^20 3- ( (^ + 2/)' + ^ + «/ = 56| 10. j a^J^y^ = 5S \ xy = 10\ \xy-x + y = 25 4. Ua^-^y^y^a^-\.y2 = so^ \ a;2 ~ / = 3 ! 6- {{x-{-yy-\-2x + 2y = 80) I (x-yy-{x-y)= 6f 6- ((^ + #-3(^ + ^) = 575) 1 icy = 150t" EQUATIONS OF TWO UNKNOWN QUANTITIES. 207 II. Two homogeneous equations of the second degree may be solved by putting y =zvx when they can not be more easily solved otherwise. ^+ xyJty'= 28 (A) ) /=-28 (B) ] UluBtration.— Solve | !I "^ « ^ ^ "^ Solution: Substitute vx tor y in (A) and (B), a;s + va:« + t;«a;« = 28 Factor (1) and (2) and divide, x^ (1 + f + v^) = 28 a;8(l_2f-v») = -28 or, 1 + V + v^ _ l-2v Clear of fractions, Transpose, Substitute in (3), Divide, Extract the >/, Substitute (10) in y = vx, l + v + v^ = — l-\-2v + v^ v= 2 7a;« = 28 x^= 4 x= ±2 y = 2x(±2) = ±4. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Solve EXERCISE 107. 3. 1= a^ -\-y^ = 6 2a^-\-xy-\-y^ = S 2. ( a^-xy-\-y'^=12) a^ — xy = — —z ^ 16 ia?JrxyJ^y^ = 2S\ \ 2a:2_|_3^2^44f j 5a:2-3y2=_63 a* + a;y = 27 x^-\-2xy-\-%y^ = b 2a^ + 5/ = 7 3/^ ] \ f-x^= 8f M 2x2-a;y-y2= _40] a^ + / = 40 f 7? — xy — y^ = — 125 ) a^-\-2xy=zl25 f ). J2:« + 3a;y + 2y2 = 40 ) ELEMENTARY ALGEBRA, Solve : Miscellaneous Examples. EXERCISE 108. 6. 8. 10. xy-y' = 11) ( x-^y= 2) x — ^yz=z x-\-y = a — h xy— — ah x^ + y^ = a^ i xy = l f I a^ + 2/=^ = 126 ) ( ^+y= 6f • ( a?-xy-\-y^= 21 u-^+^z+^z' 133 19 a;3 + 1/3 = 7 (a; + ?/) x-y=l x^-{-xy = A:% y^-\-xy = lQ 11. 12. 13. 14. 15. 16. 17. 18. 20. I S^(^ + «/) + 2/(^ + «/) ( xy{x-\-y =4 = 80 a; y x-y x-^y 2-i-Sxy x^-\-y^-l = xy(xy-^l) = x^y^x-^yY = a^-\-y^ = 1 = dx) 2xy) 42 f 2916 ) 2xy^ 1 ] 19. Ja;2 + / + 2^ + 2^ = 50) xy-i-x-{-y = 23^ 81 180 21. ^ :z;4-?/4-2;= 6 -l xy -\-xz~\-yz= 11 ( a; + 2; = ?/^ EQUATIONS OF TWO UNKNOWN QUANTITIES. 209 Examples involving Quadratic Equations of Two Unknown Quantities. EXERCISE 109. 1. The sum of two numbers is 13, and their product is 40. Find the numbers. Suggestion. — Let x equal one number and y the other. 2. Tiie difference of the squares of two numbers is 40, and their sum is 10. Find the numbers. 3. A man has two fields in the form of squares, contain- ing 16,400 square rods, and the one is 20 rods longer than the other. Required the length of each. 4. The difference between two cubical blocks of marble is 152 cubic feet, and the difference of their lengths is 2 feet. Required the length of each. 5. A has a rectangular field containing 240 perches, and a square field containing 676 perches. If the side of the square field is equal to the diagonal of the rectangular one, what are the dimensions of each ? 6. A man bought a number of horses for $3600 ; had he bought five more at 15 apiece less, they would have cost him $225 more. How many did he buy, and at what price ? 7. A and B each worked as many days as he received dollars a day, and together received $89. Had A worked as many days as B, and B as many as A, they would have received only $80. How long did each labor, and what did he receive per day ? 8. The product of two numbers exceeds the square root of the product by 30, and the quotient exceeds the square root of the quotient by 74. What are the numbers ? 9. There is a number consisting of two digits ; the sum of the squares of the digits exceeds their product by 21, and if 9 be added to the number the digits will change places. Required the number. 210 ELEMENTARY ALGEBRA. 10. A man and boy worked at one time as many weeks as the man earned dollars a week, and received $700 ; at another time as many weeks as the boy earned dollars per week, and received 1525. How much did each earn per week ? 11. A man has a rectangular lot containing 1 acre ; if it were 4 rods longer and 4 rods narrower, it would con- tain only % of an acre. What are the dimensions of his lot? 12. The fore- wheel of a carriage makes 88 revolutions more in going a mile than the hind-wheel ; but if the cir- cumference of the fore-wheel be diminished 1 foot, it will make 146% revolutions more than the hind- wheel. What is the circumference of each wheel ? 13. A and B each invested $182 in wheat, A receiving 10 bushels more than B. Had A paid 5 cents a bushel more for his, and B 5 cents a bushel less for his, they would have received the same amount. At what prices did they buy ? 14. A man sculled 24 miles down a river and back again. He found that it took him 8 hours longer to return than to go, and that his rate down was 3 times his rate return- ing. What was his rate of sculling in still water, and what was the rate of the current ? 15. The area of a rectangle is 4 feet less than the area of a square of equal perimeter, and the length is Vg of the breadth. Required the side of the square. 16. The greater of two numbers divided by their sum, added to the smaller divided by their difference, gives 3Y7, and the difference of their cubes is 4625. Required the numbers. 17. The difference between the hypotenuse and the base of a right triangle is 6, and the difference between the hypotenuse and the perpendicular is 3. What is the length of each side ? NEGATIVE RESULTS, 211 18. A man invested equal sums of money in 6^ and 7^ stocks, paying $10 a share more for the latter and receiving 10 shares less. The income of the latter was $20 more than of the former. What was the sum invested, and the price of tlie shares ? 19. A drover sold 10 horses and 7 cows for $800. He sold 5 cows more for $160 than he did horses for $198. At what price did he sell each ? 20. A and B start together on a journey of 36 miles. A travels one mile per hour faster than B, and arrives 3 hours before him. Find the rate of each. 21. Two partners gained $140 by trade. A's money was in trade 3 months, and his gain was $60 less than his stock ; B's money was $50 more than A's, and was in trade 6 months. What was A's stock and what was B's gain ? Negative Results. 221. Some questions, evidently intended to be taken in an arithmetical sense, give rise when solved by algebra to negative results. How shall these results be interpreted ? 1. A negative result may arise from an erroneous state- ment of a condition. niustration. — A certain number increased by 5 equals % of the number, diminished by 3. Required the number. Solution : Let x = the number ; Q then x-\-^=.-rX — ^i 4 or, 4a: + 20 = 3a: — 12; whence, a; = — 32. Tn an algebraic sense this result may be verified, but in an arith- metical sense it is meaningless. The condition of the question as stated is erroneous. If it be modified to read, "A certain number diminished by 5 equals 2/4 of the number, iticreased by 3," the result will be 32, which is consistent and intelligible. 212 ELEMENTARY ALGEBRA. 2. A negative result may arise from an erroneous state- ment of a question, niustration. — A man is 40 years old, and his brother is 25. When will he be twice as old as his brother ? Solution : Let x = the number of years hence ; then 40 + a: = 2 (25 + a;) = 50 + 22; ; whence, a: = — 10. In an arithmetical sense this result is not intelligible. If the question be asked, " When was he twice as old as his brother ? " the result will be 10 years, which will satisfy the question. S. A negative result may arise from an erroneous sup- position made in the solutio7i of a question. lUustration. — A man sold his horse for $80 ; had he sold him for $40 more, he would have gained 20^. lie- quired his gain or loss. Solution : Let x = his gain ; then 80 — a: = the cost, and -^ (80 — x) = the gain by second condition ; whence -^ (80 — a:) = the selling-price by second condition. |-(80-a;) = $120, and x= — $20. In an algebraic sense, gaining — $20 is equivalent to losing $20. Had X been assumed equal to the loss, the result would have been $20. J/.. Negative results in examples involving quadratics are generally the numerical equivalents of positive results in analogous examples. Illustrations. — 1. A man has a square board, such that the number of inches in length added to the number of square inches in the area equals 12. Kequired the length. Solution : Let x = the length ; then x^ = the area, and a:^ + a; = 12 ; whence x = S or —4. x = S satisfies the question as stated, a; = — 4 indicates that the number of inches should be arithmetically subtracted from (algebraic- ally added to) the area. NEGATIVE RESULTS. 213 2. If to 280 more than the square of my age you add 34 times my age, the result will be zero. Required my age. Solution : Let a; = my age ; then x^ + 280 + 34a: = 0; or, a;* + 34^= -280; whence a; = — 14 or — 20. No value of x will arithmetically satisfy the question as stated. But, since x^ is positive and 34 a; is negative for these values of a:, the analogous question, " If from 280 more than the square of my age you subtract 34 times my age, the result will be zero; required my age," will be satisfied by a: = 14 or 20. EXERCISE no. Solve and interpret the results of the following examples : 1. What number increased by 7 equals 5 ? 2. 12 diminished by what number equals 20 ? 3. A man is 40 years old and his son is 20. In how many years will the father be 272 times as old as the son ? 4. A line 40 feet long was cut into two parts, sucli that one part increased by 20 feet equaled the other part dimin- ished by 30 feet. Required the length of each piece. 5. The numerator of a fraction is 2 greater than the denominator, and if 9 be added to both terms the result will equal 2. What is the fraction ? 6. Two thirds of A's age increased by 12 years equals ^5 of his age. What is his age ? 7. The square of a number diminished by the number is 12. Required the number. 8. The product of the sum and difference of two num- bers is 17, and one of the numbers is 9. What is the other number ? 9. A garden, 40 yards long and 30 yards wide, has a gravel-walk along its perimeter that occupies "/is of the garden. Required its width. Ans., 2V2 yd. or 32 y^ yd. Interpret the meaning of 32 V2 yards. CHAPTER VI. EXPOJ^EJ^TS, RADICALS, AJfD IJfEQUALITIES. Fractional ^and Negative Exponents. Principles and Applications. 222. We learned [P. 75] that dividing the exponent of a factor by the index of a root extracts the root of the factor. If this principle be accepted as general in its char- acter, and applied when the exponent of the factor is not divisible by the index of the root, the result will be a quantity with a fractional exponent. Thus, V«^ = a^, read a, exponent two thirds. 223. A fractional exponent, from the nature of its origin, denotes a root of a power, the numerator being the exponent of the power, and the denominator the index of the root. 224. Since «» z= Vfl^"* [223] = Va X aX a X to m factors = Vax Vax VaX..., to m factors [P. 76] = (V^)", it follows that a fractional exponent may also be regarded as denoting a power of a root, the numerator still being the exponent of the power, and the denominator the index of the root. FRACTIONAL AND NEGATIVE EXPONENTS, 215 SIGHT EXERCISE. Name the value of : 1. 8^ 3. (a^)^ 5. (-ar^)^ 7. 32^ 2. (-8)^ 4. 10^ 6. 27^^ 8. {b')^ '■ ig "■ (i)' '■• (I)' Name the equivalents of the following quantities : 12. Va 15. V^ 18. X^ 21. (Va)3 13. Va 16. (Vay 19. X^ 22. (\/'a)5 14. 7^ 17. 4* 20. 3« 23. (Viy 225. Since a^^ = 'V^ = i^ Va« = V?= «^, it follows that a^ and aA are equivalent. Therefore, ^Hw. 89, — Multiplying or dividing both terms of a fractional exponent by the same quantity does not change its value. 226. Since a^ = a-^ [P. 891, and a^ = a^ [P. 89], aixai = a^Xa^ = (Vaf X (V^Y = ('V^)" = a+^ = a^+^, it follows that, Prin. 90* — The exponent of a factor in the product equals the sum of the exponents of the same factor in the multiplicand and multiplier, when the exponents are posi- tive fractions. 227. Since a? X a^ = a^ [P. 90], it follows that a^ -r- a* = at=:fljH— J. Therefore, Prin. 91, — The exponent of a factor in the quotient equals the exponent of the same factor in the dividend, minus the exponent of that factor in the divisor, wh:n the exponents are positive fractions. 216 ELEMENTARY ALGEBRA. SIGHT EXERCISE. Complete the following expressions : 1. G^i = 6?"f^, a^ = a^^, a^ = «t^, and a^ = ci^ 2. a'^ = a^ — «T^ = ^2ir — a^T = cC^ = aJ^ = a^^ 3. «'i^ = a^ ; a^ = a^; a^^ = a^ ; a^- = a^ 4. a^Xa^=? xi xx^=? x^ X x^ X x^ = ? x^ X x^ ? 6.a^.-^a^=? c-^ci=? c^i-^c^=? c^ ^ c^ = ? 6. a'^a'=? (x^ Xx^)-^x^=? {x^ -r- x^) X x^ = ? 228. We learned [P. 12, 91] that the exponent of a factor in the quotient equals the exponent of the same factor in the dividend minus the exponent of that factor in the divisor. If this principle be accepted as true when the exponent of the divisor exceeds that of the dividend, exponents will arise from its application. Thus, a^ -^ a^ = a~^, read a, exponent minus 3 ; also, -t- a^ = a~'^j read a, exponent minus one sixth. 229. «^'* -^ a'"" = a-^"" [228] ; but ^^— l-a^«[P. 53] = -^„ 1 a' a- 2" in: ^- [Ax. 1]. Therefore, JPrin. 92. — A quantity affected by a negative exponent equals the reciprocal of the quantity affected ly a numeri- cally equal positive exponent. 230. a''^ -^ a^"" = a^^ [P. 12] ; but fl^2n_____ |-^j^^ -j^-j^ Therefore, Prin, 93, — A quantity affected hy a positive exponent equals the reciprocal of the quantity affected hy a numeri- cally equal negative exponent. FRACTIONAL AND NEGATIVE EXPONENTS. 217 SIGHT EXERCISE. 1. Find the value in negative exponents of : ff* -f- a^ ; a? ^ x^ \ x -^ x^ ; a^ -7- a^ ; c^ -r- c^ 2. Express in positive exponents : a-\ a-\ x-^, x-K 2-^ S'S {xy)-^ {^~' 3. Express in the integral form : 1 1 _L J:_ i _L a^' X?' 2-2' a;-^' ar*' x— 231. Since 2 1 t. c-^d ~ 1 . , '-^- ^'^-' - ^ - ^-^ ^ rf -:^' it follows that, JPrin, 94, — A factor may be transferred from either term of a fraction to the other if the sign of its exponent he changed, 232. Since a-' X a-^ = ^4 X ^ [P. 92] = ^\ = «"« [P. 94], and a-J X a-^ - -\ X \ [P. 92] = 4i = «"** t^- ^^]» ' . a^ a* ais^ it follows that, Prin, 95, — Tlie exponent of a factor in the product equals the sum of the exponents, of the same factor in the multiplicand and the multiplier when the exponents are negative. 233. Since «-« -^ a-^= \. -- \ [P. 921 = i X f = ^3 = «-' ; and a' a^ ^ ^ a^ i a^ a %^a 5 = -.-- = -.x^=^ = a *, at flft at 1 as it follows that, 218 ELEMENTARY ALGEBRA. Prin, 96, — The exponent of a factor in the quotient equals the exponent of the same ^ factor in the dividend, minus the exponent of that factor in the divisor, when the exponents are negative. SIGHT EXERCISE. 1. Clear of negative exponents : a-^ a^ a-'^ a'^l x'"^ xy-'^ a-^h-^ a^h J-3. ^-2' ^,2 . ^-2^^ ^_|. ^-1 . 2 ' c-^ 2. Express in the integral form : a a~^ a Sa^ ax x-"^ mn^ I' J2 ? j3? j-4? 5-1^-1' 2^-4' p-^q-^ 3. 4. Express in the integral form : a «2 ^-2 an ah 1 2 2 c' r^' x-^' c-^' cd' a-'P' a-'' 2"^ Express in positive exponents : a-^ a-^W x-^if 2-^x3 {a-\-h)-^ j-2> ^-3 ^ ^2 . 5-1 ^ {a-\-h)-' 5. Find the numerical value of : 6. Find the value of : a^ X a-^ ; a^ X a~^ ; x-^ X a:~^ ; a-^ X «~* ; «-'X«-*; 2;-^X:r-^; S-^xS-^; 4-2x4^ 7. Find the value of : x^-^x-'^', a~^-^a~^', a~^ ^ a~^ ; a~^ -^ a~^ ; a-^ ^ a-^ ; x-^ -^ x'^ ; 2"* -^ 2-« ; 2"^ -^ 2"^ 8. Find the value in positive exponents of : a-^ X a-^ ; a"^ -^ «-^ ; a~^ X a^ ; a~^ -^ a^ a^ x~^y^ (a — h)~'^ {x -\- y)^ p~^ q^ FRACTIONAL AND NEGATIVE EXPONENTS. 219 2. General Principles. 234. 1. a'" = aXaXaX to m factors. a''=aXaXaX to n factors. . • . arxar={aXaXa to m factors) X(aXaXa to n factors) = axaXa to (7n-{-n) factors = «"*+* 2. af = /A [P. 89] = ('Vfl)^- [224] flT = al^ [P. 89] = (V^)-^ [224] .-. af X a* = (Vay X CVa)-^ = ('V^)'-+"^ [1] = a «» [P. 75] = a 9 ^ « 3. a-^X.:" = ^xi[P.92]=-i^.[l] = «-<•»+'•> [P. 94] 4. a-f xa-^ = -^X-^ = -^^^ [2] = -1^) [P. 94] =a-(f + T) Therefore, JPrin. 97* (f X a* = a'^^ for any positive or negative, integral or fractional, values of x and y. 235. a'^-^a*^ a""-*, since a""" X a" = a" [P. 97] p^ r_ p f_ £__!_ -L JL ai -^ a* =a — Any quantity equals the nth root of the nth power of the quantity. JL ± 1 264. Since («•)«• =a— i [P. 99], we have, Prin. 108. — The V\/~a and the "Va are equivalent. 224 ELE3IENTARY ALGEBRA, Reduction of Radicals. Problems. 1. Mixed to pure radicals, ninstrationsc — Keduce Vs a'^ b and a/ — to pure radicals. Solutions : 1. ^San = V4«^x 2ab= \/Aa^ x y\/2ab [P. 102] = ±2a\/2ab. 2- r 2^ = 4/2^ Xy=: 4/125- = y 125- >< 15 = ^125- >^ V15 ^ £P. 102] = I Vl5. EXERCISE 112. Eeduce to pure radicals : 1. V12, a/is, and \/33 3- VT6, V-54, and v^i28 2. V45, Vis, and a/75 4. ^-56, Vi08, and a/ISS 5. \/4^^ Vl6^^ and V^c* 6. vieT^ V18^, and V-8«* 7. VsOfl^, a/45^^, and V27^V 8. VTe^^, V-:?;^«/2;S and V^V^ 9. V«2(« + ^') and Va(a+^ 10. '\/x{x-\-yY and a/:z; (:ic + 1^)^ 11. V (x -\- yY {x — yf and Va;^ 5/^ (a; + yY 12. Va^ + 2 rz;^ y + a; «/^ and Vx {x -\-yY 13. VH^-y'){x^y) 14. Vfl2Z>-^(a=^-Z'^)(a-^) REDUCTION OF RADICALS, 225 17 18. 19. 8 3 /16 1 3 /27 9' 1/27' ^'^ l/re G^% x^, and a-\-b 24. 26. X 4/^^ ^"' 1/^ y)' "•i/pf-'i/lxf 2. To lower degree. niiistratioiis.- Reduce VOo^ and V64a^ to simplest form. Solution : 1. V9a« = VV^a* [P. 108] = V^oT. 2. V64o» = vVWo* [P. 108] = v^= \/4x2a = V^x \/2a= ± 2 226 ELEMENTARY ALGEBRA, EXERCISE 118. Eeduce to simplest form : 1. V9, V64, and Wl 7. Vsi, Vl25, and V729 2 iA iA i/^ 8 iA ]/^ ^//"^ 3. Vl6^V and V25^V 9- V^V^and V-^x^y'' 4. V^V^ and VcF^'^ 10. V^3_3^2^_|_3^j2_j3 ^- y ^^ ^^^ y -.:? "■ y -27? ^"^ y -^ 6. V-32a5/« and V^V^ 12. Va2 + 2a^ + Z>2 3. Bational to radical quantities. Illustrations. — 1. Reduce 2 a to a radical of the third degree. Solution: 2a = V(2a)3 [P. 107] = VSo^ 2. Free %aXf%~a of its coefficient. Solution : 2a V2a= V(2a)3 x XJ%az=. \f^ x ^^0^= Vl^a^ [P. 103]. EXERCISE 1 14. 2 1. Reduce 5, 3 x, and — o;^ to radicals of the second degree. 2. Reduce 3«, 5 5a7, and - to radicals of the third X degree. 3. Reduce a^lP', xy^, and -^ to radicals of the fifth degree. 4. Free 2^5, 3 a/3, and - Vs of coefficients. 5. Free a Vd~a, 7? Vd~Xy and - V? of coefficients. REDUCTION OF RADICALS, 227 6. Free a(a + h)^ and {a—h) Va + b of coefficients. 7. Reduce a?, y^, and 4 to equivalent expressions having an exponent of - 8. Free x{y)^ and a^{z)^ of coefficients. 9. Free a;^ {yY and 16 (2:)* of coefficient (2\l 1 + l^j ={x'^y^)i 4. Keduction to same degree. Hlxistration, — Reduce Vs, V2, and a/2 to the same degree. Solution : ^3 = 3^ = 3^ [P. 89J = 'V3« = ^l/m V2 = 3* = 2t^ [P. 89] = V2"' = 'VI6 V 2 = 2i = 2^i [P. 89] = V23 = *y8 Kote. — The operation may be shortened by remembering that both index and exponent may be multiplied by the same number [P. 89]. Reduce Vc^, Vc^, and Va* to the same degree. Solution : l/d^ = 'V(^ [P. 89] = V^* V^5 ^ ly (^« [-p, 89] = Va^" ^~a* = V^ [P. 89] = V^ Note. — The common index is the L. C. M. of the given indices. Why! EXERCISE lis. Reduce to the same degree : . VI VI ^ VI ,^,^.„,|/T 2. Vx^y vary, and Vy^ ./— e^ 6. a, va^ and Va^ 3. aK hK and c^ ^ ^'^ y^^ ^^^ jy^ 4- Va-\-h and l/a-{- b 8. Var (a; -|- y) and (a: -f- y)^ 228 ELEMENTARY ALGEBRA. Addition and Subtraction of Radicals. Illustrations. — 1. Find the sum of 4 \/8, 5 i/i, and | a/- Solution : 4 \/8 = 4 V4 x 3 = 4 x V^ x ^2^= 8 V^^ |t/| = |4T^=| X 4/1 X V2= i-V2 2. Find the value of 2 V- 81 a*+ 8 Vs^- 2 « y- 24 «. Solution : 2 V- 81 a* =2v^-27a3x3a = 3x V-27a3x V3^=-6aV3^ + 8V^ = + 8Va^x 3a = + 8 X V«^ X VSa = + 8aV3a -2a V-24a = -2a V-8x3a = -2ax V^x V^= + 4aV3a Sum = 6a v^ EXERCISE 116. v?