I T . T A- Digitized by the Internet Arciiive in 2008 witin funding from IVIicrosoft Corporation littp://www.arcliive.org/details/collegealgebraOOwellricli WELLS" MATHEMATICAL SERIES. Academic Arithmetic. The Essentials of Algebra. Academic Algebra. Higher Algebra. University Algebra. College Algebra. Plane Geometry. Solid Geometry. Plane and Solid Geometry. Plane and Solid Geometry. Revised. New Plane and Spherical Trigonometry. Plane Trigonometry. Essentials of Trigonometry. Logarithms (flexible covers). Elementary Treatise on Logarithms. Special Catalogoie and Terms on application. COLLEGE ALGEBRA. WEBSTER WELLS, S.B., Pkopessob op Mathematics in the Massachusetts Institute of Technolort. LEACH, SHEWELL, AND SANBORN. BOSTON. NEW YORK. .CHICAGO. COPTRIGHT, 1890, WEBSTER WELLS. NottoDoti ^rcBS : J. S. Gushing & Co. — Berwick & Smith. Norwood, Mass., U.S.A. :^A 5^3 PEEFAOE. This work is designed as a text-book for the use of colleges and scientific schools. The first eighteen chapters have been arranged with ref- erence to the needs of those who wish to make a revieiv oi: that portion of Algebra preceding Quadratics. While com- plete as regards the theoretical parts of the subject, only just enough examples are given to furnish a rapid review in the class-room. Attention is respectfully invited to the following : The proofs of the five fundamental laws of Algebra — the Commutative and Associative Laws for Addition and Multiplication, and the Distributive Law for Multiplication — for positive or negative integers, and positive or nega- tive fractions, Chapter II. ; Arts. 114 and 115 ; Arts. 208 to 215 ; Arts. 230 and 232 ; Chapter -XVI. ; the proofs of the fundamental laws of Algebra for irrational numbers, Chapter XVII. ; Arts. 350 and 351; Arts. 355 to 357; Chap- ters XXIV. and XXVI. ; the proof of the Binomial Theorem for positive integral exponents, Arts. 443 and 444 ; Chapter XXXI. ; the Note to Art. 469 ; the proof of the Binomial Theorem for fractional and negative exponents, Art. 483 ; Arts. 532 to 538 ; Art. 542 ; Chapters XXXVIL, XXXVIIL, and XL.; Art. 650; the proof of Descartes' Eule of Signs for Positive Roots, for incomplete as well as complete equations, Art. 653 ; Arts. 657 to 663 ; Arts. 673 and 674 ; iv PREFACE. the Graphical Eepresentation of functions, Arts. 682 to 688; Art. 689; the solution of Cubic and Biquadratic Equations, Arts. 706 to 716 ; Art. 718. In Appendix I. will be found graphical demonstrations of the fundamental laws of Algebra for pure imaginary and complex numbers ; and in Appendix II., Cauchy's proof that every equation has a root. WEBSTER WELLS. Boston, 1890. OONTEISTTS. PAGE I. Definitions and Notation 1 II. Fundamental Operations 9 III. Addition; Subtraction; Use of Parentheses 24 IV. Multiplication 29 V. Division 36 VI. Formulae 41 VII. Factoring 50 VIII. Highest Common Factor . 59 IX. Lowest Common Multiple 66 X. Fractions 70 XI. Simple Equations containing one Unknown Quantity 85 XII. Simple Equations containing two or more Unknown Quantities 98 XIII. Discussion of Simple Equations 113 XIV. Inequalities 124 XV. Involution 130 XVI- Evolution 136 XVII Surds; the Theory of Exponents .... 1G4 XVIIL Imaginary Numbers lOO XIX. Quadratic Equations 203 XX. Theory of Quadratic Equations 221 XXL Problems involving Quadratic Equations . 234 XXII. Equations Solved like Quadratics .... 239 XXIII. Simultaneous Equations involving Quad- ratics 246 XXIV. Indeterminate Equations of the First Degree 262 V vi CONTENTS. ^ PAGE XXV. Ratio and Proportion 268 XXVI. Variation 279 XXVII. Arithmetical Progression 285 XXVIII. Geometrical Progression 295 XXIX. Harmonical Progression 305 XXX. The Binomial Theorem; Positive Inte- gral Exponent 309 XXXI. CONVERGENCY AND DIVERGENCY OF SeRIES . 316 XXXII. The Theorem of Undetermined Coeffi- cients 328 XXXIII. The Binomial Theorem ; Fractional and Negative Exponents 344 XXXIV. Logarithms 352 XXXV. Compound Interest and Annuities . . . 378 XXXVI. Permutations and Combinations .... 385 XXXVII. Probability (Chance) 393 XXXVIII. Continued Fractions 407 XXXIX. Summation of Series 418 XL. Determinants 429 XLL Theory of Equations 450 XLII. Solution of Higher Equations .... 498 APPENDIX I. Demonstration of the Fundamental Laws of Algebra for Pure Imagi- nary and Complex Numbers . . . 529 APPENDIX II. Cauchy's Proof that every Equa- tion HAS A Root 54(1 ANSWERS TO THE EXAMPLES. QUADRATIC EQUATIONS. 20^ XIX. QUADRATIC EQUATIONS. 339. A Quadratic Equation is an equation of the second degree (Art. 179), containing but one unknown quantity. A Pure Quadratic Equation is a quadratic equation involv- ing only the square of the unknown quantity ; as, 2a^= 5. An Affected Quadratic Equation is a quadratic equation involving both the square and the first power of the un- known quantity ; as, 2 a-^ — 3 a; — 5 = 0. PURE QUADRATIC EQUATIONS. 340. A pure quadratic equation may be solved by reduc- ing it, if necessary, to the form x^ = a, and then extracting the square root of both members. 1. Solve the equation Sa.-^ -f- 7 = — - + 35. Clearing of fractions, 12 x- -f 28 = 5 a;^ + 140. Transposing and uniting, 7 a.-^ = 112, or, a.-2 = 16 ; which is in the form x^ = a. Extracting the square root of both members, we have a; = ± 4. (:N"otes 1 and 2.) Note 1. The sign ± is placed before the result, because the square root of a number is either positive or negative (Art. 236). ■ Note 2. It follows from Art. 267, Note, that hke roots of the mem- bers of an equation are equal. 2. Solve the equation 7a^ — 5 = 5.^'^ — 13. Transposing and uniting, 2 a;' = — 8, or, a^ = — 4 j which is in the form x^ = a. 204 COLLEGE ALGEBRA. Whence, x = ±-\/— 4 = ±2V^^ (Art. 332). Note 3. In this case the values of x are imaginary (Art. 326) ; it is impossible to find any real values of x which will satisfy the given equation. EXAMPLES. 341. Solve the following equations : J _5 L = _^. 3 _A_ = § ^. ■ 6ic2 4a^ 16* ■ 4 — a; 3 4 + a;' h 2. 4-V3ar' + lG = 6. 4. b XT — a 5. 2(a; + 3)(a;-3) = (a; + l)2-2a;. 6. (3a; - 2) (2x + 5) + (5a; + 1) (4x - 3)- 91 = 0, ^- 2~" + l2-24~^+^' g 2a;^-5 _ 3«Mi_2 _ x" - 10 ^ ^ 3 7 6* ^ *~^^ 10. 2 ^^ — = -/'3 x + b a — 2b 2;ir — 1 3 \ 3.x-^' + 2 11. (2 X - a) (a; + 6) +(2 a; + a) {x-b) = cr + b'. 12 5a.-^-l 3a;^ + l 89 ^^ a-2-3 x-2 + 2 (.r^ - 3) (ar^ + 2) 13. X -\- Var 4- 3 = ^ Var^ + 3 14 1 1 _V3 15. Vl + a; + a;- + Vl — a; + x-^ = V7 + V3. 16. M-l , |a;=^_2 R'-3 , Ix" - -4 3' ■V QUADRATIC EQUATIONS. 20^ AFFECTED QUADRATIC EQUATIONS. 342. By transposing all terms containing the unknown quantity to the first member, and all other terms to the second member, any affected quadratic equation may be reduced to the form ax^ -\-hx = c. 343. To solve a quadratic equation of the form a!(? -{-hx = c. If the coefficient of x"^ is a perfect square, the equation may be solved by adding to both members such an expres- sion as will make the first member a perfect trinomial square, an operation which is termed completing the square, and then extracting the square root of both members. 1. Solve the equation 9ar^ + 2 a; = 11. A trinomial is a perfect square when its first and third terms are perfect squares and positive, and the second term plus or minus twice the product of their square roots (Art. 125). Therefore the square root of the third term is equal to the second term divided by twice the square root of the first. Hence the square root of the expression which must be 2x 1 added to 9 x^ + 2 x to make it a perfect square, is — , or -• -J ox o Adding to both members the square of -, we have Extracting the square root of both members (Art. 126), (See Art. 340, Notes 1 ami 2.) ^"^ + 3"* 3 206 COLLEGE ALGEBRA, o 1^10 o 11 Whence, x = 1 or — — • We then have the following rule : Complete the square by adding to both members the square of the quotient obtained by dividing the coefficient of x by twice the square root of the coefficient of a^. Extract the square root of both members. Note. The values of x may be verified (Art. 175) as follows : Putting a; = 1 in the given equation, 9 + 2=11. ~ = -^. i|l_|=n. If the coefficient of x^ is unity, the rule may be modified to read as follows : Complete the square by adding to both members the square of half the coefficient of x. 2. Solve the equation cc^ — llx = — 2S. Adding to both members the square of — j ^_n. + (f)=-28 + f=2. Extracting the square root, 11 ,3 ^ — ^ = ± ^• 2 2 Whente, a; = ii ± ^ = 7 or 4. 2 2 344. If the coefficient of a^ is not a perfect square, it may be made so by multiplication. 3. Solve the equation 3 a;^ + 13 tc = — 12. The coefficient of a^ may be made a perfect square by multiplying each term by 3 ; thus, QUADRATIC EQUATIONS. 207 9a^ + 39x = -36. Completing the square by the rule of Art. 343, Extracting the square root, 2 2 3a; = -— ±- = -4or -9. 2 2 Whence, x-= or — 3. 3 If the coefficient of v? is negative, the sign of each term must be changed. 4. Solve the equation —%x^-\-15x = — 2. Changing the sign of each term, we have 8ar-15rc = 2. The coefficient of a? may be made a perfect square by multiplying each term by 2 ; thus, 16 ar'- 30 a; = 4. Completing the square, ' 4 ^ ^16 16 Extracting the square 4a; root, 15 _ 4 -?■ 4x = f* 17 4 = 8 or 1 2 Whence, X = :2 or - 1 208 COLLEGE ALGEBRA. 345. Fractions may be avoided in completing the square by multiplying both members of the given equation by four times the coefficient of a?. For consider the equation ax^ -\- hx = c. Multiplying both members by 4 a, we have 4 a^a^ + 4 dbx = 4 ac. Completing the square by the rule of Art. 343, 4 aV + 4.abx + b' = b''-i-A ac. Extracting the square root, 2ax + & = ± V&^ + 4ac. 2ax = — b± Vb^ + 4 ac. Whence, ^^-b±^b' + 4ac 2a It will be observed in the above case that the quantity required to complete the square is the square of the coeffi- cient of x in the given equation. . 5. Solve the equation 2a^ — 3x=:lA. Multiplying both members by 4 times 2, or 8, 16ar^-24a;=112. Adding to each member the square of 3, Wx" - 24a; + 3^ = 112 + 9 = 121. 4a; -3 = ±11. 4a; = 3± ll = 14or -8. Whence, x = ~ or — 2. 2 If the coefficient of x in the given equation is even, the rule may be modified to read as follows : Multi2^ly both members of the equation by the coefficient of a?, and add to each the square of half the coefficient of x in the given equation. QUADRATIC EQUATIOJ^S. 209 6. Solve the equation 9 a;- + 14 a; = — 1. Multiplying both members by 9, 81x- + 12G.T = -9. Adding to each member the square of 7, 81x2 + 126x + 72 = - 9 + 49 = 49. 9a; + 7 = ±V40 = ±2VlO (Art. 295). 9a; = -7±2VlO. Axn. - ^ ± 2 VlO Whence, x = Note. The method of complethig the square exempUfied in the present article, is known as tlie Hindoo Method. EXAMPLES. 346. Solve the following equations : 1. 4.x' + 3x = 10. 7. 11 a; + 12- 36 .^•- = 0. 2. x^ — x = 6. 8. 6.^■2-5a; = -l. 3. 25£c2-15a; = -2. 9. 9x' + 6x=19. 4. 49i»2 + 49x + 10 = 0. 10. 32rc2^ 20a; - 7 = 0. 5. 2a;2_i5^^_13 n 32 a; - 48 o.-^ = - 3. 6. 8a;2 + a;-34 = 0. 12. 12ar'+ 5a; + 1 = 0. SOLUTION OF QUADRATIC EQUATIONS BY A FORMULA. 347. It was shown in Art. 345 that if ax^ -\- bx = c, ,, —b± Vb^ + 4 ac ... X then x = - "^ — ^ (1) 2 a This result may be used as ^ for mi da for the solution of any quadratic equation in the form ax^ + bx = c. 210 COLLEGE ALGEBRA. 1 . Solve the equation 2 a^ + 5 ic = 18. Here, a = 2, b = 5, and c = 18; substituting in (1), - 5 ± V25 + 144 _ - 5 ± VT69 ^- 4 4 ^-5^13^2or-5. 4 2 2. Solve the equation 110 a^ — 21 cc = — 1. Here, a = 110, & = — 21, and c = — 1 ; therefore, 21 ± V441 - 440 21 ±1 1 1 X = = = — or — 220 220 10 11 Dividing both terms of the fraction in equation (1) by 2, we have by Art. 297, x = = ^ ; (^) a a which is a convenient formula to use in case the coefficient of X in the given equation is even. 3. Solve the equation — Sx^ + 14 a; = — 3. Here, a = — 5, 6 = 14, and c = — 3 ; substituting in (2), ^^_7±V49 + 15^ -T±8 ^_1^^3_ EXAMPLES. Solve the following equations : 4. 2x-' + 3a; = 27. 8. 6a;2 + 7a; = -l. 5. 3ar'-2a; = 5. 9. 4x2_8aj_3 = o. 6. x2-7x = -10. 10. 5x^ + 12a; = -4. 7. 5x'-^ + x = 18. 11. 0x2 -25a; + 14 = 0. QUADRATIC EQUATIONS. 211 12. S0a;-9a;2 = 16. 14. 15a;2-8a; = 16. 13. a.-2 + 39a; + 387 = 0. 15. 10 -21a;-10a;2 = 0. 348. The following equations may be solved by either of the preceding methods, preference being given to the one best adapted to the example under consideration. 1. Solve the equation 3x 2a;-3 ^ _5 2a;-3 3x ~ 6 The equation must first be reduced to the form aar+6a;=c. Multiplying both members by 6 cc (2 a; — 3), ISa^ - 2(2a; - 3)2 = - 5a;(2a; - 3). ISar^ - 8a;2 + 24a; - 18 = - 10^2 _,. -^^^ 20x'-\-9x = 18, which is in the form aar + 6a; = c. Solving by formula (1) of Art. 347, we have - 9 ± V81 -{- 1440 40 -9±39_3^^ 40 4 6 5' The artifice employed in the following example may often be used to advantage. 2, Solve the equation 2ar'-a; + 3 3a^+a; + 2 ^ 2ar'-2a; + 3 2a;-l 3a; + l a;-l The equation may be written in the form a;(2a;-l) + 3 .T(3.'r +1) + 2 ^ 2a;(a; -l) + 3 2a; -1 "^ S.T + 1 ~ a;-l Dividing each numerator by the corresponding denomi- rator. ^ !12 COLLEGE ALGEBRA. Q 9 o 2a;-l 3a; + l x-1 Whence, — ^ h — - — = "^ . 2x-l 3x+l a;-l Clearing of fractions, 9aj2 _ g^. _ 3 _,_ 4^ _ g^ _^ 2 = 18a;2 - 3a; - 3. -5ar-9a; = -2. Whence, ^.^ 9±V81+40 -10 9 ± 11 ^ 1 = = — 2 or — -10 5 EXAMPLES. Solve the following equations : 3 2 + ?=-:^. 4 2__5^^_15 ' X 2 2 ' 5 2x 4.x'' 5. 4a;(18a;-l) = (lOx-l)l 6. {3x-5y-{x + 2y=-5. 7. (x + Sy- {x-iy=19. 8. {x-iy-{3x-\-8y-(2x + 5y = 0. 9. 2a; + 3 2a; + 9 ^ 8 + a; 3a; + 4 14. X a; - 1 3 a;-l X 2 ^ 5 3a; + l 1 a; a;^ 4 15. a; 5 - a; 15 5 — a; a; 4 11. 4 a; 14 -a; ^14 a; + l 16. a- + 1 a: + 3 8 a; + 2 a; + 4 3 12. ^^^ _ E _ 34 = 0. 17. V20 + a;-a;2^2a; 13. 3ar^ l-8a; a; a; -7 10 5 18. 2V:t^+— = 5. V.C •w. 19. 20. 21. 22. 2a. 24. 25. QUADRATIC EQUATIONS. 2a; -1 3x 213 3a;-l +i = o. a;3 _ a.-^ ^ 7 ^ 11 cc2-{-3x-l '*^"'~3' 2ar' + 3a;-5 ^ 3a;^ + 4a;-l 2x2-a;-l 3x2-2x- + 7' 7 3 ^22 a;2-4 x + 2 5* 1 111 3' x^-l 3a;-3 a; + l 3x + 2 2a; + 3 ^ 2 CC4-3 a;-3 12+5a; 1 9-ar' J 2+a; = 0. 12 — 5a: 1 — 5a; x a; + l = l. 0. 26. V4X--3 27. 2V^- Vx + 5 = a;4-2 6 28 Va; + 5 ^2 ^-^ . a — 1 a;+5 '^ x + l 29. V3a; + 1 = V9a; + 4- V2a;-1. 30. (a;-3)(a; + 4) = (a;-7)V3. 31 ^ — ^ _ a; — 4 _ a; + l _ a; + 2 x — 3 x — 5 x-\-4: x + 5 32 gar' + 4a; 4a;^ + 8a; + 5 3a:^-x 3a; + 2 2a; + 3 = 0. 33. + 1 + Va^_l_9(a;.t.l) a; + l^Va;2-l 8 34. (11 4- 6V2)x'-\- (1 V2 - \))x = 7 V2 - 10. 214 COLLEGE ALGEBRA. SOLUTION OF LITERAL QUADRATIC EQUATIONS. 349. For the solution of literal affected quadratics, the methods of Art. 345 will be found in general the most con- venient. 1. Solve the equation x^ + ax — bx — ab = 0. The equation may be written ar' + (a — b)x = ab. Multiplying both members by 4 times the coefficient of x', 4ar^ + 4(a — 6)a; = 4a6. Adding to each member the square of a — 6, 4.ari-{. 4:(a - b)x + {a - by = Aab + a" - 2ab + b'' = a' + 2ab + b\ Extracting the square root, 2x-\-(a-b)= ±(a + b). 2x= -{a-b)±(a + b). Therefore, 2x=-a + b + a + b = 2b, or 2x= —a-^b — a — b= —2a. Whence, x = b or —a. Note. If several terms contain the same power of x, the coefficient of that power should be enclosed in a parenthesis, as shown above. 2. Solve the equation (m — l)a^ — 2m^cc = — 4m^ Multiplying both members by ??i — 1, (m - lyx"" - 2w?{m - l)x = -4.m\m - 1). Adding to each member the square of w?, {m-iyx^-2m-(^,n-l)x + vi'= vi*- 4mH 4m*. QUADRATIC EQUATIONS. 215 Extracting the square root, {m — l)x — m,^= ±(m- — 2m). (m — l)x = m^-\-m^—2vioTm"— m--\-2m = 2m(m— 1) or 2m. Whence, a; = 2 m or -1^^. m — 1 EXAMPLES. Solve the following equations : , 3. ar' + 2ma:=2m + l. 7. 0:^ + 2(0 + 8)0; = -32c. 4. u?-2ax={h-\-a){h-a). 8. a^ - m^a: + m^'a; = m^ 5. a? — ax + hx — ab=2 0. 9. acx^ — bcx — adx = —bd . 6. cc2 + (a + l)a;= -a. 10. (x+22)y-(x+py=37p' 11. 6ar^-h9ax-{-2bx + 3ab = 0. 12. 2a;(a-a;)^a^ ^^ a: + l = ^ + ^. 3a — 2a; 4 a; w m 13 _^ = _^. 15. ^-±1 = ^L±1. ' a; + l ?i + l Vx Va 16. V(a + 6)a; - 4a& = a; - 2&. 17. v^^r4;^=(^ + ^>(«-^). \18. 2V^3^ + 3V2^ = ' ^^ + ^^ . > Va; — m 19. ^= + L^ = l + ?. a + Va^ — a; a — V a^ — a; « ar' + l^ g + S c_ X c a + b a; — a x — b a b V 20. 21(3 COLLEGE ALGEBRA. 22 a^ + m ^ x-\-n ^5 X + ?i a; + ??i 2 oo cc — a.x + a 5a.'c — 3a — 2 ^ <*«5 . 1 = U. x+ a X — a a- — X- 24. _^ = 1 + 1 + 1. a + b -\-x a b X 25. x{x-\-b + c) = {a-\-c){a-b). 26. (c + a - 2&)a;2 -|-(a + 6-2c)a;-+-6 + c-2a = 0. 27. a^^ I 3a''a;^ 6a^ + a6-26^ 6^x c c? c 28. (3a2 + 62) (a:2 _ ^ ^ 1) ^ (3^2 j^ a"") {:i? + x + 1). 29. (2 a^ + a6) a^ - (7 a^ + 11 a6 + 4 62) a; + 3 a^ + 4 a& = 0. 30 ^ — O' — b — c x — a + b + G _ _ 6^ + c^ X — a + b + c X — a — b — c be 31 . (a + b) Va^ + 6^ + X - (a - &) Va^ + 6^ _ a; = «=> + 61 32. (a? + l){a* + a' + l) = 2x{a* + 3a' + l). 33 c[^ + ^(g + l) _ x + b _ ax — b(a-j-l) X 2ax — b 34. {6a' + llab-10b')x'-(ldd''+9ab + 13b')x = -15a' + 17ab + Ab\ 35. (a" -b^-c'-2bc) {x" + 1) = 2a;(a2 + 62 + 0^ + 26c) SOLUTION OF EQUATIONS BY FACTORING. 350. Let it be required to solve the equation (x-S){2x + 5) = 0. It is evident that the equation will be satisfied when x has such a value that one of the factors of the first member is equal to zero ; for if any factor of a product is equal to zero, the product is equal to zero. QUADRATIC EQUATIONS. 217 Hence, the equation will be satisfied when x has such a value that either a; - 3 = 0, (1) or 2a; + 5 = 0. (2) K Solving (1) and (2), we have a; = 3 or z K Therefore the roots of the given equation are 3 and In general, if A, B, ..., K are any integral expressions (Art. 65) containing x, the equation ^•x5x ••• x/r=0 may be solved by placing the factors of the first member separately equal to zero, and solving the resulting equations. EXAMPLES. 