+i/^+v? Find the value of : 1. Via -^ Vl6a -\- VS6a 2. a/2 + \/8 - iA , /- ^ /— ^V"-^W^^^ 3. V27x+Vl2x-V4.Sx \ a-b b 3 / 3/5 1 3 y , ^ 3 /o^ 3 /o^ , 3 / J 5. \/2^+ \/2F+ ^2^ 11. V7«^+ VY^- a/7? 6. 'v/(a+T)^+V'(^^^^^ 12. V«^2_|_«/^2_ 3/7^2 7. a/^- V^+a/^ 13. 2^/3+ V27 + 3 Vsi 15. (a^ ^^a^h^aW)^ -{a^-2 aH-\-ai^)^ 16. Va^ -2aH-^ab^ ± Va^ + 2aH -\- ab^ MULTIPLICATION OF RADICALS. 229 19. -i-^ a/?+^+ i/— ^ a-b ^ ^ y a + h 20. A/ -^L—^xVax — x^ \ a — x 17 18 Multiplication of Radicals. Illustrations. — Multiply 3 yi by 5 i/|, and 3 \/2 by 2 Vs. Solutions : 1. 5l/|x34/| = 5x3x/|xl/I=:15 /f^ [P. 103] = 15/| = 15i/| = 15>/|x VS" [P. 102] =5^/3: 2. 2 V3 X 3 a/2 = 2 X 3 X V3 X y^ = 6 x {/O x Vs [P. 89] = 6^72 [P. 103]. EXERCISE 117. Find the value of : 1. V3 X a/6 7. A/3rt" X vTrt X vT« 2. V3X V6 ' Z.'^txViy.ZyV'iy.zV'x 4.-4 a/2^ X 3 Vac lo i /— £— X i A + ^ /- - V a + b^V a-b ^•yfxy* u. a/5"xV2 c l/i^Vl^^ 12. 2A/3X3 V3X V3 13. 2 Vfl^ X — 3 Va^ 230 ELEMENTARY ALGEBRA, 14. (a/2 + V3 + Vb) X a/2 15. {Va + Vh){Va-Vl) 16. a/o+T X ^/a — b X A/a^ + « + ^ 3 /a —X _ 3/7 ^ 18. a ^/xy^ X 5 V?p X c V^^ 19. (3 a/2 - 4 a/6 + 5 a/10) X a/6 20. (2 a/^- 3 Vy) (2 a/^+ 3 a/^) 21. (ic - a/^+ y) {Vx-\- Vy) 22. (a; + a/^ + ^) (^ - V^+ ?/) Division of Radicals. Illustrations.— Divide 3 j/l by 2 l/|, and V% by Vs. Solutions : 1^ = 1^1 X Vl2 [P. 103] =-1^12. 2. V2-5- V3= V8-5- V9= V8T9 [P. 105] =^ = V'^^^STI^ = t^"^ = ^^ = - Vl28. EXERCISE 118. Find the value of : 1. a/8-4- a/2 2. 4 a/is -4- 2 a/3 4. 5 a/30 -t- 3 a/6 5. Va/b^ Wc DIVISION OF RADICALS. 231 l.'UVa^-^Vab 13. 6\/3 x2a/5-^ Vl5 14. (2 a/30 -^ - \/2) X V6 9. V2 ^ V2 ,- 10.2-^V2 15.aV^X^A/^^|^| 12. 2\/3x3\/2-v- 4/1 ,^- /- ^ 18. (a/6 + V^+ VTO) -J^ a/2 19. (3A/lO + 4A/5-6Vi5)-T- V5 20. (5 a VaJ^ — b^ — )i0ab's/a-\-b)^6aVa-\-h 21. (iC + 2 A/a^y + i/) -j- (V^+ \/^) 22. (o^^xy-^r y^) -^ (a^ + a/^+ y) Involution of Radicals. niustrations. — 1. Raise v5 to the second power. Solution : ( V^)' = (Si)' = 5l [P. 99] = V^* = \/25. 2. Raise Va to the third power. Solution : ( V^)« = (ai)» = at [P. 99] = ak [P. 89] = ^/~a. 3. Raise Va to the sixth power. Solution : ( V")" = iahf = aS [P. 99] = a^ [P. 89]. EXERCISE 1 19. Find the value of : 1. {V%f 6. (2 V5)« ' 9. {{a-\-h)VahY 2. (a/3)» 6. (a Vabf lo. (A/a - Z')=* 3. (2a/2)« 7. (a/2^)* 11. (a/^T^)* 4. (3 V^f 8. (V^)« 12. (V^^^f 232 ELEMENTS IRY ALGEBRA. "• (i/l)* -m' 14. {V~a+Vhf 20. (2 + V3)' ...(.vi)- "•(/i-/D" 16. {Vx — Vyf 22. (Vs-iy " m 23. (^lA/2 + iA/3)^ 18. (2 a/3 - 3 V2f 24. {-«(V^-VI)l Evolution of Radicals. Illustrations. — 1. Extract the square root of Vs. Solution : i^l/S = V 5 [P. 108]. 2. Extract the cube root of Vs. Solution : \/^8= V^ [P. 108] = VJ/S [P. 108] = V2. 3. Extract the square root of 8 V2. Solution : \^^i/2 = |/4x2 V^ = VT^H/W = V4 X i^J/TS = ± 2 V^ EXERCISE 120. ^ Find the yalue of : 1. VVSa 6. \/J/7W7 2. /W? 7. V2V2 3. 4/2 a V2 a 8. ^ - Vs EVOLUTION OF RADICALS. 233 11. Vl6V2a 18. i^27V2aa^ 12. VV(a - by 19. Vi-\/l 13. VV? + 27h^ 14. ^7(^T^^ 20. \/A:i V{a-\-b) 15. V2Vl5(a + a:) 16. V'A/(a + J)(a-J) 21. ViVi 17. yA/a3 + 3«2J + 3aZ>2^^,3 Rationalization. 255. To rationalize a quantity is to clear it of radicals. 2 niustrations. — 1. nationalize the denominator of — r= V3 Solution : —^ = —p- — ^^-r^ [P. 56] = :. = tt V 3. 2. Rationalize the denominator of — ■]= 2-^/3 Solution : 2+V3 ^ (2+\/3)(2+\/3') ^ 4 + 4V3+3 ^^ ^^ 2_/v/3 (2- V 3) (2+ a/3) 4-3 ^ EXERCISE 121. Rationalize the denominators of : \ a 2 1. VT a/F 2+V2 „ 2 a/7 3 3. V5 ^- v^» *^-V2-vT a/3" 1 g a/5" ^' 1 - a/ 2" ^' a - VT 234 10. 11. ELEMENTARY ALGEBRA. 3- V2" Vx + Vy 2-^2" 12. 13. 3 + V2 V2- V3" a/2 +V3~ 14. 15. Vic — a/^ 16. If the a/2 = 1-4142, what is the value of -7^ ^ 17. What is the numerical value of 7= ? 2 + A/2 Imaginary Quantities. . Principle and Definition. 256. _ 4 = a/4 X (- 1) = a/4 X a/^ [P. 102] = ±2a/-T. a/^=^= a/6 X (- 1) = a/6 X a/-1 [P. 1021 = Therefore, ± a/6 (a/^^). JPrin, 109, — Every imaginary quantity of the second degree may he reduced to the form of ± x ^J — 1, in which X may he rational or irrational. 257. The factor a/— 1 is the imaginary unit. — a/— 1 is equivalent to — 1 X A^— 1. 2. Examples. EXERCISE 122. Keduce to simple form : 1. V— 9, a/— 4a^ and V— 16 2. a/- 25:^2, V- 36^2 2:*, and A/-49a*2/^ 3. a/- 8, a/- 12 «, and a/- 18 «2 0^3 4. Add a/^^, V^-^, and a/- 16 IMAGINARY QUANTITIES. 235 6. Find the value of V-^a"" + V-25a^ - Vl6^' 6. Simplify {V^y, (v'^l)^ (V^i)S (V^iy ' 7. Simplify (- V^^)% (- ^/^)^ (- V^=T)S and (- V^^Y 8. Multiply V^^ by \/^ Suggestion. V— ^ = V^ x ^/^^ and v^^ = ^/^ x ^ —1; hence, V— ^ x ^/^ = aA x v^ x V— 1 x a/— ^ = V^^ x (Vin;)« = Vl8 X (- 1) = - Vl8 = - 3 V2. Multiply : 9. V^^ by V^^; V^^by V-20; 10. 2 \^^ by 3 V"^^; 4 V- 16 by 2 \/-25; a V-b^ by 5 V^^ 11. 2 + V^^ by 2 - V^; V'^^H- \/^^ by 12. Square 2 + v^^^; 3 + V^^; V^- V^^ 13. Divide V— 36 by V^^ Suggestion. V— 36= V^^ x — 1 = 6 V— 1 ; a^d V— 4 = V- 4 2V-1 ^ Divide : 14- \/^^ by V^^; V-12 by a/^^; ^Tby v^^ 15. 2 V-4a:2 |5y V-a;^; V-16a:* by V-2a:; Rationalize the denominators in : ^/^^by V^ 236 ELEMENTARY ALGEBRA. Square Root of Binomial Surds. I. Definitions and Principles. 258. A binomial one or both of whose terms are surds is a Mnomial surd ; as, a ± ^/h, Va ± V^. Note. — The discussion in this section will be limited to binomial surds of the second degree. 269. Since the rational term, if there be any, may be put in the form of a radical, and the coefficients of the terms, if there be any, be placed under the radical sign — JPHn, 110. — Every Unomidl surd of the second degree may he reduced to the form of V^ ± vh, in which one of the terms may he rational. 260. The square of {\^ ± Vh), or {Va ± Vhf = « ± 2 VoT -\-h = {a-\-h) ±2 Va/b, a binomial surd. Therefore, Frin. Ill, — A binomial surd may he a perfect square, and, when it is the square of a binomial surd of the second degree, one of the terms is rational, 261. Since {Va ± ^fhY =.(a-^h)±% Vah, {a-\-b)±2 V^ is the type of a binomial surd that is a perfect square. Therefore, Prin, 112. — A binomial surd with a rational term, and the coefficient of the irrational term reduced to ± 2, is a perfect square when the quantity under the radical sign is composed of two factors whose sum equals the ra- tional term ; and its square root equals the sum or differ- ence of the square roots of these factors. 262. {a-{-b) ±% Vab is the type of a square binomial surd. Now, {a-\-bY-{±%Vahf=a^-{-^ah-^l^- 4tab = a^ - 2ab -\- b^ = {a - bf. Therefore, SQUARE ROOT OF BINOMIAL SURDS, 237 Brin, 113, — When a Unomial surd is a perfect square, the difference of the squares of its terms is a perfect square, and is equal to the square of the difference of the two factors described in Frin. 112. 2. Examples. Ulnstrations. — 1. Extract the square root of 14 + 6 V^. Solution : 14 + 6 ^/~E= 14 + 2 y^. The two factors of 45 whose sura is 14 are 9 and 5 ; therefore, the square root of 14 + 2 ^45 is ± ( V^ + V5) [P. 112] = ± (3 + VS). 2. Extract the square root of 2 a — 2 Va^ — t^. Solution : The two factors of a' — &* whose sum is 2 a are a + 6 or and a — h\ therefore, y 2 a — 2 Va* — 6' = ^Ja + &— ^a — h ^Ja — h — ^Ja + 6 = ± {^/a + ft — ^/a — h). 3. Extract the square root of 81 — 36 Vb. Solution : 4/8I - 36 //S = \/^\ - 2 Vl620. The two factors of 1620, whose sura is 81, are not readily seen. Let x equal one and y equal the other. Then, (1) a; + y = 81 ; and (2) (X - yf = 81« - (2 VT620)« = 81 [P. 113J .-. (3) x-y = 9 Add (3) to (1) and subtract (3) frora (1), 2a; = 90 and 2y = 72 x = 45 and 3/ = 36 ••• 4/8I - 2 V1620 = V36 - V45» or V^ - V^ = 6 - 3 -/S or 3 ^/5^ 6 = ± (6 - 3 v^) EXERCISE 128. Extract the square root, when possible, of the following expressions : 1. 5 + 2\/6 3. 7-I-4A/3 5. 15 + 6 a/6 2. 9 - 4 a/5 , 4. 8 - 2 vis 6. 4 + a/3 7. 3a-2a\/2 9. 11 + 2a/30 8. 6a; + 2a;A/5 10. 13 — 2 a/42 238 ELEMENTARY ALGEBRA. 11. 16-4\/20 12. 2a; + 2 Va;2 — ^2 13. 15-2 V56 14. 2a; + 2 Va;^-4/ 15. 22-4 VSO 16. (a; + ?/) — 2 V^ '••5 + 3^ 23. 15 - V56 24. (« + 1) + A/4a 25. ^i + ^4/l| 26. 1 - I V6 o 27. 25 + 2 \/i56 28. -3 + 4\/^ 19. (l + 2a;) + 2Va:2 + a; 20. 9 + Vn 21. 23 - ^528 22. 16 + Vise 35. (x -{-yY •— 4t{x — y) \fxy 29. x — y — %v — xy 30. 42 + 36 a/2 31. 41-4 Vl05 32. + 2 a/^ 33. 6 + V35 34. 10- V 100-4 a;2 36. (2a; + l) + 2Va;2 + a;-2 Miscellaneous Examples. EXERCISE 124. Express with fractional exponents : 1. V^ 3. Va^^l* 2. V{aH(^Y 4. V(^ + Z')2 5. l/S{x-yf "v>^ 'V(a» - h^Y 7. V(a2 - ^^)" 8. Express with the radical sign : 9. x^ 11. a^h^ 13. {a-{-x)^ 15. a;^(a; + 2^)* 10. {a c)^ 12. a;^ y^ 14. (a^ - a;^)^ 16. a^ {a + J)^ MISCELLANEOUS EXAMPLES, 239 Express as mixed surds : 17. 3V3 20.^3 3/^ 22, xyV^Ty lQ.a'V¥ 21. a:tV^ 23. | VS 19. {a + b) Va-\-b 24. {a + a:) Va — x Place the coefficients of the following expressions within the parentheses : 25. 3 (3)^ 29. 7? (a.'2)-8 33. «-^ (a"')-' 26. 4 (2)^ 30. 2^ («^-2)-* 34. a;^ (« + J a;-^)* 27. 4 (4)^ 31. - 8 (a - 5)^ 35. x'^ (a + a;)"* 28. a (of 32. a (a-3 J)i gg. ^f (^-f )^ Reduce the following expressions to equivalent ones having a coefficient of 2 : 37. 6 a/3 40. 7 V2 43. - 8 XT^ 38. 5 Vo" 41. - 3 V4 44. 8 (a - If 39. V20 42. 3 (a - h)^ 45. 54 (a: + yf Complete the following expressions : 46. (8aa;)^ = 4( )^ hO, (a^ 7?)^ = w' { )* 47. (16 a; 2^)^ = 8 ( )^ 51. (a;^ y^)\ = ^^l ( )f 48. (32a# = 4( )^ 52. av(a-i)^ = ( )^ Express as pure surds : 54. Va3 + 2«2^, + aZ^ 66. """VS^ 58, 65. V(a - x) {a^ - a:^) 57. Vax^^* 240 ELEMENTARY ALGEBRA. Simplify : 62. 7 V54 + 3 Vl6 + a/432 63. {x^y^-^ay^)^-2{x'z^-^az'f 64. V(a + Z>)-^X a/(« + ^)~' 66. ( VU + A/2i - V42) ^ V7 67. ( V40 - Vl6 + V56) -s- VS 68. {a — x)-^ ( V« — V^) 4 V40 14 vT2 2 Veo 69. 3 Vl08 5 Vl4 * 3 V84 Reduce to equiyalent forms having a rational denom- inator : 70. 71. ^x V6a Vda V2 VI 73. 74. 75. a-\-Vb 4 V3 - V2 2 76. 77. 78. 3 + 2 V2 V5"- Vs 1- a/^^ 1+ V^^ V-3+ V-5 1- V5 Arrange in the increasing order of magnitude : 79. 3 V2, 2V3, Vis . 81. Ve, a/15, 2 V3 80. a/5, a/IO, 2 a/2 82. a/2, Vs, 'V20 Eesolve into two binomial factors [see P. 39] : 83. a — b 85. x^ — y^ 87. 16 — a; 89. a;^ — 5 84. x — 4^ 86. ic3 — 1/3 88. a; — 25 90. y — 2 Write the quotients of the following examples [see P. 45]: 91. {x-y)-r- (x^ - y^) 93. (x-y)^ (x^ - y^) 92. {x-{-y)-^ {x^ + 2/*) 94. {x-\-y)^ {x^ + ^^) MISCELLANEOUS EXAMPLES. 241 96. {l^x-S\y)^{2x^-^y^) 97. (16a;-8l2/)^(2a;i + 32^*) 98. (x^ - y^) -^ {x^ - y^) Reduce to lowest terms : 99 ^ ~ *^ a;— Vy 1 no """ ^x + ^y ^^^- V^+ Vy a— Vb (Va: - Vy) Va; + y x{x-\-y) 101. a/- , 8/- Va;+ Vy Va:^-y^ 104. 1~= ic Vic + y Expand : 105. \x^-]-{xy)^-\-y^ \x^ ■-(^y)* + y^l 106. (a + Va« - a:*)* 109. (« 4- 5 V- 1)^ 107. (a - * V- 1)' no. (-2-3 V- 2)3 108. (VS+V^)* 111. (2 V2- V3)* Simplify : 112. i/27 Vl35a«&* 4 , T= 115. 4/49 + 12 V5 113. V25a:* Vy 116. V2a;- V4a;2-4 '"i/i/i 117. V-3-2 V2 „0 a;4-A/^ , x-^- I_3/ _ yj^^^ p x—v — y ar+v- -2^ 119. Va; + y+Va :-y _^ Va; + y- Va :-y ^ ^^^^^ ^ 'x-{-y — "s/x — y Vx + y + Va; — y 120. Square /i+^ 242 ELEMENTARY ALGEBRA, 121. Square a/ x a/xA-- Y X Y ^ 122. Extract the square root of : x^ + 2xy^-]-Sx^y^-{-2x^y-^y^ 123. Extract, the cube root of : a;f + 3^ + 6^i+7 + 4 + § + -3 Radical Equations. Illustrations— 1. Solve X^=4: (A) Solution: Extract ^J, xi=±2 (1) Cube (1), x=±8 2. Solve V2a;2 = 2 (A) Solution: Cube (A), 2a;« = 8 (1) Divide, a;2 = 4 (2) Extract V, x= ±2 3. Solve Vx-\-14:-^Vx—14:= 14 (A) Solution : Square (A), a; + 14 + 2 V^^-196 + x-U = 196 (1) Transpose, 2 ^x^ — 196 = 196 - 2 a; (2) Divide, ^x^ — 196 = 98 - a; (3) Square, * a;^ - 196 = 9604 - 196 a; + a;^ (4) Transpose, 196 a; = 9800 (5) Divide, a; = 50 Caution. — In squaring a radical binomial, do not simply square each term. Thus, i\/x x 'x/ af is not x + a, but a; + 2 \/ax + a. 4. Solve Vx^ -{-ax — Vx = x (A) Solution : Transpose, ^/x^ + ax = x + ^/x (1) Square, x'^ -^ ax=.x^ ->r-2x ^/x + x (2) Transpose, 2 x ^/x =x{a — \) (3) Divide by x, 2 v^ = (a — 1) (4) Square, 4 a; = (a — 1)* (5) Divide, X = - — T — - RADICAL EQUATIONS, 5. Solye a^ - 3 x - 6 V2^^-3x-S = -2 (A) Solution : Subtract 3 from both members, (a;8_3a;_3)_6(a;2-3rc-3)i = -5 (1) Complete the square, (a;«-3a;~ 3)-6( ) + 9 = 4 (2) Extract V» ^x^-dx-3- 3 = ± 3 (3) Transpose, y\/x^ — dx — 3 = 5 or 1 (4) Square, 2^ - 32; - 3 = 25 or 1 (5) Transpose, a;» — 3 a; = 28 or 4 (6) 9 121 25 Complete the square, x^^Sx + ■j = -j- or -j- (7) Extract V, a- |= ± ^ or ± | (8) Transpose, a: = 7, — 4, 4, or — 1 EXERCISE 128. Solve : 1. V4^=16 2. (a; + 2)* = 36 3. (3 x)i = 9 4. V?=l 7. ^/x — x-3 Va; + "4 = 5 8. : 2 + Va; - 12 9. 10. flj = 2 Va; — a 11. = 7-V4cX-K 6. V(a;-a3)« = 4a* 6. V(a;* + «T = «* 12. V?+2"= Va;-4 13. Va;4-2-f- Va;-2 = 2 14. Vx-\-Vx + 2=Vx-{-3 15. Va; — 10 + Va; - 9 = 5 16. Va; + 3-f Vx-3 = 2 a/x-^^ ,- ^ + 1 , ^ /- v^-f 2 ^ 17. — T— +5=Va; 19. /- =2 V a; Va; — 2 18. ^7=- - a = — W 20. /- = i» Vx Vx a — Vx 2U ELEMENTARY ALGEBRA. 21. 22. 25. 26. 27. 29. 30. l-\-^/l- 1 _ Vl - a; ic — 4 = 5 Va;-2 = 4 23. 24. X — a \fx — Va 2 2Va Vx + l-{ Vx-i-1 3 xJf-^x + 1 Vx-i-l Vax-\-a r- . r- —j= '—^ = ^x -{- Va x-\-a v^ — V« 28. -T + -^i ^ + o; 31. -7= Vx — Va 32. 33. 34. 35. 36. 41. 42. 43. 44. 45. Vx + Va x-\-a 4:X — 9a = 2Vx-\-SVa x-a= Vx-\- Va 37. a; + 4+ Va: + 4 = 20 Vl — a; = 1 — Vx 3Q.xi — x^ = d xJ^x^ = Q 39. 3xi + x^=z2 « Of* + J a;"2' = c 40. a;^ + 0:4 = 2 2x^-^Sx-4.V2a^-i-Sx-2 = ll 2ax — x^ = 2a^ — aV2ax — x^ a^-4:X^-2Va^-4:a^-^4: = dl (x -SY = 13-{x^-ex-{- 16)^ x^ + 5 Va^-lQx=: 16 ic + 300 RADICAL EQUATIONS. 245 lsVx-2Vy= 0) ^^' \x- Vy = Vy) Vx-\-y + Vx — y = S Vx-{-y — Vx — y = 2 Vx 4- Vy = 1 ) \x4- Vx + V = 11 SO. i r^ ^ y 51. ^ ^ 49. 4cx-{- V 9 y = 4:) ' ly-{- ^^ -i-y = 4 62. 2^ - ^^ + ViT - 2^2 _ 20 xy{xy-{-l) = 240 53. x2 + 2a: = 6 + 4 V3 Character of the Roots of Equations. Definitions. 263. A root containing one or more imaginary terms is an imaginary root j as, x = a ±b V— 1. 264. A root containing no imaginary terms is a real root ; as, x = a-{-b, or x = S ± a/2. 265. A real root that contains one or more irrational terms is an irrational root j as, x = 3 ± V2, 266. A real root that contains no irrational term is a rational root ; as, x = a-\-b, or a; = 3 ± a/J. SIGHT EXERCI SE. Tell which of the following roots are real and which imaginary : 1, x= vT+5 6. x= V53 - 7* 2. x= V8-2 n. x= V5 (7 - 12) 3. x= V5-9 Q. x= V- 3 (6 - 9) 4. x= V12-3X5 9. x= V(- 2)2-1 6. x= V42-8 10. X = Vb"^ - 4« 246 ELEMENTARY ALGEBRA, If a and h are positive, and a is greater than h 11. X = wa — h 14. a; = V— a{a — b) 12. x= 's/'b — a 15. a; = a/^^ — a J 13. a: = -Ja(a — V) 16. a; = wc^^—cFb Tell which of the following roots are rational, which irrational, and which imaginary : 17. x= V18 + 7 18. x= VSO — 5 19. X = V15 — 3 20. a; = V32 - 2 X 5 22. a; = a; = a: = x = :V8- -32 23. : ^/72- -13 24. : V4 X 5 - 6X3 25. :V5^- -32 21. x=Vs^ — 20 26. a; = V25 - 4^ Give the sign of each of the following roots : 27. a; = 4+ V12 30, x=-4:+^/5 28. a; = 8 — V96 31. a; = 8 — Vlb 29. a; = - 2 - ^ Vio 32. a; = - 10 + - Vi20 If ^ and g' are positive, and p^ is greater than q : 33. x=p-{- ^p^-^q 36. a; = —p — V^^ _|_ ^ 34. a;=^ — VpH-^ 37. J9 4- V^i?^ — g! 35. a; = —p + v5M-~^ 38. — ^ -}- Vp^ — q The roots of the equation a;^ +^ a; = g' are in which j^^ is the square of -^ the coefficient of x, and q is the absolute term. CHARACTER OF THE ROOTS OF EQUATIONS. 247 Tell the character of the above roots, whether real or imaginary : 1. If p and q are positive. 2. If p is negative and q positive. 3. If p is positive and q negative, and q -rp^y nu- merically. 5. If p is negative and q negative, and q<-TP^, nu- merically. 6. If p is negative and q negative, and q> -rp^y nu- merically. Tell whether the above roots are rational or irrational : 7. If -7P^-\-q is a perfect square. 8. If ^ = 0. 9. If q is negative, and numerically equal to -rp^, 10. If jt? = 0, Give the signs of the above roots, and tell which root is numerically the greater : 11. If j9 and q are both positive. 12. If p is negative and q positive. 13. If ^ is positive, q negative, and -tP^> q, numeri- cally. 14. If p and q are negative, and -tP^> q, numerically. 16. If j9 and q are negative, and -p^ = qj numerically. What are the values of p and q in the following equa- tions : 16. a:2_|_4a.^7 xg, a?-^xz=-b 17. ar-9a; = 5 20. ea:^ — 9 = 18. a^ + 6a;=-4 21. 2a:« + 5a; - 3 = 248 ELEMENTARY ALGEBRA. In the following equations, are the roots — 1. Real or imaginary ? 2. Rational or irrational ? 3. Positive or negative ? 4. What are their relative values ? 22. x^-\-Q>x=l 28. a;2 + 6 a; = - 9 2Z. x^ — 4:X = b 29. ic^ — 5 a: = 10 24. x^-\-bx= — Q zo. x^-]-^x=—6 25.a^ — Sx=—2 3l.a^ — 6x=—8 26. x^-\-7x= —16 32. 4:X^ — l!x= — l 27.^ + 1^ = 1 33, 0^-^=-^ Inequalities. I. Definitions and Principles. 267. An expression denoting that two quantities are unequal in value is an Inequality. 268. The symbol of inequality is >, read greater than; or <, read less than. 269. The quantities compared in an inequality are the memhers of the inequality. 270. Two inequalities are said to subsist in the same sense, when the first members are both greater or both less than the second members. 271. Two inequalities are said to subsist in an opposite or contrary sense, when the first member of the one is the greater and the second member of the other. 272. A negative quantity is considered less than a posi- tive quantity, whatever their absolute values. 273. The process of changing the form of an inequality without changing its sense is transformation. INEQUALITIES. 249 274. The followiug principles of transformation may readily be illustrated : Prin, 114, — 1. The same or equal quantities may he added to both members of an i^iequality. 2. The same or equal quantities may be subtracted from both members of an inequality. 3. Both members of an inequality may be multiplied by the same or equal positive quantities. J^ Both members of an inequality may be divided by the same or equal positive quantities. 6. Two unequal positive members may be raised to the same power. 6. Two unequal positive members may have the same root extracted, provided the positive results only are comr pared. 7. The sum of two inequalities^ subsisting in the sam^ sense, may be taken member by member. 276. {a -bf>0 whether a>b ox b>a[V. 27]. Expanding, a^ -%ab-{-b^ > (1) Add 'Zab to both members [P. 114, 1] a^ -\- b"^ > 2 a b. Therefore, Prin, 115, — The sum. of the squares of two unequal quantities is greater than twice their product. 276. a^-\-b^>2ab\V. 1161 (1) a2 + c2>2ac '' (2) b^-]-c'>2bc " (3) Adding member by member [P. 114, 7], 2a^-\-2b^-[-'-Zc^>2ab-\-2ac-\-2bc (4) Dividing by 2 [P. 114, 4], a'^-^b^-}-c'>ab-\-ac-\-bc. Therefore, Prin, 116, — The sum of the squares of three unequal quantities is greater than the su7n of their products taken two and two. 250 ELEMENTARY ALGEBRA, 2. Examples. Ulustration. — Which is the greater, a^ -{■ ¥ ov a^ h -\- a W, for any positive values of a and h ? Solution : a^ + b^> = = = <:, ab Adding a & to both members, a^ + &^ > = < 2 a 6. But a' + b^>2ab [P. 115], . • . a^ + b^^a^b + ab^, since no operation has been perf onned to change the sense of the inequality. EXERCISE 126. Prove the following statements true for unequal positive values of the letters : 1. a^ + ab^>2aH 4. {a + hy> 4:an-i-4.ab^ 2. aH-{-a¥>an^ + ab^ 6. a^ + db^ > 2ab + 2b^ 3. {a-\-by>4tab 6. a^ > a^ -i-a -1 7. a^ + an^-ira^c^>aH + a^c + aHc 8. a^>2a-l 9. U a^-{-4:X> 12, show that x>2 10. If 3a^-^6x> 42, show that x>3 11. If 7x^ — 3x< 160, show that x<6 12. - + ^>2 13. -2 + ^>- + - 14. 