351. 1. Solve the equation 4a;^ — 2 a; = 0. Factoring the first member, 2a;(2a;-i) = 0. Placing the factors separately equal to zero (Art. 350), 2a; = 0, or a; = ; and 2.T — 1 = 0, or x = — 2 Therefore the roots of the given equation are and — A 2. Solve the equation x'^-|-4a;2 — a; — 4 = 0. Factoring the first member (Art. 120), ix-\-^){7?-\) = ^. Therefore, a; -}- 4 = 0, or a; = — 4 ; and a;^ — 1 = 0, or a; = ± 1. 218 COLLEGE ALGEBRA. 3. Solve the equation a:^ — 1 = 0. Factoring tlie first member, Therefore, x — 1=0,otx = 1; and a^ + x + l = 0. Whence, by Art. 347, x = -^^Vl^^ ^ -l±V-3 ^ Z L Note. The above examples are illustrations of the principle (Art. 634) that the degree of an equation (Art. 179) indicates the number of its roots ; thus, an equation of the third degree has three roots ; an equation of the fourth degree, four roots ; etc. It should be observed that the roots are not necessarily unequal ; thus, the equation x^ — 2x + l = may be -written (x — 1) (x — 1) = 0, and therefore the two roots are 1 and 1. Solve the following equations : 4. (Zx + l){^a?-4Q) = 0. 6. 3:x? + l^a? = 0. 6. 2a:3_i8a; = 0. 7. {x" -^){x' + 4.) = 0, 8. a?-\-^x'-24.x = Q. 9. {x-2){2a? + 13x+2Q) = Q. 10. 24.x' -2a? -127? = 0. 12. x3_27=:0. - 11. a;* + x = 0. 13. a;*-16 = 0. 14. (a;2-a2)(4a^-4aa;-15a2) = 0. 15. (x2-5cc + 6)(ic2+7a; + l2)(ar'-3a;-4) = 0. 16. 0^-1 = 0. - 18. 27x3 + 640=^ = 0. 17.\8a;3^125 = 0. "- 19. x^-a^.g^j^. 9 = 0. 20.'^2a;3 - 3x5+ 2x - 3 c= 0. 21. 3x* + x*-3Gx-2-12x = 0. QUADRATIC EQUATIONS. 219 22. Solve the equation V2 — 3 a; + Vl -f 4 a; = V 3 + a;. Squaring both members, 2 - 3a; + 2V2 - 3a;vT+4^ + l + 4a;= 3 +05, Whence, 2 V2 - 3 xVl + 4 a; = ; or, V2-3a;Vl+4aj = 0. Squaring, (2 - 3 a;) (1 + 4 a;) = 0. 2 Therefore, 2 — 3a; = 0, or x = ~; o and 1 4-4a; = 0, or a; = — -• 2 23. Solve the equation a; + 8 x + 9 a; + 10 a; + 6 Adding the fractions in each member, we have 7a; + 58 ^ 7a; + 58 • (a; + 8)(a;+9) (a; + 10) (a; + 6) ' Since 7a; + 58 is a factor of each member, we may place it equal to zero ; thus, 7a; + 58 = 0, or a; = -— • The remaining roat is given by 1 1 {x + 8) (a; + 9) (a; + 10) (a; + 6) or, (a; + 8) (a; + 9) = (a; + 10) (a; + 6). That is, af-hnx + 72 = x^ + 16x-\-60, or, a; = — 12. Therefore the roots of the given equation are — 12 and _58 7' 220 COLLEGE ALGEBRA. Solve the following equations ; 24. Vx + a + Vxl^b = V2ic + a + 6, v 25. -±- + -^— + -^— = 3. a; + 1 a- + 2 . .r + 3 26 ^~^' = a;^-4ar^ + 9 ' 2 + x^ ar' + 4a^ + 9' 27. V2 — 3 cc - V7 + a; = V5 + 4a;. V 28. V(2 + a;)(a;-f-l)+V(2-a;)(a;-l)= V6^ a; +8 a; — 8 x + 6 x — 6 30. Vo/*^ + aa; — 6 a- — Va;^ — ax — 6 a^ = V^a;-— 12 a^. 31. V4 + ox — a.-2 = 2V2a; + Va;- + 3a; — 4. 32 x^+5^'+l 2x-^-2x + l 3x^-15.^ + 1 ^ 6x^+6x--? . x + 5 X— 1 X— 5 x + 1. 1.1.1.1 = 0. x-\-a + b'x — a — b x + a — b x — a + b 36. Vx^ - 8x + 15 + Vx-' 4- 2x - 15 = V4x^ - 18x + 18. 36.* (a-xy + (b-xy = {a + b-2x)\ 3^ x^-K2x + 4 ^ 4 x^ — 2 X -f- 4 x^ 38. (3x + 2)3-(2x-l)3 = (x+3)l * Factor the first member by the method of Art. 130. THEORY OF QUADRATIC EQUATIONS. 2-21 XX. THEORY OF QUADRATIC EQUATIONS. 352. A quadratic equation cannot have more than two different roots. Every quadratic equation can be reduced to the form 3!^ -\-px = q. If possible, let this equation have three different roots, Vi, rg, and r^. Then by Art. 174, r^^ +2^n = Q> (1) r/ +i)?-2 = q, (2) and ri-\-pr^ = q. (3) Subtracting (2) from (1), we have I"! — t'i + i> (^1 — rg) = 0. Whence, (ri + r^ (rj — rj) + p (rj — r^ ) = 0, or, (ri - rj) {r^ + r2 + i?) = 0. Therefore, either r^ — rg = 0, or else ^i + rg + i> = 0. But ri — ?*2 cannot be zero, for by hypothesis r^ and r, n + ^-2 +P = 0. (4) In like manner, by subtracting (3) from (1), we obtain ^i+r3+iJ = 0. (5) Subtracting (5) from (4), rg — rg = 0. But this is impossible, for by hypothesis r^, and r^ are different; hence a quadratic equation cannot have more than two different roots. Note. It follows from the above that an expression cannot have more than two different square roots. 22 COLLEGE ALGEBRA. 353. Let Ti and rg denote the roots of the equation a:r -\-px = q. By Art. 347, r, = -P+^P' + ^q , (1) and ^^^-P-^P' + ^Q. (2) Adding (1) and (2), Multiplying (1) and (2), we have P'-i -4g ^^,^^ f-(p'^ + ^g) (Art. 108) 4 =-^- Hence, if a quadratic equation is in the form x^+pa; = q, the sum of the roots is equal to the coefficient of x with its sign changed, and the product of the roots is equal to the second member with its sign changed. EXAMPLES. 1. Find the sum and product of the roots of the equa- tion 2a:2_7a._i5^0. Transposing — 15, and dividing by 2, the equation becomes o 7x 15 7 15 Hence, the sum of the roots is - and their product Find by inspection the sum and product of the roots of : 2. x=' + 5a; + 2 = 0. 6. 8x'-x + 4. = 0. 3. a^-7x + ll = 0. 7. 6a;-4ar' + 3 = 0. 4. x' + c>x-l = 0. 8. 7 -12x-Ux' = 0. 5. 22ty'-3x-2 = 0. 9. 4x2- 4ax + a^ - 6- = 0. THEORY OF QUADRATIC EQUATIONS. 223 10. If rj and rj are the roots of the equation x^ -f-^jx = ^, (a.) Prove that 1 + 1=-^. (6.) Prove that r/ + ri =p- + 2q. 354. By aid of the principles of Art. 353, a quadratic equa- tion may be formed which shall have any required roots. For, let ri and r^ denote the roots of the equation x'+px-q^O. (1) Then by Art. 353, p = — (ri + r^) and —q = nr.^. Substituting these values in (1), we have a^— (^1 + ^2)^ + '"1^2 = 0, or, {x — ri) (x — r2) = 0. That is, any quadratic equation may be written in the form {x-r,)(x-n) = 0, (2) where Ti and rj are its roots. Hence, to form a quadratic equation which shall have any required roots. Subtract each of the roots from x, and place the p)'roduct of the resulting expressions equal to zero. EXAMPLES. 1. Form the quadratic equation whose roots shall be 4 and 4 By the rule, (x - 4) ( a; + ^ ) = 0. Multiplying by 4, (x - 4) (4a; + 7) = 0. That is, 4 ar' - 9 x - 28 = 0. 224 COLLEGE ALGEBRA. Form the quadratic equations whose roots shall be -. 2. 4, 5, 4. 3, _|. 6. I I 8. -f, 0. 3.,. 3. a.-.-f. 7. -|i 9. -I 4 10. a-6, a + 2&. 12. 2 + 5 VS, 2 - 5 Va 11. ?n(l + m), m(l-m). 13. ? ^ FACTORINQ. 355. A quadratic expression is an expression of the form aoi? -\-hx •\- c. The principles of Art. 354 serve to resolve such an ex- pression into two factors, each of the first degree in x. We have aar^ + 6x + c = a^ar + -| + ^Y (D Now let rj and rg denote the roots of the equation a? + *^ + °=0. ^ a a By Art. 354, (2), the equation can be written in the form (x — rj) (x — Ti) = 0. Hence, the expression a? -\ ' -\-~ can be written in the form {x — ri) (x — r^) . Substituting in (1) , we have aa? + hx + c = a{x — r-^){x-ri), (2) where rj and r^ are the roots of the equation ar^+ — + - = 0, or aoi? + hx + c = 0\ which, we observe, is obtained by placing the given expression equal to zero. THEORY OF QUADRATIC EQUATIONS. 225 EXAMPLES. 1. Factor 6 0^ + 7 a; -3. Placing the expression equal to zero, we have Whence by Art. 347, ^ ^ - 7 ± V49 + 72 ^ -7±ll ^l _3 12 12 3 °^ 2* 1 3 Here, a = 6, rj = -, rg = — - ; substituting in (2), o Z = (3a;-l)(2a; + 3), 2. Factor 4 + 13 a;- 12 a:2_ Solving the equation 4 + 13 k — 12 a;^ = 0, we hare a.^ -13±Vl<39 + li)2 ^ -13±19 ^ 1 4 _24 -24 4°^ 3* Whence, 4 + 13a; - 12a.-2 = - 12 ("a; + -) fx - ^"j = (1 + 4a;) (4 -3a;). Note, It must be reraembered that a, in the formula a(x—r{) («— rj"), represents the coefficient of x- in the (jiven expression ; thus, in Ex. 2, we have a = — 12. 226 COLLEGE ALGEBRA. Factor the following : 3. ar' + 13a; + 40. 13. 12x'-4:X-5. 4. x'-Ax-GO. 14. 6-ic-2a^. 5. a^- 11a; + 18. 15. Sa^^^igx-S. 6. 2x2 + 7a; -15. iq 10x^-233; + 6. 7. 5a;2 + 36a; + 7. 17. l-Sx-a:^ 8. 8a;2-18a; + 9. 18. 15 + 26a; - 24 a;^^ 9. 39-10a;-a;2^ 19. Gx" -llax-SBa". 10. 2 + x-6ar'. 20. 20x^ + 41mx + 20m\ 11. x' + 4.x + l. 21. 12a^-{-7xy-10y-. 12. 9ar-6a;-4. 22. 21x' -mm7ix + 21mV. 23. Fa,ctoT2x^-3xy-2y^-7x + 4:y + 6. Placing the expression equal to zero, we have 2a^ - x(^3y + 7) = 2?/^ -4y-6. Solving by the formula of Art. 347, ^_ 3y + 7±V(3y + 7)2+16y^-32y-48 X- - ^ 3y + 7±V25r + lQy + l 4 ^ Sy + 7±(5y + l) 4 8y + 8 -2v + 6 o . o — V + 3 = ^^ or ^ ^ = 2^/ + 2 or — ^ ^ ■ . Therefore, 2x2_33,y _ 2^2 _ 7^_l_4y ^g = 2[x-(2y + 2)]^a:-:::^J = (u;-22/-li)(:i.« + y-3). THEORY OF QUADRATIC EQUATIONS. 227 Factor tlie following : 24. x^ + xy—- + 5a -Z>- 12. 29. X? — bxy -\-Qy^ — 5xz + l-iyz-^r ^z^. 356. Quadratic expressions may also be factored by the artifice of completing tlie square (Art. 343), in connection with Art. 128. 1. Factor 9 ar^— 9a; — 4. By Art. 343, the expression 9 a;" — 9aj will become a per- fect square by the addition of [ - | ; thus, 9a;2 _ 9x - 4 = 9ar - 9a; + (~^'- - - 4 = ("'^~2 + l)(^''-|-2) (^^-^-l^S) = (3.^' + l)(3a;-4). Note 1. If the x^ term is negative, tlie entire expression sliould be enclosed in a parenthesis preceded by a — sign. 2. Factor 3-12a;-4.^•2. 3 - 12a; - 4ar = - (4a;2 + 12 a; — 3) = _(4a;2 + 12a; + 9-9-3) = - [(2 a; + 3)2- 12] = (2a- + 3 + Vl2)-(-l)(2.T + 3-Vl2) = (2V3 -1-3 + 2:i;) (2V3 - 3 - 2a;). 228 COLLEGE ALGEBRA. Note 2. If the coefficient of x^ is not a perfect square, the expres- sion should be divided by this coefficient. 3. Factor 6 ar'- 19 a; + 10. 6si^ - 19x + 10 = Qfx'-— + ^ = 3lx--\.2fx-- 3J \ 2 = (3a; -2) (2a; -5). Note 3. Tlie coefficient of a;- may sometimes be made a perfect square by dividing it by one of its factors. Tlius, in factoring 8x- — 20 a; — 7, the coefficient of x^ may be made a perfect square by dividing it by 2. EXAMPLES. Factpr the following : 4. a;2 + 12a; + 35. 9. 36ar + 24.t;-5. 5. 4ar'-9a;-a 10. 8 :^r + 38 a; + 35. 6.' 16ar-31a;+15. 11. 42 + 23.^- lOa.-^. 7. 3a;-' + 7a; -6. 12. 15a;-'- 14a; + 31 8. 9ar-12a;+L 13. 25 x-' - 20 a; - 2. Note 4. In factoring expressions of the form ax" +bx + c, the nietliod of Art. 35(3 is preferable when the coefficient of x^ is a perfect square ; if the coefficient of a;^ ig not a perfect square, the method (if Art. ;J55 is shorter. THEORY OF QUADRATIC EQUATIONS. 229 357. Certain expressions of the fourth degree may be factored by completing the square. 1. Factor a* + a-b' + h\ By Art. 125, the expression will become a perfect square by the addition of a-lr to its second term ; thus, a' + a~W + h^ = {a* + 2 arb" + b') - a-b^ = {a' + b-Y-a-b'' = {a- + &- + ab) (a^ + &2 _ ab) (Art. 128) = {a- + ab + b-) (cr — ab + b-).. 2. Factor 9 a;* - 39 o^ + 25. 9a;* - 39.^2 + 25 = (9x* - 30ar + 25) - 9^^ = (3ar^-5)2-(3a^)2 = {Sx" + 3x - 5) {3af - Sx - 5). Note. The expression may also be factored as follows : 9x^-39x2 + 25= (9x4 + 30 x2 + 25)-69x2 = (3x2+5)2-(xV69)2 ' = (3x2 + xV69 + 5)(3x2-xV(39 + 5). Several of the expressions in the following set may be factored in two different ways. 3. Factor a;-* + 1. x* + l = {x'-\-2xF + l)-2x' = {a^ + iy-(xV2y = (af + xV2 + l){x'-xV2 + l). EXAMPLES. Factor the following : 4. a;* + a.-2 + l. 7. m' + nv'n' +25 n\ 5. x* — 7x- + l. 8. l-13b^ + 4.b\ 6. 4:a^-8a-b^+b\ 9. x'-12xy + Ay\ 230 COLLEGE ALGEBRA. 10. 4a^ + 8a- + 9. 15. IGx* -4.9m-x^ + 9m\ 11. 4.m' + 7m' +16. 16. 9a;*-G.^- + 4. 12. a*-5aV + a;*. 17. 9a'' + 14aW + 25m^ 13. .'c^ + 81. 18. 4-32w2+497i^ 14. 4a^ + loa-&- + 16&^ 19. 16x* -Adxh/ + 25y\ 358. The equation o;^ + 1 = may be solved, as in Art. 351, by placing the factors of the first member (Ex. 3, Art. 357) separately equal to zero, and solving the result- ing equations ; thus, ar + xV2 + 1 = 0; whence, a; = ~ ^^ * V2-1 2 -V2±^ /-2 2 V2±V- > 72 and X' — xV2 + 1 = 0; whence, x = - EXAMPLES. Solve the following equations : 1. .r^ + 16 = 0. 4. a;* + «* = 0. 2. a;^-Ga;^ + l = 0. 5. a;-* - 8.^- + 4 = 0. 3. x'--x^-hl = 0. 6. x^-^ + l=0. DISCUSSION OF THE GENERAL EQUATION. ' 359. The roots of the equation axr + 6.x + c = are given 2a It is evident that : I. Ifb^ — 4:ac is positive, the roots are both real (Art. 318), beivr/ rational or irrational according as 6' — 4ac is, or is not, a perfect square. THEORY OF QUADRATIC EQUATIONS. 23:1 II. I fir — 4, ac is zero, the roots are equal. III. Iflr — 4i ac is negative, the roots are both imaginary. "' 360. The roots of the equation ocr +px = q are ^ -p+-Vir + 4q ^^^ ^ -p--Vp^ + iq^ 2 2 We will now discuss these values for all possible real values of p and q. I. Suppose q positive. Since p^ is essentially positive (Art. 109), the expression under the radical sign is positive and greater than p"^. Therefore the radical is numerically greater than p. Hence r^ is pjositive, and r^ is negative. If p is positive, rg is numerically greater than r-^; that is, the negative root is numerically the greater. If J) is zero, the roots are numerically equal. If p is negative, ?"i is numerically greater than r, ; that is, the positive root is numerically the greater. II. Suppose q~0. The expression under the radical sign is now equal to p'^. Therefore the radical is numerically equal to p. If p) is positive, r^ is zero, and n is negative. If p is negative, r^ is positive, and r.2 is zero. III. Stcppose q negative, and 4g numerically Kp"^. The expression under the radical sign is now positive and less than p-. Therefore the radical is numerically less than p. If p is positive, both roots are negative. If p is negative, both roots are positive. IV. Suppose q negative, and 4 q nunierically equal to p-. The expression under the radical sign is now equal to zero. Hence y-j is equal to r^. (Compare Art. 359, II.) 232 COLLEGE ALGEBRA. If p is positive, both roots are negative. If p is negative, both roots are positive. V. Suppose q negative, and 4,q numerically >jp^. The expression under the radical sign is now negative. Hence both roots are imaginary. (Compare Art. 359, III.) The roots are both rational or botli irrational according as J)- + 4 g is or is not a perfect square. EXAMPLES. 1. Determine by inspection the nature of the roots of the equation 2.x-- — 5a; — 18 = 0. The equation may be written o? ^ = 9. Since q is positive and p negative, the roots are one positive and the other negative ; and the positive root is numerically the greater. In this case, p- -(- 4 g = — -f 36 = — ^ ; a perfect square. 4 4 Hence the roots are both rational. Determine by inspection the nature of the roots of the following : 2. .r-' + 2.^-15 = 0. 8. 4ar^-7 = 0. 3. a;2-10.T = -25. 9. 9 x-^ + 30 .'k = - 25. 4. 3a.-2-4a; = 0. 10. 9ar + 8 = 24ic. 5. a.- + 5a; + 3 = 0. 11. 2x- + x = 0. 6. 3ar'-5a; + 4 = 0. 12. 11 - 27.x- - 18.x- = 0. 7. 6x2 -7:k- 10 = 0. 13^ 4x2 + 13a; + 11 = 0. 361. It sometimes happens that on solving, by the ordi- nary process of clearing of fractions, an equation involving the unknown quantity in the denominator of a fraction, certain values of the unknown quantity are obtained which will not satisfy the given equation. THEORY O^ QUADRATIC EQUATIONS. 233 Thus, let it be required to solve the equation ^ox-2_2x±l J2_^^ (1) Multiplying each term by x^ — 4, we have (3a.- - 2) (re + 2) - (2a; + 1) {x - 2) + 12 = 0. 3ar' + 4x - 4 - 2ar= + 3a; + 2 + 12 = 0. ar' + 7a; + 10 = 0. (2) ^Tru - "^ ± ^-^9 - 4U Whence, x = — 2 -7±3 = -2 or The value — 2 does not satisfy the given equation ; for, 2i±l and -12_ x-\-2 ar — 4 with this value of o:, the fractions and -^; are infinite (Art. 212). The equation may be solved in such a way as to obtain only one root, as follows : It is evident from (2) that the sum of the fractions in the first member of (1) is a^ + 7a; + lQ ^ (a;4-2)(a; + 5) _cf + 5. a^_4' (a; + 2)(a;-2) a;-2* tc + 5 Then (1) may be written = 0. Clearing of fractions, a; + 5 = 0. Whence, a; = — 5. It follows from the above that every value of x obtained by solving an equation which involves x in the denominator of a fraction, should be verified in order to make sure that it satisfies the given equation. 234 COLLEGE ALGEBRA. XXI. PROBLEMS. INVOLVING QUADRATIC EQUATIONS. 362. 1 . A man sold a watch for .$ 21, and lost as much per cent as the watch cost him. Kequired the cost of the watch. Let X = the cost in dollars. Then, x = the loss per cent, and X • -^, or -^ — the loss in dollars. 100 100 By the conditions, -^ = .r — 21. ^ 100 Solving, X = 70 or 30. That is, the cost was either $70 or § 30 ; for either of these answers satisfies the conditions of the problem. 2. A farmer bought some sheep for ^72. If he had bought 6 more for the same money, they woidd have cost him % 1 apiece less. How many did he buy ? Let X = the number bought. Then, — = the price paid for one, X and '"^ = the price if there had been more. x + G 72 7'^ By the conditions, — = — - — V 1. ^ X x + 6 Solving, X = 18 or - 24. Only the positive value of x is admissible, for the negative value does not satisfy the conditions of the problem. Therefore, the number of sheep was 18. Note 1. In solving problems which involve quadratics, there will usually be two values of the unknown (juantity ; and those values only should be retained as answers which satisfy the conditions of the problem. PROBLEMS. 23.f Note 2. If we should modify the given problem so that it shal read: "A farmer .bought some sheep for $72. If he had bought G feiver for the same money, tliey would have cost him $ 1 apiece more. How many did he buy ? " " we should find the answer 24. (Compare Art. 207.) PROBLEMS. 3. Find two numbers whose difEerence is 11, and whose sum multiplied by the greater is 513. 4. Find three consecutive numbers whose sum is equal to the product of the lirst two. 6. Divide 20 into two parts such that one is the square of the other. 6. Find two numbers whose sum is 13, and the sum of whose* cubes is 637. 7. Find four consecutive numbers such that, if the first two are taken as the digits of a number, that number is the product of the other two. 8. A merchant bought a quantity of flour for f 96. If he had bought 8 barrels more for the same money, he would" nave paid f 2 less per barrel. How many barrels did he buy, and at what price ? 9. A merchant sold a quantity of wheat for -f 39, and gained as much per cent as the Avheat cost him. What was the cost of the wheat ? 19. If the product of three consecutive numbers is divided by each of them in turn, the sum of the thrCe quotients is 74. What are the numbers ? 11. A crew can row 5i miles down stream and back again in 2 hours and 23 minutes ; if the rate of the stream is 3i miles an hour, find the rate of the crew in still water. 236 COLLEGE ALGEBRA. 12. A man travels 9 miles by train. He returns by a train which runs 9 miles an hour faster than the first, and accomplishes the entire journey in 35 minutes. Required the rates of the trains. 13. At what price per dozen are eggs selling when, if the price were raised three-pence per dozen, one would receive four less for a shilling ? 14. A merchant sold goods for $ 16, and lost as much per cent as the goods cost him. Required the cost of the 15. A certain farm is a rectangle, whose length is twice its breadth. If its length should be increased by 20 rods, and its breadth by 24 rods, its area would be doubled. Of how many acres does the farm consist ? 16. A man travelled by coach 6 miles, and returned on foot at a rate 5 miles an hour less than that of the coach. He was 50 minutes longer in returning than going. What was the rate of the coach ? 17. A square court-yard has a gravel-walk around it. The side of the court lacks one yard of being six times the width of the walk, and the number of square yards in the walk exceeds the number of yards in the perimeter of the court by 340. Find the area of the court, and the width of the walk. 18. The circumference of the hind-wheel of a carriage is greater by 4 feet than that of the fore- wheel. In travelling 1200 yards, the fore-wheel makes 75 revolutions more than the hind-wheel. Find the circumference of each wheel. 19. A cistern can be filled by two pipes running together in 2 hours and 55 minutes. The larger pipe by itself will fill it sooner than the smaller by 2 hours. What time will each pipe separately take to fill it ? TROBLEMS. 237 20. The telegraph poles along a certain railway are at etxual intervals. If there were one more in each mile, the interval between the poles would be decreased by 8| feet. Find the number of poles in a mile. 21. A and B gained in trade $1800. A's money was in the firm 12 months, and he received in principal and gain $ 2600. B's money, which was $ 3000, was in the firm IG months. How much money did A put into the firm ? 22. The sum of f 100 was divided among a certain num- ber of persons. If each person had received $ 4.50 less, he would have received as many dollars as there were persons. Required the number of persons. 23. My income is f 1000. After deducting a percentage for income tax, and then a percentage, less by one than that of the income tax, from the remainder, the income is reduced to f 912. Find the rate per cent of the income tax. 24. If f 2000 amounts to $ 2163.20, when put at com- povmd interest for two years, the interest being compounded annually, what is the rate per cent j)er annum ? 25. A man travelled 102 miles. If he had gone 3 miles more an hour, he would have performed the journey in 5| hours less time. How many miles an hour did he go ? 26. A man has two square lots of unequal size, together containing 15,025 square feet. If the lots were contiguous, it would require 530 feet of fence to embrace them in a single enclosiire of six sides. Find the area of each lot. 27. A man has a cask full of wine, containing 72 gallons. He draws a certain number of gallons, and then fills the cask up with water. He then draws out the same number of gallons as before, and finds that there are 50 g'allons of pure wine remaining in the cask. How many gallons did he draw each time ? 238 COLLEGE ALGEBRA. 28. A set out from C towards D at the rate of 3 miles an hour. After he had gone 28 miles, B set out from D towards C, and went every hour -^ of the entire distance ; and after he had travelled as many hours as he went miles in an hour, he met A. Eequired the distance from C to D. 29. Find a number such that the sum of its cube, twice its square, and the number itself, is twenty times the next higher number. 30. A courier travels from P to Q in 14 hours. Another courier starts at the same time from a place 10 miles the other side of P, and arrives at Q at the same time as the first courier. The second courier finds that he takes half an hour less than the first to accomplish 20 miles. Find the distance from P to Q. 31. A and B start at the same time, from two places 180 miles apart, to meet each other. A travels 6 miles a day more than B ; and the numbej of miles travelled each day by B was equal to twice the number of days which elapsed before he met A. How many miles did each travel in one day? 32. A man bought a number of $20 shares, when they were at a certain rate per cent discount, for $1500; and afterwards, when they were at the same rate per cent pre- mium, sold them all but 60 for f 1000. How many shares did he buy, and hoAV much did he give apiece ? 33. The first digit of a certain number is the square of the second. The number exceeds that formed by reversing its digits by twice the amount by which the number next greater than the given number exceeds that formed by reversing its digits. Eequired the number. EQUATIONS SOLVED LIKE QUADRATICS. 239 XXII. EQUATIONS SOLVED LIKE QUADRATICS. EQUATIONS IN THE QUADRATIC FORM. 363. An equation is said to be in the quadratic form when it is in the form as?"- + &.x" = c, where n is any rational number (Art. 269). For example, x^ — Qo?= 16, and x-3 + x' '^ = 72, are equations in the quadratic form. 364. Equations in the quadratic form may be readily solved by the rules for quadratics. 1. Solve the equation x'^ — 6.^" = 16. Completing the square by the rule of Art. 343, a;6_6a;'' + 9 = 16 + 9=25. Extracting the square root, a;'' - 3 = ± 5. or'- = 3 ± 5 = 8 or - 2. Extracting the cube root, » = 2 or - ^2: Note. There are also four imaginary roots which may be obtained by the method of Art. 351. 2. Solve the equation 2x + 3V;« = 27. Since ^x is the same as x^i, this is in the quadratic form. 240 COLLEGE ALGEBRA. Multiplying by 8-, and adding 3- or 9 to both, members, 16x + 2Wx + 9 = 21G + 9 = 225. Extracting the square root, 4 V^ + 3 = ± 15. Wx = - 3 ± 15 = 12 or - 18. ■y/x = 3 or 2 cj . n 81 Squaring, ic = 9 or — • 3. Solve the equation 16 x~ ^ - 22a;~^ = 3. Completing the square by the rule of Art. 343, (f) = 10.-._22.-* + (^l=3 + f = ^'. . -I 11 ^13 4a; 5- — = ±— . 4 4 . -3 11 13 1 ^ Ax ^ = — ± — = — - or b. 4 4 2 Extracting the cube root. S 9. 1 f3\i or - 2 V2; Raising to the fourth power, _i 1 X ^ = — or 10 Inverting both members, cc = IC or ( ^ Note. In solving equations of the form x^ = a, first extract the root corresponding to the numerator of the fractional exponent, and afterwards raise to the power corresponding to the denominator. Particular attention must be paid to the algebraic signs ; see Arts. 109 and 2;3G. EQUATIONS SOLVED LIKE QUADRATICS. 241 EXAMPLES. Solve the following equations : 4. x'-2rjx' = -lU. 12. 4a; -15 = 17 Vx 5. 8x« + 37.r^ = 216. 13. a.- + a;* = 702. 2 4 6. .r^ - 97 x--" + 1296 = 0. 14. 2;c'' + 3a;" - 5G = 0. 7. 12a;-- + a;-i = 35. 15. 3x^-94x^ = 64. 8. 69-20.T-3-x-''=0. 16. 3x' -f-26 = -16.x~^. V 9. x-^-21x-- = -10S. 17. 2x-^+61x"^-96 = 0. y_10. 32.x^ + ^ = -33. 18. 6x-^-5x~ 3 =-1184. 11. .^•3- 3x^ = 88. 19. Sx"^ - 15- 2x^ = 0. V 365. An equation may sometimes be solved with refer ence to an expression, by regarding it as a single quantity. 1. Solve the equation (.t — 5)^ — 3(.x — 5) ^ = 40. Completing the square by the rule of Art. 343, we have (._5).-3(.-5)i + (|J=40 + |=f. Extracting the square root, (x-5)^ = ~±l^ = 8 or -5. 2 2 Extracting the cube root, (x-5)' = 2 or -a/5. Squaring, a; — 5 = 4 or -^25. Whence, a; = 9 or 5 + ^^^25. 242 COLLEGE ALGEBRA. Certain equations of the fourth degree may be solved by the rules for quadratics. 2. Solve the equation x*+ 12ay^ + 34ar— 12a; — 35 = 0. The equation may be written (x' + 12 x^ + 36 x^) - 2 .^2 - 12 a; = 35. That is, (x- + 6xy--2 {x- ■j-6x) = 35. Completing the square, {x^ + 6xy- 2(x" + 6x) + l= 36. Extracting the square root, (x^ + 6x)-l = ±6. a^ -{-6x = 7 or — 5. Completing the square, x--\- 6x' + 9 = 16 or 4. Extracting the square root, .r + 3 = ± 4 or ± 2. ^Vlience, a; = - 3 ± 4 or - 3 ± 2 = 1, ~ 7, — 1, or — 5. Note 1. In solving equations like the above, the first step is to complete the square with reference to the a;* and x^ terms. By Art. 343, the third term of the square is the square of the quotient ohtaincd by dividing the x^ term by twice the square root of the x* term. 3, Solve the equation x^ — 6.r + 5^xr — 6x-\- 20 = 46. Addine: 20 to both members, (a^ _ 6a;+ 20) + 5 Var - 6a; + 20 = 66. Completing the square, 289 (x"-Gx+ 20) +5Var'-6.x-+20-h=r = 66 + ^ = ^ 4 4 4 Extracting the square root, Va;2-6a; + 20 + ^^ = ± ^. Vx--— 6 a; + 20 = 6 or — 11. (juaring, a;- — 6a; + 20 :^ 36 or 121 EQUATIONS SOLVED LIKE QUADRATICS. 243 Completing tlie square, x2_6ic + 9 = 25 or 110. Extracting the square root, a;-3=±5 or ± VllO. Whence, x = 8, - 2, or 3 ± VIlO. Note 2. In solving equations of the above form, add sucli a quantity to both meUibers tliat the expression without tlie radical in the first member may be the same as that within, or some multiple of it. 4. Solve the equation 2a^ + 5cc — 2 xy/x^ + 5a; — 3 = 12. The equation may be written x^ + Bx -2x^%^ + 5x - 3 + a.-2 = 12. Subtracting 3 from both members, (ar + 5a; — 3) — 2a; Va;- + 5a; — 3 + a;^ = 9. Extracting the square root, Va;^ + 5a; — 3 — a; = ± 3. Va;2 + 5a;-3 = a;±3. a;2 -f- 5» — 3 = ar ± Ct + 9. Therefore, -a; or 11a; = 12. Whence, . = -12orl|. 5. Solve the equation —^ 1 '- = — XT — X X- — 3 2 a."^ — 3 Representing — by y, the equation becomes y 2 or. 2f + 2 = 5y. Solving, y = lov2. 244 COLLEGE ALGEBRA. That is, ■ 4^ = J or 2. x- — X 2 Taking the first value, 2ar — 6 = ay^ — x. Or, a^ + a; = 6. Solving, x=2 or —3. Taking the second value, ar — 3 = 2 a^ — 2 cc. Or, -ar^4-2a; = 3. Solving, a; = 1 ± V— 2. EXAMPLES. Solve the following equations : -- 6. (x2 - 2 a;)2 - 18 (ar^- 2 a;) = -45. 7. a:'' + 8ar''-10.T2-104x + 105 = 0. — 8. 2a;2_,.i^V2a:2+l = 12. 3a; + 6V2ar^-3a; + 2 = 14. 10. (3 x~ + x-iy-26{3x^-^x-iy' = 27, 11. ar' + 7 + V^+T = 20. 12. (2a.-2 + 3x--l)- + 2a;- 4-3.1; -3 = 0. 13. -^+^+1 = 1 x' + l X 2 14. 2ar'-3a;-21 = 2a;Va;'-3a; + 4. 15. Va; + 10 + v a — b = c. The same notation will be used in the case of two or more equa- tions, each involving double signs. S^x-+ 42/2 =7G. (1) (3/-llx-= 4. (-0 9 a;- + 12?/- = 228. 12r-44./;-= 16. 53x-^ = 212. a;-' -4. .r = ± 2. (3) SIMULTANEOUS EQUATIONS. 247 Thus, the equations a; = ± 2, ?/ = ± 3, can be read either a; = + 2, 2/= + 3, or x=— 2, ?/ = — 3. And tlie equations x = ± 2, ?/ ;= =p 3, can be read either a; = + 2, 2/ = — 3, or x = — 2, y = + 3. 367. Case I. When each equation is in the form aa? + hy- = c. 1. Solve the equations Multiplying (1) by 3, Multiplying (2) by 4, Subtractin.o- Whence, Substituting from (3) in (1), 12 + 4 ?/- = 76. 2-f = 16. Whence, y = ± 4. Therefore, x = 2, ?/ = ± 4 ; or, x = — 2, y = ± 4. Note. In tliis case there are four possible sets of values of x and y whicli satisfy the given equations : 1. x = 2, 2/=4. 3. x = -2, y = 4. 2. x = 2, y = — 'i. 4. x = -2, y = -4:. It -would be incorrect to leave the result in the form a; = ±2, y — ±4:; for, by Art. 366, Note, this represents only the first and fourth of the above sets of values. EXAMPLES; Solve the following equations : 2 (4:x'+ f-= 61. 4 f 8a.-2-lly2= 8. X x- + 6r = 159. ■ (I2ar + 13r = 248. . 3_ 55a;2-9?/- = -121. ^ j x-- + ?/- = o(«- + ?>-). l7y- — 3x-= 105. ' iix- — y- = ria{r>a — ih). 248 COLLEGE ALCiEBUA. Q)x-y G5 a; + 2/ ((a; + 6t/)^-(5a; + 32/)(32/-a;) = 162. * l(3a;-4?/)2-(6a;-2?/)- = -195. 368. Case II. When one equation is of the second degree^ and the other of the first. Equations of tliis kind may always be solved by finding the valiie of one of the unknown quantities in terms of the other from the simple equation, and substituting this value in the other equation. 1. Solve the eouations . \'^^'' ~ ^*^ = ^^- (^) 1 X +2y = 7. (2) ■ From (2), "^ = ^~^- (^) Substituting in (1), 2x^-x (^-^^ = G f^^^^Y Clearing of fractions, 4 x-^ — 7 x + x-^ = 42 — 6 aj. 5ar-x = 42. Solving, '^ x = 3 or 5 7+li 7 — 3 5 Substituting in (3), y = ^ or — —L- Therefore, x = 3, ?/ = 2; or, x = -—, y = —- Note 1. In this case there are only two possible sets of values of x and y which satisfy the given equations : Note 2. Certain cxanipk'.s where one equation is of the third degree, and the otiier of the first, may be solved by the metluxl of Case IL SIMULTANEOUS EQUATIONS. 249 3. EXAMPLES. Solve the following equations : f2a;--3/ = -10 \3x + y = l. ^x + y = -3. \xy = - 54. 9. x^y ^x-y= 1. U^ + 2/^ = 113. 10 ix+y=2a. ■ lx' + y' = 2(a' + b'). f ar + xy — y- = — 19. il. (2a; -2/ =-1. \x-y = -7. ( x^ -f = - 117. 12. (x' + 3a;2/-r = 23. {x + 2y = 7. 'Ix -y =- 3. (a)^ + 2/3 = 217. la; +?/ = 7. 13. < [x y^lO y^x 3 3x-2y = -12. f.T-2/ = l. 14. <27r-3xy=15a-10a''. l3x+2y =12a-13. ( a;y = a^ + a. 369. Case III. When the given equations are symmetrical (Art. 74) with respect to x and y, and one equation is of the second degree, and the other of the second or first. Equations of this kind may always be solved by combin- ing them in such a way as to obtain the values oi x + y and x — y. \x + y = 2. (1) 1 xy = -15. (2) Squaring (1), x' + 2 xy -{- y- — 4t. (3) Multiplying (2) by 4, Axy =-60. (4) Subtracting, x^ — 2xy -\-y- = 64. Extracting the square root, x — y = ± 8. (5) 1. Solve the equations 250 COLLEGE ALGEBRA. Adding (1) and (5), 2 a; = 2 ± 8 = 10 or - 6. Whence, x = 5 or — 3. Subtracting (5) from (1), 2y = 2 q: 8 = - 6 or 10. Whence, y = — 3 ov 5. Therefore, cc = 5, y = — 3; or, a; = — 3, y = 5. Note 1. In subtracting ±8 from 2, we have 2:^:8, in accordance with the notation explained in Art. 3G6, Note. In operations with double signs, ± is changed to ip, and qp to ±, whenever + would be changed to — . Note 2. The above equations may also be solved by the method of Case II. ; but the symmetrical method is shorter, and more elegant. 2. Solve the equations \ lx^ + xy+f = 2S. (1) (2) Dividing (1) by (2), x-y = 2. (3) Squaring (3) , ar — 2 xy + ?/" = 4. (4) Subtracting (4) from (2), 3 a;?/ = 24, a;^ = 8. (5) Adding (2) and (5) , a.-' -f 2 xy + y- = 36. Whence, x + y = ±6. (C) Adding (3) and (G), 2 a; = ± G + 2 = 8 or -4. Whence, a; = 4 or -2. Subtracting (3) from (G), 2?/ = ±G-2 = 4or -8. Whence, ?/ = 2 or -4. Therefore, a; = 4, y =2 ; or, a; = — 2, y'~ — 4. Note 3. The above equations are not symmetrical according to the definition of Art. 74 ; but the method of Case III. may often be used in cases where the given equations are symmetrical except with respect to the signs of the terms. Note 4. Certain examples in which one equation is of the third degree, and the other of the first or second, may be solved by the method of Case III. . SIMULTANEOUS EQUATIONS. 251 « o, 1 . . ( ar -r ?/- = 50. (1) 3. Solve the equations - ( xy = ~7. (2; Multiplying (2) by 2, 2 a;?/ = -14 (3) Adding (1) and (3), cc^ + 2,x\j + 1/ = 36. Whence, x + ?/ = ± 6. (4) Subtracting (3) from (1), a? -2 xy + y' = 64. Whence, x — y = ±?>. (5] Adding (4) and (5), 2a; = 6 ± 8, or - 6 ± 8. Whence, x = 7, — 1, 1, or — 7. Subtracting (5) from (4), 2?/ = 6 ip 8, or - 6 rp 8. Whence, ?/ = — 1, 7, — 7, or 1. Therefore, x=±T, ?/ = :p 1 ; or, x=±l, ?/ = ip 7. EXAMPLES. Solve the following equations : ^ + 2 .x-7/2 + /• = - 21. 44 I i>j' - 2/' + X 4- y = - 2. (a; + ?/)(2.x- + 32/-l) = 3. Vx-^ + ?/2 4- x — y = 12. 45. V(a; - ?/)-(.x-' + /) = 35. r Va^ — Va;-2/ = ll. 46. ^ l^ ^x'y — y'-x = 60. ( X--2 xy + 3xz = —lG. \ {y-\- 1>) (2 + c) = a^. 47. -] 2 .T - 3 2/ = 7. 49. \ {z + c) (x + « ) = 6-. [ 3.r 4- r>z - - 14. [ (X 4a) (.'/ 4 ^) = c-. r a;^4r + z^=14. ( (y + z) {x + y + z)=2. 48. -j 2:«- 3.7 + 2 = 11. 50. } {z +a:){x + y + z)=3. x + 2y-z=:-G. [{x + y){x + y + z)=i. SIMULTANEOUS EQUATIONS. 259 PROBLEMS. " 372. 1. The sum of the squares of two numbers is 106, and the difference of their squares is ^ the square of their difference. Find the numbers. 2. The difference of the squares of two numbers is 55, and the product of their squares is 576. Find the numbers. 3. If the length of a rectangular field were increased by 2 rods, and its breadth by 3 rods, its area would be 108 square rods ; and if its length were diminished by 2 rods, and its breadth by 3 rods, its area would be 24 square rods. Find the length and breadth of the iield. . cJ 4. The sum of the cubes of two numbers is 407, and the sum of their squares exceeds their product by 37. Required the numbers. 6. If the product of two numbers is multiplied by their sum, the result is 520 ; and the sum of the cubes of the numbers is 6.37. Find the numbers. 6. A man bought 6 ducks and 2 turkeys for $ 15. He bought four more ducks for $14 than turkeys for $9. What was the price of each ? 7. Find a number of two figures, such that, if its digits are inverted, the sum of the number thus formed and the original number is 33, and their product 252. n 8. The sum of two numbers exceeds the product of their square roots by 7 ; and if the product of the numbers is added to the sum of their squares, the result is 133. Find the numbers.. 