7?y-\-xy^-\-Q?z-\-x^-\-y^%-\-yz^>^xyz a-\-b a , ^ a-\-b ^ a . 15. — h^ > -, when ac c-\-b c c-\-b c a — b ^ cc 1 17. 7 > = < -, when a < = > c c — b c 18. Which is the 2rreater, , or „ ,p , if a and h ... ^ ^ 'a — b a^ — W are positive ? 19. What integral value of x will satisfy 3 ic^ + ^a; > 64 and 3 ic2 + 4 ir < 132 ? CHAPTER VII. RATIO, PBOPOBTIOJf, AKB PROGRESSIOJ^. Ratio. I. Definitions and Principles. 277. A relation of values exists between two similar quantities — that is, one of them is a number of times or a part of the other. 278. The relation which the value of one quantity bears to that of another is the Ratio of the quantities, and is obtained by dividing the quantity compared by the quan- tity with which it is compared. niustration. — The ratio of 3 apples to 5 apples is — , 3 since 3 apples = ;^ of 5 apples. 279. A ratio is expressed by writing a colon between the quantities compared, or by a common fraction. niustration. — The ratio oi a to h i^ a : h, or -r. 280. The quantity compared, or the first term of a ratio, is the antecedent ; and the quantity with which the comparison is made, or the second term of the ratio, is the consequent, 281. Since the ratio is obtained by dividing the quan- tity compared by the quantity with which it is compared, it follows that, 252 ELEMENTARY ALGEBRA. Prin. 117 > — The ratio equals the antecedent divided hy a trie consequent ; or r c 282. Since r = — , a=.c X r, and c = —. Therefore, c r ' Prin, 118. — The antecedent equals the ratio times the consequent, Prin, 119, — The consequent equals the antecedent divided hy the ratio, 283. Since — = r, — =^nr, and = nr (Ax. 4). c c c-^ n ^ ' Therefore, Prin, 120, — Multiplying the antecedent or dividing the consequent multiplies the ratio, 284. Since - = r, = -, and — = - (Ax. 5). c en nc n ^ ' Therefore, Prin, 121, — Dividing the antecedent or multiplying the consequent divides the ratio, 285. Since - = r, — = r, and — '- — = r. Therefore, c nc c-^n Prin, 122, — Multiplying or dividing loth terms of a ratio hy the same quantity does not alter its value, 286. The product of two or more simple ratios is a compound ratio. Thus, ] ^ ! ^ [ ^^ f X ^ is a compound ratio. 287. To duplicate a ratio is to use it twice in a com- pound ratio. Thus, the duplicate of a-.l is ?X^ = t^. 0^ 288. To triplicate a ratio is to use it three times in a compound ratio. Thus, the triplicate of a-. I is -^X^X^ = t^. 0^ RATIO, 253 289. When the antecedent and consequent of a ratio are equal, it is called a ratio of equality ; as, a : a, or 1:1. 290. When the antecedent is greater than the conse- quent, the ratio is greater than one, and is called a ratio of greater inequality. 291. AVhen the antecedent is less than the consequent, the ratio is less than one, and is called a ratio of lesser in- equality, 292. When the ratio of two quantities can he exactly expressed hy a rational number or fraction, it is said to be commensurable, 293. When the ratio of two quantities can not be exactly expressed by a rational number or fraction, it is called an incommensurable ratio ; as, a/2 : a/3 — i/ ~, 2. Examples. EXERCISE 127. 3 5 1. Find the ratio of 4 to 20 ; 16 to 12 ; -^ to 9 ; 8 to - ; 2,3 56 2. Find the value of ab^ : a^b ; a^ — a^ : a-\-x; {a + by-.a^-b^; a^ -^7? : a"" - ax-^3? 3. Find the value of : m-\-n m^^n^a^ — y^x — y 1 ^ a m — n' (m — ny x-^ y ' a^-{- y^^ 7? — y^' x — y 4. Reduce to their lowest terms : 25:75; aH^ \ aV \ a^^ab-.ab + b^ 6. Cleai* of fractions and reduce to lowest terms : j.3-,1 ^n3„^l X z , a , c 2-r:7^; 18-i-:31-i-; -: — ; a -\- -r : c -{- j- 4: 2 4 4:^ y xy^ b b 12 254 ELEMENTARY ALGEBRA. 6. Compound the ratios '4:5, 5 : -6, and -03 : 40 7. Compound the ratios 2*5 : -32, -08 : 1*5, and -12 : -016 8. Which is the greater, the ratio of 2—: 7— or the 1 ..1 ^ ^ duplicate ratio of 2 — : 7 - ? 9. What must be subtracted from both terms oi a;h to make it c: d^ 10. Compound the ratios of : a-l'' {a^yf^^^ a-\-l)'' {a- If 11. If 5 horses and 8 cows cost as much as 8 horses and 2 cows, what is the relative value of a cow to a horse ? 12. Find the ratio of 2 to a/2 to within one thousandth; also, the ratio of 3 to a/3. 13. The side of a square is 4 feet. What is the approxi- mate ratio of the side to the diagonal ? 14. If the same number be added to both terms of a ratio of lesser inequality, will it be increased or diminished ? Which, if the same number be subtracted from both terms ? 15. If the same number be added to both terms of a ratio of greater inequality, what will be the effect ? What, if the same number be subtracted from both terms ? 16. The ratio of A's money to B's is the same as the ratio of 5 to 6, and they together have $1320. How much has each ? 17. The sum of A's and B's ages bears the same relation to A's age as A's age bears to 8 years, and the difference of their ages is 10 years. Eequired the age of each. 18. The fore-wheel of a wagon makes 128 revolutions more in going a mile than the hind-wheel, and their cir- cumferences are in the ratio of 5 : 6. What is the circum- ference of each ? PROPORTION, 256 Proportion. Definitions. 294. The equality of two or more ratios may be ex- pressed by writing between them a double colon, or the symbol of equality. Thus, the fact that the ratio of 2 to 3 equals the ratio of 4 to 6 may be expressed : 1. 2 : 3 : : 4 : 6 \ 2 _ 4 >- read 2 is to 3 as 4 is to 6. ^* 3 "^ 6 ) 295. The expression of the equality of two or more equal ratios is called a Proportion. 296. A proportion of two simple ratios is a simple pro- portion ; one of three or more ratios, a multiple proportion. 297. The ratios of a proportion are called couplets. 298. If, in a multiple proportion, the consequent of each couplet is the same as the antecedent of the following couplet, it is called a continued proportion. Thus, a'.hwh'. c\\ c'.d is a continued proportion. 299. Every simple proportion has four terms. The first and fourth are called the extremes ; the second and third the means ; the first and third the antecedents ; and the second and fourth the consequents. 300. A mean proportional between two quantities is a quantity to which the first bears the same relation that the quantity bears to the second. Thus, J is a mean proportional between a and c, when a :b : : b : c. 301. A third proportional to two quantities is a quan- 256 ELEMENTARY ALGEBRA, tity to which the second bears the same relation that the first bears to the second. Thus, c is a third proportional to a and I, when a : h : : 1) : c. 302. If a proportion contains one or more compound ratios, it is a compound proportion. When the word " proportion " is used alone, it designates a simple proportion. Propositions. I. In any proportion, the product of the extremes equals the product of the means. Given a-.h :: c : d : (A) Prove aX d=bx c Demonstration : T = T [another form for (A)], Clear of fractions, a x d = b x c. CorcUary 1, — Either extreme equals the product of the means divided ly the other extreme. dyr, 2, — Either mean equals the product of the extremes divided hy the other mean. II. If the product of two quantities equals the product of two other quantities, either pair may ie made the ex- tremes, and the other pair the means, of a proportion. Given m X n=p X q (A) Prove, 1. m: p '.'. q: n 6. p : m: : n: q 2. m: q\: p \ n 6. p : n: : m: q 3. n:p :: q:m 7. q:m: : n : p 4. n: q\: p'.m 8. q : n: : m :p Demonstration: 1. Divide (A) by P X g (1) Divide (1) by jp, m q p-n (2) PROPOSITIONS. 257 Write in another form, m: p :: q: n. Let the pupil derive the remaining seven. Exercise. — Write the eight proportions deducible from : 3X4 = 2X6; aXd=zhXc\ xXz=vXy 303. A proportion is taken by alternation when the means or the extremes are made to change places. III. If four quantities are in proportion, they are also in proportion by alternation. Given a:h :: c -. d (A) Prove, 1. a: c'.'.h : d 2. d : b : : c : a Demonstration : a x d = c x b [P. I], Exercise. — Write by alternation : 3:4::9:12; x : y :: m: n; x : a:: y :b 304. A proportion is taken by inversion when the means are made the extremes and the extremes the means. IV. If four quantities are in proportion, they are also in proportion by inversion. Given a: b :: c : d Prove, 1. b : a : : d : c 3. c: a:: d:b 2. b : d : : c : a 4:. c : d:: a:b Demonstration : a x d = b x c, ll'-V'-'^'-'l and \''-y-^'-l\ [P.iq. (b :d:: a:c S ic:d::a:b ) *- -* Exercise. — Write by inversion : 5 : 10 : : 15 : 30 ; X : m : : n : y ; a-^-b : a — b : : c-\- d : c — d 305. A proportion is taken by composition when the sum of the two terms of each couplet is compared with either the antecedent or the consequent of that couplet. 258 ELEMENTARY ALGEBRA. V. If four quantities are in proportion, they are also in proportion by composition. Given a: b : : c : d (A) Prove, 1. a-{-b : b : : c-\-d : d 2. a-{-b : a : : c-\-d : c Demonstration : -r = -^ [another form of (A)] Add 1 to both members, -r- + 1 = -r + 1 Reduce to improper fractions, — j— = , ; or a + b :b : : c + d : d Let the pupil prove the second part. Exercise. — Write by composition : 2:3::6:9; 8:2::16:4; x:a::y:b 306. A proportion is taken by division when the differ- ence of the two terms of each couplet is compared with either the antecedent or the consequent of that couplet. VI. If four quantities are in proportion, they are also in proportion by division. Given a-.bwc'.d Prove, 1. a — b\b::c — d'.d 2. a — b'.awc — d'.c Demonstration : 'b~'d [^^^ther form of (A)] Subtract 1 from both members, ^ — 1 = -y — 1 ' d -r,! a — he — d Reduce, — v — = —i — ; or a — h'.h'.-.c — d-.d Let the pupil prove the second part. Exercise. — Write by division : 3:9::6:18; 6:3:: 12:6; a-\-X'. xwb^y : y PROPOSITIONS, 259 307. A proportion is taken by composition and division when the sum of the two terms of each couplet is compared with the difference of these terms. VII. If four quantities are in proportio7i, they are also in proportion hy composition and division. Given a:b w c \ d (A) Prove a-\-b:a — h'.'.c-\-d'.c — d Demonstration : Take (A) by composition, — r— = —-r- (1) Take (A) by division, ^^^ = ^^ (2) Divide (1) by (2), ^ = ^^d 5 ^^ a + b : a — b : :c + d :c — d Exercise. — Write by composition and division : 3:8:: 12: 33; x : y : : m, : n ; x — y : x-^ y : : 3 : 6 VIII. If two proportions have a couplet in each the same, the remaining couplets form a proportion. b::c:d (A) ^ :c:d (B) ) :e:f Given , ^ r,; Prove a : b Demonstration: -r- = y = ^(A)and^ = j(B) -r — -f (Ax. 1); whence o : 6 : : e :/ Exercise. — Prove that, Cwr, 1, — If two proportions have the antecedents alike, the consequents form a proportion ; or, Given a-.bw c\ d and a\x\\ c.y Prove b '. d :: x: y Cor, 2. — //' two proportions have the consequents alike, the antecedents form a proportion ; or. Given a:b\:c:d and x:b\\y \d Prove a: c'.'.x: y 260 ELEMENTARY ALGEBRA, Car. 3, — If two proportions have a QOuplet in propor- tion, the remaining couplets form a proportion ; or. Given a : h : \ c : d, e -. f : : g : h, and a:h::e:f Prove c\ dwg \h 308. Equimultiples of two or more quantities are the products obtained by multiplying each of the quantities by the same number. IX. Equimultiples of two quantities are proportional to the quantities themselves. Given the two quantities a and 5 and their equimul- tiples 7na and mh, Prove ma'.mh'.-.a'.h _ ... ma a Demonstration : — r = -r mb ma : mb : : a : b Exercise. — JProve that equal parts of two quantities are proportio7ial to the quantities themselves ; or that, a h -, — : — :: a:o m m X. If four quantities are in proportion, equimultiples of the first couplet are proportional to equimultiples of the second couplet. Given a\l\\c\d (A) Prove ma: mh :: nc: nd Demonstration : x = t [another form of (A)] a ma ^ c nc T- = — r> and -=- = —-5 b mb a nd ma_nc mb~ nd^ ma : mb : : nc : nd Exercise. — Prove that, 1. Equal parts of the first couplet are proportional to equal parts of the second couplet. PROPOSITIONS. 261 2, Equimultiples of the antecedents are proportional to equimultiples of the consequents, S. Either extreme may he multiplied and the other divided by the same quayitity, 4, Either mean may he multiplied and the other divided by the same quantity, XI. If two quantities are increased or diminished by like parts of themselves, the results are proportional to the quantities themselves. Given the two quantities a and h, to prove, 1. a-\ — a : h-\ — h : : a : h n n m ^ m , 2. a a: h h :: a:b n n Demonstration: a\\ ± -) \ n) a a ±. — a n a ^{^-t)~' n a±^o:ft± ™6, n ::a:b Exercise. — Given a: h :: c : d Prove a±—a:b±—h::c±-c:d±-d n n a a XII. If four quantities are in proportion, like powers and like roots of them are also in proportion. Given a\h\\c\ d (A) Prove, 1. «" : J" : : c" : d^ 2. Va : Vh : : ^/c : Vd . ^ Demonstration : a x d = b x c [F. I] (1) liaise lx)th members to the nth power, arxd* = h*xr* (2) Then, oT : h"" : : c* : dT [P. II] (3) Extract the nth root of (1), V » x V^ = V^ x aA (4) Then, V» • V^ : : Vc : Vd (5) 2G2 ELEMENTARY ALGEBRA. XIII. The corresponding memhers of two equations form a proportion. Given a — I (A) and c = d (B) Prove a\ c :\i : d Demonstration : Divide (A) by (B), — = -^ ; (1) or a : c : : b : d XIV. The products or quotients of the corresponding terms of tivo jjroportions form a proportion. \a:h'.:c\d ( A) ) Given ' ^ ' e:f'.:g:h (B) S Prove, 1. a X e:i Xf: : c X g : dXh ^ a h c d ^ f g h Demonstration : a x d = b x c (1), and e x h—.fx g (2) [P. I] Multiply (1) by (3), {a x e) x (d x h)z={b x f) x {c x g) (3) Therefore, a x e : b x f : : c x g : d x h [P. IIj Let the pupil prove the second part. XV. If two proportions have three terms of the one equal to three terms of the other y each to each, the fourth terms are also equal. (B)i and X = --^ [P. I, Cor. 1] Given < , a : : c : ; c : d X Prove X - = d Demonstration l: d: _b X c CL Therefore, X : = d XVI. In any multiple proportion the sum of the ante- cedents is to the sum of the consequents as any antecedent is to its consequent. Given a :h -.: c : d :\ e :f Prove a-\-c-{- c :h-[-d +/ '.'. a-.h PROPOSITIONS. 263 Demonstration : Let r equal the ratio of each couplet, Then, t- = ^> Z"**' ^^^ 7~^ Clear of fractions, a = br (1), c = dr (2), and e=fr (3) Add (1), (2), and (3), a + c + e = {b + d + f)r (4) Divideby6 + rf+/, ^±±±j=r = ^ (5) Therefore, a + c + e : b + d + f : : a : b XVII. A mean proportional between two quantities equals the square root of their product. Given h, a mean proportional between a and c, to prove h = Va c. Demonstration : a:b ::b -.c [317] . • . J'* = a 6 [P. I] ; whence b = ^^/o^ Additional Propositions. EXERCISE 128. If a-.h'.'.C'.dy prove that : 1. 2fl:3J::2c:36? Z. na\ ml) w nc.md 2. 3a:4^::6c:8c? ^ a-\- c\ a\\h-{-d'. c 5. 2a + 3*:2c + 3c?::2a:3J 6. w a -|- wi 5 : w c -f- ''I fi? * : « : c 7. a: d::hc: d^ 9. (a + c) 2r : (a — c) a: : : (J + c a w X -\- a -rl 7. Va:* — c? : ^/x — a : : Vx -\-a : x Vx Vx + V2 V2 4-1 11. {x-i-y:x::5:4:) Vx-j-S-{-Vx — S 1 12. {x-\-y:x~y::3:l) Vx-[-a+Vx-a _ ^ 13. a^-^f -, x^-f : : 35 : 19 Vx-i-a-Vx-a ar-f/ = 52 4 Examples involving Proportion. Ulnstration, — 1. A's age is to B's as 2 to 3 ; but in 10 years their ages will be to each other as 3 to 4. Required the age of each. Solution : Let 2 rr = A's age ; then will Sx = B's age ; and 2a; + 10 = A's age 10 years hence; and dx + 10 = B's age 10 years hence ; then 23; + 10 : 3 j: + 10 : : 3 : 4 (A) 8a; + 40 = 9a; + 30 (1) X = 10 2 2: = 20, A's age; 3 a; = 30, B's age. 2. The sum of A's and B's capital is to the difference of their capitals as 9 to 5 ; but if A withdraws $100 and B adds 1100, their capitals will be to each other as 4 to 3. Required the capital of each. Solution : Let x = A's capital and y = B's ; then X ■{- y : x — y : :9 : 6, (A) and a;- 100 : y + 100 : : 4 : 3 (B) Solre (A) and (B), x = $423 ^3 , y = $107 ^ 266 ELEMENTARY ALGEBRA. EXERCISE ISO. 1. The length of a room is to its width as 4 to 3, and the floor contains 588 square feet of boards. What are the dimensions of the room ? 2. A man's age is to his wife's as 6 to 5 ; but 30 years ago his age was to hers as 7 to 5. Kequired the age of each. 3. The difference of two numbers is to the difference of the squares of the numbers as 1 to 13, and the product of the numbers is 42. Find the numbers. 4. A and B are in partnership. A's capital is to the whole capital as 5 to 8 ; but if A withdraws $2000 and B adds $2000, A's capital will be to the whole capital as 3 to 5. Required each man's share of the stock. 5. The length of a rectangular field is to its width as 5 to 4 ; but if 4 rods be added to the length and 5 rods to the width, they will be to each other as 6 to 5. Find the area. 6. Two thirds of A's money is to ^4 of B's as 5 to 6, and 2/3 of A's + % of B's is $1500. Required the fortune of each. 7. The rate of a fast train is to that of a slow train as 5 to 3, and if it is 60 miles behind the slow train it will overtake it in 3 hours. What is the rate of each train ? 8. I have a cubical box, such that if each of its dimen- sions be increased by one foot the contents will be to the entire surface as 1 to 2. Required the contents. 9. The circumferences of circles are to each other as their diameters. If the circumference of a circle whose diameter is one is 7r = 3*1416, what is the circumference of a circle whose diameter is ^ ? 10. The areas of circles are to each other as the squares of their diameters. If the area of a circle whose radius is EXAMPLES INVOLVING PROPORTION. 267 one is tt, what is the area of a circle whose radius is r ? What, when r = 4 ? 11. The surfaces of spheres are to each other as the squares of their diameters. If the surface of a sphere whose radius is one is 4 tt, what is the surface of a sphere whose radius is r ? What, whea r = 5 ? 12. The volumes of spheres are to each other as the cubes of their radii. If the volume of a sphere whose radius is one is Y3 tt, what is the volume of a sphere whose radius is r ? What, when r = Q^ Surfaces and volumes that have the same shape are similar. To have the same shape, they must have their corresponding angles equal and their corresponding dimensions proportional. 13. Similar surfaces are to each other as the squares of their like dimensions. If a field a rods long contains m acres, what will a similar field c rods long contain ? 14. Similar volumes are to each other as the cubes of their like dimensions. If a keg whose bung diameter is c inches holds n gallons, what will a similar keg d inches in bung diameter hold ? 15. The quantities of water that flow through circular pipes are to each other as the squares of the diameters of the pipes. If c gallons flow through a pipe m inches in diameter in one minute, how many gallons will flow through a pipe n inches in diameter in the same time ? Limiting Ratios. Definitions and Principles. 309. A quantity that retains the same value throughout an operation or discussion is a constant. 