9. The sum of the terms of a fraction is 17. If the numerator is increased by 5, and the denominator dimin- ished by 5, the product of the resulting fraction and the original fraction is •^. Required the fraction. 260 COLLEGE ALGEBRA. 10. A rectangular garden is surrounded by a walk 7 feet wide ; the area of the garden is 15,000 square feet, and of the walk 3696 square feet. Find the length and breadth of the garden. 11. A rectangular field contains an acre. If its length were increased by 4 rods, and its breadth by 3 rods, its area would be increased by 100 square rods. Find the length and breadth of the field. 12. A man rows down stream 12 miles in 4 hours less time than it takes him to return. Should he row at twice his usual rate, his rate down stream would be 10 miles an hour. Find his rate in still water, and the rate of the stream. 13. A distributes' $ 180 equally amongst a certain num- ber of persons. B distributes the same amount amongst a number of people less by 40, and gives to each person $ 6 more than A does. What amount does A give to each person ? 14. A, B, and C together can do a piece of work in one hour. B does twice as much work as A in a given time ; and B alone requires one hour more than C alone to per- form the work. In what time could each alone perform the work ? 15. Two couriers, A and B, start at the same time from wo towns, P and Q, respectively, and travel towards each other. «k When they meet, it is found that A has travelled 72 miles more than B ; also, that A will arrive at Q in 9 days, and that B will arrive at P in 16 days. Required tlie distance between P and Q, and the rates of the couriers. 16. If the product of two numbers is added to their sum the result is 47 ; and the sum of their squares exceeds thei sura by 62. Required the numbers. Note. Keprescut the numbers by x + y and x — ij. SIMULTANEOUS EQUATIONS. 261 17. The sum of two numbers is 7, and the sum of their fourth powers is 641. Find the numbers, 18. The difference of two numbers is 2, and the differ- ence of their fifth powers is 242. Find tlie numbers. 19. A sets out to walk to a town 7 miles off, and 20 min- utes afterwards B starts to follow him. When B has over- taken A, he turns back and reaches the starting-point at the same instant that A reaches his destination. If B walked at the rate of 4 miles an hour, what was A's rate ? 20. Three vessels ply between the same two ports. The first sails half a mile an hour faster than the second, and makes the trip in li hours less time. The second sails three-quarters of a mile an hour faster than the third, and makes the trip in 2|- hours less time. Eequired the dis- tance between the ports. 21. A and B run a race of four miles. A reaches the half-way post five minutes before B ; he then diminishes his speed 3 miles an hour, while B increases his speed 4 miles an hour, and beats A by seven minutes. Eequired the rates of A and B at first. 22. A cistern can be filled by three pipes, A, B, and C, when opened together, in 6 hours. If A filled at the same rate as B, it would take 8^ hours for A, B, and C to fill the cistern; and the sum of the times required by A and C alone to fill the cistern is double the time required by B alone. What time will each pipe alone require to fill the cistern ? 262 COLLEGE ALGEBRA. XXIV. INDETERMINATE EQUATIONS OP THE FIRST DEGREE. 373. It has already been shown that a single equation containing two or more unknown quantities is satisfied by an indefinitely great number of sets of values of these quantities (Art. 189) ; and, in general, that a set of m in- dependent equations containing more than m unknown quantities is satisfied by an indefinitely great number of sets of values of the unknown quantities involved in it (Art. 204). Such equations are called indeterminate. If, however, the unknown quantities are required to sat- isfy other conditions, the number of solutions may be finite. 374. We shall consider in the present chapter the solu- tion of indeterminate equations of the first degree, containing two unknown quantities, in which the unknown quantities are restricted to positive integral values. Every such equation can be reduced to one of the forms ax ±by = c, or ax ±by = — c, where a, b, and c represent positive integers which have no common divisor. The equation ax -\-by = — c cannot be solved in positive integers ; for, if x, y, a, and b are positive integers, ax + by must also be a positive integer. Again, the eqxiations ax ±by = c and ax — by = — c can- not be solved in positive integers if a and b have a common divisor. For, if x and y are positive integers, this common divisor must also be a divisor of ax ± by, and consequently of c ; which is contrary to the hypothesis that a, b, and c have no common divisor. INDETERMINATE EQUATIONS. 263 SOLUTION OF INDETERMINATE EQUATIONS IN POSITIVE INTEGERS. 375. 1. Solve 7x -{- 5y = 118 in positive integers. Dividing tlirougli'by 5, tlie smaller of the two coefficients, we ho.ve 5 5 or, ^^ — = 23 - a; - y, o Since by the conditions of the problem x and y must be positive integers, it follows that ~ '- must be an integer. Let this integer be represented by iJ. Then, ^'^"'^ = p, or 2 x - 3 = 5p. (1 ) o Dividing (1) by 2, a; - 1 - ^ = 2^. +1 ; or, x-1- 273 = Vj±1, ^ 2 Since x and p are integers, cc — 1 — 2^9 is also an integer ; and therefore - ^ — must be an intes:er. 2 Let this integer be represented by q. Then, ^^^ = q,ovp = 2q- 1. Substituting in (1), 2x — o = y)q — 5. Whence, x = 5q — l. (2) Substituting this value in the given equation, 35g-7 + 5?/ = 118. Whence, y — 25 — 7q. (3) 204 COLLEGE ALGEBRA. Equations (2) and (3) form Avhat is called the geneial solution ill integers of the given eq^uation. By giving to q the values zero, or any positive or negative integer, we shall obtain sets of integral values of x and y which satisfy the given equation. Now if q is zero, or any negative integer, x will be nega- tive ; and if q is any positive integer greater than 3, y will be negative. Hence the only jwsitive integral values of x and y which satisfy the given equation are those. arising from the values 1, 2, 3 of q. That is, a; = 4, 2/ = 18 ; x = 9,y = ll; and a; = 14, ?/ = 4. 2. Solve 8 a; — 13?/ = 100 in positive integers. Dividing through by 8, the coefficient of smaller absolute value, we have x — y ^ = 12 + -: ^ 8 8' -IO 5y + 4 or, a; — w — 12 = ^ — . y 8 Then '^^ J^ must be an integer. 8 Multiplying by 5, ' ^ ^ '" must also be an integer. That is, 3?/ + " + 2 + - must be an integer ; and therefore 8 8 " must be an integer. 8 ^ , . Let this integer be represented by p. Then, -^ = p, or y = 8|) — 4. 8 Substituting in the given equation, 8x- 104i) 4- 52 = 100, or x = 13p + 6. In this case ;:> may be any positive integer. INDETERMINATE EQUATIONS. 265 Thus, if p = 1, x = 19 and ?/ = 4 ; if p = 2, a; = 32 and y = 12; etc. The number of solutions is therefore indefinitely great. Note. The artifice of multiplying '^^T" by 5 saves considerable work in the above example. The rule in any case is to multiply the numerator of the fraction by such a number that the coefficient of the unknown qliantity shall exceed some multiple of 'the denominator by unity. If this had not been done, the last part of the solution of Ex. 2 would ' have stood as follows : Let ^]L±A = p^ or 5y + i = 8p. (1) 8 Dividing by 5, 2/ + ^ = p + ^• 5 5 yhcn ^~ must be an integer. 5 Let ^£-~ = q, OT 3p-i = 6q. ' (2) Dividing by 3, p - \ -1 = q + "11. Then ^^ must be an integer. Let 2g|J: = r, or 2g + l = 3r. ^ (3) 1 r Dividing by 2, g -f .- = r + — Then ^~ must be an integer. Let tZL^^s, orr = 2s+l. Substituting in (3) , 2 g + 1 = 6 s + 3, or g = 3 s + 1. Substituting in (2), 3p — 4 = 15 s + 6, or p = 5 s + 3. Substituting in (1), 5 ?/ + 4 = 40 s + 24, or ?/ = 8 s + 4. . Substituting in the given equation, 8a; -104s- 52 = 100, or a; = 13s +19. The values of x and y differ in form from those obtained above ; but it is to be observed th^t 13s +19 and 8s + 4, for the values 0, 1, 2, etc., of s, give rise to the same series of positive integers as 1.3p + (? and 825 — 4 for the values 1, 2, 3, etc., ol p. 266 COLLEGE ALGEBRA. We will now show how to solve in positive integers two equations involving three unknown quantities. 3. In how many ways can the sum of $14.40 be paid with dollars, half-dollars, and dimes, the number of dimes being equal to the number of dollars and half-dollar^- together ? Let X = the number of dollars, y = the number of half-dollars, and 2 = the number of dimes. Then by the conditions, 10a; + 5?/ + ^ = 144, and x + y = z- (1) Adding, llx -{-(J7j+z = lU + z, or, llx + 6y=lU. (2) Dividing by 6, x + ^^ + y = 24. Then — must be an integer ; or, x must be a multiple of 6 6 Let X = Gj?, where p is an integer. Substituting in (2), QGp + Gy = 144, or ?/ = 24 - lip. Substituting in (1), z = 6p -f 24 - lip = 24 - 5^^ The only positive integral solutions are when p = l or 2. Therefore the number of ways is two ; either 6 dollars, i:> half-dollars, and 19 dimes ; or 12 dollars, 2 half-dollars, and 14 dimes. EXAMPLES. Solve the following in ])ositive integers : 4. 2.T-f3?/=21. -6. 7;« + :5S?/ = 211. 5. 7x + 4y = S0. 7. .".1 ..; -h '>// - 1222. INDETERMIXATE EQUATIONS. 2G7 8. 24a; + 7?/ = 422. 10. A6x+lly = 1117. 9. 8x + G7y=WS. 11. 8x + 19?/=700. Solve the following in least positive integers : 12. 4.x- 3y = 5. 15. 21a; - 8^/ = - 25. 13. 5x-Ty = ll. 16. 13a;- 30i/ = 61. 14. 19a; -42/ = 128. 17. 17a; - 58 7/ = - 79. Solve the following in positive integers : jg f 2a; 4- 3?/ -52 = -8. ^g f 3a;-22/ - 32 = - 65. 1 5a;- 7/ + 42= 21. ' \8x + 5y + 2z= 177. 20. In how many different ways can the sum of $ 3.90 be paid with fifty and twenty cent pieces ? 21. In how many different ways can the sum of 19 s. 6d. be paid with florins worth 2 s. each, and half-crowns worth 2s. 6d. each? 22. Find two fractions whose denominators are 9 and 5, respectively, and whose sum shall be equal to J^V-. 23. In how many different ways can the sum of ^5.10 be paid with half-dollars, quarter- dollars, and dimes, so that the whole number of coins used shall be 20 ? 24. A farmer purchased a certain number of pigs, sheep, and calves for 1 160. The pigs cost $ 3 each, the sheep $ 4 each, and the calves $ 7 each ; and the number of calves was equal to the number of pigs and sheep together. How many of each did he buy ? 25. In how many different ways can the sum of £8 2s. be paid with half-crowns, florins, and shillings, so that twice the number of half-crowns together with five times the num- ber of florins shall exceed three times the number of shillings by 11? X 268 COLLEGE ALGEBRA. XXV. RATIO AND PROPORTION. 376. The Ratio of one number to another is the quotient obtained by dividing the first number by the second. Thus, the ratio of a to 5 is -; and it is also expressed a:b. ^ 377. A Proportion is an equality of ratios. Thus, if the ratio of a to 6 is equal to the ratio of c to d, they form a proportion, which may be written in either of the forms : a : b = c : d, = -, or a : b : : c : d. b d 378. The first term of a ratio is called the antecedent, and the second term the consequent. Thus, in the ratio a : b, a is the antecedent, and b is the consequent. The first and fourth terms of a proportion are called the extremes, and the second and third terms the means. Thus, in the proportion a:b = c:d, a and d are the extremes, and b and c the means. 379. In a proportion in which the means are equal, either mean is called a Mean Proportional between the first and last terms, and the last term is called a Third Proportional to the first and second terms. A Fourth Proportional to three quantities is the fourth term of a proportion whose first three terms are the three quantities taken in their order. Thus, in the proportion a : b = b : c, b is a mean propor- tional between a and r, and c is a third proportional to a and b. RATIO AND rKOrORTION. 269 In the proportion a : b = c : d, d is a fourth proportional to a, b, and c. 380. A Continued Proportion is a series of equal ratios, in which each consequent is the same as the following antecedent ; as, a : b = b : c = c : d = d : e. PROPERTIES OF PROPORTIONS. 381. In any proportion the prrodiict of the extremes is equal to the product of the means. Let the proportion be a:b = c: d. Then by Art. 377, a _c. b~d Clearing of fractions, ad = be. 382. A mean proportional between two quantities is equal to the square root of their jrroduct. Let the proportion be a : b = b : c. Then by Art. 381, b' = ac. Whence, b = -Vac. 383. From the equation ad = be, we obtain be -, r ad a =— , and b = — • d c That is, in any proportion either extreme is equal to the product of the means divided by the other extreme; and either mean is equal to the product of the extremes divided by the other mean. 384. (Converse of Art. 381.) If the product of two quan- tities is equal to the product of two others, one pair may be made the extremes, and the other pair the means, of a pro- portion. ad = --be. ad bd =i-'- a b^ c d a:b = --c:d. 270 COLLEGE ALGEBRA. Let Dividing by bd, Whence, In like manner we may prove that a : c= b : d, c : d = a : b, etc. 385. In any proportion the terms are in proportion by Alternation; that is, the first term is to the third as the second term is to the fourth. Let a : b = c : d. Then by Art. 381, ad = be. Whence by Art. 384, a:c=b:d. 386. In any proj)ortion the terms are in proportion by Inversion ; that is, the second term is to the first as the fourth term is to the third. Let a:b = c:d. Then, ad = be. Whence, b : a = d: c. 387. In any proportion the terms are in jwoportion by Composition ; that is, the sum of the first two terms is to the first term as the smn of the last two terms is to the third term. Let a:b = c:d. Then, ad = be. Adding both members to ac, ac + ad = ae + be, or, a(e + d)=^ c{a -\-b). Whence (Art. 384), a + b:a = e -\- d: c. Similarly we may prove that a -j- b : b = c + d : d. RATIO AND PROPORTION. 271 388. In any proportion the terms are in proportion by Division; tJiat is, the difference of the first two terms is to the first term as the difference of the last two terms is to the third 'erm. Let a : b = c : d. Then, ad = be. Subtracting both members from ac, ac — ad = ac — be, or, a{c — d) = c{a — b). Whence, a — b:a — c — d:c. Similarly, a — b -.b =c — d : d. 389. In any proportion the terms are in proportion by Composition and Division ; that is, the sum of the first two terms is to their difference as the sum of the last tioo terms is to their difference. Let a:b = c:d. Then by Art. 387, a + b ^c + d^ ,^. a c And by Art. 388, ^~^ = ^ ~ 'I (2) a c Dividing (1) by (2), ^_^. a — b c — d Whence, a -]- b : a — b = c -}- d : c — d. 390. In a series of equal ratios, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents. Let a:b = c: d = e :f. Then by Art. 381, ad = be, and af= be. Also, ab = ba. Adding, a(b + d +f) = b(a-\- c + e). Whence (Art. 384), a: b = a + c + e:b -\-d-\-f 272 COLLEGE ALGEBRA. In like manner the theorem may be proved for any num- ber of equal ratios. '391, To jyrove that if b~d~f~"'' then each of these equal ratios is equal to fpa" + gc" + re" + \2^b"+qd'^+rf^+ Let ^ = '-='-=....=k. b d f Then, a = bk, c = dk, e=fk, etc. Whence, pa" + gc" + re" +... =;)(6A:)" + q{dky + r(fk)" + = k\2^b" + qd'^ + rf'^ +■■■)■ Therefore, ^» ^ j?a" + gc" + re" + -. i?&" + gcZ" +'//" + ••. Or, k = (Pd'' + qc^+re- + \2ib'' + qd"+rf'^ + If 2', g, r, etc., are all equal, and n=l, we have f^_<^_e^_ _ a + c + e+ ••• 6 ~ d~/ ~ ■" ~ ftT^zTTT^' (Compare Art. 390.) 392. /7i any number of proportions, the 2>roducts of the cor- respjonding terms are in p)roportion. Let a:b = c:d, and e:f=fj:h. Then, ^* = ",and^ = 2. b d f h Multiplying these equals, a e c ^a ae cq - X ~ = - X ■', or — = -^. 6 / d h' bf dh Wlience, ae : bf= eg : dh. RATIO AND PROPORTION. 273 In like manner the theorem may be proved for any num- ber of proportions. 393. In any proportion, if the first two terms are multiplied by any quantity, as also the last tivo, the resulting quantities '■joill be in proportion. Let a:b = c:d. Then, Therefore, Whence, ma : mb = nc : nd. In like manner we may prove that ah c d Note. Either in or n may be unity ; that is, either couplet may be multiplied or divided without multiplying or dividing the other. 394. In any proportion, if the first and third terms are midtiplied by any quantity, as also the second and fourth terms, the resulting quantities will be in proportion. Let a:b = c:d. a 6" c ~d ma mb nc " nd Then, a c b~d Therefore, ma _ mc nb nd Whence, ma : nb = mc : % In like manner we may prove that a b c d m' n~ m' n Note. Either m or n may be unity. 274 COLLEGE 'ALGEBRA. 395. In any proportion, like powers or like roots of the terms are in proportion. Let a : b = c : d. Then, '^ = '-. b d Therefore, " ^i! = ^. Whence, a" : 6" = c" : d". In like manner we may prove that Va : 'Vb = Vc : Vd. 396. If three quantities are in continued proportion, the first is to the third as the square of the first is to the square of the second. Let a:b = b:c. Then, ^^ = ^. ' b c rpt, p a b a a a a^ Therefore, - x - = - X -, or - = • b c b b c b- Whence, • a: c = a- : &'. 397. If four quantities are in continued jiroportion., the first is to the fourth as the cidje of the first is to the cube of the second. Let a : b = b : c = c: d. a _b_c bed Then, rpv. n a b c a a a Therefore, _x-X- = 7X-X-- c d b b b Or, Whence, a : d = a^: b^. a _ a" d~b^ Note. The ratio a^ : b- is called the duplicate ratio, aud the ratio «' : h^ the triplicate ratio, of a • h. n RATIO AND PROPORTION. 275 PROBLEMS, 398. 1. Solve the equation 2.?;+3:2a;-3 = a+2&:2& — a. By Art. 389, 4 .t : G = 4 6 : 2 o. Dividing the first and third terms by 4, and the second and fourth terms by 2 (Art. 394), we have X : o = b : a. Whence, x = — (Art. 383). 2. If x:y = (x-\-z)' -.(y + zy', prove that z is a mean proportional between x and y. From the given proportion, by Art. 381, y{x-\-z)- = x{y + zy. Or, a^y -\-2 xyz + yz- — xy--j- 2 xyz + xz^. Or, _ ary — xy^ = xz- — yz-. Dividing hj x — y, xy = z^. Therefore z is a mean proportional between x and y. 3. Given the equations j 4x - Sy + 5z = 0. (1) l3x-5y-4.z = 0. (2) To find the ratio of x to y, and the ratio of x to z. Multiplying (1) by i, 16x -12y + 20z = 0. Multiplying (2) by 5, 15a; -25y-20z = 0. Adding; 31 a; - 37 ?/ = 0, or 31 a; = 37 y Whence by Art. 384, ' x:y = 37: 31. Multiplying (1 ) by 5, 20 a; - 15 y + 25 2 = 0. Multiplying (2) by 3, 9x-16y-12z = 0. Subtracting, 11 a; + 37 2 = 0, or, 37z = -llx- Whence, a; : 2; = 37 : — 11. Note. The result may be written in the form — = -■'- = ^_. 37 31 -11 276 COLLEGE ALGEBRA. 4. Prove that if " = -, then b d a^ - 62 : a2 - 3a& = c^ - d' : c^ - 3cd. Let — = - = x: whence, a = bx. b d Then, 6 V - 62 a'- -3ab b-x^-Sb'x x^-Sx t-1 d' c' - d' c" 3c c2-3cd d' d Whence, or— b- : a?— 3ab = c^ — d^ : c^ — o cd. 5. Find a fourth proportional to |, f, and |' - 6. Find a third proportional to f and f • 7. What is the second term of the proportion whose first, third, and fourth terms are 5|, 44, and If ? 8. Find a third proportional to a^ — 9 and 3 — a. 9. Find a mean proportional between 5| and IS-j^- 10. Find a mean proportional between and a; + 2 Solve the following equations : 11. 5x-3a:5a;-|-3a^9a-25:21a-25. 12. 2cc-5:3a; + 2 = x-l:7a; + l. - 13. x'-4:x--9=:x'-5x + 6:x"-\-4:X + 3. 14. x + Vl-X':x-Vl- X- =a+ V6- -a-: a- ^l)^—d\ ( X + y: X — y z= a + b : a — b. 15. J 1 I cc2 + 7/2:a2=a2 + 6^i. ' RATIO AND PROPORTION. 277 ~ 16. Find two numbers in the ratio of 16 to 9 such that, when -each is diminished by 8, they shall be in the ratio of 12 to 5. 17. Divide 36 into two parts such that the greater dimin- ished by 4 shall be to the less increased by 3 as 3 is to 2. 18. Find two numbers such that, if 4 is added to each, they will be in the ratio of 5 to 3 ; and if 11 is subtracted from each, they will be in the ratio of 10 to 3. 19. There are two numbers in the ratio of 3 to 4, such that their sum is to the sum of their squares as 7 is to 50. What are the numbers ? 20. Divide 12 into two parts such that their product shall be to the sum of their squares as 3 is to 10. 21. Divide a into two parts such that the first increased by b shall be to the second diminished by 6, as a + 3 6 is to a -3b. 22. If 5a + 4c:9a + 2c = 46 + 5c:2& + 9c, prove that c is a mean proportional between a and b. 23. If (a + 6 + c + d) (a - 6 - c + d) = (a - 6 + c - d) (a -t- 6 — c — cZ), prove that a:b = c:d. 24. If ax — by: ex — dy = ay — bz:cy — dz, prove that y is a mean proportional between x and z. 25. If a-c:b- d = Va- -f c^ : V6' + d'', prove that a : b = c : d. '26. If 8 cows and 5 oxen cost four-fifths as much as 9 cows and 7 oxen, what is the ratio of the price of a cow to that of an ox ? 27'. Given (a^+ab)x-\- {b--ab)y z= (a--{-b-)x - {a^-b-)y ^ find the ratio of x to y. - 28. Given l^^" y + 4.z = 0. (2x + 5y-3z = 0. Fiud the ratio of x to y, and the ratio of x to z. 278 COLLEGE ALGEBRA. 29. Giy^,, c,y-bx ^cx-az^hz-cy^ c b a Find tlie ratio of x to y, and tlie ratio of x to z. 30. Divide $ 564 between A, B, and C, so that A's sliaie may be to B's in the ratio of 5 to 9, and B's share to C'e in the ratio of 7 to 10. 31. Each of two vessels contains a mixture of wine and water, A mixture consisting of equal measures from the two vessels, contains as much wine as water ; another mix- ture consisting of four measures from the first vessel and one from the second, is composed of wine and water in the' ratio of 2 to 3. Find the ratio of wine to water in each vessel. 32. The population of a town increased 2.6 per cent from 1870 to 1880. The number of males decreased 3.8 per cent during the same period, and the number of females increased 10.6 per cent. What was the ratio of males to females in 1870? 33. The sum of four quantities in proportion is 30. The third term exceeds the sum of the first and second by 2, and the sum of the fourth and second terms exceeds the first term by 6. What are the quantities ? 34. If a c b~d' prove that (a .)«' + 2ab:3ab- Ab- = c- -{-2cd:ocd- ■4(Z-. (b .) ap- ab + b'-: — -W = c^ /.3 - cd + d- : — -d" 35. It - = - = -, prove that b d f ' (a.) a' + r + e' : b' + d' +f = ace : bdf. {b.) (a- + c^ + e') (b'- + (Z- +f-) = (ab + cd + e/y. 36. If a, b, c, and d are in continued proportion, prove that 2 a + od: 3 a - 4 (Z = 2 (t^ + 3 b'^ : 3 tv' — 4 UK VARIATION. 279 XXVI. VARIATION. 399. One quantity is said to vary directly as another when the ratio of any two values of the first is equal to the ratio of the corresponding values of the second. Note. It is customary to omit the word "directly," and say simply that one quantity varies as auotlier. 400. Suppose, for example, that a workman receives a fixed sum per day. The amount which he receives for m days will be to the amount which he receives for n days as m is to 7t ; that is,, the ratio of any two amounts received is equal to the ratio of the corresponding numbers of days worked. Hence the amount which the workman receives varies as the number of days during which he works. 401. One quantity is said to vciry inversely as another Avhen the first varies directly as the reciprocal of the second. Thus, the time in which a railway train will traverse a fixed route varies inversely as the speed; that is, if the speed is doubled, the train will traverse its roitte in one-half the time. 402. One quantity is said to vary as two others jointly when it varies directly as their product. Thus, the wages of a workman varies jointly as the amount which he receives per day, and the number of days during which he works. 403. One quantity is said to vary directly as a second and inversely as a third, when it varies jointly as the sec- ond and the reciprocal of the third. Thus, in physics, the attraction of a body varies directly as the quantity of matter, and inversely as the square of the distance. 280 COLLEGE ALGEBRA. 404. The symbol oc is used to express variation ; thus, ace 6 is read "a varies as &." 405. If xcr, y, then x is equal to y multiplied by a constant quantity. Let a;' and y' denote a fixed pair of corresponding values of X and y, and x and y any other pair. • Then by the definition of Art. 399, X y x' — = ^, or x — -y. x' y' ^ y' Denoting the constant ratio - by m, we have X = my. 406. It follows from Arts. 401, 402, 403, and 405 that : 1 . If X vanes inversely as y, x = — 2. If X varies jointly as y and z, x = myz. 3. Ifx varies directly ns y and inversely as z, x=—- Note. The converse of each of the statements of Arts. 405 and 406 is also true ; that is, if x is equal to y multiplied by a constant quan- tity, then XX y; and so on. 407. To 2irove that ifxccy, and yccz, then xccz. By Art. 405, ii xccy, then x = my ; (1) and if yc^z, y = nz. Substituting in (1), x = mnz. Whence by Art. 406, Note, xooz. 408. To jyi'ove that if xckz, and yccz, then x±yccz, and ■\/xyccz. By Art. 405, x = 7nz, and y = nz. Therefore, x±y= mz ±nz = (m ±n)z, and, -y/xy = Vmz • yiz = z^mn. Whence, x±y (jzz, and Vx?/ oc z. VARIATION. 281 489. To prove that if xccy, and z x ?i, then xz x yu. We Lave, x = my, and z = nu. Therefore, xz = mnyu. Whence, xz cc yri. 410. To prove that ifxccy, then a?" cc y". We have, a; = my ; or, a;" = m^'y*. Whence, x" cc ?/". 411. To prove that, if xocy when z is constant, and xccz when y is constant, then xccyz ivhen both y and z vary. Let y' and z' be the values of y and z, respectively, when X has the value x'. Let y be changed from y' to y", z remaining constantly- equal to z', and let x be changed in consequence from aj'to X. Then by Art. 399, ^ = llL. (1) X y" Now let z be changed from z' to z", y remaining constantly equal to y", and let x be changed in consequence from X to x". Multiplying (1) by (2), ^=^- (3) Now if both changes are made, that is, y from y' to y" and 2 from z' to 2", a; is changed from x' to x", and ?/2; is changed from y'z' to i/"z;". Then by (3),. the ratio of any two'values of x is equal to the ratio of the corresponding values of yz. Therefore by Art. 399, x cc yz. In like manner it may be proved that if there are any number of quantities a;, ?/, z, u, etc., such that xccy when 2, u, etc., are constant, a; cc z when y, u, etc., are constant, etc., 282 COLLEGE ALGEBRA. then if all tlie quantities y, z, u, etc., vary, x varies as their product. The following is an illustration of the above theorem : It is known, by Geometry, that the area of a triangle varies as the base when tlie altitude is constant, and as the altitude when the base is constant. Hence, when both base and altitude vary, the area varies as their product. '412. Problems in variation are readily solved by con- verting the variation into an equation by aid of Arts. 405 or 40G. EXAMPLES. 413. 1. If a; varies inversely as y, and is equal to 9 when y = S, what is the value of 'x when ?/ = 18 ? If X varies inversely as y, we have by Art. 406^, m x = —• y Putting £c = 9 and y = S, we obtain Whence, Hence, if 2/= 18, we have ' , 18 2. Given that the area of a triangle varies jointly as its base and altitude, what" will be the base of a triangle whose altitude is 12, equivalent to the sum of two triangles whose bases are 10 and 6, and altitudes 3 and 9, respectively ? Let B, H, and A denote the base, altitude, and area, respectively, of any triangle, and B' the base of the required triangle. VARIATION. 283 Since A varies jointly as B and H, we have A = mBH (Xrt.iOG). Then the area of the first triangle is vixlOxo, or 30 m, and the area of the second is mxGxO, or Bim; and hence the areaof the required triangle is 30 m + 54 m, or 84 m. But the area of the required triangle is also m X B' X 12. Therefore, 12 mB' = 84 m. Whence, B' = 7. 3. If oj varies inversely as y, and is equal to 4 when y = 2, what is the value of y when .^• = |- ? 4. If y cc 2-, and is equal to 15 when z — o, what is the value of y in terms of z'^ ? 5. If 2 varies jointly as x and _?/, and is equal to 90 when a; = 3 and ?/ = 6, what is the value of z when x = 2 and ?/=7 ? 6. If a; varies directly as y and inversely as z, and is equal to 4 when ?/=2 and z=o, what is the value of x when ?/ = 35 and 2; = 15 ? 7. If 2a; — 3 cc 3?/ + 7, and x=o Avhen y=l, what is the value of X when y — — l? 8. If ar' cc ?/-, and .r = G when 2/ = 3, what is the value of y when x — 2? 9. The distance fallen by a body from a position of rest varies as the square of the time during which it falls. If a body falls 257i feet in four seconds, how far will it fall in seven seconds ? 10. Two quantities vary directly and inversely as x, re- spectively. If their sum is equal to 7 when x = 2, and to — 13 when x = ~3, what are the quantities ? 11. The area of a circle varies 'as the square of its diam- eter. If the area of a circle whose diameter is 2^ is lO/^, what will be the diameter of a circle whose area is 34|| ? 284 COLLEGE ALGEBRA. 12. Given that y is equal to the sum of two quantities which vary directly as a? and inversely as x, respectively. If ?/ = — 1- when x=i, and y = ^'^- when x = — 2, what is the value of y when x = — |- ? 13. Given that y is equal to the sum of three quantities, the first of which is constant, an'd the second and third vary as X and a.-^, respectively. If ?/ = — 19 when x = 2, ?/ = 4 w^hen x = l, and ?/ = 2 when x = — l, what is the expression for y in terms of a; ? 14. If the volume of a pyramid varies jointly as its base and altitude, what will be the altitude of a pyramid whose base is 12, equivalent to the sum of two pyramids whose bases are 5 and 8, and altitudes 12 and 6, respectively ? 15. Three spheres of lead whose diameters are 3, 4, and 5 inches, respectively, are melted and formed into a single sphere. Find its diameter, having given that the volume of a sphere varies as the cube of its diameter. 16. The volume of a cone of revolution varies jointly as its altitude and the square of the radius of its base. If the volume of a cone whose altitude is 3 and radius of base 5 is 784, what will be the radius of the base of a cone whose volume is 47^ and altitude 5 ? 17. If 5 men in 6 weeks earn $57, how many weeks will it take 4 men to earn $ 7G, it being given that the amount earned varies jointly as the number of men, and the number of weeks during which they work ? 18. If the volume of a cylinder of revolution varies jointly as its altitude and the square of its radius, what will be the radius of a cylinder, whose altitude is 18, equivalent to the sum of two cylinders whose altitudes are 5 and 12, and radii 6 and 9, respectively ? 19. If the illumination from a source of light varies inversely as the square of the distance, how much farther from a candle must a book, which is now 15 inches off, be removed, so as to receive just one-third as much light ? ARITHMETICAL PROGRESSION. 285 XXVIL ARITHMETICAL PROGRESSION. 414. An Arithmetical Progression is a series of terms, each of which is derived from the preceding by adding a constant quantity called the common difference. Thus, 1, 3, 5, 7, 9, 11, ... is an increasing arithmetical progression, in which the common difference is 2. Again, 12, 9, 6, 3, 0, —3, ... is a decreasing arithmetical progression, in which the common difference is — 3. 415. Given the first term., a, the common difference, d, and the number of terms, n, to find the last term, I. The progression is a, a + d, a -{-2d, a-\-3d, • • • • It will be observed that the coefficient of d in any term is one less than the number of the term. Hence, in the nth or last term, the coefficient of d will be n — 1. That is, I = a-\-(n - l)fL (I.) 416. Given the first term, a, the last term, I, and the number of terms, n, to find the sum of the series, S. ^ = a + (a + d) + (a + 2d) + ••• + (Z - d) + Z. Writing the series in reverse order, S = I -\- (I - d) -h {I - 2d) -\- ■■■ -{- {a -{- d) + a. Adding these equations, term by term, • 2;S' = (a+0 + (a+0 -f- (a + + • • • + (a+0 + (« + = n(a + 0- Therefore, ^ = '^ (a + 1) • (H.) 417. Substituting in (II.) the value of I from (I.), we have >S = |[2a+(n-l)d]. 286 COLLEGE ALGEBRA. EXAMPLES. 418. 1. In the series 8, 5, 2, -1, -4, ••• to 27 terms, find the last term and the sum. In this case, a = 8, cZ = — 3, n = 27. Substituting in (I.) and (II.), Z = 8 + (27-l)(-3) = 8-78 = - 70. ^ = ?I(8-70)=27 X (-31) =-837. 2 Note. The common difference may be found by subtracting any term of the series from the next following term. Thus, m the pro- gression 5 1^ 1 J 1 5 11 _. —-, —2, •", we have a = - = r" 3 b 3 b In each of the following, find the last term and the sum of the series : 2. 3, 11, 19, ••• to 19 terms. 3. _ 5, - 11, - 17, ••• to 22 terms. 4. _ 69, — 62, — 55, ••• to IG terms. 5. ^, _§,_?,... to 13 terms. 4' 8' 2' 6. -,—,••• to 25 terms. 7. _i 1 ... to 38 terms. 3' 2' 8. —-,—'-,••• to 55 terms. 4' G' 9. — -, -•^, ... to 17 terms 5' 2' 10. 2a -5&, 7a -2 6, ••• to 9 terms. 11. ^ - y^ ?/,... to 10 terms. ARITHMETICAL PROGRESSIOX. 287 419. If any tliree of tlie five elements of an arithmetical progression are given, the other two may be found by sub- stituting the known values in the fundamental formulae (I.) and (11.), and solving the resulting equations, 5 5 1. Given a = — *-, 7i = 20, S = — -; find d and L 3 3 Substituting the given values in (I.) and (II.), we have Z = -^ + 19rf. (1) 3 3 \ 3 J' 6 3 ^ ^ From (2), ; = --- = -. ^ ^ 3 6 2 SulBstituting in (1), ^ = — 1^ + 19 cZ ; whence, d = — 2 3 6 2. Given d = - 3, l = -39, S = -264.; find a and n. Substituting in (I.) and (II.), -39 = a + (7i-l)(-3); or a = 37i-42. (1) - 264 = ^ (a - 39) ; or an - 39 w = - 528. (2) Substituting the value of a from (1) in (2), 3^2 _42u- 3971 = -528, or, n- — 27 n = — 176. wi.. .. 27±V729-704 27 ± 5 -.^ .. Whence, n = = = 16 or 11 2 2 Substituting in (1), a = 48 - 42 or 33 - 42 = 6 or - 9. 'riierefore, a = Cj and n = 16 ; or, a = — 9 and n = 11. 288 COLLEGE ALGEBRA. Note 1. The significance of the two answers is as follows : If a = 6 and n = 16, the series is 6, 3, 0, -3, -0, -9, -12, -15, -18, -21, -24, -27, -30, _33, -3G, -39. If a = — 9 and 7i = 11, tlie series is _9, _i2, -15, -18, -21, -24, -27, -30, -33, - 3G, -39. - In each of these tlie last term is — 39, and the siun — 2G4. 3. Given a = -, d = , S — — '-: find I and ?i. 3 12 2 Substituting in (I.) and (II.), i = | + (»-i)(-i);o.-' = ^- (1) Substituting the value of I from (1) in (2), 4 Solving this equation, n = 12 or — 3. The second value is inapplicable, for the number of terms in a progression must be a positive integer. Substituting the value n = 12 in (1), ,5-12 7 12 12 Therefore, I — —— and n = 12. Note 2. A negative or fractional value of n is inapplicable, and must be rejected together with all other values dependent upon it. Note 3. The series con-espondiug to the value n = 12 in Ex. 3 is 1 1 i i_ - ^ _1 _1 _1 _^ _1 _J. 3' 4' G' 12' ' 12' G' 4' 3' 12' 2' 12* ARITHMETICAL PROGRESSION. 