310. A quantity that continuously changes its value — 268 ELEMENTARY ALGEBRA. that is, passes from one value to another by successively assuming all values lying between them— is a variable. lUnstration. — A line a foot long is a constant. A line traced by a point moving according to some well-defined law is a variable. 311. A finite unit is a unit of comprehensible size or value. 312. A quantity that can be expressed in finite units is afi?iite quantity. 313. A quantity too small to be expressed in finite units is said to be infinitely small. An infinitely small variable is called an infinitesimal, and may be expressed by the character o , read an infinitesimal or zeroid. 314. A quantity too large to be expressed in finite units is said to be infinitely large. An infinitely large variable is called an infinite, and may be expressed by the char- acter a , read an infinite. 316. The entire absence of quantity is called zero, and is expressed by the character 0, read zero. 316. The unlimited whole of quantity, or rather un- limited quantity, is called infinity, and is expressed by the character oo, read infinity. 317. If, in the fraction — , x decreases by a constant ratio until it becomes an infinitesimal and a remains a finite constant, the value of the fraction decreases in the same ratio [P. 55], and becomes an infinitesimal. Therefore, Brin. 123, — = o . An infinitesimal divided ly a finite constant is an infinitesimal. LIMITING RATIOS, 269 318. Since — = o , it follows that, a JPHn. 124, o X a = o . An infinitesimal multiplied by a finite constant is an infinitesimal, 319. Since o x « = o , it follows that, PHn, 125, — = a. An infinitesimal divided by an infinitesimal may be any finite constant, 320. If, in the fraction -, x increases by a constant ratio until it becomes an infinite and a remains a finite constant, the value of the fraction increases in the same ratio [P. 54], and becomes an infinite. Therefore, JPrin, 126, — = a . An infinite divided by a finite constant is an infinite. 321. Since — = a , it follows that, a Prin, 127, oc X a= cc. An infinite multiplied by a finite constant is an infinite. 322. Since a X a = oc , it follows that, Prin, 128, — -=. a. An infinite divided by an infinite mxiy be any finite constant. 323. If, in the fraction -, x decreases by a constant X ratio until it becomes an infinitesimal and a remains a finite constant, the value of the fraction increases in the same ratio [P. 54], and becomes an infinite. Therefore, Prin, 129, — = a . A finite constant divided by an infinitesimal is an infinite. 324. Since — = a , it follows that, o Prin, 130, o X oc = a. The product of an infini- tesimal and an infinite may be any finite constant. 270 ELEMENTARY ALGEBRA. 325. Since o x oc = ^5, it follows that, Prin, 131, — = o . A finite constant divided hy an infinite is an infinitesimal, 326. Since — , — , and a X o are each satisfied by any finite constant, they are symbols of indetermination, 327. The limit of a variable is a value which the vari- able continually approaches but which it can never reach, but may be made to differ from it by less than any assign- able quantity. Illustration. — If a point starts at A in the direction of B, and goes Yg the distance the first second, % the remaining distance the next, Yg the remaining distance the third, A \ j — j — B and so on, the distance passed over constantly approaches the distance from A to B, and will eventually differ from this distance by an infinitesi- mal, but it can never equal this distance. From A to B is therefore the limit of the distance the point can go. 328. The limit of a variable that decreases by a con- stant ratio is zero. niustration. — If Ys a line be cut off, then Ys the re- mainder, and so on indefinitely, the part retained continu- ally approaches zero, from which it will eventually differ by less than any assignable quantity. Therefore, zero is the limit of the remainder. 329. A variable quantity that increases by a constant ratio has no limit. This fact is sometimes expressed by saying that its limit is infinity. 330. A function of a variable quantity is any expression that contains the variable. Thus, ax^ -{-b is a function of x. LIMITII^O RATIOS. 271 331. A function of a variable is generally a variable also. It is then called the dependent variable, and the variable upon which it depends the independent variable. 332. The limit of a function, when the independent variable approaches its limit, may be zero, infinity, or a finite quantity. Illustration. — 1. If x approaches a as a limit, the func- tion approaches — , or as a limit. 2. If X approaches a as a limit, the function X — a a approaches — , or oo as a limit. X 3. If X approaches a as a limit, the function — , — ^^ x-j-a approaches , or - as a limit. 333. Sometimes a factor whose limit is zero is common to both terms of a fraction. The limit of the fraction will then assume the irreducible form — • The true limit is then found by removing the common factor before passing to the limit. Illustration. — If x approaches a as a limit, the func- 2r* ^3 Q tion "2 2 approaches — as a limit. This form results, because the common factor x — a has for its limit. Re- moving this factor, we have — - -^ , which has for its limit ' ; ' = X — = - «. a-\-a 2a 'Z x^ _ ^3 3 334. To express that the limit of ^-^i is „ ^ ^^^^ the limit of a; = a, we write : Lim. ^--2 (a: = «) = -«. 272 ELEMENTARY ALGEBRA, Examples. /v3 __ ^^3 ninstrations. — 1. Find Lim. ' (x = a), ^ (X^ Q^ q3 Solution : =:x'^ + ax + a^, . • . Lim. (x = a) = x—a x—a^' Lim. x^ + ax + a\ {xz=a) = a^ + a? + a^ = 3 ^^2, % Find Lim. ~ (:r = 00 ). X «... x — a.a x — a. Solution : = 1 , . • . Lim. te = 00 ) = X X X ^ ' Lim. 1 - -, (ic = 00) = 1 - ^- = 1 - = 1. X ^ ' 00 EXERCISE 181. Find: /y;* //<* ly 1. Lim. (x = «) 9. Lim. — -— (a: = 00 ) X — a ^ ' x-\- 1 ^ ^ 2. Lim. ^~ (a: = 1) 10. Lim. f^^ ix = 0) a; ^ ^ ?72 a;^ + w a; ^ ' 3. Lim. p , — (a; = 1) 11. Lim. (x = a) XT -\- x ^ ' X — a 4. Lim. 7 r^ (x = a) 12. Lim. (x = 1) (x — af^ ' x — \^ ' 5. Lim. {x = 1) 13. Lim. -7— (^ = — «) 6. Lim. 0^2 _^ (^ ^ 0) 14. Lim, -^ (a- = 0) x — a axA-bx , . T-. rr^ — a^ , , T . ax-\-x4-l, . 8. Lim. — ; — (a; = a) 16. Lim. ' — (^ = go ) x-^a ^ '^ X ^ ' X — V'^ — a^ , 17. Lim. / „ (x = a) ,0 T- x^ + a^:^ + a\ . ARITHMETICAL PROGRESSIONS, 273 Arithmetical Progressions. Definitions and Principles. 335. Any number of quantities that increase or decrease according to a law constitute a Series, or Progression. 336. The quantities which compose a series, or progres- sion, are call^ the terms, 337. A progression in which each term after the first is derived from the preceding term by the addition of a con- stant quantity, is an arithmetical progression, 338. The constant quantity added to any term of an arithmetical progression to produce the next term, is called the common difference. 339. If the common difference is positive, the series, or progression, is an ascending one ; if negative, a descending one. Thus, «, 3 a, 5 a, 7 a, etc., is an ascending series ; and, 7 a, 5 a, 3 a, a, etc., is a descending series. 340. In the general discussion of arithmetical progres- sions, a represents the first term, d the common differ- ence, I the last term, n the number of terms, and S the sum of the terms. 341. If we represent the first term by a and the com- mon difference by dy the 2d term = a-\-d 4th term = a + 3 c? 3d term = a -f- 2 c? 5th term = a + 4 S=(Z + a)| ^=(39 + 3) X 5 = 210 2. The first term of an arithmetical progression is 25, the number of terms is 6, and the sum of the terms is 102. Required the last term. Solution : Substitute the values a = 25, n = 6, and yS' = 102 in formula B: 102 = (Z4-25)3 = 3Z + 75 dl = 27 1 = 9 ARITHMETICAL PROGRESSIONS. 275 3. Given / = 31, ^ = 4, and >S'= 136, to find w. Solution : Substitute these values in formulas (A) and (B) : 1. Since l=za + {n—\)d, 31 = a + (n— 1)4, or a + 4?» = 35 (A) 2. Since >S' = (/ + a)^, 136 = (31 + a)^, or 31;i + an = 272 (B) Transpose (A), a = 35 — 4 n (1) Substitute (1) in (B), 31w + 35n-4n« = 272 (2) Transpose (5), 4 ;i2 _ 66 » = - 272 (3) Complete the square, ^ 9 «P . /33\' 1088 1089 1 4n^-66n+ (^^j =___ + __ = _ (4) Extract the V, ^" ~ ^ ~ ^ ^ ^^^ Transpose, 2n = 16 or 17 Divide, n — ^ or 8-^ Note. — Since the number of terms is a whole number, 8 is the true answer. EXERCISE 132. 1. Find the 10th term and the sum of 10 terms of the series : 4, 8, 12, etc. 2. Find the 12th term and the sum of 12 terms of the series : 27, 25, 23, etc. 3. Find the 9th term and the sum of 9 terms of the 1,5,7 , series: -+g+^, etc. 4. Find the nth term and the sum of the n terms of the series : 1, 2, 3, etc. 6. Find the rth term and the sum of r terms of the series : 2, 4, 6, etc. 6. Given a = 3, Z = 28, and w = 6, find d. 7. Given >S'=112, w = 7, and a = 25, find I and d, 8. Given w = 8, a = 8, and 6? = 5, find S and L 9. Given (i=l-, >S=58, and a = 2, find I and n. 276 ELEMENTARY ALGEBRA. 10. Show that d = ii. Show that n = l + a 12. Show that n = — -, — f- 1 d 13. Show that 1 = a 14. Show that a = n n 15. Show that Z = — + ^ ~ x d n 2 16. Show that /S=^w[2a + (?^ — 1)^] 17. Show that 8 2 2 '^ 2d 18. Show that a = ^ — d n 2 19. Show that d = -^ r^^ n (n — 1) 20. Giyen d = -^, ? = 6-, aud 8= 45, to find a and w. 21. Given (? = 4, >S'= 190, and a = 1, to find / and n, 22. Given ^Z = 3, / = 35, and /S = 220, to find n and a. 23. Show that n = ^i-^''±^(^'^-dr + SdS 2d Concrete Examples involving Arithmetical Progressions. Blustrations. — 1. Insert three arithmetical means be- tween 3 and 11. Solution : Since there are to be three arithmetical means, the num- ber of terms is 5, the first term is 3, and the last term is 11. Take l=:a + (n-l)d Substitute, 11 = d + 4cd d = 2 , • . The means are 5, 7, and 9. CONCRETE EXAMPLES. 277 2. Find the series whose nth. term is 4 w — 1, Solution : Since the nth term may be any term, Let n = 1, then nth term = 1st term = 4 — 1 = 3 Let n = 2, then /ith term = 2d term = 8 — 1 = 7 Let n = 3, then 7tth term = 3d term = 12 — 1 = 11 Etc., etc., etc. 3. Find the series the sum of n terms of which is n^-\- n. Solution : Let /S'=n« + 7i Let w = 1, then S = first term = 3 Let n = 2, then S = sum of two terms = 4 + 2 = 6 Let n = 3, then S — sum of three terms = 9 + 3 = 12 Let n = 4, then S = sum of four terms = IG + 4 = 20 Etc., etc., etc. Since the sum of two terms is 6 and the first term is 2, the second term is 6 — 2 = 4. Since the sum of three terms is 12 and the sura of two is G, the third term is 12 — C = 6. Similarly the fourth term is 20 — 12 = 8. And the series is 2, 4, G, 8, etc. 4. The sum of five numbers in arithmetical progression is 30, and the difference of the squares of the extremes is 96. Required the numbers. Solution : Let x equal the middle term and y the common differ- ence, then will the numbers be x — 2y,x — y,x,x-\-y,x-\-'Zy (A) Since their sum is 30, 5 a; = 30 (1) a;= 6 (2) Since the difference of the squares of the extremes is 96, (a; + 2y)«-(x-23/)» = 9G (3) Expand and collect terms, 8 x y = 9G (4) a;y = 12 (5) Substitute (2) in (5) and reduce, y = 2 Substitute the values of x and y in (A), the series becomes 2, 4, 6, 8, 10. 5. The sum of four numbers in arithmetical progression is 24, and the product of the means is 35. Find the num- bers. 13 278 ELEMENTARY ALGEBRA. Solution: Let x — y and x + y h& the two means, the common difference being 2y, then will the series be x — ^y, x — y, x-\-y, x-\-^y (A) Since the sum is 24, 4 a; = 24 (1) a:= 6 (2) Since the product of the means is 35, x^ — y^ = 35 (3) Substitute (2) in (3) and reduce, 2/ = ± 1 (4) Substitute the values of x and y in (A), the series becomes 8, 5, 7, 9, or 9, 7, 5, 3 343. Any arithmetical series of an even number of terms may be formed by putting x^ y and x-\-y for the two middle terms, making 2 y the common difference. 344. Any series of an odd number of terms is more conveniently formed by putting x for the middle term and y for the common difference. EXERCISE 133. 1. The sum of three numbers in arithmetical progres- sion is 30, and their product is 910. Required the num- bers. 2. The amounts of $100 for 1, 2, and 3 years respect- ively are 1105, $110, and $115. What is the amount of the same sum for 15 years ? 3. What is the amount of $200 for 10 years at 6^, simple interest ? 4. Insert three arithmetical means between 2 and 22. 5. Find the sum of all the whole numbers from 1 to 100, inclusive. 6. There are four numbers in arithmetical progression whose sum is 38, and the product of the extremes is 70. Find the numbers. 7. If the nth. term of a series is 2 ^ -— 1, what is the series ? CONCRETE EXAMPLES, 279 8. If a body falls through 16yi2 feet the first second, three times as far the next second, five times as far the next, and so on, how far will it fall in half a minute t / ^^y^^' 9. A man walks 1 mile and back the first day, 2 miles and back the second, 3 miles and back the third, and so on. In how many days will he walk 72 miles ? 10. A man put out at interest II at the end of each month for 10 years. What did the interest amount to at 6^ simple interest? 11. The sum of five numbers in arithmetical progression is 15, and the sum of their squares is 55. Find the num- bers. 12. If the sum of n terms of a series is , find the series. 13. A travels 2 miles the first day, 4 the second, 6 the third, and so on. Five days later B starts out and travels uniformly 24 miles a day. In how many days will he overtake A ? 14. The product of two numbers is 28, and the product of the two arithmetical means between them is 60. Find the numbers. 16. A man increased his capital stock $500 at the end of each year for 10 years, and then had invested 16500. What was his capital at first ? Geometrical Progressions. Definitions and Principles. 345. A series in which each term after the first is de- rived from the preceding one by multiplying it by a con- stant quantity, called the ration is a geometrical progres- sion. 280 ELEMENTARY ALGEBRA. 346. If the ratio is greater than one, the series is an ascending one ; if less than one, a descending one. Thus, a, 3 a, 9«, 27 a, etc., is an ascending series ; and, 27a, da, 3a, a, etc., is a descending series. 347. If we represent the first term by a, the ratio by r, the number of terms by n, and the last term by I, the 2d term = ar 4th term = ar^ 3d term = ar^ 5th term = ar^ Here we observe that each term equals the first term multiplied by the ratio raised to a power whose exponent is one less than the number of terms ; hence, the nth. term = a r""^. But the ^tlx term is L Therefore, Pfin. 134. I = ar"~^ [Formula A.] 348. If we represent the sum of a geometrical series by S, we have, S=za + ar-\-ar^-\-ar^-\- ....- + 1 (1) Multiply (1) by r, j'S=ar-\-ar^-{-ar^-\-....l-{-lr (2) Subtract (1) from (2), (r-l)S=Ir-a, Therefore, (3) Prin. 135, 8= ^Z\ ' [Formula B.] Cor. S = r— J since lr = ar''. Examples in* Geometrical Progression. Illustrations. — 1. Find the 9th term of the series : 2, 4, 8, etc. Solution : Here a = 2, r = 3, and n = ^. Substitute these values in formula (A), Z = ar"-i =r 2 X 28 = 29 = 512. EXAMPLES IN GEOMETRICAL PROGRESSION, 281 2. Find tlie sum of 7 terms of the series : 3, 9, 27, etc. Solution : Here a = 3, r = 3, and n = 7, to find I and S, Substitute in formulas (A) and (B), 1. Z =^ ar*-i = 3 X 3« = 3' = 2187 2.5 = "-- r — .a^2187x 3-3^33^^ 3. Given a = 2, r- = 2, and I = 256, find n. Solution : Substitute these values in formula (A), l = ar*-^ .'. 256 = 2 x2«-i (1) Divide by 2, 128 = 2—1, (2) But 128 = 2', (3) .*. 2«-i = 2\ (4) or n-l = 7, (5) and n = 8 (6) 4. Given I = 320, r = 2, and n = 7, to find a and /SI Solution : Substitute these values in formulas (A) and (B), 1. l = ar^-^ 320 = ax2« = 64a and a = 5 2. ^^^r-a^3a0.2-5^g3g EXERCISE 134. 1. Find the 8th term of the series : 2, 6, 18, etc. 2. Find the 7th term of the series : 4, — 12, 36, etc. 3. Find the 8th term of the series : 162, 54, 18, etc. 4. Find the 10th term of the series : ^11 1 . 5. Find the nth term of the series : 1, 2, 4, etc. 6. Find the sum of 6 terms of the series : 3 + 12 + 48, etc. 7. Find the sum of 7 terms of the series : 1, ^, -, etc. 8. Find the sum of n terms of the series : 1, — , j, etc. 282 ELEMENTARY ALGEBRA. 9. Given a = 3, r = 2, and n = 6, find I and S 10. Given a = 3, r = 3, and ;S' = 363, find I and w 11. Given /* = ^, S= r^r^, and 7^ = 6, find I and a 12. Given a = 2 r, r = r, and n = 10, find ? and S 13. Show that a = -—r 14. Show that /• — ""^Z 'V^: 15. Show that S = r* — r*~* 16. Show that a={l-S)r + S 17. Show that ^= ^^^~^) + ^ r 18. Show that ar"" — Sr = a — 8 19. If the first term of a geometrical progression is 2, the number of terms 4, and the sum of the terms 80, what is the series ? 20. Show that the following series are in geometrical progression : 1. 7?, xy, y^ 3. -, X, y, | 2. X, ^fxy, y 4. x, xy, xy^, xf Concrete Examples involving Geometrical Progressions. Illustrations. — 1. Insert three geometrical means be- tween 3 and 764. Solution : Since there are to be three means, the number of terms in the series will be 5, the first term 3, and the last term 768. Take Z = ar«-i Substitute, 768 = 3 r* r4 = 256 r = 4 .*. The means are 13, 48, and 193. CONCRETE EXAMPLES, 283 2. Find the series whose wth term is 2". Solution : Since the nth term may be any term, Substitute » = 1, n = 2, w = 3, etc., in nth term = 2" 1st term = 2' = 2 2d term = 2« = 4 3d term = 2^ = 8 4th term =24 = 10 Etc., etc., etc. 3. Find the series the sum of n terms of which is 3" — 1. Solution : Given >S = 3» — 1, Let n = 1, then S = first term = 3 - 1 = 2 Let n = 2, then S = sum of two terms = 3* — 1 = 8 Let n = 3, then S, = sum of three terms = 3^ — 1 = 26 Let n = 4, then S = sum of four terms = 3* — 1 = 80 Since the sum of two terms is 8 and the first term is 2, the second terra is 8 - 2 = 6. The third term is 26 - 8 = 18. The fourth term is 80 - 26 = 54. .-. The series is 2, 6, 18, 54, etc 4. The sum of three numbers in geometrical progression is 63, and their product is 1728. Find the numbers. Solution : Let x^, xy^ and y' be the numbers, then x^ + xy + }/ = QZ (A) and _ 2^3^ = 1728 (B) Extract the Vb, xy = n (1) Add (1) to (A), a;« + 22:y + 2/' = 75 (2) Extract V(2), a; + y = ± 5 ^/~^ (3) Subtract 3 times (1) from (A), a;«_2a;y + 3/2 = 27 (4) Extract V(4), a; - y = ± 3 \/3 (5) Add (5) and (3), 2a; = ± 8 \/3 (6) a;=±4V3 (7) a;« = 48 Subtract (5) from (3), 2y = ± 2 \/3 (8) The numbers are 48, 12, and 3. y=±\/3 y« = 3 284 ELEMENTARY ALGEBRA, EXERCISE 133. 1. A man increases his capital stock at the end of each year by Vs of itself. If he begins with $100, what will his stock be at the end of 8 years ? 2. If a rangeman begins with 256 head of cattle, and increases his herd each year by 25^ of itself, in how many years will he have 625 cattle ? 3. A has twice as much money as B, B twice as much as 0, C twice as much as D, and D twice as much as E, and they together have 16200. How much has each ? 4. Insert two geometrical means between Yg and Yig. 5. Insert three geometrical means between 2 Yg and *^y5i2. 6. An elastic ball is thrown up 10 feet, then falls and rebounds 5 feet, then falling rebounds 2Y2 feet, and so on. How many times must it rebound to pass over 38 Ys feet ? 7. Show that the geometrical mean between a and h is Va/b. Find the geometrical mean between 2 and 8. 