289 It will be observed that if we count backwards three terms, beginnmg with the last term, we have a series — :^, —-,——, whose sum is — ^- This series has the same common difference, last term, and sum as the given series ; but it has not the same first term, and hence does not satisfy aU the given conditions. It will always be found that, when a, d, and 8 are given, and one of the values found for n is a positive and the other a negative integer, the series obtained by counting backwards from the last term of the series corresponding to the positive value of n, as many terms as are indicated by the negative value of n, will have its sum equal to the given sum. In like manner, it will be fomid that, when Z, d^ and S are given, and one of the values fomid for n is a positive and the other a negative integer, the series obtained by counting forwards from the first term of the series corresponding to the positive value of n, as many terms as are indicated by the negative value of ?«, will have its sum equal to the given sum. EXAMPLES. 4. Given d = 5, 1 = 11, n = 15 ; find a and S. 5. Given d--4.,n = 20, >S' = - 620 ; find a and 6. Given a = — 9, 7i = 23, 1= 57 ; find d and S. 7. Given a = — 5, n = 29, S = — 2175 ; find d and 8. Given a = -, Z = — , ^ = ^ ; find d and n. 9. Given l = — -,n=19,S = 0', find a and d. o 10. 1 n^ o Given d = ^, S = -^, a = ^', find Z and n. 12 3 u 11. Given a = l, l = - ^, d = -^; find n and S. 12. Given d = ^, n = 17, S = 17 -, find a and I 13. Given I = ^^'', d = ^^, S=^^^ : find a and n. 15 ' 15' 15 ' LOO COLLEGE ALGEBRA. 14. . Given ? = — 5J, n = 21, *S' = — 38-} ; find a and d. 15. Given a = — 5, / = — 47, S = — 1118 ; find d and n. 0()3 16. Given a = G, n = 14, /S = — ^^ ; find d and Z. o 17. Given J = — -, d = 3, aS' = ; find a and n. 3 15 o 18. Given a = - ? d = ^, >&' = 120 : find 71 and I. 4' 4' From (I.) and (II.) general formulse for the solution ol cases like the above may be readily derived. ^ 19. Given a, d, and S ; derive the formula for n. Substituting the value of I from (I.) in (II.), 2S= n\2a + {n - l)f/], or dn' + (2a - d)n = 2S. This is a quadratic in n, and may be solved by the method of Art. 345. Multiplying by 4 d, and adding {2 a — dy to both members, 4cZV+ 4d(2a -d)n + (2a-dy = 8dS + (2 a - df. Extracting the square root, 2dn + 2a-d = ± VSdS + {2a-dy. ,^, d — 2a ± VSdS + (2a — dy Whence, n = -^-^^ ^• 2d 20. Given a, I, and n ; derive the formula for d. 21. Given a, n, and S ; derive the formulse for d and /. 22. Given d, n, and S; derive the formulas for a and I. 23. Given a, d, and l; derive the formulae for n and S. 24. Given d, I, and n ; derive the formulse for a and S. 25. Given I, n, and S ; derive the^ormuhie for a and d. 26. Given «, d, and S ; derive the formula for /. ARITHMETICAL PROGRESSION. 291 27. Given a, I, and S ; derive the formulas for d and n. 28. Given d, I, and S ; derive tlie formula3 for a and n. jj 420. To insert any number of arithmetical means between two given terms. Let it be required, for example, to insert 5 arithmetical means between 3 and — 5. This means that we are to find an arithmetical progression of 7 terms, whose first term is 3, and last term — 5. Putting a = 3, ? = — 5, and n = 7, in (I.), we have — 5 = 3 + G d, or d = — -. 3 Hence the required series is Q 5 1 ^ 7 11 w ^' 3' 3' -^' -3' --3' -'■ 421. Let X denote the arithmetical mean between a and b. Then, by the nature of the progression, X — a = b — X, or 2x= a -{-b. Whence, x = " + ^. 2 That is, tJie arithmetical mean between two quantities is equal to one-half their sum. EXAMPLES. 422. 1. Insert 6 arithmetical metins between 3 and 8. 2. Insert 8 arithmetical means between - and • 2 10 3. Insert 7 arithmetical means between and — 2 2 4. Insert 8 arithmetical means between — - and — 5. 4 292 COLLEGE ALGEBRA. 3 5. Insert 9 arithmetical means between - and —11. 6. If m arithmetical means are inserted between a and b, what are the first and last means ? 7. Find the number of arithmetical means betAveen ^ 9 9 '' and , when the sum of the first two is — 7 35 Find the arithmetical mean between : 8. 21 and -If. 9. (a + &)'and -{a -by. 10. ^^ + ^ and '±^- a — b a -\-b PROBLEMS. 423. 1. The sixth term of an arithmetical progression is -, and the fifteenth term is — . Find the first term. 6 3 By Art. 415, the sixth term is a + 5 d, and the fifteenth term is a+lid; hence, («+ 5d = | (1) ■^ I a+14cZ= Y (2) Subtracting (1) from (2), Substituting iu(l), a +■-=-_; whence, a = — ^- 2 o 2. Find four quantities in arithmetical progression such that the product of the extremes shall be 45, and the product of the means 77. Let the quantities he x — Zy, x — y, x+ ?/, and x + Sy. Then by the conditions, < x2_02/2=45. 2- »/2=7/ Solving these equations, jc = 0, y—±2; or, x = — 0, 7y=±2(Art. 307). Hence the quantities are 3, 7, 11, and 15 ; or, — 3, — 7, — 11, and — 15. ARITHMETICAL PROGRESSION. 293 Note. In problems like the above it is convenient to represent the imknowu quantities by symmetrical expressions. Thus if five quanti- ties had been required, we should have represented them by aj — 2 ?/, x — y, X, X + y, and x + 2 ?/. — 3. Find the sum of the even integers beginning with 2 and ending with 500. -T 4. The 7th term of an arithmetical progression is 27, and the 13th term is — 3. Find the 21st term. 5. Find four numbers in arithmetical progression such that the sum of the first two shall be 12, and the sum of the last two — 20. 6. The 19th term of an arithmetical progression is 9 a — 2 &, and the 31st term is 13 a — 8 6. Find the sum of the first thirteen terms. 7. Find the sum of the first n positive integers which are multiples of 7. 8. Find four integers in arithmetical progression such that their sum shall be 24, and their product 945. 9. Find the sum of all positive integers of three digits Avhich are multiples of 11. 10. The 7th term of an arithmetical progression is — i, the 16th term is 2i, and the last term is 6|-. Find the number of terms. 11. Find five quantities in arithmetical progression such that the sum of the first, third, and fourth shall be 3, aiid the product of the second and fifth — 8. 12. A body falls 16 Jj feet the first second, and in each succeeding second 32|- feet more than in the next preceding one. How far will it fall in 16 seconds ? 13. Find three quantities in arithmetical progression such that the sum of the squares of the first and third exceeds the second by 123, and the second exceeds one-third of the first by 6. 294 COLLEGE ALGEBRA. 14. A man travels 3 miles the first day, 6 miles the second day, 9 miles the third day, and so on. After he has travelled a certain number of days, he finds that his average daily distance is 46|- miles. How many days has he been travel- ling? 15. The mth term of an arithmetical progression is p, and the nth term is q. What is the (m + ?i)th term ? 16. Find the number of arithmetical means between 1 and 31, when the seventh mean is to the one before the last as 5 is to 9. 17. After A had travelled for 4^ hours at the rate of 5 miles an hour, B set out to overtake - him, and travelled 3 miles the first hour, 31 miles the second hour, 4 miles the third hour, and so on. In how many hours will B overtake A ? '^' '' '' '■'.'' ■ . ^ 18. Find three numbers in arithmetical progression such that the sum of their squares is 347, and one-half the third number exceeds the sum of the first and second by 4|^. 19. If a person saves f 100 a year, and puts this sum at simple interest at 5 per cent at the end of each year, to how much will his property amount at the end of 20 years ? 20. The digits of a number of three figures are in arith- metical progression ; the first digit exceeds the sum of the second and third by 1 ; and if 594 is subtracted from the number, the digits will be inverted. Find the number. 21. There are two sets of numbers, each consisting of three terms in arithmetical progression whose sum is 15. The common difference of the first set exceeds by vinity the common difference of the second set ; and the product of the first set is to the product of the second as 7 is to 8. Required the numbers. GEOMETRICAL PROGRESSION. 295 XXVIII. GEOMETRICAL PROGRESSION. 424. A Geometrical Progression is a series of terms, each of which, is derived from the preceding by multiplying by a constant quantity called the ratio. Thus, 2, 6, 18, 54, 1G2, ••• is an increasing geometrical progression in which the ratio is 3. Again, 9, 3, 1, -, -, ••• is a decreasing geometrical pro- 3 9 ^ gression in which the ratio is - • Negative values of the ratio are also admissible ; thus, — 3, 6, — 12, 24, — 48, ... is a geometrical progression in which the ratio is — 2. 425. Given the first term, a, the ratio, r, and the number of terms, n, to find the last term, I. The progression is a, ar, ar^, ar^, .... It will be observed that the exponent of r in any term is one less than the number of the term. Hence, in the 7ith or last term, the exponent of r will be w — 1. That is, l = ar"^\ (I.) 426. Given the first term, a, the last term, I, and the ratio, r, to find the sum of the series, S. S = a-{- ar -\-ar^ + '•• -\- a?-"^^ + ar"'~-\- ar'*^^ Multiplying each term by r, rS = ar + aj" + a?-^ + • • • + or""- + a?-""^ + ar". Subtracting the first equation from the second, rS — S = ar" — a; or, S = — ~ • r — 1 But by (I.), Art. 425, rl = ar'\ Therefore, 6^ = !izL^. (H.) 296 COLLEGE ALGEBRA. EXAMPLES. 427. 1. In the series 3, 1, -, ••• to 7 terms, find the last term and the sum. In this case, a = 3, r = - n = 7. Substituting in (I.) and (11.), ^ 2186 729 1093 - 3- -i 2 243 3 Note. The ratio may be found by dividing any term by tlie next preceding term. 2. In the series — 2, 6, — 18, 54, ... to 8 terms, find the lasf term and the sum. In this case, a = — 2, r= — - = — 3, n = 8. Hence, ^ = -2(-3)" = -2 x(- 2187) = 4374. ^ -3x4374 -(-2) ^ - 13122 + 2 ^ ^ogQ -3-1 -4 In each of the following, find the last term and the sum of the series : 3. 1, 3, 9, ••• to 9 terms. 4. 6, 4, ^, ••• to 7 terms. 5. -2, 10, -50, ... to 5 terms. 6. -3, 1 -| ...to 8 terms. 7. -?, -5, -10, ... to 10 terms. GEOMETRICAL PROGRESSION. 297 25 8. ^^, — 5, 2, ••• to 6 terms. 2' ' ' 1 1 _3 3' 2' 4' to 7 terms. 10. 5, A, 2^ ... to 5 terms. 11. - 4 16 64 3 3 -, — 6, • • • to 6 terms. 2' ■ 428. If any three of the five elements of a geometrical progression are given, the other two may be found by sub- stituting the known values in the fundamental formulae (I.) and (II.), and solving the resulting equations. But in certain cases the operation involves the solution of an equation of a degree higher than the second ; and m others the unknown quantity appears as an exponent/ the solution of which form of equation can usually only be effected by aid of logarithms (Art. 519). In all such cases in the present chapter, the equations may be solved by inspection. 1 . Given a = — 2, n = 5, I = — 32; find r and S. Substituting the given values in (I.), we have - 32 = - 2r*; whence, r* = 16, or r = ± 2. Substituting in (II.), If r = 2, S = 2(^ -32) -(-2) ^ _ 54 _,_ 2 = _ 62. If r = -2, ^^(-2)(^32_)-l-2^^64+_2^_,, -2-1 -3 Therefore, r = 2 and ^= -62 ; or, r = - 2 and S= -22, Note 1. The significance of the two answers is as follows : If r=z 2, the series is —2, —4, —8, —16, —32, in which the sum is —02. K r — — 2, the series is —2, 4, — 8, 16, —32, in whicli the sum is —22. 298 COLLEGE ALGEBRA. 2. Given a = 3, r = -|^ = ^ ; find n and I Substituting in (II.), -1^-3 1640 ^ 3 ^ Z + 9 729 _l_i 4 ■ 3 Whence, ^ + 9 = ^; or, Z = -^. Substituting in (I.), Whence, by inspection, w — 1 = 7, or ?* = 8. EXAMPLES. - 3. Given r = 2, w = 9, Z = 256 ; find a and S. 4. Given a = - 2, ?i = 6, Z = 2048 ; find r and S. 9 '^O'lO ^ 5. Given r = ^, w = 7, >S = ^^ ; find a and Z. 3 24o 6. Given a = 2, r = - -, Z = - — - ; find n and aS'. 2 256 1 /*047 — 7. Given r=^,n= 11, aS = 7^7-7^ ; find a and Z. 2 2U4o 8. Given a = | ?i = 9, Z = ^-^ ; find r and ^. 1 511 ^ 9. Given a = - 8, Z = - -^, ^ = - ':^; find r and n. 10. Given a = -, r = - ^, /S = -^ ; find Z and w. 4 3 162 11. Given I = 192, r = -2, S = 129; find a and n. 12. Givena = -|z = -^, ^ = -|||; and r and n. GEOMETRICAL PROGRESSION. 299 From (I.) and (II.) general formulae may be derived for the solution of cases like tlie above. 13. Given a, r, and S ; derive the formula for I. 14. Given a, Z, and S ; derive the formula for r. 15. Given r, I, and S ; derive the formula for a. 16. Given r, n, and I ; derive the formulae for a and S. - 17. Given r, n, and yiS*; derive the formulae for a and I. 18. Given a, n, and I ; derive the formulae for r and S. Note 2. If the given elements are n, I, and 5, equations for a and r may be found, but there are no definite fonnulce for their values. The same is the case when the given elements are a, n, and S. The general formulae for n involve logarithms ; these cases are dis- in Art. 519. "^ 429. The limit (Art. 209) to which the sum of the terms of a decreasing geometrical progression approaches, as the number of "terms increases indefinitely, is called the sum of the series to infinity. The value of S in formula (II.), Art. 426, may be written ci a — rl It is evident that, by sufficiently continuing a decreasing progression, the absolute value of the last term may be made less than any assigned positive quantity, however small. Hence, as the number of terms increases indefinitely, I approaches the limit 0, and therefore rl approaches the limit 0. Then the fraction '''' ~^ approaches the limit 1 —r 1 — r That is, the sum of a decreasing geometrical progression to infinity is given by the formula S = ^^- (III.) 1 — r 300 COLLEGE ALGEBRA. EXAMPLES. S 16 1. Find the sum of the series 4, , — , ••• to infinity. o 9 2 In this case, a = 4, r = 3 Substituting in (III.), S 4 _12 1 + ^ 5- Note. This signifies that, the greater the number of terms taken, * 1'' the more nearly does their sum approach to -^ ; but no matter how 5 many terms are taken, tlie sum will never exactly equal this value. Find the sum of the following to infinity : 2. 3,.|.... 6. 7 21 63 4' 32' 256' " •• 3. 16, -4, 1, .... 7. 2 1 5 * 5' 3' 18' " 4. _1 1 _1 .... ' 5' 25' 8. 1 1 8' 18' 2 '81' 5. 5 10 20 3' 9' 27'"'" 9. 40 .5 5 49' 7' 8' ■■ 430. To find the value of a repeating decimal. This is a case of finding the sum of a geometrical pro- gression to infinity, and may be solved by the formula of Art. 429. 1. Find the value of .85151 .... .85151 ... = .8 + .051 + .00051 H The terms after the first constitute a decreasing geometri- cal progression, in which a = .051 and r — .01. GEOMETRICAL PROGRESSION. 30I Sabstitutin ig in (III.), ^ .051 ^ = 1-.01 .051 .99 _ 51 _ 990 17 '330' Hence the value of the given decimal is 8 10 ^1' = 330 281 330 EXAMPLES. Find the values of the following : 2. .7272.... 4. .G9444.... 6. .11003003.... 3. .407407 .... 5. .58686 .... 7. .922828 .... 431. To insert any number of geometrical means between tivo given terms. Let it be required, for example, to insert 5 geometrical 1*^8 means between 2 and -^^• 729 This means that we are to find a geometrical progression of 7 terms, whose first term is 2, and last term -^^^ 729 128 Putting a = 2, 1 = - — , and n= 7, in (I.), we have — ^ = 2?"^ ; whence, r^ = , and r = ± -• 729 729 3 Hence, the required result is 9 ±^ §. ±1^ 32 J>4 128 3' 9' 27' 81' 243' 729' 432. Let X denote the geometrical mean between a and b. Then by the nature of the progression, X b ■> , - = -, or or = ab. a X Whence, x = Va6. 302 COLLEGE ALGEBRA. That is, the geometriccd mean between two quantities is equal to the square root of their product. EXAMPLES. 433. 1. Insert 6 geometrical means between - and — 6 3 2. Insert 4 geometrical means between 4 and — 972. 3. Insert 5 geometrical means between — 2 and — 128. 2 125 4. Insert 3 geometrical means between and —• 7 5. Insert 4 geometrical means between and 3584. 243 2 6. Insert 7 geometrical means between - — and -^« 7. If m geometrical means are inserted between a and h what is the first mean ? Find the geometrical mean between : 8. llf and 2f 9. 4 a.-^ -f 12 xy + 9 y- and 4 x- — 12 xy + 9 y\ 10. «'-^^^ and '''+''^- ah + ly" ah - W PROBLEMS. 434. 1. Find three numbers in geometrical progression, such that their sum shall be 14, and the sum of their squares 84. Let the numbers be a, ar, and ar- ; then, by the conditions, ^ a-\- ar-\-ar'^ = 14. (1) \ a?- + a2>-2 + a2)-* = 84. (2) Dividing (2) by (1), a- ar + af^ = G. (3) Subtractuig (3) from (1), 2ar=8, or r = -- (4) a GEOMETRICAL PROGRESSION. 303 Substituting iu(l), a Or, a^- 10a = -16. Solving tliis equation, a = 8 or 2. Substituting in (4), -t-i- Therefore, the numbers are 2, 4, and 8. 2. The fifth term of a geometrical progression is 48, and the eighth term is — 384. Find the first term. 3. The sum of the first and second of four numbers in geometrical progression is 15, and the sum of the third and fourth is 60. What are the numbers ? 4. Find three numbers in geometrical progression, such that the sum of the first and second is 20, and the third exceeds the second by 30. 5. The fourth term of a geometrical progression is —108, and the tenth term is — 78732. What is the first term ? 6. The elastic power of a ball, which falls from a height of 100 feet, causes it to rise .9375 of the height from which it fell, and to continue in this way diminishing the height to which it will rise, in geometrical progression, until it comes to rest. How far will it have moved ? 7. The sum of four numbers in geometrical progression is 30, and the quotient of the fourth number divided by the sum of the second and third is 1^. Find the numbers. 8. The third term of a geometrical progression is J^, and the sixth term is -gf^. Find the eighth term. 9. Divide the number 39 into three parts in geometrical progression, such that the third part shall exceed the first by 24. 10. The product of three numbers in geometrical progres- sion is G4, and the sum of the squares of the first and third is G8. What are the numbers ? 304 COLLEGE ALGEBRA. 