8. There are three numbers in geometrical progression. The product of the first two is 75, and the product of the last two is 225. Find the numbers. 9. There are four numbers in geometrical progression. The product of the first and third is 9, and the product of the second and fourth is 81. Find the numbers. 10. The sum of three numbers in geometrical progres- sion is 84, and the quotient of the third and first is 16. Find the numbers. 11. Find the series whose nih term is 2 X 3*. 12. The sum of three numbers in geometrical progres- sion is 42, and the sum of their squares is 1092. Find the numbers. 13. The three digits of a number are in geometrical progression ; the sum of the digits is 14 and their product is 64. Find the number. INFINITE SERIES. 285 14. The sum of n terms of a series is ^2 (3" — 1). Find the series. 15. A milkman drew a gallon of milk from a can con- taining 40 quarts, then put in the can a gallon of water ; he then drew off a gallon of the mixture and put in its place a gallon of water. He did this five times. What part of the contents then was water ? 16. Find the sum of n terms of the series — , x, y^ — , etc. y "^ 17. Insert three geometrical means between 3 and 243, and find their sum. Infinite Series. Definitions and Principles. 349. Any series of an unlimited number of terms is called an Infinite series. 350. When the sum of n terms of a series constantly approaches some definite value as n increases indefinitely, the series is said to be convergent. Thus, 1 + 2"^4"^8'^16"^ ^^^'' ^^ ^ convergent series, since the greater the number of terms taken, the nearer will their sum approach to 2. 351. A series that is not convergent is called divergent 352. The limit of a convergent series is the value whicli the sum of n terms continually approaches as w is in- creased indefinitely, but which it can never quite reach, though it may be made to differ from it by less than any assignable quantity. Thus, 2 is the limit of l-h^ + T + 3 + ^+ etc. /« 4 o lo 286 ELEMENTARY ALGEBRA. 353. In any geometrical progression, ar"" — a a — af a af &■ r — \ 1 — r 1 — r 1 — r Suppose r < 1, then r* approaches as a limit as n is increased indefinitely, and hence the limit of z is also ; and the limit (L) of ^S' is • ^ ' 1 — r Therefore, rHn. 136. L - — ^ (0) 1 — r ^ ' Examples involving Infinite Series. Illustrations. — 1. Find the limit of the series : 3, 1, -, -, etc. Solution: Here a = 3, and »* = -q-. Substitute these values in formula (C), "^ J « 3 1 3 , 2. Find the value of the circulating decimal -36. solution : -36 = -363636, etc. = ^ + j||g + ^^ + etc. 36 1 Here a = ^kk and r = j^. Substitute these values in (C), L = 36^ 100 36 4 1-r . _ J_ 99 11 100 3. Find the value of the circulating decimal '24. solution : .24 = -24444, etc. = ^^^(^^^^^^^ etc. ) rpu T V 4^ / 4 , 4 . 4 , \ a 100 4 The limit of (^— + _ + j^Q+ etc.j=^^^ = — ^ = ^ 10 10 "^ 90 "~ 90 "^ 90 ~ 90 ~ 45 EXAMPLES INVOLVING INFINITE SERIES. 287 4. A hound is 20 rods behind a fox, and runs 2 rods to the fox's one. Ilow far must the hound run to catch the fox ? Solution : While the hound runs the 20 rods the fox is ahead, the fox runs 10 rods ; then, while the hound runs these 10 rods, the fox runs 5 rods ; then, while the hound runs these 5 rods, the fox runs 2 Vx rods, etc. Therefore the hound runs in all the sum of 20 rd. + 10 rd. + 5 rd. + 2 g- rd. + etc. ; or, since ^ a 20 ._ , L = z = = 40 rods. 2 EXERCISE 136. 1. Find the limit of the series : 1. 2 + l+l + etc. 3. 10 + 1 + 1 + etc. 2. 9 4-3 + 1 + etc. 4. 12^ + 6^ + 3^ + etc. /* 4 o 2. Find the limit of the series : 1. a + ^ + ^ + etc. 3. a-^b + - + etc. or 3j cti cty 2. ax-{-x-\ f- etc. 4. - + -« + -^ + etc. 3. Find the value of : 1. -45 3. i728 5. -012 2. -124 4. -36 6. -012 4. A ball is thrown up 10 feet, falls and rebounds 5 feet, then falls and rebounds 2 72 feet, and so on. How far does it move before it stops ? 5. An oflScer is 100 rods behind a thief, and goes 3 rods while the thief goes 2 rods. How far must the oflBcer go to catch the thief ? 6. At what time after 4 o'clock are the hour and minute hands of a watch together ? CHAPTER VIII. Miscellaneous Examples. EXERCISE 137. 1. It x = l, y = d, Z = 6, U = 0, find the value ot x^ -^ 2 y^ -\- 3 z^ -\- 4t u- 2. If a; = 2, y = 0, z=i-l, u = l, find the value of {xy — uz){yz — ux){uy ^ xz) 3. Find the value of : 2x^-^2f-2z^ + ^xy 1 Zx'-'dy'-Zz^^Qyz '^ ^ ~ *' ^"2' ^^ 4. Add 3«^+4^+ g^. 4^""3^ + i^' ^^^ 6^-4^ + 3^ 5. Sax-{-6I?x — 2cx + 7dx — 4:ax-\-6dx-{- llcx ~ 6clx equals how many times x ? . T? 3 4,2^,1,3 4 6. From -x--y-i--z take _:z: + -^--2; 7. Simplify a— [{ — a — {a-{- a) — a] — a — (a-\- a)] 8. From o^m^ + ^mTe + cw^ take (b — c) m^ — (a — c)mn — (b — a) n^ 9. What is the value of 4:(mq — np) — {{m — n) — {p-q)V, if ??i = 0, /^ = 2, ^ = — 3, (7 = 4? 10. Multiply x^ -\-x^ y^ -\- y^ by x~i + ^~^ y~^ + 2/~^ 11. Expand (.i + i)(.i-i)(.+ l)(.^ + fg) MISCELLANEOUS EXAMPLES. 289 12. Divide x^-\- (a-\-h-\-c) xr-\- {ab-\- ac-\- bc)x-{-ahc hy x-\-b 13. Write the quotient of a}"" + b'"" divided by a^ + b- 14. Factor x^- -\- y^^ 16. Factor x^ -\- x!^ y^ -\- y^ 16. Find the H. C. D. of ax-\-ay -{-bx-\-by and cx-\-cy — dx — dy 17. Find the L. C. M. of 9 a;^ -^ 12 a: + 4 and Zx^^llx-\-(j 18. Simplify -^ ^-\-^y-{-y^ ^^-{-y^ X xy-\-y^ x' — f 19. Reduce >^-y^^2yz-z^ ; to its lowest terms. 20. Simplify ^ + ^ P-^ ^,M, ' p-q p^-q q^-p"" 21. Simplify : qr pr pq (p-q){p-r) {q-p){r-q) (r-q)(r-p) 22. Simplij 1 a « + 1 a^ — 1 '' 1 1 - fl2 i-^a 23. Solve *^ + 4,7.-3 ^^^ + ^1, 9 7 ' 2 + (ix~ 7 ' 28 i x-\-y-z = 3) 24. Solve < x — y-^z = 5> i-x-\-y-^z = 7) . [ 1 + 1-1 = 1^ 26. Solve 1-1+1=3 X y z -1+1+1=5 26. Square Vz-\- V^ — Vz 27. Cube 1 ~ V2 290 ELEMENTARY ALGEBRA. 28. Cube a-\-Vax-{-x 29. (x-^-) ^ -^ (a;^') » = ? 30. Expand (^^"^ + -^13) and express the result with- out negative exponents. 31. Multiply 2 Vis + 2 V288 - 3 V32 - VT28 by a/2 32. Simplify {x + V^^)^ {x — V—yY 33. Solve a^-\-{a-\-b-\-c)x= —i{a-^c) 34. Solve ^ , = = ^ a; — V 2 — a;2 3 35. Solve 2x^ + dV2x^-\-3x = 18-3x 36. Solve i a o o o 2 00 r 37. Extract the square root of : x — 2x^y^-\-3x^y^ — 2x^i/^-\-y 38. Extract the cube root of : a^-dx^+6x^-7x-{-6x^-3x^-\-l 39. Divide x!^ -{- a^ y^ -\- y^ by x-{- Vxy + y 40. Simplify : 1 1 . x(x—y){x-z) y{y-x){z-y) z{x-z){y-z) 41. Find the value of 7? — xy -\-y^ when a — l) T a-\-'b X = — —7 and V = 7 a-{-b ^ a — h 42. Multiply ma^-{-nx —p by ax — b, and inclose the coefficients of the different powers of x in parentheses. 43. Factor ^-\ and a^ ^3aH - 4.al)^ -121)^ 44. Put y for a; + - in the following expressions, and X simplify: ^' + -3; ^' + ^5 ^ + ^ MISCELLANEOUS EXAMPLES. 291 45. Factor 32 3^-]-z^ and x^ + xt/-\-y^ 46. Raise 1 — V— 3 to the fourth power. 47. Place the monomial factors of the following expres- sions within the parentheses : 48. Simplify 1 - [- 1 - {- 1 - (- 1) - Ij - 1] - 1 49. Simplify [](-2)-2|-2]~' 50. Solve x-^-\ : = x-^ 2 x-^ x~^ 61. Solve x = y and y = x X ^ ^ y ^ , a: + 2 x — % rr + 4 x — 4: 52. Solve — hs -o = — —^ n x-\-Z x — d x^5 x-^6 53. Develop _ into a series by division. By the law of the series, what will the 10th term be ? What the wth term ? 64. Simplify and clear of negative exponents : ^l!jfr:!. ^iHjr^. i^-y^)-^ «"* — y~*' x-^-\-y-^' {x -yy 56. Multiply 2 a/SO - Vis + Vll by a/S + V^ - V^ 56. Rationalize the denominators of ; 57. Solve -7=-' — = —7^ — Va:-|-6 Va; + 9 68. Find the H. C. D. of 2ax-{-%hx-\-l a + lh and 2cx — 2dx-\-lc — ld 69. From my^-\-nyz-\-rz^ take (;i — r)y^ — {m — r)yz — {n — m) z^ 60. Find the value of a; in a;:a — 1::1 — Va : 1 + Va 292 ELEMENTARY ALGEBRA, 61. Find the value of ^ X — y when X = V2 and y = — a/2 62. Find the yalue of : 1 -\- X -\- x^ -{- x^ -\- etc., ad infinitum, when x = - 63. Find the equation whose roots are 1 + a/^ and 1 - a/^ 64. Find the equation whose roots are a, l, c, and d, 65. Solve ?>ax^ -\-%'bx^^c 66. Solve ic — ^ x = a-\- Va 67. Solve J- + ^+V. + ^= ^0 ^'r+ ^^ =3660) 68. If x = y^-{-y-{- 1, what is the value of x^-\-x-{-l? 69. Substitute ay^ — hy for ic in x^ -{- x y -\r y^ , and bracket the coefficients of the like powers of x. 70. Solve / „ = 7 by proportion. 71. Solve —J- T= = a— ^Ta V X— V a { x^-{- xy -^y^ = 21 ) 72. Solve ] ; .\ ; *^ ^ [ i x -i-y =35) ^ . {x4-xy4-xy^ = 26) 74. Solve -^ ' ^3^ ^ ^^ ^ i X — xy^ = — 62 ) 75. Solve j x^ -\- y^ -\- X y = 4t9 49 I = 31 ( \xy-\-2x-\-2y 76. If ^i is integral, what kind of number, odd or even, is represented by 2^ + 1? 2^^ — 1? 2n? 77. Find V + 1, V — 1, V+l, vl6 78.. Solve "'"V^ = - 1, V^ = - 1 MISCELLANEOUS EXAMPLES, 293 EXERCISE 138. 1. John and James together had $6800. John spent Vs of his money, and James Yi of his, and each had the same sum remaining. How much had each at first ? 2. John is V3 as old as his father, but in 20 years he will be Yi3 as old. How old is each ? 3. Divide 100 into two such parts that the quotient of the smaller part divided by the difference between the parts may be 12. 4. The sum of the two digits of a number is 10, and if 36 be added to the number the order of the digits will be reversed. Find the number. 5. Find a fraction such that if 2 be added to the numerator the fraction equals 1, but if 5 be added to the denominator the fraction equals Yg. 6. If a rectangle had its width increased by 4 feet and its length diminished by 8 feet, it would become a square inclosing an equal area. Find the dimensions of the rectangle. 7. A can do a piece of work in 2 hours 45 minutes, and B can do it in 3 hours 15 minutes. In what time can they do it working together ? 8. A man was engaged to work for 40 days on condition that for every day he worked he was to receive $3, and for every day he was idle to forfeit $1Y2- At the end of the time he received $75. How many days was he idle ? 9. At an election there were three candidates for sheriff. The whole number of votes polled was 5325. B received 662 votes more than C, and A's majority over B and C was 1 vote. How many votes had each ? 10. One man rides a mile on a bicycle in 5 Ye minutes ; another, a mile in 6^3 minutes. If they start at the same time from two towns 18 miles apart and approach each other, in what time will they meet ? 294 ELEMENTARY ALGEBRA. 11. A man can row 4 miles an hour in still water, and 11 miles down a river and back again in 6% hours. What is the velocity of the current ? 12. At what time between 5 and 6 o'clock are the hour and minute hands of a watch together ? At right angles ? Opposite each other ? 13. Find two numbers in the ratio of 2 to 3, such that if each be diminished by 12, they shall be in the ratio of 1 to 2. 14. A boat steaming Yg a mile an hour above its ordi- nary rate gains 17 Y7 minutes in going 60 miles. What is its usual rate ? 15. Six silver and 4 gold pieces are worth as much as 16 silver and 2 gold pieces, and 10 of each are together worth 130. What is the value of a gold piece ? What of a silver piece ? 16. Two numbers are in the ratio of p to q, and the sum of their squares is jt?* — q^. What are the numbers ? 17. Show that any square exceeds a rectangle of equal perimeter by the square of Y2 the difference of the length and breadth of the rectangle. 18. A man bought 12 apples. Had he bought 3 less for the same sum, they would have cost him 1 cent apiece more. What did he pay apiece ? 19. A and B together have 110 sheep, A and together 100 sheep, and B and together 90 sheep. How many has each ? 20. Twelve men agreed to do a piece of work in a given time, but 4 men did not report for work, in consequence of which the time had to be extended 4 days. What was the time agreed upon ? 21. Two trains pass a station at an interval of 3 hours, moving respectively at the rate of 20 and 32 miles an hour. In what time will the fast train overtake the slow train ? MISCELLANEOUS EXAMPLES, 295 22. What principal will in 8 years at 6^ produce as much interest as 1800 in 9 years at 8^ ? 23. The population of a Western city increases annually 10^ ; it is now 29,282. What was it 4 years ago ? 24. The sum of the fourth powers of three consecutive numbers is 962. Find the numbers. 25. A number of two digits is to the number formed by interchanging the digits as 4 to 7, and the difference of the two numbers is 27. What is the number ? 26. A man saved $1026 in 3 years. How much were his annual savings if they increased in geometrical progres- sion, and he saved $18 the first year ? 27. The sum of 7 numbers in arithmetical progression is 56, and the sum of their squares is 560. Required the numbers. 28. The volumes of two stones are to each other as 3 to 4, and the weights of equal volumes as iVg to 1. What is the weight of each if their united weight is 340 pounds ? 29. The diagonal of a rectangle is 10, but, if the length be increased by 4 and the width by 3, the diagonal will be 15. Required the length and width of the rectangle. 30. A man has three horses. The value of the first is 160, the value of the second equals the value of the first and Vs the value of the third, and the value of the third equals the value of the first two. Required the value of the three together. 31. A man owns $15,000 worth of stock, part of which is 5 j^ and part 6 ^ stocks ; his annual income is $830. How much of each kind has he ? 32. I sold a horse for as many per cent above $150 as I lost per cent on the cost, and lost $45. What was the cost and loss per cent ? 296 ELEMENT AR Y ALGEBRA. 33. A and B can do a piece of work in 5 days, working 10 hours a day ; A and in 6 days, working 9 hours a day ; and B and in 8 days, working 8 hours a day. In what time can each alone do it, working 8 hours a day ? 34. A is 50 yards in advance of B, and goes 1 yard the first minute, 3 yards the second, 5 yards the third, and so on ; B goes uniformly 15 yards a minute. In how many minutes will he overtake A ? 35. A passenger-train, after running one hour, was par- tially disabled, and could run only at Vg of its usual rate, which caused it to be one hour late at its destination. Had the accident occurred 15 miles farther on, the train would have been only 52 Yg minutes late. What was the usual rate of the train ? 36. The sum of the two means of a geometrical progres- sion of four terms is 36, and the sum of the extremes is 84. Find the series. 37. The product of five numbers in arithmetical progres- sion is 23,040, and their sum is 40. Find them. 38. The sum, the product, and the difference of the squares of two numbers are equal. Find the numbers. 39. A is a feet behind B, and goes m feet in t seconds, while B goes n feet in p seconds. In how many seconds will A overtake B ? 40. If from a number whose four figures are in arith- metical progression, be subtracted the number formed by reversing the order of the digits, the remainder will be 6174 ; the sum of the digits is 24. Required the number. 41. Two numbers are in the ratio of 2 to 3, and if 2 be added to each of them, they will be in the ratio of 3 to 4. Find them. 42. A's age 2 years hence is to his age 3 years ago as 9 times his age 3 years ago is to 4 times his age 2 years hence. Required his age. MISCELLANEOUS EXAMPLES. 297 43. The capacities of two cubical cisterns are to each other as 1 to 8 ; but if 2 feet were added to each of their dimensions, their capacities would be to each other as 27 to 125. Required the capacity of each. 44. The sum of the squares of three numbers is 29, the sum of the products of them taken two together is 26, and the first is 5 less than the sum of the other two. Find the numbers. 45. A general drew up his army in the form of a square and found he had 615 men over ; he then increased the side of the square by 5 men, and lacked 60 men to com- plete the square. How many men were in the army ? 46. A man has a farm of 150 acres, in the form of a rectangle, whose length is to its breadth as 5 to 3. A road of uniform width, containing 3^Y4o acres, surrounds the farm, and is a part of it. How wide is the road ? 47. The fore-wheel of a carriage makes 88 revolutions more in going a mile than the hind-wheel, but if the cir- cumference of i\i6 fore-wheel were diminished 2 feet the fore-wheel would make 220 revolutions more than the hind-wheel. Required the circumference of each wheel. 48. A merchant gains annually 20 per cent of his capi- tal ; of this he spends $1000, and adds the balance to his capital for the next year ; at the end of 4 years his stock is $25,736. What was his original stock ? 49. A man starts at the foot of a mountain to walk to its top. During the first half of the distance he walks Yg a mile an hour faster than during the last half, and he reaches the top in 4 hours 24 minutes. Returning, he walks Y2 a mile an hour faster than during the latter half of his ascent, and completes the descent in 4 hours. Find the distance to the top of the mountain. 50. A lump of gold 22 carats fine contains 36 ounces of alloy. How many ounces of alloy in a lump of the same weight only 16 carats fine ? 298 ELEMENTARY ALOEBRA. 51. In a mile walk, A gives B a start of 1 minute and overtakes him at the mile-post. In a second trial, A gives B a start of 60 yards, and beats him 10 seconds. At the rate of how many miles an hour does each walk ? 62. If the cost of an article had been 8 ^ less, the gain would have been 10 ^ more. Find the gain per cent. 53. A railway-train, after traveling for 1 hour, has an accident which delays it 60 minutes, after which it pro- ceeds at % of its former speed, and arrives at its desti- nation 3 hours behind time. Now, had the accident occurred 50 miles farther on, the train would have arrived 1 % hour sooner. What is the length of the line ? 54. Two men, A and B, engaged to work for a certain number of days at different rates. At the end of the time, A, who had been idle 4 days, received 75 shillings ; but B, who had been idle 7 days, received only 48 shillings. Now, had B been idle only 4 days, and A 7 days, they would have received the same sum. For how many days were they engaged ? 55. Three pipes. A, B, and 0, can fill a cistern in one hour. B delivers twice as much water per minute as A. alone will fill it in one hour less than B alone. How long will it take each to fill it ? General Definitions. 1. Quantity is anything that may be increased, dimin- ished, and measured. 2. Quantity is estimated by assuming some definite por- tion of it as a standard of measure, and finding how many times it contains this standard. 3. Any definite portion of quantity assumed as a stand- ard of measure is a unit, 4. Number is that which denotes how many units a quantity contains. GENERAL DEFINITIONS, 299 6. A quantity that contains a definite number of units is a specific quantity ; as, five pounds. 6. A quantity that may contain any number of units is a general quantity ; as, a fiock. 7. The number of units in a specific quantity is ex- pressed by one or more of the figures of arithmetic. 8. The number of units in a general quantity is ex- pressed by one or more of the letters of the alphabet, or by both figures and letters. 9. By a figure of speech, the representation of the num- ber of units in a quantity, by figures or letters, is also called a quantity. 10. When the number of units in a quantity is denoted by figures, the expression is called a numerical quantity. 