11. The sum of three numbers in arithmetical progression is 12. If the first number is increased by 5, the second by 2, and the third by 7, the results form a geometrical pro- gression. What are the numbers ? 12. The product of three numbers in geometrical progres- sion is 8, and the sum of their cubes is 73. What are the numbers ? 13. Divide f 700 between A, B, C, and D, so that their shares may be in geometrical progression, and the sum of A's and B's shares equal to $ 252. 14. There are four numbers, the first three of which form an arithmetical progression, and the last three a geometrical progression. The sum of the first and third is 2, and of the second and fourth 37. What are the numbers ? 15. What is the ratio of the geometrical progression, the sum of Avhose first ten terms is 244 times the sum of its first five terms ? 16. The sum of the first three terms of a geometrical progression is one-fourth the sum of the third, fourth, and fifth terms ; and the seventh term is 384. Find the first term and the ratio. 17. There are three numbers in geometrical progression whose sura is 57. If the first is multiplied by i, the secoud by I, and the third by |-, the results form an arithmetical progression. What are the numbers ? 18. If the mth term of a geometrical progression is ^>, and the nth. term is q, what is the (m + w)th term ? 19. If a, h, c, and d are in geometrical progression, prove that (a -cy + {h- cY + ih- df = (a - cZ)l HARMONICAL PROGRESSION. 305 XXIX. HARMONICAL PROGRESSION. 435. Quantities are said to be in Harmonical Progression when tlieir reciprocals form an arithmetical progression. Thus, 1, -, -, -, -, ••• are in harmonical progression, 3 5 7 9 l)ecause their reciprocals, 1, 3, 5, 7, 9, ..., form an arith- metical progression. 436. Any problem in harmonical progression, which is susceptible of solution, may be solved by taking the recipro- cals of the terms, and applying the formulae of the arith- metical progression. There is, however, no general method for finding the sum of the terms of a harmonical series. 437. If any three consecutive terms of a harmonical series are taken, the first is to the third as the first minus the second is to the second minus the third. Let the terms be a, b, and c. Then since -, -, and - are in arithmetical progression, ah c 1 _ 1 ^ 1 _ 1^ c b h a b — c _ a — b be ab Mviltiplying both members by -- — , we have b — c a _a — b c b — c 438. Let X denote the harmonical mean between a and b. Then - is the arithmetical mean between - and • X a b 306 COLLEGE ALGEBRA. Whence, by Art. 421, Therefore, 1 a b a+b X 2 2ab 2 ah X = a + h 439. Let A, G, and II denote the arithmetical, geonietri cal, and harmonical means, respectively, between a and h. Then by Arts. 421, 432, and 438, and // = a + h ^ = ^^+-^(^ = Vr^,and//= 2«& But, — ■ — X = ah — i^aby. 2 a+b ^ ^ Whence, A K 11= G-, or G = VA x H. . That is, tJie geometrical mean between two quantities is also the geometrical mean hetioeen their arithmetical and harmonical means. 440. Let a and b be two positive real numbers. By Art. 439, the positive value of their geometrical mean is intermediate in value between their arithmetical and harmonical means. B t g + & 2 ah _ (a + b)- — 4ft6 2 a + b" 2{a + b) ^ a'-2ab + b- ^ (a-b)\ 2{a + b) 2(a + &)' a Xiositive quantity. Hence, of the three means, the arithmetical is the greatest, the geometrical next, and the harmonical the least. to 3G terms, find the EXAMPLES: 441. 1. L. 9 9 the series 2, ", ", • iast ten 11. HARMONICAL PROGRESSION. 307 Taking the reciprocals of the terms, we have the arith- ,. , .13 5 metical series -, -, -, ••• • Zi A Li In this case a = -, cZ = 1, and n = 36. Substituting in (I.), Art. 415, we have Z = | + (36-l)xl=^. 2 Then — is the last term of the given harmonical series. 2. Insert 5 harmonical means between 2 and — 3. We have to insert 5 arithmetical means between - and • 2 3 Substituting a = -, 1 = , and n = 7, in (I.), Art. 415, 11 ^ -- = - + 6fZ: or, d=-— . 3 2'' 36 Then the arithmetical series is 1 13 2 J_ _J^ _1_ _1 2' 36' 9' 12' 18' 36' 3* Therefore the required harmonical series is 2, ?^ 9 ^2, _18, _§6 _3. ' 13' 2' ' ' 7' Find the last terms of the following : 3. ?, A 1, ... to 11 terms. 4. ? -, 1, - to 17 terms= 4 11 6 5 7 5 5 5. , 10, -, ••• to 26 terms. 3 4 6. 6, - 2, --,••• to 23 terms. 7. --, -— , -— , ••• to 31 terms. 7 23 16 308 COLLEGE ALGEBRA. 3 15 8. Insert 7 harinonical means between - and — 2 26 9. Insert 5 harmonical means between and . 5 13 8 10. Insert 6 harmonical means between 4 and — —• 19 Find the harmonical mean between : 11. If and - IvV 12. ^^+^ and ^'^^. 13. Find the last term of the harmonical series a, ^, ... tan terms. 14. If m harmonical means are inserted between a and b, what is the second mean ? 15. The first term of a harmonical series is x, and the second term is y ; continne the series to three more terms. 16. The arithmetical mean between two numbers is — \, and the harmonical mean is 42. What are the numbers ? 17. The fourth term of a harmonical series is — f, and the ninth term is — i. What is the seventh term ? 18. The geometrical mean between two numbers is 12, and the harmonical mean is 9f . " What are the numbers ? 19. There are three numbers in harmonical progression whose sum is ff. If the second and third numbers are multiplied by 5 and 16, respectively, the three numbers form a geometrical progression. What are the numbers ? 20. If the mth term of a harmonical progression is 2h '^i^d the nth. term is q, what is the (m + «)th term ? 21. Prove that if a is the arithmetical mean between b and c, and b the geometrical mean between a and c, then c is the harmonical mean between a and b. THE BINOMIAL THEOREM. 309 XXX. THE BINOMIAL THEOREM. POSITIVE INTEGRAL EXPONENT. 442. The Binomial Theorem is a formula by means of which any power of a binomial may be expanded into a series. We shall consider in the present chapter those cases only in which the exponent is a positive integer. PROOF OF THE THEOREM FOR A POSITIVE INTEGRAL EXPONENT. 443. By actual multiplication, we obtain : (a + x)^ = a- + 2 ax + x- ; (a + aj) 3 = a" + 3 a-.^• + 3 ax^ + x^ ; (a + x) ^ = a^ + 4 a^x + 6 a-x^ + 4 ax^ + x^ ; etc. In the above results we observe the following laws : I. The number of terms is greater by 1 than the exr ponent of the binomial. II. The exponent of a in the first term is the same as the exponent of the binomial, and decreases by 1 in each succeeding term. III. The exponent of x in the second term is 1, and in- creases by 1 in each succeeding term. IV. The coefficient of the first term is 1 ; and of the second term, is the exponent of the binomial, V. If the coefficient of any term is multiplied by the exponent of a in that term, and the result divided by the exponent of x increased by 1, the quotient will be the coefficient of the next following term. 310 COLLEGE ALGEBRA. 444. We will now prove by induction (Art. 114, Note) that these laws hold for any positive integral power of a+x. Assume the laws to hold for (a + x)"; where n is any positive integer. Then, (a + re)" = a" + na"-^x + ^^^^^— ^avV H (1) Z Let P, Q, and R denote the coefficients of the terms in- volving a"~''a;'', a"~''"^ic''+\ and a"~''~^a;'"+^, respectively, in the second member of (1) ; thus, (a + a;)" = a" + na'^-'^x + ••• + Pa"-''a;'-+Qa"--^x'-+i-f 7?a"-'-V+2-| (2) Multiplying both members by a + x, we have ya + xY^^ = a"+i + ncCx + ••• -f Qa"-''a;'-+i + Pa"-''-ia;'-+- + ••• + a"a; + .-. + Pa"-''a;'-+^ + Qa""""-' x'-+2 + ... = a"+^ + (n + 1) cC'x + • • • + (P + Q)a«-'-x-'-+i + (Q + P)a"--i.T'-+2+... . (3) This result is in accordance with the second, third, and fourth laws of Art. 443. Since the fifth law of Art. 443 is assumed to hold with respect to the second member of (2), we shall have Q = -P<"-'-),a..J7i=. ?• + 1 r + 2 Therefore, ^ ^ Q{n-r-\) Q{n + 1) Q±Ji^ r + 2 _ r + 2 ^n-r P+Q Q{r + 1) J ^ Q(n + 1) r + 2' ?i — 7" ■ n — r Whence, Q + /j = ( /> + Q ) !izil". r-h2 THE BINOMIAL THEOREM. 311 But n — r is the exponent of a in that term of (3) whose coefficient is P + Q, and r+2 is the exponent of x increased byl. Therefore the Jifth law holds with respect to (3). Hence, if the laws of Art. 443 hold for any power of a 4- X whose exponent is a positive integer, they also hold for a power whose exponent is greater by 1. But the laws have been shown to hold for (a + x)'^, and hence they also hold for {a -\- xy ; and since they hold for (a + xy, they also hold for (a + x^; and so on. Therefore the laws hold when the exponent is any positive integer. By aid of the fifth law, the coefficients of the successive terms after the third, in the second member of (2), may be readily found ; thus, (a + x)" = a" + na^^^x -i — ^^ ^ a''~'-x- , n(n — l)(n—2) „_t q , ,,. — rrivs — ^-a'''^+--- (4) This result is called the Binomial Theorem. Note. In place of the denominators 1-2, 1 • 2 • 3, etc. , it is cus- tomary to write |2, |^, etc. Tlie symbol \n, read ^'■factorial ?i," signi- fies the product of the natural numbers from 1 to n inclusive. 445. Putting a = 1 in equation (4), Art. 444, we obtain (1 + xy = l + nx-\- ^-— — '- x-+ -^ ^ '- of^ H EXAMPLES. 446. In expanding expressions by the Binomial Theorem, it is convenient to obtain the exponents and coefficients of the terms by aid of the laws of Art. 443, which have been proved to hold for any positive integral exponent. 312 COLLEGE ALGEBRA. 1. Expand (a + a:) ^ The exponent of a in the first term is 5, and decreases by 1 in each succeeding term. The exponent of x in the second term is 1, and increases by 1 in each succeeding term. The coefficient of the first term is 1 ; of the second term, 5 ; multiplying 5, the coefficient of the second term, by 4, the exponent of a in that term, and dividing the result by the exponent of x increased by 1, or 2, we have 10 as the coefficient of the third term ; and so on. Hence, (a + xy = a' + 5a'x + lOaV + lOa'x^ + Saa;" + af. Note 1. The coeflBcients of terms equally distant from the begin- ning and end of the expansion are equal. Thus the coefficients of the latter half of an expansion may be written out from the first half. 2. .Expand (1 + 2 a;'')''. (l + 2a;'')«^ = [l+(2x^)]« = 1«+ 6 -1^.(20;^) +15-l*-(2x^)2+20-13.(2a;^)'^ + 15.12.(2x0^-1- 6-1 ■{2xy + (2xy = 1 + 12 x^ + GO or^ + 160 x'' + 240 x"" + 192 xr'> Note 2. If the first term of the binomial is a number expressed in Arabic numerals, it is convenient to write the exponents at first with- out reduction. The result should afterward^ be reduced to its simplest form. Note 3. If either term of the binomial has a coefficient or exponent other than unity, it should be enclosed in a parenthesis before apply- ing the laws. 3. Expand (Sm'^ - Vny. = [(3m-^) + (-nO]* = (;]m"^)* + 4(3m-2)^'(_„') + 0(3m"^)-(-w^)- + 4(3m-^) (-n^)^ + (-7t^)* = 81 ?u - — 1 08 vr '^ ?(^ +54 wi ' n'' — 12 m~ - n + n^. THE BINOMIAL THEOREM. 313 Note 4. If the second term of the binomial is negative, it should be enclosed, negative sign and all, in a parenthesis before applying the laws. In reducing afterwards, care must be taken to apply the prin- ciples of Art. 109. Expand tlie following 4. {a + xf. 3\5 5. {a-xy. - 13. (^a^h-'+a-"-h")\ 6. (1 +»;)'• 14. (Va^^ + 4v''a)*. -7. (ct+d-r- . 15. ri^-^Y- 8. (m-*-n-)«. 16. U'^-lx^. 9. (x--2^2'')^ -^ 17. {x'^ + ^y-'y. 10. (a« + 5V^)'. 18. (3a-W&-6"^^a)'- A trinomial may be raised to any power by the Binomial Theorem if two of its terms are enclosed in a parenthesis and regarded as a single term. 20. Expand (ar-.2a;-2)l {x'-2x-2y = [(x2-2x) + (-2)y = {x^ -2xy + 4(x2 -2xy{- 2) + 6(x-^ - 2xy{-2y + 4.{x^-2x){-2y + {-2y = a;8 _ 8 xJ + 24 .r« - 32 x' + 1 G .x-^ - 8 (a;« - 6 .r^ + 1 2 .i;^ - 8 x"') + 24 (a;^ - 4 x"" + 4 .i;-) - 32 (x- - 2 x) + IG = a;» - 8 .t' + IC x' + IG :c-' - 5G a;' - 32 .r' + G4 x' + C4 x + 1 G. 314 COLLEGE ALGEBRA. Expand the following : 21.' (x^ + x + iy. 24. {2x' + x + 3y. 22. {3a--ax-4.x'y. 25. (l-x + x'y. 23. {l+2x-c^y. 26. (x' + 2x-2y. 447. To find the rtli or general term in the exjyansion oj (a + a;)". The following laws will be observed to hold for any term in the expansion of (a + x)", in equation (4), Art. 444 : 1. The exponent of x is less by 1 than the number of the term. 2. The exponent of a is n minus the exponent of x. 3. The last factor of the numerator is greater by 1 than the exponent of a. 4. The last factor of the denominator is the same as the exponent of x. Therefore, in the rth term, the exponent of x will be r— 1. The exponent of a will be n — (r — 1), or n — ?• -|- 1. The last factor of the numerator will be n — r-\ 2. The last factor of the denominator will be r — 1. Hence, the rth term ^ 71 (n - 1) (n - 2) ■ ■ ■ (n - r + 2) ^^„_.^j ^^._, 1.2.3."(r-l) EXAMPLES. 1. Find the eighth term of (3a^ - 6-^)". In this case, r = 8, and ti = 11 ; hence the eighth term ^11. 10.9.8. 7. 6-5 X . 1.2.3.4.5.6.7 ^ ^ ^ ' = 330 . (81 a-) ( - ?> ') = - 26730 d'b'. THE BINOMIAL THEOREM. 315 Note. Notes 3 and 4, Art. 446, apply with equal force to the examples in the present article. Find the 2. Seventh term of (a -|- xy\ 3. Sixth term of (1 + m)'«. 4. Eighth term of (c - ciy-. — 5. Fifth terra of (1 - a-y^ 6. Seventh term of /'-+- \b a 7. Tenth term of (x™ - V^)". 8. Seventh term of ( a^^ — ~ab 2 9. Eighth term of (x-' + 2iJy^ 10. Sixth term of (a" ^- - 3x^y\ 9 \ 13 V«i + -^^ ) • -\/mJ 12. Find the middle term -e-r 13. Find the term involving x^^ in far + — ^ 14. Find the term involving x" in ( 2ar= =- ) '. 816 COLLEGE ALGEBRA. XXXI. CONVERGENCY AND DIVERGENCY OF SERIES. 448. A Finite Series is a series having a finite number of terms. An Infinite Series is a series the number of whose terms is unlimited. The progressions, in general, are examples of finite series; but in Art. 429 we considered infinite geometrical series. 449. Infinite series may be developed by the processes of Division and Evolution. Let it be required, for example, to divide 1 by 1 — x. l-x)l{l-\-x-^x' + •■• 1-x X X — a? The quotient is obtained in the form of the infinite series l-\-x + x?-\ Again, let it be required to find the square root of 1 + a;. l+x\l+-^--^ + ^ o 1 2 + 1 X XT ^ + ^ 4 ~^' ar' 2 -{-x- ~ 8 4 1 + The result is obtained in the form of the infinite series X 0" , CONVEKGENCY AND DIVERGENCY OF SERIES. 317 Infinite series may also be developed by other methods, one of the most important of which will be considered in Chapter XXXII. 450. A series is said to be convergent when the sum of the first n terms approaches a certain fixed quantity as a limit (Art. 209), when n is indefinitely increased; this limiting value is calleS. the Sum of the Series. A series is also said to be convergent when the sum of all its terms is equal to a fixed finite quantity. A series is said to be divergent when the sum of the first n terms can be made to numerically exceed any assigned quantity, however great, by taking n sufficiently great. 451. Consider, for example, the infinite series l+x + x- + x^-\ I. Suppose X = Xi, where Xi is numerically < 1. The sum of the first n terms is now 1 + a^i + a;,- + ••• + a;i"-^ = ^ ~ ^'" (Art. 115). When n is indefinitely increased, x^' decreases indefi- nitely in absolute value, and approaches the limit 0. 1 — a;" 1 Therefore '-^ approaches the limit • 1 — Xi 1 — Xi That is, the sum of the first n terms approaches a certain fixed quantity as a limit, when n is indefinitely increased. Hence the series is convergent when x is numerically < 1, II. Suppose X = 1. In this case, each term of the series is equal to 1, and the sum of the first n terms is equal to n; and this sum can be made to exceed any assigned quantity, however great, by taking n sufficiently great. Hence the series is divergent when x = 1. 318 COLLEGE ALGEBRA. III. Suppose a; = — 1. In this case the series takes the form 1-1 + 1-1 + ...; and the sum of the lirst n terms is either 1 or according as n is odd or even. Hence the series is neither convergent nor divergent when X- = - 1. IV. Suppose x= Xi, wlaere Xi is numerically > 1. The sum of the first n terms is 1 + Xi + xi^ + ... + xr' = "^LJzl (Art. 115). X'l — 1 x" — 1 By taking n sufficiently great, the expression — — can Xi — 1 • be made to numerically exceed any assigned quantity, how- ever great. Hence the series is divergent when x is numerically > 1. 452. Consider the infinite series 1 + x + .^•^ + a.-^ H , developed by the fraction (Art. 449). Let x = .l, in which case the series is convergent (Art. 451). The series now takes the form 1 + .1 + .01 + .001 + ..., while the value of the fraction is — , or -— . .9 9 In this case, however great the number of terms taken, their sum will never exactly equal — ; but it approaches this value as a limit (Art. 430). Thus, if an infinite series is convergent, the greater the number of terms taken, the more nearly, does their sum approach to the value of the expression from which the series was developed; and the sum of the series (Art. 450) is (Hiua.l to tlic value of this expression. CONVERGENCY AND DIVERGENCY OF SERIES. 819 Again, let x — 10, in which case the series is divergent. The series now takes the form 1 + 10 + 100 + 1000 + •••, while the value of the fraction is , or 1-10 9 In this case it is evident that, the greater the number of terms taken, the more does their sum diverge from the value 9 Thus, if an infinite series is divergent, the greater the number of terms taken, the more does their sum diverge from the value of the expression from which the series was developed. It follows from the above that an injinite series cannot be used for the inirposes of demonstration, unless it is convergent. Note. It will be understood hereafter that, in every expression in- volving a convergent infinite series, the sum of the series is meant. For example, the product of two convergent infinite series will be understood as signifying the product of their sums. ELEMENTARY THEOREMS ON THE CONVERGENCY AND DIVERGENCY OF SERIES. 