11. When the number of units in a quantity is repre- sented wholly or partially by letters, the expression is called a literal quantity. 12. Quantities which are opposed to each other in char- acter — that is, which tend to destroy each other when combined — are positive and negative quantities. 13. Of two opposite quantities, it does not matter which is considered positive and which negative, if consistency is maintained throughout the operation or investigation into which they enter. 14. The number of units in a positive quantity is char- acterized by placing before it the symbol + (p^^s) ; and the number of units in a negative quantity, by placing before it the symbol — (minus). This peculiar notation gives rise to symbolized numbers. 16. Arithmetic is the science of numbers, irrespective of their character as positive or negative. Arithmetic based on the literal notation is Literal Arithmetic. 16. Algebra is the science of symbolized numbers as the representatives of positive and negative quantities. 300 ELEMENTARY ALGEBRA. Principles. 1. The algebraic sura of two or more similar terms with like signs equals their arithmetical sum with the same sign (page 16). 2. The algebraic sum of two similar terras with unlike signs equals their arithmetical difference with the sign of the greater (page 16). 3. The algebraic sum of two or more dissimilar terms equals a polynomial composed of those terms (page 17). 4. The algebraic difference of two quantities equals the algebraic sum obtained by adding to the minuend the sub- trahend with the sign changed (page 23). 5. The product of two quantities with like signs is positive (page 27). 6. The product of two quantities with unlike signs is negative (page 27). 7. The exponent of a factor in the product equals the sura of its exponents in the multiplicand and multiplier (page 28). 8. Multiplying one factor of a quantity multiplies the quantity (page 28). 9. Multiplying every term of a quantity multiplies the quantity (page 31). 10. The quotient of two quantities with like signs is positive (page 33). 11. The quotient of two quantities with unlike signs is negative (page 33). 12. The exponent of a factor in the quotient equals the difference of the exponents of the factor in the dividend and divisor (page 34). 13. Any quantity with an exponent of zero equals unity (page 34). 14. Dividing one factor of a quantity divides the quan- tity (page 35). PRINCIPLES. 301 15. Dividing every term of a quantity divides the quan- tity (page 37). 16. If the same quantity or equal quantities be added to equal quantities, the results will be equal (page 39). 17. If the same or equal quantities be subtracted from equal quantities, the results will be equal (page 39). 18. If equal quantities be multiplied by the same or equal quantities, the results will be equal (page 39). 19. If equal quantities be divided by the same quantity or equal quantities, the results will be equal (page 40). 20. A term may be taken from one member of an equa- tion to the other, if its sign be changed (page 40). 21. If both members of a fractional equation be multi- plied by a common denominator of its terms, it will be cleared of fractions (page 40). 22. If the sign of every term of an equation be changed, the members will still be equal (page 41). 23. If a number of terms are inclosed by a parenthesis preceded by plus, the symbol and the sign before it may be removed without altering the value of the expression (page 50). 24. If a number of terms are inclosed by a parenthesis preceded by minus, the symbol and the sign before it may be removed, if the sign of every term inclosed be changed (page 51). 25. Any number of terms may be inclosed by a paren- thesis and preceded by plus, without changing the value of the expression (page 51). 26. Any number of terms may be inclosed by a paren- thesis and preceded by minus, if the sign of every term inclosed be changed (page 51). 27. An even power of a positive or a negative quantity is positive (page 62). 28. An odd power of a quantity has the same sign as the quantity (page 63). 14 302 ELEMENTARY ALGEBRA. 29. Multiplying the exponent of a factor by the ex- ponent of a power raises the factor to that power (page 63). 30. Raising every factor of a quantity to a given power raises the quantity to that power (page 63). 31. The square of the sum of two quantities equals the square of the first, plus twice their product, plus the square of the second (page 65). 32. The square of the difference of two quantities equals the square of the first, minus twice their product, plus the square of the second (page 65). 33. The cube of the sum of two quantities equals the cube of the first, plus three times the square of the first into the second, plus three times the first into the square of the second, plus the cube of the second (page 66). 34. The cube of the difference of two quantities equals the cube of the first, minus three times the square of the first into the second, plus three times the first into the square of the second, minus the cube of the second (page 35. The product of any even number of factors with like signs is positive (page 67). 36. The product of any odd number of factors with like signs has the same sign as the factors (page 67). 37. If the signs of an even number of factors be changed, the sign of their product will remain unchanged (page 67). 38. If the signs of an odd number of factors be changed, the sign of their product will be changed (page 67). 39. The product of the sum and difference of two quan- tities equals the square of the first minus the square of the second (page 69). 40. The product of two binomials having a common term equals the square of the common term, and the alge- braic sum of the unlike terms times the common term, and the algebraic product of the unlike terms (page 70). 41. The product of any two binomials equals the prod- PRINCIPLES. 303 net of the first terms, and the algebraic sum of the prod- ucts obtained by a cross-multiplication of the first and second terms, and the algebraic product of the second terms (page 71). 42. The difference of the equal even poTvers of two quan- tities is divisible by both the sum and the difference of the quantities (page 73). 43. The sum of the equal odd powers of two quantities is divisible by the sum of the quantities (page 74). 44. The difference of the equal odd powers of two quan- tities is divisible by the difference of the quantities (page 75). 45. The laws of the quotient in exact division (page 76). 46. A divisor of a quantity is one of the two factors of the quantity, and the quotient is the other (page 78). 47. A factor of every term of a quantity is a factor of the quantity (page 78). 48. The highest common divisor is the product of all the common prime factors (page 87). 49. The lowest common multiple of two or more quan- tities equals the product of all their different prime factors, each taken the greatest number of times it occurs in any one of them (page 90). 50. Dividing one quantity and multiplying another by the same factor does not alter their product (page 93). 51. Multiplying the dividend or dividing the divisor multiplies the quotient (page 93). 52. Dividing the dividend or multiplying the divisor divides the quotient (page 94). 53. Multiplying or dividing both dividend and divisor by the same quantity does not alter the quotient (page 94). 54. Multiplying the numerator or dividing the denom- inator multiplies the value of a fraction (page 101). 55. Dividing the numerator or multiplying the denom- inator divides the value of a fraction (page 102). 304 ELEMENTARY ALGEBRA. 56. Multiplying both terms of a fraction by the same quantity does not alter its value (page 103). 67. Dividing both terms of a fraction by the same quan- tity does not alter its value (page 103). 68. Changing the signs of both terms of a fraction does not alter its value (page 104). 69. Changing the apparent sign and the sign of either term of a fraction does not change the value of the fraction (page 104). 60. Any common multiple of the denominators of two or more fractions is a common denominator of the fractions (page 108). 61. The lowest common multiple of the denominators of two or more fractions in their lowest terms is the lowest common denominator (page 108). 62. The sum of two or more similar fractions equals the sum of their numerators divided by their common denom- inator (page 110). 63. The difference of two similar fractions equals the difference of their numerators divided by their common denominator (page 110). 64. The product of two fractions equals the product of their numerators divided by the product of their denom- inators (page 115). 65 Canceling a factor common to the numerator of one fraction and the denominator of another does not alter the product of the fractions (page 115). 66. The quotient of two fractions equals the dividend multiplied by the inverse of the divisor (page 116). 67. Canceling a factor common to the numerators or the denominators of two fractions does not alter their quo- tient (page 117). 68. Eaising both terms of a fraction to any power raises the fraction to that power (page 120). 69. Principles of transformation of equations (page 125). PRINCIPLES. 305 70. Any term of an equation may be transposed from one member to the other if its sign be changed (page 125). 71. An equation with fractional terms may be cleared of fractions by multiplying both members by a common multiple of the denominators of the fractions (page 126). 72. The binomial theorem (page 161). 73. The square of a polynomial equals the sum of the squares of its terms, and twice the product of each term into all the following terms (page 163). 74. The cube of any trinomial equals the sum of the cubes of its terms, and three times the square of each term into all the other terms, and six times tlie product of the three terms (page 164). 76. Dividing the exponent of any factor by the index of a root takes that root of the factor (page 165). 76. Taking a root of every factor of a quantity takes the root of the quantity (page 165). 77. Any even root of a positive quantity may be either positive or negative (page 166). 78. Any odd root of a quantity has the same sign as the quantity (page 166). 79. An even root of a negative quantity is impossible (page 166). 80. Extracting a root of both terms of a fraction ex- tracts the root of the fraction (page 167). 81. If a number be pointed off into terms of two figures each, beginning at the units, the unit of each term will be a perfect square (page 172). 82. If a number be pointed off into terms of three fig- ures each, beginning at the units, the unit of each term will be a perfect cube (page 177). 83. Every pure quadratic equation of one unknown quantity may be reduced to the form oi as? = by in which a and b are integral and a positive (page 182). 84. Every pure quadratic equation of one unknown 306 ELEMENTARY ALGEBRA. quantity has two roots, numerically equal, but opposed in sign (page 183). 85. Every complete quadratic equation of one unknown quantity may be reduced to the form of aoi? -{-hx = c, in which tty hy and c are integral, and a positive (page 185). 86. Every complete quadratic equation of one unknown quantity may be reduced to the form of x^ -\-px = q, in which p and q may be integral or fractional, positive or negative (page 185). 87. The sum of the two roots of an equation of the form of x^-\-px:=q equals the coefficient of x, with the sign changed (page 194). 88. The product of the two roots of an equation of the form of x^-\-px = q equals the absolute term with the sign changed (page 194). 89. Multiplying or dividing both terms of a fractional exponent by the same quantity does not change its value (page 215). 90. The exponent of a factor in the product equals the sum of the exponents of the same factor in the multipli- cand and multiplier, when the exponents are positive frac- tions (page 215). 91. The exponent of a factor in the quotient equals the exponent of the same factor in the dividend, minus the exponent of that factor in the divisor, when the exponents are positive fractions (page 215). 92. A quantity affected by a negative exponent equals the reciprocal of the quantity affected by a numerically equal positive exponent (page 216). 93. A quantity affected by a positive exponent equals the reciprocal of the quantity affected by a numerically equal negative exponent (page 216). 94. A factor may be transferred from either term of a fraction to the other if the sign of its exponent be changed (page 217). PRINCIPLES, 307 95. The exponent of a factor in the product equals the sum of the exponents of the same factor in the multipli- cand and the multiplier when the exponents are negative (page 217). 96. The exponents of a factor in the quotient equals the exponent of the same factor in the dividend, minus the exponent of that factor in the divisor, when the exponents are negative (page 218). 97. fl' X a' = «*"^*' for any positive or negative, integral or fractional, values of a; and y (page 219). 98. a' -^ rt" = a'"" for any positive or negative, integral or fractional, values of x and y (page 219). 99. («')" = «*' for any positive or negative, integral or fractional, values of x and y (page 220). 100. {a b)' and a' X If are equivalent for any positive or negative, integral or fractional, values of x (page 220). 101. (x) and t; are equivalent for any positive or negative, integral or fractional, values of x (page 220). 102. Any root of the product of two quantities equals the product of the like roots of those quantities (page 223). 103. The product of the equal roots of two quantities equals the like root of their product (page 223). 104. Any root of the quotient of two quantities equals the quotient of the like roots of those quantities (page 223). 105. The quotient of the equal roots of two quantities equals the like root of their quotient (page 223). 106. No fractional radical is pure (page 223). 107. Any quantity equals the wth root of the nth power of the quantity (page 223). 108. The Wa and the "Va are equivalent (page 223). 109. Every imaginary quantity of the second degree may be reduced to the form of ± a; a/— 1, in which x may be rational or irrational (page 234). 308 ELEMENTARY ALGEBRA. 110. Every binomial surd^of the^ second degree may be reduced to the form of V a ± Vh, in which one of the terms may be rational (page 236). 111. A binomial surd may be a perfect square, and, when it is the square of a binomial surd of the second degree, one of the terms is rational (page 236). 112. A binomial surd with a rational term, and the coefficient of the irrational term reduced to ± 2, is a per- fect square when the quantity under the radical sign is composed of two factors whose sum equals the rational term ; and its square root equals the sum or difference of the square roots of these factors (page 236). 113. When a binomial surd is a perfect square, the dif- ference of the squares of its terms is a perfect square, and is equal to the square of the difference of the two factors described in Prin. 112 (page 237). 114. Principles of transformation of inequalities (page 249). 115. The sum of the squares of two unequal quantities is greater than twice their product (page 249). 116. The sum of the squares of three unequal quantities is greater than the sum of their products taken two and two (page 249). 117. The ratio equals the antecedent divided by the consequent (page 252). 118. The antecedent equals the ratio times the conse- quent (page 252). 119. The consequent equals the antecedent divided by the ratio (page 252). 120. Multiplying the antecedent or dividing the con- sequent multiplies the ratio (page 252). 121. Dividing the antecedent or multiplying the con- sequent divides the ratio (page 252). 122. Multiplying or dividing both terms of a ratio by the same quantity does not alter its value (page 252). PRINCIPLES. 309 123. An infinitesimal divided by a finite constant is an infinitesimal (page 2G8). 124. An infinitesimal multiplied by a finite constant is an infinitesimal (page 2G9). 126. An infinitesimal divided by an infinitesimal may be any finite constant (page 269). 126. An infinite divided by a finite constant is an in- finite (page 269). 127. An infinite multiplied by a finite constant is an infinite (page 269). 128. An infinite divided by an infinite may be any finite constant (page 269). 129. A finite constant divided by an infinitesimal is an infinite (page 269). 130. The product of an infinitesimal and an infinite may be any finite constant (page 269). 131. A finite constant divided by an infinite is an infini- tesimal (page 270). 132. In an arithmetical progression, l = a-^{n-l)d (page 273). 133. In an arithmetical progression, >S'=(^ + «)| (page 274). 134. In a geometrical progression, lz=iar*-^ (page 280). 135. In a geometrical progression, 136. In an infinite series, S = z (page 286). APPEIfDIX. Highest Common Divisor by Successive Division. Definitions and Principles. 1. If one quantity be divided by another, then the divisor by the remainder, then the next divisor by the next remainder, and so on, until the division terminates, the process is called Successive Division, 2. Since a is a divisor ot ax and also of nax, it fol- lows that, Trin, 1, — Any divisor of a quantity is also a divisor of any number of times the quantity, 3. Since « is a common divisor of at and aCy and also a divisor of ai ± ac, it follows that, JPrin, 2. — A common divisor of two quantities is also a divisor of their sum and of their difference, 4. Theorem, — The last divisor obtained by the succes- sive division of two quantities is their highest common divisor. Demonstration : Let A and B A) B (q represent any two quantities. Di- ^ ^ vide B by J., and let the quotient — —. . , be q and the remainder R ; divide ^ ) ^ \Q A by E, and let the quotient be q' ^^ Q and the remainder B' ; divide i^ by R' ) R { q" R', and let the quotient be q" and R' a" the remainder zero. Prove R' the — pp" H. C. D. of A and B. " APPENDIX. 311 1. R is a divisor of R, since the division has terminated ; hence it is also a divisor of R q' [P. 1] and of R' + R q', or A [P. 2] ; and, therefore, ot Aq [P. 1], and of i2 + ^ ^, or ^ [P. 2] ; therefore, R' is a common divisor of A and B. 2. There can be no higher common divisor of A and B than R' ; for if there could be it would be a divisor of A and B, and hence, too, ot Aq [P. 1], and ot B — Aq, or R [P. 2], and, therefore, of R q' [P. 1], and ot A — Rq\ or R [P. 2] ; that is, a quantity of a higher degree than R would be a divisor of R', which is impossible. There- fore, R is the H. C. D. of A and B. 6. Since the highest common divisor equals the product of all the common factors, it follows that, PHn. 3. — Either of two quantities may he multiplied or divided by a factor not found in the other without changing their highest common divisor. Problem. To find the highest cominoii divisor by successive division. niustratioiis.— 1. Find the H. C. D. of : %3^-^a?-14.x-{-Z and ^x'' -14:3? - ^x-^% Foniit 2a:* --9a;3- 14a; + 3)6ar*- 28 a;3_ 13^^4(3 6^*j-2?V--42^-l-9 -l )-x^-\-%^x-6 a;»-24a; + 5)2ic*- ^a? -l^x-\-Z{'Zx-^ - 9a?-\-AS3?- 24a;+ 3 - 9a? +216 a; -45 48 )48 3:^-240 a: + 48 a?- 5x-^ 1 H. 0. D. Q^-bx-\-l)a?-24:X +5(2;-h5 3? — 53?-\-X 5ic2_25a; + 5 6a?-'25x-\-6 312 ELEMENTARY ALGEBRA. Solution : Taking the second polynomial for the dividend, we ob- serve that the first term of it is not divisible by the first term of the divisor; we therefore multiply the dividend by 2 [P. 3], and then divide and obtain for the first remainder — a;^ + 24 a; — 5. We now divide the remainder by — 1 [P. 3], and divide the divisor by the result, obtaining for the next remainder 48 x^ — 240 x + 48. We reject the factor 48 from this remainder [P. 3J and divide the previous divisor by the result, and obtain no remainder. The last divisor, x^ — ^x + \, is the H. C. D. (theorem). 6. If one polynomial can be factored, its factors may be made ayailable in factoring the other by trial. The first term of a factor is always a divisor of the first term of the polynomial, and the last term of a factor a divisor of the last term of the polynomial. 2. Find the H. 0. D. of : ^4_3^_28 and x'>-'2:^-\-1 x^-lO^-^l'^x-^ Solution: The factors of x^ — ^x^ — 2S are x^ — l and a;2 + 4; x^ — 1 is not a factor of x^ — 'Z a^ + 11 oi^ — 10 x^ ■\- V^ x — %, since 7 is not a divisor of 8 ; if the two polynomials have a common divisor, it must therefore be x'^ + 4. By trial we find that a;^ + 4 is a divisor of the second, and is therefore the H. C. D. of the two. 7. Since each remainder is a number of times the H. 0. D., it is sometimes more convenient to use the re- mainders, or a remainder and one of the quantities, than to use the polynomials themselves in the progress of the work. 3. Find the H. C. D. of :^-Qx^-x-\-m and Form. ^_6a;2-a: + 30):z;3 + 9a:3 + 26iz; + 24(l a^-Qx^- g; + 30 3)152;^ + 27a;- 6 ^7?-^ ^x- 2 = {5x-l){x-\- 2) H. C. D. = a: + 2 APPENDIX. 313 EXERCISE 1. Find the H. C. D. of : 1. a:^ + 3 7?^ 3a; + 2anda:3_^2_^__2 2. 7? — b'jr-\-Zx-\-4i and 7:^-{-bx^—'7x — Q 3. 2rr*-a;3 + 2a;2_j_^_l and %a^ -^7? -\-bx -% lbo^-^OQ^-4.7^-\-Qx-^ 5. 2 a;* - 3 a:^ + 4^:2 + 3 a; -6 and 2a;*-3a:3_|_8a:2_3^_|_6 6. 6a* + 7a'4-7a2 + 3a + l and 14a* + a3-j_8a2_^_^2 7. a:24-8a; + 15 and a:^ - 3a;2- 10a; + 24 8. 6a;- + 5a;-6 and 8 a:^ - 22 a;^ __ 21 a; + 45 9. 7?-^^x'-\-^x-^l and3a:3-a;2_ii^_.j' 10. 7^-Qx^-\-Qx'-Zx-10 and 3 a:* - 13 a;3 - 11 a;2 4- 8 a; - 15 11. 42^3 + 8y2 + 8?^4-4 and 7«/3-14/ + 21 12. a^2 + ^>»2 and a^^ _^ «» ^s + 5^6 13. a^-f 2a^ + J2_^2 ^ud a2-«J_2^>2_^4«c + Jc + 3c2 14. a:* + a:«/ + y*and xf -\-^7?y -^-^^x"" y"" -\-dxy^ -\-f 15. 3«^ + 201fl2_|_i98and Sa^ + lOa^ + lOa^-j-iOa + S 16. Qax^ — ^axy-l%ay^2iTidiQhx^ — lQhxy-\-^hy^ 8. T^e highest common divisor of three quantities may he obtained hy finding the highest common divisor of two of them, and then the highest common divisor of that and the third quantity. Demonstration. — Let x, y, and z be three quantities, and m the H. C. I), of X and y, and n the H. C. D. of m and z ; then will m be the product of the factors common to x and y, and n the product of the factors common to m and z, or n will be the product of the factors common to x, y, and ^, which is their H. C. D. 314 'ELEMENTARY ALGEBRA. EXERCISE 2. Find the H. C. D. of : 1. x^ — xy^, ^y-\-y^i and x^y — xy^ 2. x^ -\-xy -\-y^, ^ -\-x^y^ -\- y^, and x^ -\- xf^ y^ -^ y^ 3. 2 a:2 + 3 a; + 1, 2 2;2 + 5 2: + 2, and 2ic3_j_5a;2_4^_3 4. 3ic3-17a; + 10, ^t^ -^x"" -Zx^^, and 3:z:*-2a:3 4-3a;-2 5. 2a:3 + 7a:2 + 8aJ + 3, 2ar^ - a;^ - 4rz; + 3, and 2:c5 + 3ic* + 2a;3 +3a;2 + 2a; + 3 6. 4a;3 + 4a;2-36rr-36, 4rc^ - 4a;2 - 36a; + 36, and 2a;3 + 6a:2 — 2a;— 6 7. ic* + a;3 — 8ar — 9a; — 9, a;5 + 3a;* + a:-^ + 3a;2_^^^3^ and a:5 + 22;* + a.'3 + 2a;2 + a: + 2 8. 12a;3-2x2-3a; + 2, ISa;^ - 9a;2 - 8a; + 4, and 36 ic* - 25 a;2 + 4 Lowest Common Multiple of Quantities not readily factored. 9. To find the lowest common multiple of quantities not readily factored. Theorem, — The lowest common multiple of two quan- tities equals their product divided hy their highest common divisor. Demonstration.— Let c equal the H. C. D. of A and B. Suppose — = a:, and — = y ; then A = c X X, and B = c y. y. Now X and y are prime to each other, since c is the product of all the common factors. T n Tit A . B A y. B .'. Li. ij. M. =:cxxxy = Axy = Ax— = APPENDIX. 315 lUnstration.— Find the L. C. M. of 6 ar^ -f 13 a; + 6 and 6a:3_|.9a^_|_3^_j_12. Solution : We find the H. C. D. to be 2 a; + 3. Therefore, the 3a;» + 4 l^ ^ ^l _ (ga;*^1-4^^3^a^(6a:* + 9a;* + 8a; + 12) _ ' ^ ' ~ 2^ir-K3 (3rK« + 4)(6a:» + Oa;' + 8a; + 12). EXERCISE 3. Find the L. C. M. of : 1. 6a:2_^i4a._j_8 a^^ 8a:2_|_6^_20 2. a;3 + 6ar^4-6a: + 5 and ^7? -\-^x^ -^^x-^1 3. o^ - 2 a - 1 and fl3 + 2 «2 4- 2 « + 1 4. a*H-2a2-^.9 and 1 a^ -11 aJ" -\-l^ a -^^ 5. 2 a^ — a:* y + ^ y^ + 3/^ a^d 2 o^ -{-^ x^ y -{-^ x if -\- tf' 6. a:^ + 2a:^y4-2a;^^ + y^ and a;3 + 3:r2|^ + 3.r/ + 2?/=* 7. 3a;3 4-5a:2 + 3a; + l and 3a:* + 2a:3 + 4a;2 4- 2a;+ 1 8. 2a:3_^^a;2_j_^2^_^3 and 3a;3 + 4aa:2 _^ 4^2^^^3 9. Za? — l^a?-\-14.x-Q and 6a;3 + a;2 _ g.^^ 6 10. 4««-4a*-29a2_21 and 4 a'^ + 24 a* + 41 «2 _|_ 2I 11. 20;z* + ;z2_i ^nd 25;2* + 5^^ - ;2 - 1 12. 3a:* + 5a;3^52.2_^5a._^2 and 3 re* - a;^ + a:^ __ ^ _ 2 13. 6a:* + 17a:3-10a; + 8 and 9a;* + 18a:3_^2_9^^4 14. 2a-^-4a2_ 13^-7 and Ga^ - 11 a^ _ 37a - 20 15. 6m3+15m2-67?i4-9 and 9^^+ Gm^- 51 w + 36 16. ?i* — an^ — a^n^ — a^n — '^a*' and 37i3-7a?i2-|-3a2 7i-2a3 ANSWEES Exercise 1. 1. 9 units, 9 tens, 9 fives; 9 times the number. 2. 9 times a, 13 times a 3. 15 a, 19 a 4. 9 J, 155 5. 27, 36 6. 10 m, 20, 50 7. 13^, 39, 78 8. 10^, 10 n 9. 5 tens, 5 twenties; 5 times tlie number; 5a, 5m 10. 8x, 7y li. 5a, 15, 35 12. 4a, 20, 32 13. 12, 35" 14. 48, 90 15. mn, pq, xyz 16. pqr, 24, 60 17. 210, 108 18. 2, 3, 6 19. 3, 4 20. 3, 7 21. 4, 3 22. 8, 14 23. 2. 5; 3, 5; 3, 7; 3, a; 5, a;; a, y; x, z 24. 2, 2, 3; 2, 3, 3; 2, 3, 5; 2, 5, a; 5, a, 6; x, y, z 25. 2, 7; 3, 7; 5, m; c, d 26. 2, 5, x\ 5, a, y; p, q, r 27. 4, 8, 16 28. 9, 64, 16 29. 16, 27 30. 8, 27, 64 31. 36, 16 32. a, a; x, x, x; m, m, m, m; x. x, y, y, y 33. 2, 3, 4, a, x 34. 2, a-, 3a 35. 4, 5, a, ax 36. 3, 4, a, ax 37. 4, 7 38. 3, 4 39. x, a, c 40. 6, 4 41. w^, n^, mn,-,^ Exercise 2. 1. 8, 17 2. rc + y, a: + y + 2, 2a + 3^ 4j: + 5// + 62 3. 5, 6 4. m-n, 2a-3 6, 5x2-72/*, a:^-?/' 6. 9, 3, 24, 27, 9, 189, 54, 21 7. 20, 8, 12, 0, 18, 22, 18, 46, 40 8. 2a6«, 4a6«, 7o6«; 3a'b, GaH, SaH, 9a«5; 5aH\ Qan\ 9a«6« 11. 32 12. a + b 13. a-b 14. ab 15. J 16. (a + b)^ 17. a2 + 5-' 18. (a + Jf 19. a» + b^ 20. (a + 6)(a-6) 21. ab{a-b) 22. ^ 15 318 ELEMENTARY ALGEBRA. Exercise 3. 2. 5 a ct. 3. $3« 4. (3 a + h) ct. 5. 10 re mi. 6. (Sa qt. 7. (10-c) ct. 8. %{y-m) 9. %{^x + 4:y) 10. %Zm 11.3a 8. 12.?^^^^ X 13. f |, 14. ^ 15. 1»« 16. 1 6 M 6 17. %m{r-n) 18. I*'"^ 19. "f 20. $a a (^-) -^- ^ 22. (2 a + 20) yr., (2a-10)yr. 23. {m—7i)c or {n—m)c ^- *ll ^^- w ^^■<^-m ^'•*(— 1^)'' ^'-''^ -2 ex 29. ^ — ct. a + c 3°- / .xJ"-- 31. j3 32.^ a ^^•C-i'')i»".(^)'«'> '^^e^'- 3^-(^-^)««'(^-l)««^-' '^■^"^ 37. (2;.v-s2) sq. rd. 3£ 1. — tons. 39. V^'^ ft. 4C ^ 172Spqr x^ 43.$(.+ ^gi„-a 41. V«'+^'^ yti. 42. Vc'-tt' Exercise 4. 14. +4 lb. 16. -15 bii. 17. +(a— 5), —{b—a) 18. +{:x + y) 19. -{a + h) 20. + a & sq. rd. Exercise 5. 2. +61 lb. 4. -45 cows 6. -30 lb. 7. +8 a 9. +5 10. 11. + 4:X 12. — 4wi Exercise 6. 1. +20 a 2. -25a; 3. - -9.ry 4. —4a 6 5. +2aa: 6. -(Sx^y 7. - -3wn 8. —Spq ANSWERS. 319 9. 2{a + b) 10. 2(:m-?i) 11. -{x^ + i/) 12. -8(.r + v/)2 13. Uan^ 14. -4a: 2/2 15. —dia-m) 16. 8« 17. 18. -7a 6 19. -(Sm^n^ 20. -4{a + b) Exercise 7. 1. ;i-i-i/ + 2 2. 2x—dy + 4:Z 3. 5a— 36— 2c 4. 7r+6»— 5m 5. 4h + cd—ab 8. 4xy—3ab 6. 7ab—4cd+5ac—Gbd + 4:am 9. 4a + 86 7. 7y0— a:y— 4a;2 + 9m^— 6wa: 10. a; + 7/ + 142; 11. 6/?i2 + „i;i_3;i2 12. Exercise 8. 1. (a + b + c)x 2. {a—d + m)yz 3. (2a + 3 6-4c)?/ 4. (2-a-6)a:2; 5. (2a+3 6 + 4c— rf)a:y 6. {n—m+p—q + r)ab 7. (— a + 26— 3c)a:y 8. (3a6— 4&c + 6cc?— 5a + 6y) 13. 1. {2m-{Sn-4:a)} — \(jb-{7c-2d)} — {4:e-(g-2h)} 2. J4a-(26 + 3c)i — |4c^-(5e + 6/)i + ]7^-(2A-4?)} 3. \2p-{Sq-ir)\ — \2s-{5t + 6u)\ — \7v-{2w-Gij)\ 14. ■|.T_(2/_£)} — |m-(?i-^)} + ]g-(r + s)} Exercise 31. 1. a2_j2 2. a2 + 2a6 + &2 3. a^-2ab + b^ 4. 4a2_9^,2 5. a;2 2/2 + 6 a; 2/- 72 6. a;2-aa:— 6a: + a6 7. &C + 61C— ca;— a;^ 8. m^ + m^n^ + m^n^ + n* 9. 12a*-10an-12b^ 10. 63a4_23a2 J2_56^,4 n. abcy—acdx—b^dy+ bd^x 12. a:6 + l 13. a^-b^ 14. .r«— 8 15. a'2^1 16. 729 + c« 17. b^ + Mc^ 18. 6m4 + 4m3n-9m'^w2— 6m7i3 19. 36a.-4y4_49^24 20. 27a3_i728 21. a^—y^ 22. a^-^s 23. 9x^ + dx''y^ + 4:y'' 24. 25 a'^-ga^js^ 16^,4 25. 81a:4-16;c2 2/'-^ + 8a-y='-2/^ 26. 4 a;^ + 56 a.-^ + 324 27. a6_32&5 28. 243x^ + 32 jf 29. a;8 + a^i/'+2/*' 30. 6a;'' + 13a.6y + 6a;\'?/' + 17a-4.y3_i3^;3,/4^.n^2y5_9^^4 + 9^7 ANSWERS. 323 Exercise 82. 1. a + 6 2. x + 4: 3. x-5 4. .t^ + G 5. .r + y 6. x^ + xij + if 7. a;*-3a:y + 9?/2 8. a-'-a^i^ + i* 9. a^ + an + ab^ + h^ 10. 4a:«-6.xi/ + 9?/ 11. 2x-Q 12. 2a:-3a 13. 2a:-5a 14. ma^+20x^y^ + 25y* 15. 16;i-4-8.x»7/ + 4x2/-2a:y3 + 2/4 15. 4:7n^ + ()m^n^ + dn'^ 17. a;«-a:2/ + !/^ 18. 3:^ + 2 a; + 4 19. a'^-aH^ + b^ 20. 4w27t44.0m3n3 + 9m4w2 21. x^+x+1 22. 3a.-» + 2^2 + l 23. a;«-.r-3 24. 4:a* + Qa^-2 25. 2a:— 2/ + 2 26. x—2y—Sz 27. 2a:— 3?/— 2 28. 2a: + 3// + 2 29. x^—xy—if 30. a + &— c— 2/'*2»<> \0. Q2^ mf^ 'n}^ z^ 11. {a + hfic + df 12. m4.A:«(a + 6)»« 13. 21{a-\-bf{x-yf 14. d^^b^c*{m-{-nf 15. 8 a« &» (x* + yY 16. m^{x-yy^{x+yf 17. 108a'9 6»4ci6 18. 6y*2 19. IQ'^j^y^^z^ 20. 13a:4y4 2.6 21. 22. d2x^y^z-* 23. -250a:"y"2»8 24. -48 a:" 7/1*2" 324 ELEMENTARY ALGEBRA. Exercise 37. 1= x'^ + 2xy + y'^ 2. x'^—'^xy + y^ 3. m'^ + 2mn + n^ 4. m^—2mn + n^ 5. a;2 + 8a;+16 6. x^—Ux + i9 7. x^ + 2ax + a^ 8. x'^—2ax + a'^ 9. 4:X^ + 4:Xy+y^ 10. 9x^-Qxy + y^ ll. 4:a^ + 12ab + 9b^ 12. 25a^-d0ab + 9b^^ 13. 4tx^ + S2x + G4: 14. 9a;2-30a; + 25 15. 25 + 20x + 4:X^ 16. dQ-d6x + 9x^ 17. x'^y^ + 2xy + l 18. a;2y2 + i0xi/ + 25 19. l-2c-2.7:5+l 20. Not divisible. Why? 21. l-9a;« + 81a:* 22. 23. 125a»-25rt» + 5a-l 24. a»»-a»i« + a«6'«-a3 6i8 + 6«* 25. a^^-a' b"^+b^^ ELEMENTARY ALGEBRA. 26. «5 + 3fj4y + 9«3y24.27«2 2/3 + 81a?/4+2432^6 27. a5_3a4y + 9^3^2_27a-i2/3 4.8i^2/4_243^ 29. 8a3-12a2J + 18a^'2_27^;3 30. a;4_4a;2y2 + i6y4 31. 4.i;4-6a;2 2/3 + 9/ 32. a;8-2 a:^ 2/* + 4 a^ 2/^-8 a;^^/!^ + 102/^6 33. x^7f + x^y'^ + .r'^y^ + x^i/ 34. a^x^^—abx^f + b^i/"^ 36. «4_9 f^2 yi + 81 y4 37. 64 ^2 ^,2_8 « 6 c + c^ 38. 4:X^-G.i^7f + 9y^ 39. ^8-2 a,-« y^ + 4: .i:^ ij^-8x^ y^^ + 16 2/" Exercise 43. 1. a{a + b) 2. b{a—c) 3. a: (.?• + «?/) 4. a;(x2 + 3a;-2) 5. 3a(a-2 6 + 3^/2) 6. 2a^x{l + 2ax~Sa'^x^) 7. 6x.v(;r-^2/- 2:?^ 2/^^-3) 8. 5a:2(2.r2 + 3x-4) 9. 7r2(l-2r+3?-2) 10. (a + &)(2a + 3 6) 11. {a-x){a—b) 12. (m + 7i)(c + c?) 13. 12a3&2c(a6c-2^/2c3 + 3a) 14. 5pq{2pq^ + np'^q'^-4r) 15. 12 ^ / (2 2:^-3.^2/3 + 4 y") 16. {a'^-rb'^)i4c-od) Exercise 44. 1. (a + 2 6)(a-2 6) 2. (2a + 5&)(2a-5&) 3. (32:+72/)(3x-72/) 4. {ay + 2){ay-2) 5. (4+2) (4-2) 6. (^+8) (a:- 8) 7. (a; 2/ + 10) (a; 2/- 10) 8. (9 + 2) (9-2) 9- (abc + Q){abc- 6) 10. 2/^ (.^- + z) (x—z) 11. (a2 + 22)(^^2)(a-2) 12. (a^+b^){a^ + b){a^-b) 13. (4a24.922)(2a + 3 2)(2a-3 2) 14. (92/''' + 1622)(32/ + 42)(3 2/-42) 15. x^y^{x + y)(x-y) 16. (;?:3 + 2/2)(a:3_^2) 17. (25 + 22) (5 + 2) (5-2) 18. (^4^.2^)(a;2 + 2/2)(a: + 2/)(a;-2/) 19. (x^ + 2/2) (a;3 + y) (x^-y) 20. (m4 + M8)(^2^^4)(„j + ^i2)(,^_,^2) 21. {a + b + c){a + b—c) 22. (a—x + y){a—x—y) 23. (w— ?i+l)(m— w— 1) 24. (2 + .'r + 2/)(2— a;— 2/) 25. (c + a + &)(c— a— &) 26. {c + a—b)(c—a + b) 27. (5a + a;— 2/)(5a— a: + 2/) 28. (4 + 2— a:) (4— 2 + a;) 29. {l+x-y){l-x + y) 30. (7 + 2a- + 22/)(7-2a;-22/) ANSWJERS. 327 Exercise 45. 1. {x-y)ix^+xy + i/) 2. (x + y){x^-xy + 9f) 3. (a-l)(a2 + a+l) 4. {a + l){a^-a+l) 5. (x-2)(^* + 2j: + 4) 6. {x + 2){x^—2x + 4) 7. (2a + 6)(4a2-2a6 + J2) 8. (2a-6)(4a2 + 2a6 + ^;2) 9. (3a-2&)(9a2 + 6a6 + 4i2) lo. (8a + 26)(9a2-Ga& + 462) 11. (a« + l)(a*-a2 + l) 12. (a + l)(a-l)(a2_a+i)(rt2 + a + l) 13. (a;« + 2)(2:4-2a;2 + 4) 14. {x^ -2) (0^ + 2x^ + 4) 15. (a:+y)(a;*— a^'y + ^'y'— ^y^ + y*) 16. {x-y){x* + x^y+x^if+xy^ + y^) 17. (.r + y) {x^—a^ y+x^ y^—x^ y^ + x'^ y*—x y^ + y^) 18. (2 x-y^} (16 2:^ + 8 x^ y^ + 4: x^ i/ + 2xy^ + y^) 19. 20. (2a:«-3y3)(4a.-4 + 6a;2//3 + 92/«) 21. 22. {x—y){x^ + xy + ^f){x^ + x^y^-\-y^) 23. x{x+\){x^-x-\-\) 24. x^(x-2){x^-\-2x-\-4) 25. (a:y + 2)(a;2 2^— a:i/£ + 22) 26. 27. (4 w* + 5 n*) (16 m4-20 m^ n^ + 25 n'*) 28. a:y(a:2/-l)(^2 3/2+a;2/ + l) 29. x{x-[-y) {x^-x y + y^) {x^-x"" y^ + y'') 30. {x + y-z){{x-\-yf + {x-^y)z-\-z^ 31. (a;+2/ + 2){(a;+y)«-(x + 2/)2 + 2*f 32. {x-y-z)\x^-k-x{y + z) + {:y + zf\ 33. (.r + 7/ + 2)|a:2-a:(2/ + £) + (2/ + 2)2f 34. (a + 6_c_tZ) \{a + bf-^{a + b){c-^d)-\-{c^-dy\ Exercise 46. \.{x-\-yf 2, {x-zf Z. {x+\f 4. (a--2)2 5. (.e + 9)« 6. (2a:-3)2 7. (3a: + 2y)2 8. (5x + l)2 9. {x^-Qf 10. (a;+2)2(x-2)« 11. {x + yf{x'*-xy + y'^f 12. {ah-cdf 13. (2a;+33/)«(2a:-32/)« 14. 15. (a; + l)«(j;*-a;3 + a:'-a:+l)2 16. x^y^(x*-\-2f 17. (a;« + y»)2(a: + 2/)»(x-i/)« 18. 19. (2a;» + 3)« 20. -f)^(-4:)(«4)K)= /x ^\(^ "\ («-f)(-f)(«'-y-3(«'-if%") Vy xJ\y^ a;2/' Vwi* 7i^/\m* m^n^ 7i*J 29. a:4-a;' + l- ^, + ^ 30. | 336 ELEMENTARY ALGEBRA, Exercise 69. 1. a;=3 2. x=4: 3. x=\ 4. x^l'Z 5. 2^=24 6. x=U 7. x=^% 8. x=.'^X o 2 9. x=0 10. x=7 11. x=3 12. x=7 14 5 4 13. ic=ljx 14. a;=-y 15. a;=8 16. a;=-y 22. x=-3 — ^_ ..3,^ -^. ^_„ ^ 4 82 5 5 5 17. x=l-i^ 18. a;=^ 19. x=8 20. a;=3 21. a;=-r 22. a:=-3 23. a;=-l;^ 24. a;=5^ Exercise 70. 1. x= 1 2. x=l 3. x=c—d a + b , „ _ «c ^ c+bm—an 4. a;=:m2 + ?i2 5. a;=-; 6. a:= — — b—c m—n _ 12 o _ 2ffl + & _m{m + n) 18— 17 a 9 {m—nf &2_a2 aH-cd^ ,^ ar^-c2 10. a:=: ^ 11. rr=— -T- 12. a:=-— , 2b ad+ac md—nc _- a&c .^ bcd^ 13. a;=— 5 5— 14. a:= ab + ac + bc acd + b^d—bc^ ,_ 1 ,_ {a—b)c __ cd—ab 15. a:= r 16. 2;=^^ r^^ 17. x=- a + b ab a + b—c—d Exercise 71. 1. a:=ll 2. a;=f~ 3. a;=25a + 24J 06 2 _ , « 457 „ a 6-1 4. a:=— — 5. a;=4 6. ^=-7^^ 7. x= 9 102 -bm + ii 8. a;=2 9. x=3 10. a^^:^ 11. x=4.-r 19 4 11 17 12. x=l-^ 13. a;=-=-a 14. x=:^ — - 15. x=zr7i 2 7 2— a 10 16. a;=-4 17. a;=0 18. a:=~ 19. a;=4 16 2 c 20. x=——- 21. 0;=-'^ 7— 22. a:=.8 a a + o 4a62_l0a «^ « «^ o 1 23. a;= — . ^^— 24. a;=3 25. x=B ^ 4a— 3 2 ANSWERS. 337 Exercise 72. 1. 10 2. $5, $50, $200 3. . 18 4. 80 yr. 5. 300 bu., 200 bu. , IGO bu. 6. < 7. 24, 60 8. $80, $r rs 9. 28 87 10. $2000, $9000, $5000 11. 72, 91 12. 90 yd., 75 yd., 35 yd. 13. 400 14- $00 15. $20,000 16. 196 lb. 17. 84 18. $162, $118, $104 19. 20 yr., 16 yr. 20. $3000, $2500 21. 300 bu., 200 bu. 22. 84, 96 23. GO ct. 24. 66 25. $75 26. $12,500, $10,000 27. $133 4 28. $1000. $700 29. $2.91 1 30. 30^ 31. 14^^ 32. 7^^ 33. 87^^ 34.20^ 35. $5000 36. $800 37. $500, $300 38. $720 2 200 39. 5^ yr. 40. 15 yr. 41. — yr. 42. G% 100 2 43. ^^% 44. 677^ 45. $75 46. Q% n 4n. $100 48. 75^ 49. $4800 50. 22 ^ mi. 61. 24 mi. 52. 2 mi. 53. I5? da. 54. 13 ^ hr. 37 o 56. 17^ hr. 56. 2 da. 57. $300 58. $800, $500 59. $200, $280 60. 1740 61. 15 yr. 62. $900 63. 15 lb. 64. $60 65. 6^ hr. P. m. 66. 2 p. m. 67. 29— min. past 4; 5:J^ min., or 38 tt- min. past 4; 54-j- min. past 4. gg^ ^g ^^ ^,j.^ gg_ 2Q ^^^ rjQ^ ^c) ct. Exercise 73. 1. a:=3, y=l 2. x=4, y=\ 3. x=o, y=5 4. a;=-, y=- 5. x=6, y=5 6. x=5, y=A 7. x=i, y=S 8. x=-2, y=-S 9. x=-l, y=0 10. x=5, y=4 11. a;=4, 3/= -4 12. x=2^, y=^'^ 338 ELEMENTARY ALGEBRA. Zxercise 74. 1. ir=4, 2/=3 2. 3—2, y=-2 3. a:=4, 2/=4 4. a;=-5, 2/-2 5. r^=7, 2/=2 6. a;=:3, 2/=10 12 11 7. a;=l^, 2/=i7 8. a;=^, y= ^ 9. rr=3, 3/==5 1 1 10. x=^, y=z~ Exercise 75. 1. a;rrl4, y—U 2. x=10, y=12 3. a:=10, 2/=3 4. a;=3G, .^=90 5. a;=13, y=17 6. a;=5,^, y=4~ lo 13 7. a;=9, y=lo 8. a:=-60, 2/=5 9. a:=10, y=2^ o j^ 10. x=TA-, y=m^ 11. a^=G, 7/=-9 100 40 o .- ^ 12. ^=-j^, y=;32i 13. a:=la, 2/=8 14- a:=:-27, ?/=13 15. a:=3, 2/=2 16. x=^, y=-5 Exercise 76. 2. x=2, y=3 3. a;=4, 2/=6 1 _1. 2' ^~3 11 oil Exercise 77. c+d c—d ^ n—hm am—n 2- ^=-77— IT' 2/= 2a ' -^ 26 a-6 ' -^ a-& 2^ ^' "^ 2 ^ ^ ms—nr' ^ nr—ms an—hm bm—nn ^ c + n bn—ac cn—bd'' "^ mc — ad a + b^ ^ m{a + b) an+b b—am 7. a;= , y= 8. x=l, y=0 m+n m+n ^ b 71— bd bm—bc nc—md' ^ nc—d m 10. a;= 7 / , y= ^ mn{c7i,—dm)' ^ 7n7i{cm—d7i) ANSWERS. 339 Exercise 78. 1. $168, $175 2. $()00, $400 3. 4, 3 4 ^ ^ *• IG' 8 5. CO bu., 40 bu. 6. $100, $20 "■l B-l 9. 8, 3 10. 34 11. 39 12. 12 13. 24 14. 12 4 da., 21 i da. 15. 8 da. 16. 15-^ da., 17 da. 17. 42 da., C3 da. 18. 10, 18 19. 16 ft, 10 ft. 20. 15 rd., 9 rd. 21. 15 yd., 10 yd. 22. 16 rd., 10 rd. 23. 45 in.," 03 in. 24. 150 yd., 30 yd. per min., 20 yd per min. 25. $3 26. 8 ft., 6 ft. 27. 3500 28. 8 persons, 50 s. 29. 32 ft., 21 1^ ft., 691 i sq. ft. 30. $600, 6% 31. $800, 5 yr. 32. 25 mi., 15 mi. 33. 15 mi., 5 mi. 34. $500, $400 Exercise 79. 1. x=2, 2/=60, 2=26 2. x=S, ?/=5, 2=4 3. x=2, y=3, z=\ ' 4. .t=12, y=U, 2=36 1 1 1 6. x=^, y=3, z=-^ ,111 6. x=^. y=^, 2=^ m + n—r m—7i + r n + r—m '= 2c 8. a;=4, y=5, 2=6 9. ;r=10, 2/=8, 2=6 10. a:=8, y=16, 2=24 11. x=l, y=-l, 2=0 12. x=ry, y=4, 2=3, u=2 13. x=-^, y= ^, z= ^ 14. j:=12, y=24, 2=36 15. x=a, y=b, z=c Exercise 80. 1. 00 ct, 35 ct., 65 ct. 2. 100 A., 90 A., 120 A. 3. $600, $400, $200 4. 10 da., 20 da., 30 da. 13 1 5 20 yr., 35 yr., 42 yr. ^' '^ij ^*'' ^13 ^^'' ^'^7 ^^ 7. 20 hr., 30 hr., 40 hr. 8. $252 ^ $200, $47 ^ 9. 346 10. $600, $800, $1000 340 ELEMENTARY ALGEBRA. 11. $450, $225, $237|, $87|- 12. 2 ct, 3 ct, 5 ct. 13. $40, $60, $80 14. 5 gal., 3 gal., 2 gal. Exercise 81. 1. 14, 6 2. 36 yr., 12 yr. 3. $90, $80 4. 3^ da. 5. $700, $300, $800 6. 40 7. 5 8. 120 A., 160 A. 9. $4, $1 10. i(c + ^), \{c-d), 22 yr., 14 yr. 12. 5, 6, 7 ,, 100— flfc? ac— 100 _ „ ,_ d—am d—an ,, ^^ 11. -p-, -j—; 6, 8 13. -, ; 14, 10 c—d c—d n—m n—rn ' ,^ bn—dm cm— an ^^^ ^„^ ,_ ad ad ^. ^^ 14. -T :j, -t tt; $25, $35 15. -^ — , -^ — ; 50, 75 be— ad be— ad d—c a—d + c ,^ bd—ac—ab{b—a) be—ad—ab{b—a) .. .. 1^- p=^r- — ' pz^2 ; 14, 10 ,_ lla+& _-, __ ce + bd be— ad ._ .. 17. — ^ — , 85 18. r^, j^; 20, 10 2 ' ae + b^ ' ae + b^ ' ' 19. ^(a + b-c), ^(a + c-b), ^{b + c-a); 40, 54, 36 63(aQ?-5c) 6S(ad-bc) ^ ^°- 63^-17 6 ' 17a-6ac '■ ^' ^^ Exercise 82. 1. c'^ + 4:C^d + Gc^d^ + 4:ed^ + d^ 2. a'^-4:a^d + 6a^d^-4ad'^ + d* 3. a;5 + 5a?*2/ + 10a;3 2/2+10a;2 2/^ + 5;r2/^+?/5 4. x^-5x^z + 10 x^ 2^-10 x^z^ + nxz'^-z^ 6. 7n^ + 6m5n + 15m*n2 + 20m3 7i3 + 15m2n* + 6mw* + w^ 6. m®— 6 TO^ n + 15 m* /i^— 20 m^ w^ + 15 m^ w^— 6 m n^ + n^ 7. c'-7c«a; + 21c5x2-35c4.'r3 + 35c3a;4-21c2a:6 + 7ca;8-a;' 8. x^ + ^x'^z + %%x^z'^ + mx^z^->t'70x^z^ + bQx^z^-\-2Sx^z^-¥%xz'^+z^ 9. a;8-8a;'^ + 28a;V-56xV + Wa:*2/^-56a;V + 28a;V-8a:2/' + 2/« 10. c9 + 9 c8 2 + 36 c' 2^ + 84 66^3 + 126 c5^4 + 126 c4 25 + 84c3 ^6 + 36c2 0' + 9c28 + 29 11. yo- 10 a: 2/» + 45 0:2 2/8_ 120 ^:3y7 + 210 a.-4y6-252a;5y5 + 210.^6^4 - 1 20 a;' / + 45 0:8 2/2 _ 10 a:9 y + a;io 12. 2" + 11 2^0 y + 55 ^9 i/2 + 165 28 ?/3 + 330 ^t _,y4 + 4(32 ^6 ^5 + 4^2 z^ y^ + 330 £4 2/7^.165 23^8 + 5522^ + 112 2/10 + ^1 ANSWERS. 341 17. x* + ij:^ + Qx^ + 4x + l 18. a^-4.x^+Gx^-4x+l 19. a:* + 5u;4 + 10a;»+10a;« + 5a:+l 20. a;'-5a;* + 10a:«-10a:« + 5a;— 1 21. l + 6z + loz^ + 20z^ + 15z* + Qz^ + z* 22. 1-62 + 1522-2023+1524-0^5 + 26 Exercise 83. 1. a* + 8an + 2Aan^ + 'S2ab^ + 16b* 2. 81a4-21G«3^ + 216a2 62-96a63 + i664 3. 32a:5 + 80^^ + 80a;« + 40a:2 + 10a: + l , , Sa^x 10a3a:2 lOa^a;' Saa;* rc» 4. o^ — — + s s — + — 7 J a* 5a2 + 15 a« 616-6 a3 ^'^0^.^24 lO..--12.- + 60^-1604-^-^ + ^ 81 6^ Oi*""'' a* "^ 16 a* ^_200 5 500 625 3125 ^ 3125 ,^, 243 243 ^ "^ 243 ^ 243 ^ "^ 1944 ^ ~ 7776 ^ 13. 64 ai« + 576 a'o a;3 + 21 60 a8a;« + 4320 a«a:» + 4860 a* a:'2 + 2916 a« .«»«• + 729 a;'8 14. a:i»-6a:'<'2/*+15j*y»-20a:6^«+15a:4 2/^-6a;2yo + yi2 15. -32 a:«-240x*y-720a:«2/«-1080 a:* 2/3_8lOa: 3/4-243 3/5 16. ^^-^^y + 6a:^y'-9a:3/3+^y4 17. flJo ^io_5a6a4+ 10 a«a:«- 4^ + -.-.- ^ a«a:« a'®a;'« 810 a8 243 18. 32x'0-240a«a;' + 720a4a.4-1080a«a; + X Exercise 84. 1. x^-\-y^ + z^ + 2xy + 2xz-i'2yz 2. a:» + y« + 2«— 2a-?/— 2a:2 + 2y2 16 342 ELEMENTARY ALGEBRA. 3. x^ + y^ +1 + 2 X 7j + 2 x + 2 y 4. a'^ + ¥ + i-2ab + 4:a~4b 5. a'' + 2aH + dan^ + 2ah^ + b^ 6. 4:a^ + d¥ + c^+12ab-4:ac-Qbc 7. 3^4^.2:^2 + 3+4 + 4 X X/ 8. 4a;2 + 252/2 + 9c4 + 20a:i/-12c2^-30c2 2/ 9. -„ + — +6+ -^ +--^ 10. a;2 + 2/2 + 22 + 4_2x2/ + 2a;2— 4a;— 22/e + 4y— 4^; 11. -2-2:c3 + 2;2/ + 2a;2/-22/3+^^ 2/ "^ 12. m6 + 2m^ + 3m'* + 4m3 + 3m2 + 2m+l 13. a3 + ^,3 + i^.3^2j.^3^2 + 3^j2 + 3j2.,.3^.,.3^^.(j^5 14. a;3-2/3— 23_3a;2y_3a;2 2 + 3a;2/2-32/22 + 3a:22_3^22 + 6a-y^ 15. a;3 + 8 + 2/3 + 6a^2.^3^2y + 12a: + 12y + 3a;2/2 + 6?/2+i2xy 16. 8a;3-272/3 + 125-36a;2 2/ + 60a;2 + 54a:2/^ + 135i/2+150a: -225?/ -180 a: 2/ 17. a:« + 3a;5?/ + 6a-'4_iy2 + 7a;3^3_^g^2^4.,.3^.^5.^^6 ^«- ^7«^+ i^^+ 1^^+^^^4'^^^-^ 1-^^+ 1^^^+ 1«^^+ 25^ 5 ^ ^6c2+-a&c 19. 2;3 + 82/3-2723 + 6a;2 2/-92;2 2 + 12xy2_362/2 2 + 27x22 + 542/22 -36 a; 3/ -2 20. a;9 + 3x6 + 6a;3 + 7+ -o + -« + \ X^ x^ x^ 21. x«+6a^ + 9a;2-4--, + ^--i a;2 a;^ a;^ 22. l + 15a; + 84a;2 + 215a;3 + 252a;4+i352.5^27a;« no « 3 . 11 ^ 17 , 11 , 2 8 23. 2/«- 2-2/^+ -4 2/^- "8-2/^+-^/- 3 2/ + 27 24. a;6 + 3x5-12x4-29a;3 + 60a;'^ + 75a;-125 5. 15 Exercise 85. 1. ±18 2. ±36 1 3. ±48 4. 8 5. 6. 18 7. ±8 8. ±12 Exercise 86. 9. 12 1. ±ab^c^ 2. ±2a4j3c5 3. xy^z* 4. 2mn^ 5. -Sx^y^ 6. ±mn^p* 7o -2aH^c^ 8. ±12a4a;3y5 9. -9(a+a;) ANSWJERS. 343 10. ±4(a-a-)« 11. -(a + b)c^ 12. ±10x^x+y)* 13. -3(m + n)« 14. ±2(a:»-/)2 15. ±20244 16. ±21000 17. 1 18. 184 19.1125 20. ±1, ±|, ±f, ±|, ±1 21. ±r,, ±- r , ±^-4t. ±77 ^ 22. 2 1 a: 2.rgy3 g-^g + x) 3' ~ 2 ^ y' ~ 3¥F' c^rfs ^(a-6)2' (a;+y)3' '^'(a-i)*' 2a» QA. 2 2 3 "^3' 3' 4' ax^ .1 ^(a -a-)3' ^2 Exercise 87. 1. ±(a + 6) 2. ±(a:-y) 3. ± (.r + 4) 4. ± ix-S) 5. ±{x + y + 2) 6. ±{x-y-z) 7. ±(a+26 + 30 8. ±(2a:-3/ + 3) 9. ±{.c' + y+{) 10. ±{Sx-4y) 11. -(^-^0 12. ±(2x' + ^y + Exercise 88. 1. ±{x^+x+l) 2. ±(a;«-2a:+l) 3. ±(a;»+2x+3) 4. ±{x^-Sxy + 2y^) 6. ±{3^-2x^ + 'Sx) 6. ±{2x'^-5xy + 3y*) 7. ±(^a:«+ia;+i) 8. ±(^x' + 2 + ~'^ 9. ± {x?^+x'-x + l) Exercise 89. 1. ±17 2. ±20 3. ±35 4. ±43 6. ±52 6. ±09 7. ±71 8. ±84 9. ±127 10. ±245 11. ±324 12. ±408 15. ±.07 16. ±.25 17. ±.012 18. ±29.7 19. ±.0004 20. ±.324 21. ±5.82 22. ±3.38 23. ±10.42 24. ±32.01 25. ±9.999 26. ±89.5 27. 3.1022, 3.3106, 3.4641, 3.6055 28. 1.4142, 1.8165, .9354, .5590 29. 6.324, 6.403, 6.480, 6.557 344 ELEMENTARY ALGEBRA. 1. x+\ 4. 2a:«+3 7. ax—b 10. -1 Exercise 90. 2. a-6 5. x'^+x-\ 8. 2aa;— 3&y 11. ax- — ax 13. CH-5 + C Exercise 91. 2. 27 3. 35 6. 84 7. 88 10. 222 11. 305 16. .12 17. .03 20. .50 21. 3.4. 3. rr + 4 6. y^-y-\ 9. a;2-2a: + 3 12. x^ + l+K 1. 14 2. 27 3. 35 4. 67 5. 72 . 6. 84 7. 88 8. 98 9. 122 10. 222 11. 305 12. 420 15. .2 16. .12 17. .03 18. .25 19. .31 20. .50 21. 3.4. 22. 2.4 23. 4^ 24. 1.442, .854, .646 o Exercise 92. 1. ±{x^ + y^ 2. ±(:x + y) 3. ±4, ±3, 5, ±.2 Exercise 93. 1. (a;2 + «.r + a2)(a:2-aa: + a2) 2. (a: + 5)Cr + l) 3. (2,r + 52/)(2a;+?/) 4. {^x^ + 1 xy ^-4:y^){^x'^-l xy ■^4:y^) 5. {2p + q){p^Zq){2p-q){p-^q) 6. (8«'-^ + 4a6 + 962)(8a2-4(i& + 962) 7. (.2;2 + ic + 3)(2:2 + x-l) 8. {2x■\-^y){x^-^J){2x'^ + ^xy-y'^) 9. (2a; + l)(4a;2 + 28:r + 61) 10. (3a:-y)(9a:2-15a:2/ + 7y«) 11. (x + y){x—y){x^-^1lx^y^^-\^y^) 12. (2a2_3 62)(4a4 + 3 54) 13. (a: + 5)(a;2 + 25rK+175) 14. (2a3 + i)(4a6_8a3 + 7) 15. (^s + 3 ^,3) (^^e ^_ 3 ^,6) 16. (a2 + 2 6-^) (a2_2 &2) («8 + «4 ^,4 + 7 ^,8) 17. (2a + 46 + 5c)(2a + 26 + 3c) 18. (3a— &)(9a2— 42a6 + 61 62) 1 9. (a2 x-h'^y) (a^ ;c2 + 4 ^2 i^ .r 2/ + 7 6^ y^) Exercise 94. 1. a;=±6 2. a:=±2 3. a-=±2 4. a:=±i\/lO 2 6. a;=±2V^ 6. a:=±l, i^V^ ANSWERS. 345 8. x=±a/'_^^ 9.x=±^a^ 10. a:= ± >/6, ± V^ 11. a;=±3 12. a:=±\/2, ±oV^ 13. a:=2, -10 14. x=±V^iK^, +l/_2— 15. 5 rd. 16. 15, 12 10 "^ 50 17. 50 A. 18. $90 19. 9, 21 20. 9 in. 21. 10 da. Exercise 95. 1. x=2, -4 2. a:=6, -4 3. x=-2, -3 4. x=4, 5 5. x=G, -3 6. a:=5, -4 7. x=i, 7 8. a:=6, -10 9. x=7, -8 10. x-W, -10 11. x=-l, -li 12. a:=3, -| 13. a=G, -2^ 14. rr=3, -4^ 15..:=-7, -1 16. j^=9, -li 17. ^=-i|, -4 18. ;r=7, -2^ 19. ..-2^,-1 20. x=2^, -Ij 2 2 21..= -|, -l| 22. a:=2i -si 23. a:=2, -3| 24. a:=3, -2j 25. x=2±\^ 26. a;=3±2'v/5 27. 2;=4, i 2S.x=6, -1 29. a:=l±i\/2 30. a:=8, -lo| 3l.a-=|±iA/l3 32. a;=8, 3 Exercise 96. 1. a;=a, —3 a 2. x=2b, b 3. a:=2m, —3m a a 5. x——ab, —b 6.:r=|(l±V-3) 7. a:=3a, -a 8. x=|(1±a/5) 9. a:=&± V«^ 10. 2^=±\/i>? 1 11. a:=2a, -« 1 12. 2:=a, \ 13. T=l(-ft±2v^ -be) 1^^=2< [n±2^mn) 15. a:=~, a' c 16.0:=^ b a 346 ELEMENTARY ALGEBRA, la X 4. X-. 7. X 10. X = ±2, ± Exercise 97. ^6 2. a;=l, 2 1 3. x= ± ^2, ± y^ij :±3, ±2a/^ 5. a;=2, ^ 6. x=l, -2 8. a;=±7, ±5 9. a:=±l, ±V^ 11. a;=-2±A/3, -2±V-^ ±1, ±^ 2a 12. a:=3, -5 13. x=±2 1 1 14. xz 16. CC: 19. a;: 21. X-- ,0, -1, _1(1T a/-1») 15- ^=3, 3, 4(-13±\/T53) =3, -5, 2, -4 17. a;=7, 3, ±2 18. x=d, -1 :±1, ±2 20. x=j^{ym ±\/c + 171^— a) = V2, V^ 22. x=-2, 1, -^ (IT a/5) 1 1 23. a,-=2, -^, -^(-7±a/33) 2' 4 = ±4 :3, -6 -i -2 -4' 1. X-. 4. a:: 7. x. 10. a;: 13. a;: 15. x: 17. a; 19. X 21. a:=±3, ±3^/^ 23. X Exercise 98. 2. ^=±4 5. a;=— 2, —5 1 8. a;=2, -1 11. x=5, — 1 3. x=±^a 6. x=2, 4 9. a;=r-li -1- 12. a;: 2 3 '3' 2 14. a;=2 g, -5 =2, -1±V^ = -2, liV^ r-3, i(3±V'^=27) 16. a;=-l, _(l±V-3) 18. a;=3, ^(-3±a/^^) 20. a:=±a, ±a\/^ 22. x=±l, ^{Tl±V^) ±a, ^a(Tl±A/-3) 25. a;=±2, -6 24. :r=^-a, |(-3±a/=27) 26. a;=±l, +1 ANSWERS. 347 27. 2:=J(Tl±V^ 28. r=-l, -1, -1 30. a:=^(l±A/^) -2 32. a:=2±A/3, 2±'v/^ 33. x=a, b, b—1 Exercise 99. 1. x^-6x=-S 2. x^-2xr=l5 3. x^ + 5x=^24: 4. a:« + 9a:=-20 5. 3;2-3aa:=-2a« 6. x*-px=Qp^ 7. a;« + 7aa;=8a« 8. x*-2ax=b^-a^ 9. a;»-2a«a;=6*-a* 10. a:2-2a:=l 11. a;«-6a;=-7 12. a;*— 4aa;=6«— 4a« 13. x^—2ax=b-a^ % A 9 1" ^ tea ^^ * 16. x^—2ax=4?n^—a* Exercise 100. 1. a;»-9=0 2. a;2 + 2a;-35=0 3. a;«-2a;-35=0 4. a:« + 12a; + 35=0 5. a;»-12a; + 35=0 6. 12a:«-17x + 6=0 7. 2a;»-5x-25=0 8. 0a;« + 13a:+6=0 9. 6a;« + 13a:-15=0 10. x^ + ax-2a^=0 11. a:»-2a:«-9x + 18=0 12. a^«-4a;«-9a; + 36=0 13. a:3-5a;« + 8a;-4=0 14. a:3-4a:» + 3a:=0 15. 12a;3-4a:«-3a; + l=0 16. 10a.-«-39a;«4-39a;-10=0 17. 16a;'-16a:* + 3a:=0 18. 60x»-133a;» + 98a;-24=0 Exercise 101. 1. x=^a(l±V2a* + l) 4. x=l, I 5. a:=-3, -4 7. a:=±5 8. a;=±9 10. a:=i(9± a/145) H- a:=7, ^ 13. a;=±'v/mn 14. a;=-, - c c^G, 6 3. a:=14, -10 -4 6. x=±l 9. a:=21, 5 12. x=4, 6 15. a;=a± — a 348 ELEMENTARY ALGEBRA. 16. x=h, -a 17. a:=9.477, -1.477 18. a;=2.108, -2.608 19. a;=5.236, .764 20. a:=1.148, -0.348 21. a:=:± 1.095445 22. a; =±4.54923 23. a:=30.716, -0.716 24. a;= 7.464, 0.536 2b.x=±a, ±- 26. x=±^-l, ±Y^{l±^/-^) a 27. x=2, i i(-13± Vl53) 28. x=4:, -3 2' 4 29. x=l, -1(1^^/^) 30. a;=-l, i(l±v^ 31. X=±l, ±V^ 32. (x2-a2)(a;2_&2)(a,2_c2)^0 Exercise 102. 1. 12, 13 2. 3, 14 3. 7, 15 4. 20 rows. 5. 30 yd. 6. 5, 25 7. 8 rd., 6 rd. 8. 40 mi. an hr. 9. 4 da., 6 da. 10. 5 hr. 11. 36 12. 12 ft. 13. $50 14. $80, $120 15. $2000 16. 3 in. 17. 20 ct. 18. $24, $30 19. 2+^8 rai. 20. 24 mi., 16 mi. 21. 6 ft, 4 ft. 22. $9 23. $41.83 + , $33.83+ 24. 3 ft. 25. $2, $3 26. 6, 8 27. 24, 18 10. 13. 14. . \ x=S, 4 I ( y=4, 3 f ANSWUBS. 849 Exercise 104. 2. {x=5, -4) ly=4, -5f 3. ja:=3, 3, -3, -3^ 4. {x=2, 2, -2, -2 iy=d, -3, 3, -3 6. {x=z4, -4, 3, -3 I 6. \x=2, -2, 1, -1 iy=3, -3, 4, -4) (y=-h 1, -2, 2 1. U=3, 2) I y=2, 3 f a:=3, 3, -3, -3 } y=l, -1, 1, -1 f . j x=Q, 3 ) < y=l, 2 f 9. 10. ra:=3, 3, -3, -3 ) 11. r A 1 _ 1 1 _1 _1 i _l[ J'^-^3' 3' ^3' ^3 ^~2' 2' 2' 2) 1 1^ _1 J. 1 12. r .1 .11 1^ '^^~ 3' ^2' ^2; ^2 ^=%' -^2' 2' -2 (y=l, -1, 3, -3 > Exercise 105. x=±i) 2. j.r=±3) 3. {x=±2l y=±2) (y=±S) iy=±l) 4. \x=±Sl 6. ja:=±2A/6{ 6. ja:=±5/ 7. ja;=±5) 8. ja;=±5} 9. {x=±4 y=±3) iy=±5) (y=±l (y=±4^ iy=0 ) U=±5) iy=±5) (.= ±1) iy=±l) 10. 3. ja;=2, 5, -4±\/G } 4. (a;=±2 ly=5, 2, _4T'v/G^ <2/=±1 *• fa;=3, 5^, -6, -3l| ^' (x=15, 10, i(-23± V^=^) [y=5, 2^, -4, -eij |y=10, 15, 1(-23t \/=^) 350 ELEMENTARY ALGEBRA. 5.0-../— ^^f <2/='J', -3, -1±2V y=6, 8, ±|(3q:v^ Exercise 107. 2/= ±2, T^a/sJ [2/= ±4, ±^Vl3 b2V'7_ J. 5. -4 4^1 '■ h^^'^Vfl^ 7. r .5 /— ir^ 8. ( a;=0, ±2 ) Exercise 108. 2. ra:=7, -18 ^ 3. r . o ^3 1. (.T=5) 2. ^a:=:7, -18) d. r_3 _^±y y=±i(Va' + 2&qF V^'-^^) 6. {x=5, 1} 7. jx=±5, ±1) 8. ja^=±3, ±2 2/=±l, ±5i ^, V9, V^ 3. aW, 6A, ctV *• r 16' y 729' y 64 7. Vo*^. 'Vo^, 'V^« Exercise 115. 2. v^ v^, v^ 6. V"^ V^ V«^ 1. 12 Vo" 2.|V2 8. Va^(a:+y)», (a;+y)A Exercise 116. 3. A/3a; 4. 6 V^" 354 ELEMENTARY ALGEBRA. 5. 8. 10. 12. 15. 18. {a + l + 0)^/2 \a b cj ^ 6. 2a\/x 7. a {'\/x — \/y + ^/^ ^a^-b^ b{a—b) 11. {a^-b-c)^l 13. V^ 2a^b 19 16. 2aya or — 2&\/a 2a 14. f ^ 17. 2^4 a2_62 Y a^ + a 6 20. ^~—^/ax-x^ 12. 15. 18. 20. ±3v^ 1 / — c iv^ 2 ^ 18 a—b Exercise 117. 2. Vl8 3. ±^6" 4. Tl2a\/2bc 7. ±2a\/6a 8. G;^^^^ 11. Vsoo 13. TQ a ^aWc^ 14. 2+^6'+ \/iO 17. Va-a; 6. IVs 10. ± n\/c{a—b) Cb U 16. Va^-64 ±abcxy'^^X/^^ 19. 6\/3"-24 + 10 Vl5 4x—9y 21. x-y/x+y^/y 22. x^ + xy + y"^ Exercise 118. 1. 5. 9. 13. 17. 20. ±2 — x/ac c ^ ±12 2. 2v^ 1 3.^V5 7. 2^6" 11. \/x — \^y 4.|VT 8. aV^ 10. V4 14. 12^/10 15. ±63 18. V3"+3+a/5" 'a—b^2b 21. ^/x-\-^Jy 12. 12^3 16. V^-l 19. 3^2"+ 4-6 Vs" 22. x—^\/xy + y Exercise 119. 1. l/I 2. 9V^ 3. 8 4. 108 5.4^5" 6. a^x/ab l.Aa^b^ Q. aH ^/oFb 9,ab{a + bf 10. (a-b)^^^ 11. {x+yy 12. 15. 18. 21. 23. 1. 6. 9. 13. 16. 19. 6(5-2a/6) -^yV2x 13 ANSWERS. 4 355 9 14. a + 2\^alf + b 16. x—2\/xy + y 17. i» Vl2a& 19. p 20. 20+15^3" 22. 2 + 3V3'-3V9' 24. a2(a-2V«^ + 6) Exercise 120. 2. V2a 3. \/2a 6. Va*^ 7. Vs" 11. 2!t/Sa 10. I^VSa 14. \^{a-\-b)x 17. \/a + 6 20. ^j^^^W 4. ^^a^x 8. -v^ 12. Va^ 15. V60(a+a;) 18. 3V2aa;« 21. -XJ^ 1. 4. 7. 10. 12. 16. 2-V2 1(11-6^2) a;+2Va;y+y Exercise 121. 2.|V5 6. -r's/ab 8. -3(V2'+ v^) 3.1^/15 6. -I-/V/2" a«-6 11. 3 + 2V^ 13. 2^-h 16. .7071 14. a^—2a^/b^b a^-b 17. .1716 Exercise 122. ±3V^. ±2aV— 1. ±4V-1 ±5x^/^, ±6aa;»V^» ±7a«y'V^ ±2v^xV^, ±2V3a'xy^, ±Sax^^/2x X ^/-i 9^/^ 5. 3aV^ ^- -1, -V^» +1» +\/-l 356 ELEMENTARY ALGEBRA. 7. 10. 12. 14. 16. -1, +a/-1, +1, -V-l 9- -6, -10, -18, -160, -an^ 11. 7, 1 S^/~-i, 8 + 6V^, -7 + 2^10 V^ V3, ±2, ±- 15. ±4, ±2x^/2^, ±\\^ ^/i o /' — o\ a^—h + 2a^/—b 2's/ah—a—h _(l + 2V-2), -^^^ , j3^— 1. 4. 7. 10. 13. 15. 17. 19. 22. 25. 27. 30. 33. 35. 1. 4. 7. 10. 13. 16. Exercise 123. 2. ±(2-V5) 5. ±(3+V6) 8. ±(^5 2; + V^) 11. 12. ±{^x + y+^x—y) 3. ±(2+V3) 6. Not a square. 9. ±(V6'+V5) ±(V3"+V3) ±(y2~« — -nA) ±(V^-V6) ±(2^2"- VT) 14. ±(V'a;+22/+V^-22/) ±(2v^_VlO) 16. ±{^-^y) ±(V^+1 + aAJ 20. ±(^6+ a/3) 21. ±(2'v/3'-Vi1) ±(Vl3+V3) 23. ±(1-/^14) 24. ±(l+A/a) ±(|-Vl5 + iV6") 26. ±1(^15- VlO) ±(2'\/3'+V^) 28. ±(1 + 2V^) 29. ±{^/^-^^) 31. ±(2\/5"-V'21) 32. ±(1 + V-1) 34. iCV^ + .-r- V^— x) ±{2^^—{x—y)\ 36. ±(V^ + 2+ V^-1) ±i(Vl4+VlO) at 07 (a4-6)t Exercise 124. 2. af &f c-^^ 5. 2(:c-# tn 8. (a«— S")^ 11. 'a2j 3. Onb'n 6. a«— 1 Jn— 1 9. V^ 12. v^v 14. V(a^-«T 17. V^ 15. ^x{x + yf 18. Va«^ ANSWERS. 357 20. V^ 23.1/4 5 26. (128)i 29. 0)"* 32. (6)i 35. (aa;+a;*)— i 38. 2V^3l4 4 21. V^ 41. 2 v: -13- 52. (a «t )"• 44. 2(2a-2 6)« 47. 8(4a:y)i 50. a»(a-*a;«)i 24. \/{a+xy{a—x) 27. (32)1 30. (y-i)~^ 33. (l)-i 36. ix—^)p 39. 2V^ 45. 2(3a;+3y)» 48. 4(ay)i 51. z^(xy^i 55. ± (a— 2^)^/0+^ 56. a*tya2 54. ±{a + b)\/a 57. -, Va 58. ± r\/a^ + ab a + o 59. ± -v/o^^ a—x CO. x\/o^ 61. ± -\\/a^-b^ 62. 33 V^" 63. (y-2£)(a;2-3a)^ 65. ±i'V^«6T^V 67. y5"-V2"+V7' 64. (a + 6)-2 V(a + &)* 66. V2'+V3'-V6' 68. a^+aibh + aibh + a^bi+ahbi + b^ -41 70. ^x\/lE o ,X.iV4 a'— aVT 74. 4\/S+4\/2 75. 72. -g^VlOO 4(1 +V5) a*— 6 76. ^(3V5'+3V3'+2yi0+2v^) 77. -^(1 + 2^/^) 78. -4+ Vl5 79. 2V^ ^/T5, 3^3^ 80. VlO, a/s; 2V2' 81. 2V^ V^ Vl5 82. \/2, V^. VS" 83. (V« + V^X^/a-V^ 84. (V^+2)(y^-2) 85. (xi + yi)(a;i-yi) 358 ELEMENTARY ALGEBRA. 86. {a^-\-yh){x^—y^ 88. (V-^ + 5)(V^-5) 90. (V^+ VsJcV^- V2j 92. a;f— a:i?/i + ?/i 87. (4+ Va;)(4-Va^) 89. (a;+ V5)(a^-V5) 91. :^;f + a:ii/^ + ^t 93. X5+x5y5+xi-yi + :Ay5+yi 95. 4n;l-G:ci2/^ + 9yf 98. 2:l + iC9 2/9+2/^ 94. xi—xiy^ + xiyi—xiys+yi 96, 8xi+12xiy\ + 18xiyi+27yi 97. 8a;f-12ari2/i + 18xi2/*-272/f 99. V^-VV 100. a+V^ 101. a;(V^2_^ 102. v^-V^ 103. {^x + y){^x + ^y) 104 105. xi + {xy)^+y^ 106. 2a^—x^ + 2aA/a^^ 107. a3_3aj2_(3a2 6_j3)^/ZY 108. a'^ + {4ta + 4:b)^/ab + 6ab + b^ 109. a3_3^j2_(^,3_3a2^,)yC:Y 111. 217-88 V'6 112. 9aV564 3 114. ^V243^ 116. ±(^yx+i-^x-i) 2 (x^-y) 118. x^ + y 119. 2a; 121. 2a: -^/x^-1 X 110. (100 + 18 V -2) 113. x^Q25y 115. ±(2 + 3^5) 117. ±(V^+V^ 120. -+2+^ 122. ±{x^ + xiy^+yl) l.'a:=:±32 5. x=da^, 8. x=16 11. a;=4 123. xi + l + — xi Exercise 125. -218 3. x=±9 2. a;=214 ■7a^ 6. xz=±a^/±a^—l 9. a:: 6 15. 2;=15 18. -r^y 21. .=1 12. x=l-r- 13. a;=2 4 16. a;=-5 19. a:=36 22. x=4 23. a;=l 17. x= 4. .'r=±l 7. a:=29 10. x=^a o 14. 2^=7 1 25 20, -Kl^y '6-l\2 24. a;=a ANSWERS, 359 25. x=S, - J^ 26. 2=2, -3 27. x=2, -1 1 28. a:=::^(l±\/4a'+l) 29. x=ia, a 30. x=a, 1—a 32. x=j{d\^a+l)^ 35. a:=4, 9 37. a:=21, 12 31. a;=-,-(l-a±\/l-<^« + «'^) 33. .?— rt 34. a;=l, 36. j-= I ~(-6±a/4«c + 6^)[^ 38. a:=5±2\/l3 39. x=l, ^Vl2 40. a;=VX V^ 41. 2:=3, -4^, i(-3±V33) 4L2. x=a, a{l±'\/^) 2' 4 43. a;=±3, ±a/7, ±a/-5, ±a/^ 45. a:=25, -9, 8 ±4^/29 47. x=4, 2/ =9 49. rr=17, 3/=8 50. a:=l, y=4 44. x=0, 6, 3(l±\/2) 46. x=2 ± V^, hi ± V-3) 4 4 48. x=0, ^ ; 2/=0, ^j 51. x=Sy 16; ?/=l, 9 62. (^=±5, ±3a/^1, etc, 53. a:=l + V3; -(3+^3) ^^•=±0, ±dv— ^> ^i^^' < y=±S, tS-v/— 1» etc. Exercise 126. .8. » + * a— ^» 19. x=5 Exercise 127. 11 13 8 2: ^' 5' ^3' 15' •'S' 9' y 2. A, a-x, a-6' «^^ 3. 1, a;*+x«y« + y^, *^ •^^ ^ ax + ay 4. 1:3; aft :1; a:b 6. 11 : 30; 3:5; ;r«: 2; o:c ^" 2000 7. 3 i 8. The first. ^ be— ad 5- :r- 10. a«- -&» 11. 1 :2 12. 1.414, 1.732 13. .707 14. Increased, diminished. 15. Diminished, increased. 16. $000, $720 17. 20 yr., 30 yr. 18. 6^ ft, 8^ ft 14. 2a; : 5 : : 1 : 2 Exercise 128. 15. 2x:dy ::21o:206 360 ELEMENTARY ALGEBRA, Exercise 1S9. 1. x=% 2. rr=10 3. x=26 4. x=^'- ...^4 6. x=da 7. x=\ 8. a;=4 9. x=5 xo. .=«<*;;» 11. U=±13 [ 12. i;: =4) 13. \x=±Q) =2f U=±4f Exercise 130. 1. 28 ft., 21 ft. 2. 72 yr., 60 yr. 3. 7, 6 4. 150,000, $30,000 5. 2000 sq. rd. 6. $1022^, 11090 Y^ 7. 50 mi., 30 mi. 8. 8 cu. ft. 9. ird, or 3.1416xcZ 10. Trr^, 16^, 16x3.1416 12. -^irr^, 288ir 13. mx-^ A. 11, 4irr2, 100 IT 14. d^ gal. 1. 2a 2. ( 7. 2A/a 11. wa«-i 15. 3 15. ex— 5 Exercise 131. 3. a 4. 00 5. 5 6. oo 8. 9. 1 10. 12. m 13. 14. a 16. a + 1 17. 1 18. a* Exercise 132. 1. Z=40, >S'=:220 2. Z=5, >Sr=192 3. /=3|, >S'=16i 4. ^=w, S=hn^ + n) 5. Z=2r, >S=r2 + r 7. 1=1, d=-^ 8. /=43, >S'=204 6. d: =5 9. 1 = -4' n=\ 21. 1= =37, n: =:10 20. a=l, -i; w=12, 15 22. w=ll, a=5 Exercise 133. 1. 7, 10, 13 2. $175 3. $320 4. 7, 12, 17 5. 5050 6. 5, 8, 11, 14 7. 1, 3, 5, 7, etc. 8. 14475 ft. 9. 8 da. 10. $35.70 11. 1, 2, 3, 4, 5 ANSWEKS. 361 12. 2, 5, 8, 11, etc. 13. 16. 3 da., 10 da. $1500 14. 2, 14 Exercise 134. 1. 4374 2. 2916 3.1 ^' 513 6. 2»-' 6. 4095 -?^^ 8. 2"-^ 9. 96, 189 10. 243, 5 ''• m='^ 1 4^ =a 12. Z=2r»»; >S'=2 (r" + r9 + ;•«+... .r) 19.2, 6, 18, 54, etc. Exercise 135. 1. $429.98 2. 4 yr. 3. $3200, $1600, $800, $400, $200 8. 5^3, 5v^, 15V3 9. 1, 3, 9, 27 10. 4, 16, 64 11. 6, 18, 54, etc. 12. 2, 8, 32 13. 248, 842 14. 3, 9, 27, etc. 15. (fo)' 16. ^_3y(x-i+rr« Exercise 136. 17. 9, 27, 81; 117 1. 3, 13^, 11-1, 25 « aft 2- 6-1' a«x ' a-] .' a-6' 3/- ^ 3. 5 124 192 IV 999' 1111' 11 2 11 30' 165' 900 4. 40 ft. 6. 300 rd. Exercise 6.21^ 137. min. 1. 94 2. -4 3.| 4: 7 6« + 11^ 21 i2* + T2^ 6. Ub + dc + d-a .\. 31 22 -20^+15" 7. 8a 8. {a—b + c)m^+{a + b—c)mn+ {b + c—a)n ,9 9. -1 10. S + 2x-iyi+x- -lyi+2xhy- ■i + .r*2/- 1 11. X*- 1 256 362 ELEMENTARY ALGEBRA. 14. 15. 17. 20. 24. 26. 28. 29. 32. 35. 37. 39. 42. 43. 44. 45. 46. 47. 48. 51. 63. 54. {x^ + y^) {a^—a^ y* + y^) (a^—x^ y^ + y^) {x^ + xy + y^) (x^—x y + y^) (3a; + 2)2(a: + 3) ^pq 21. 1 18. 22. x—y 1 o?-aA-\ 16. a: + ; 19, ^+y + ^ ' x—y+z 23. a;=l 1 1 ^=3' ^=4 a;=4, 2/=^? '2:=6 '25. ic=^ x\y^z->t^{^Jxy—^xz—^/yz) 21. 7— Sy'S a3 + 6a2a; + 6a2:2+a:3 + (3a2 + 7aa: + 3x2) Vo^ 31. 20 x^ + 2x^y + y^ x=l~, -3, -^{iTVm 30. a:i^ + 4:c6 + 6+4 + 4i 33. a;=-&, a;" a:;' -{a + c) 34. a;=±l 36. xt=±3, y=±2 38. a;3— ica + l 1 a4 4-14a2i2 + 64 x^—x^y\ + yi a^—x^y^ + xty^—xiv^ + y^ 40. -^^ 41. , „ ,„,, - amx^-\- {a n—h m) x^—{ap+b n) x + bp (a2 + l)(:r+i)(x-l); {a + 2b){a-2b)(a+db) y^-2, y^-dy, y*-4:y' + 2 {2x+z){lQx^-8a^z + 4:X^z^-2xz^ + z^); (x+\/^+y) 8(^i:y_l) {x-^/xy+y) (16«3a;2-24a3y2)|^ (xV^-xfif)^, {aa^ + bx^)—z 1 49. ^ 50. 2;= ± a/^ -^(1 T VS) a:=±^V2; 2/= ±2 a/2" l+x + x^ + x^+ etc., a:», a;* 52. ^^=±-^^265 + xy {x + yy 55. 30 57. 69. 60. 56, x + y—2\/xy 1 3 , ±V9; v^ -V-i rc=144 58. 2a; + 7 {m~n + r)y^ + {m + n—r) yz + {r+n—m) 2* x=-{^^a-l)^ 61. Vr 62. 2 63. a;2-2a;=-2 ANSWERS. 3Ga 64. x'*—{a + b + c + d)x^ + iab + ac + ad + bc + bd + cd)x^— {ab c + abd + acd + bcd)x + abcd=0 65. x=^ i-b± ^12 a c + b^) 66. x=a, {1 + \/a)^ 67. a:=10, 6, 8±5a/5^ |-(25 ± a/385), i(25±V809), y=6, 10, 8t5V5^ ^(25^^^). ^(25^^869) 68. y» + 22/3 + 42/'^ + 3i/ + 3 69. a2.y4 + (a-2a&)2/' + (i2-64-l)/ 70. ^=±"7^^^^ 71. a:=(a-2\/a)2 72. :r=4, 1; 2/=l, 4 73. a:=27, 8; y=8, 27_ 74. -r=2, 2/=3 75. a:=3, 5, -5±V^^; 2/=^, -"i, -5ta/-26 _ 76. Odd, odd, even. 77. +1, -1, ±1, V±4 78. x=-\, x-1 Exercise 138. 1. $3600, $3200 2. 15 yr., 45 yr. 3. 48, 52 4. 37 -I 6. 16 ft., 4 ft. 7. 1 hr. 29 rain. 22^ sec. 8. 10 da. 9. 2663, 1662, 1000 10. 56 rain. .1. Ig mi. 12 . 27y^ rain., 10— min., or 43^^ rain., 60 rain. .3. 24, 36 14 . 10 mi. 15. $2^, 50 ct. 16. pvp^—q^i qyp^—ff 18. 3 ct. 19. 60, 50, 40 20. 8 da. 21. 5 hr. 22. $1200 23. 20,000 24. 3, 4, 5 25. 36 26. $126, $882 27. 2, 4, 6, 8, 10, 12, 14 28. 180 lb., 160 lb. 29. 8, 6 30. $480 31. $7000, $8000 32. $225, 20^ 910 454 413 33. 10^^ da., 14^ da., 17j^ da. 34. 10 rain. yoU 16\3 oil 36. 30 rai. 36. 3, 9, 27, 81 37. 4, 6, 8, 10, 12 38. i(l+ VS), i(3+V^ 39. ^^\ seconds. 40. 9753 41. 4, 6 42. 13 yr. 43. 64 cu. ft, 512 cu. ft. 44. 2, 3, 4 45. 4840 46. 1 rd. 47. 10 ft, 12 ft. 48. $15,000 49. 12 rai. 50. 144 oz. SC)4: ELEMENTARY ALGEBRA. 51. 2^ mi., 2jj mi. 52. 15^ 53. 106-^ mi. 54. 19 da. 55. 6 hr., 3 hr., 2 hr. APPENDIX. Exercise 1. 1. a;2+a; + l 2. x'^-x—\ 3. 2 a:— 1 4. 5a;2-3 5. 2x^—^x + Q 6. 2a^ + a + l 7. x + S 8. 2a; + 3 9. a:2 + 2a: + l 10. x-5 11. 2/ + 1 12. a8-a4 54^58 13. a + b + c 14. a;2 + a:3/+/ 15. a^ + 1 16. a:-2y Exercise 2. 1. x + y 2. 3;2 + a;y + 2/2 3. 2a; + l 4. 3x-2 5. 2a; + 3 6. 2x+Q 7. a:2+a; + l 8. dx—2 Exercise 3« 1. 2(3a:2 + 7rc + 4)(4a;2 + 3a:-10) 2. (a;2+a; + l)(a;+5)(2a:+l) 3. (a+l)(a2_a-l)(a2 + a + l) 4. (a2-2a + 3)(a2 + 2a-3)(7a + 3) 5. (2a;+y)(a;2+a:3/+y2)(^2_^2/+2/2) 6. (a-2+a:i/ + 2^«)(a: + y)(a;+2y) 7. (3a;2 + 2a:+l)(a:+l)(a;2 + l) 8. (a;2 + aa; + a2)(2a:-a)(3a; + a) 9. (3a;2-4a:+2)(a;-3) (2x4-3) 10. (2a4+5a2 + 3)(2a2-7)(2a2 + 7) 11. (5 22-1) (422^1) (522 + 2+1) 12. (Sx + 2)(x^-x^ + x-l){a^ + x^+x + l) 13. (3a: + 4)(2a:3 + 3a;2-4a: + 2)(3a;3 + 2a;2-3a: + l) 14. (2a2_6a-7)(6a'-lla2_37a_20) 15. 3(w + 3)(2m2-m + l)(3m2-7w + 4) 16. (n-2 a) {n^ + an^ + a^n + a^) (3 w^-a n + a^ OF THE { UNIVERSITY M V OF / V- THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW NC ANINITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETTURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. '''*' -nn tA lO'-ti* W§ APR 14 IJJ' fEB i9mw I rf£C 2b m B' ^> ^ ^p ME3DC»C DBC .IW *'*^^ ■-ii «^W^I|^Mtm. LD 21-100ot-7,'33