453. The infinite series a + hx + cxr + dx"' + ••• is convergent when .'^ = ; for the sum of all the terms is equal to a when a; = 0. 454. If all the terms of an infinite series are of the same sign, the series must he either convergent or divergent. Note. An infinite series may be neither convergent nor divergent; an example of this has been given in Art. 4-51, III. Such series are called indeterminate or neutral. In this case, the greater the number of terms taken, the greater will be the absolute value of their sum. 320 COLLEGE ALGEBRA. Therefore, either the sum of the first n terms approaches a certain fixed quantity as a limit, when n is indefinitely increased ; or else the sum of the first n terms can be made to numerically exceed any assigned quantity, however great, by taking n sufficiently great. Hence the series must be either convergent or divergent. 455. By Art. 454, if all the terms of an infinite series are of the same sign, the series must be either convergent or divergent ; hence, I. If the sum of the first n terms is numerically less than a certain fixed quantity, however great n may be, the series is convergent. II. If each term of the series is numerically less than the corresponding term of another infinite series whose terms are all of the same sign, and which is known to be convergent, the first series is convergent. III. If each term of the series is numerically greater than the corresponding term of another infinite series whose terms are all of the same sign, and which is known to be divergent, the first series is divergent. 456. If all the terms of an infinite series are of the same sign, and each term is numerically greater than some assigned finite quantity, however small, the series is divergerit. For if each term of the series is numerically greater than a, the sum of the first n terms is numerically greater than na, and hence can be made to numerically exceed any assigned quantity, however great, by taking w sufficiently great. Therefore the series is divergent. 457. It follows from Art. 456 that, i^H;l7he terms of an ^infinite series are of the same sign, the series is divergent unless its 7ith term approaches the limit when n is indefi- nitely increased. CONVERGENCY AND DIVERGENCY OF SERIES. 321 458. If the terms of an ivjinite series are alternately iiositive and negative, and each term, is numerically less than the lire- ceding term,ihe series is convergent. For let tlie series be Ui — ti^ + tu — Ui+ ••• • It may be written in. either of tlie forms {u^ - Ma) + (t(3 - W4) + {ih - Ue) + •", (1 ) or, «i — (W2 — W3) - (M4 — 7/5) (2) By hypothesis, each of the expressions u^ — Uo, u^ — ■?'« etc., is positive ; that is, all the terms of series (i) are pos- itive ; and it is evident from (2) that the sum of the first n terms of the series is less than ii^, however great n may be. Therefore, by Art. 455, I., the series is convergent. As an example of the above, the infinite series 1-1+1-1+ 2 3 4 is convergent. 459. If all the terms of an infinite series are of the same sign, and the ratio of each term to the preceding term is less than a certain quantity which is itself less than unity, the series is convergent. Let the series be % + «2 + «3-l hw,_i + ?(^H ; (1) and suppose ~ n, ^ is positive and < 2 ; and hence Vl^L± _ 1 is numerically < 1. , .\t-f V CONVERGENCY AND DIVERGENCY OF SERIES. 325 II. Suppose n negative, and numerically < 1. 11 A- 1 Then, whatever the value of r, is positive and < 1 ; 4-1 ^ and hence ^^^^t 1 is numerically < 1. r ni. Suppose n negative, and numerically > 1 ; and let w = — n', where n' is positive and > 1. Then, n-r + l^^-n'-r + l^^i;^-l r r — 1 r "■ +iV-fi-i)- 1 If r is taken sufficiently great, — [- 1 may be made to r — 1 differ from 1 by less than any assigned quantity, however small. Hence, if r is taken sufficiently great, f— hi )a; may be made numerically less than 1, since x is numerically less than 1; and as r increases beyond this value, (— |-l)a; decreases indefinitely in absolute value. v ~ J Also, 1 is less than 1. r Thus, in the first and third cases after a certain assigned term, and in the second case always, the ratio of the (?•+!) st term to the rth term is numerically less than a quantity which is itself numerically less than 1. Therefore by Art. 462, the series is convergent in each of the above cases. IV. If n = — 1, the series takes the form 1 — a; + x- — x^ + •••, which is convergent by Art. 458. 32G COLLEGE ALGEBRA. 467. To irrove that the infinite series l + | + | + j; + - (1) is convergent tvhen n is > 1, and divergent lohen ?i = 1 orn < 1. I. If n is > 1, the second and third terms are together 119 4 < — + — , or < -^^ ; the next four terms are together < — ; 8 the next eight are together < — ; and so on. 8" Therefore the series is less than the series which is known to be convergent by Art. 451-, I. Hence tlie given series is convergent (Art. 455, II.). II. If n = 1, the series becomes i+l+i+h <2) 2 1 The third and fourth terms are together > -, or > - •, the next four terms are together > -, or > - ; and so on. 8 2 Hence, by taking a sufficiently great number of terms^ their sum may be made to exceed any assigned quantity, however great. Therefore the series is divergent. III. If n is positive and <1, or negative, each term of (1) is greater than the corresponding term of series (2). Hence the given series is divergent (Art. 455, III.). CONVERGENCY AND DIVERGENCY OF SERIES. 327 EXAMPLES. 468. Expand each of the following to four terms : . . 2-5aj a. 2-5. T + 6a; '^ „ ,-— — 1 + ar 1 + 4 a; — o a;- 1 2-5.T + 6a;'^ 3_7x-2_x-^ 5. Vl-2a^ 8. Vl + ar^. -. _ 6. Va--4a& + 6l 9. -^8a^-'6b. 10. Prove that the infinite series 1 + 1+1+1 + ...., is convergent. The series is less than the series 14.1,14.1-L which is known to be convergent by Art. 451, I. Therefore the series is convergent (Art. 455, II.). Examine whether the following infinite series are con- vergent or divergent : - il (3 11^ ' 1.2^2-3 3.4^4-5 13. l+| + | + ^. + -- 16. l+f3 + | + J + -- '+ "• l + r2 + 2^ + 3-^ + 4-^5+-- . 328 COLLEGE ALGEBRA. XXXII. THE THEOREM OF UNDETER- MINED COEFFICIENTS. 469. An important method for expanding expressions into series is based on the following theorem, known as the Theorem of Undetermined Coefficients : If the series A + Bx +Cx^-\-Daf+ ••• is always equal to the series A'+ B'x+C'u:^+ D'af+ •••, when x has any value which makes both series convergent, the coefficients of like powers of x in the two series will he equal; that is, A=A', B = B', C = C', etc. For since the equation A + Bx + Car + Dx^ + ■•• = A'+ B'x+C'x' + D'x^ + ... is satisfied when x has any value which makes both mem- bers convergent, and since both members are convergent when x = (Art. 453), it follows that the equation is satis- fied when x = 0. Putting a; = 0, we have A = A'. Subtracting A from the first member of the equation, and its equal A' from the second member, we obtain Bx+Cx" + Z>x-3 + ... = B'x+C'x- + D'x" -\ Dividing through by x, B +Cx + Dx" + -•■ = B' + C'x + D'x" -i This equation also is satisfied when x has any value wliich makes both members convergent ; and putting x=0, we have B=B'. In like manner, we may prove C = C', D = D', etc. Note. The above demonstration is the one usually giveti in text- books on Algebra; it is, however, open to olijection in one respect. UNDETERMINED COEFFICIENTS. 329 It is demonstrated rigorously that 5x + Cic^ + 2>x3 + ... = B'x + (7'x2 + D'oc^ + -, or, x{B+CxfDx''+-) = x{B<+C'x-\-D'x^+-), (1) when X has any value which makes both members convergent, including the value zero. But when we divide through by x, and put the result m the form 5 + Cx + Dx^ +••• = £'+ C'x + D'x:^ + •••, (2) all that we know about this equation is, that it is satisfied by every value of X, except zero., which makes both members convergent. We cannot assert that the equation is satisfied when x = ; for equation (1) is satisfied when x = 0, even if B+ Cx-\- Dx:^-\ and B' + C'x + D'x^ + ••• are unequal for this value of *. In order to demiinstrate rigorously that B = B\ we may proceed as follows : Both members of (2) are convergent, and the equation therefore satisfied, when x is taken sufficiently small (Art. 463), and for all smaller values of x, except zero. But if X is taken sufficiently small, the first member may be made to differ from 5, and the second member from J3', by less than any assigned quantity, however small. ■ Hence B and B' cannot differ by any assigned quantity, however small, and are therefore equal. 470. A finite series being always convergent, it follows from the preceding article that if two finite series A + Bx+Cx--\ + Ax" and A' + B'x + C'x- -f • • • + K'x"-, are equal for every value of x, the coefficients of like powers of X in the two series are equal. - , EXPANSION OF FRACTIONS INTO SERIES. 9 _ 3 3j2 ^.3 471. lo Expand -^ '—, in ascending powers of x. 1 — 2x + 3a^ We have seen in Art. 449 that a fraction of the above form may be expanded into a series by dividing the numerator by the denominator ; we therefore know that the proposed expansion is possible. 330 COLLEGE ALGEBRA. Assume then A + Bx-{-Ca^ + Dx' + Ex* + — ; (1) 1 — 2a;H-3ic^ where A, B, C, D, E, ..., are quantities independent of x. Clearing of fractions, and collecting the terms in the second member involving like powers of x, we have »*+.... (2) Bx+ C ar'+ D .T^+ E 2A -2B -20 -2D + 3.4 + 35 + 3C The second member of (1) must express the value of the fraction for every value of x which makes the series con- vergent (Art. 452). Hence equation (2) is satisfied when x has any value which makes both members convergent, and by the Theorem of Undetermined Coefficients, the coefficients of like powers of X in the two series are equal ; that is, A= 2. B-2A= 0; whence, 5 = 2^ =4. C-2B + 3A = -3; whence, C = 2B-3A-3=-l. lj_2C-\-3B = -l; whence, D = 2C-3B-1= -15. E-2D+3C = 0; whence, E=2D-3C =-27; etc. Substituting these values in (1), we have l^zl^l^l^ = 2 + 4:x-x''-15x'-27x' l-2a; + 3ar Tlie result may be verified by division. Note 1. A vertical lino, called a bar, is often used in place of a parenthesis ; thus, + Ji\x \s equivalent to (B — 2A) x. -2a\ Note 2. The result expresses the value of the given fraction only for such values of x as make the series convergent (Art. 452). UNDETERMINED COEFFICIENTS. 331 If the numerator and denominator contain only even powers of x, the operation may be abridged by assuming a series containing only the even powers of x. Thus, if the fraction were ^ ^^ ^ — — -, we should assume 1 — 3ar+5a;"* it equal to ^ + B^^ + Cx* -|- Dx^ + Ex^ -\ In like manner, if the numerator contains only odd powers of X, and the denominator only even powers, we should assume a series containing only the odd powers of x. If every term of the numerator contains x, we may assume a series commencing with the lowest power of x in the nu- merator. EXAMPLES. Expand each of the following to five terms, in ascending powers of x : 2 ^-^ 6 ^-^-^^ 10 2-3a; + 4a.-^ ' 1 + x ' ^ ■-■ • - ^ 3 - + 5a; l-'dx 4. ^-^^\ 8. ^-■■^^■•^ . 12. l + bx" 2x 1+x + x^ X -Sx^-x" 1 -2x-x- 2 — x + x^ 1-x' l-2ar^ 11. 13. l+2x-5x^ a;^ + 2ar^ 2-x-x'' 3 + a!-2a;3 3 - a^ + a;3 l-3a)2 3-2ar^ I + 2a;-3a,-2 2-3i»-2ar^ If the lowest poAver of x in the denominator is higher than the lowest power in the numerator, we may determine by actual division what power of a; will occur in the first term of the expansion ; we should then assume the fraction equal to a series commencing with this power of x, the exponents of X in the succeeding terms increasing by unity as before. 14. Expand — - — — - in ascending powers of x. 3ar —or Dividing 1 by 3x^, the quotient is ^- 33i COLLEGE ALGEBRA. We then assume, . — = Ax-' + Bx-''+C+Dx + Ea^ + -- (3) Clearing of fractions, l = 3^ + 35|a; + 3C|a;- + 3i)ja;' + 3^|a;*+---. - a\ - b\ - c\ - d\ Equating the coefficients of like powers of x, 3A = 1 3B-A = 0: 3C-B=0 3D- C=0 3E-D = 0; etc. Whence, ^ = | i? = | C'=l., D = l^, E = ^^, etc. Substituting in (3), we have _J— — ^"-4- — -I-— -l-^ + — + • • • Sx'-x^" 3 "^ 9 ^27 "^81^243 Expand to five terms in ascending powers of x 15. 3a;--4ar^ 16. 1+x-o^ x-2x^-\-3x' 17. 18. l-2a.-^-.r\ X^ + X^ — X* 3-2a; + .^•^ 2x'-x*-'2x'^ EXPANSION OF SURDS INTO SERIES. 472. 1. Expand Vl — x in ascending powers of x. We have seen in Art. 449 that the square root of an imper- fect square can be expanded into a series by the process of Evolution ; we therefore know that the proposed expansion is possible. Assume then Vr^^ = ^ + Bx + Cx' + D.r + E,i^ + ■■• - (1) UNDETERMINED COEFFICIENTS. 833 Squcaring both members, we have by Art. 230, l-x = A' + 2AB x-^ B' x' x'+ C- + 2 AC + 2 AD + 2AE -{-2BC + 2BD 0)' + Equating the coefficients of like powers of x, A^ = 1; whence, -4=1. 2 A 2 ^ = _L 2 A 8° BC^_\^ A 16" 2AB = — 1 ; whence, 5= — J5^ -f 2 ^C = ; whence, (7= - 2AD^2BC= 0; whence,Z)=- C-^2AE+2BB= 0; whence, £= — 2A 128' etc. Substituting these vakies in (1), we have VT^^' = 1 X X- X' ox^ 2 8 16 128 The answer may be verified by the method of Art. 449. Note 1. The result expresses the value of the given surd only for such values of ic as make the series convergent. Note 2. The equation A^ = 1 gives ^ = ± 1 ; and taking the nega- tive value of ^4, we should find B = -, C= -, -D = — , etc. 2 8 15 Thus another answer to the example is — 1 + ^ + ^ + — + •••• 2 o lu EXAMPLES, Expand each of the following to five terms, in ascending powers of x : 2. Vl-f-2cK. 4. VI - 2a; + ax''. 6. (1) x{x + iy X x + i {x + iy {x + i) Clearing of fractions, 3x + 2 = A{x + lY-{-Bx{x + ly + Cx{x + 1) + Dx = {A + B)x' + {^A + 2B+C)x' + {'5A + B+C + D)x + A. Equating the coefficients of like powers of x, A + B = 0, 3^ + 25 + C=0, ^A + B + G + D = d, and A = 2. Solving these equations, we have yl = 2, 5= - 2, C= - 2, and i) = 1. Substituting in (1), _3^_-f2_ ^ 2 _ _2 2 1 x{x + iy' X re 4-1 {x + iy {x + iy Note. It is impracticable to give an illustrative example for eveiy pdSsible case ; but the student will find no difficulty in assuming the proper partial fractions if attention is given to the following general rule : A fraction of the form should be put equalto (x + a)(x + 6)...(x + m)-.. x+a x + b x + m {x + my {x + m)'^ Single factors like x + a and x + h having single partial fractions corresponding, arranged as in Case I. ; and repeated factors like (x + my having r partial fractions corresponding, arranged as in Case II. UNDETERMINED COEFFICIENTS. 339 EXAMPLES. Separate each of the following into partial fractions : x{x + 2y' ' x{x-l){x-'2y 3 3.T-1 g IB-Tx + Sx'-Sx' ar(a; + l)- * x*-\-5a^ ^ 3a^ — 7x + 3 - ,^ 5x^-{-3x-{-2 (2a;-3)(2a^ — Tx + O) x^{x-\-iy 478. If the degree of the numerator is equal to, or greater than, that of the denominator, the preceding methods are inapplicable. x'^ — 3 x^ — 1 Let it be required, for example, to separate ^ into partial fractions. '*' ~ If we proceed as in Case I., we should assnme 0? — X X x — 1 Clearing of fractions and uniting terms, or^ - 3ar - 1 = (^1 + B)x - A. Equating the coefficients of x^, we have 1 = 0; a result which shows that the method of Case I. is inapplicable. But by actual division we obtain ar — X X- — X We can now separate — '^ into partial fractions by XT — X the method of Case I. ; the result is 1 3_ X x—i Substituting in (1), we have X- — X X X — 1 340 COLLEGE ALGEBRA. EXAMPLES. Separate each of the following into an integral expression and two or more partial fractions : J 8x^-36x^-2 g 5x^ + 5x^-2x^ + 3 (2.r-5)(2a; + l)" ' x* + cc' ' 2 3ar^ + 19a;^ + 35a; ' ^ 3af -2x' + 22x^ + 9x {x + 2y ' • (ar-ir 479. If the denominator of a fraction can be resolved' into factors partly of the first and partly of the second, or all of the second degree, in x, and the numerator is of a lower degree than the denominator, the Theorem of Unde- termined Coefficients enables us to express the given frac- tion as the sum of two or more partial fractions, whose denominators are factors of the given denominator, and whose numerators are independent of x in the case of frac- tions corresponding to factors of the first degree, and of the form Ax + B in the case of fractions corresponding to fac- tors of the second degree. The only exceptions occur when the factors of the denom- inator are of the second degree and all equal. 1. Separate — into partial fractions. The factors of the denominator are x + 1 and xF — x + 1. Assume then, —^ = -^— + ^^+G . (V) ' x' + l x + l^x'-x + l ^ ' Clearing of fractions, we have 1 = A{x' -x + l) + (Bx + C) (:r + 1) = (.1 + B) .r + ( - yl + B +C)x + A + C. UNDETERMINED COEFFICIENTS. 341 Equating the coefficients of like powers of x, A + B = 0, -A + B-\-C=0, and A+C=l. Solving these equations, A = -, B = — ~, and C= -• o O 3 Substituting in (1), Ave have 1 1 x-2 a;3 + l 3(a;-f-l) ^{x'-x-\-l) EXAMPLES. Separate each of the folloAving into partial fractious : x^-1 x^+x' + l ^ ar^4-2a;-2 " g 45 + 36a;-a;^ (x2 + 2)(a;2 + a; + 2)' ' a-" - 6ar'- 27* 4_ 20a;^-2a;^ ^ a;^ _ 2.t^ + .r^- a; + 1 REVERSION OF SERIES. Note. To revert a given series y = a + bx"' + ex" + ••• is to express X in tlie form of a series proceeding in ascending powers of y. 480. Example. Revert the series y = 2x + x^-2a?-3x'-\ Assume x = Ay+ By- + (7/ + Dy"^ -\ . (1 )- Substituting in this the given value of y, we have x = A{2x + x- -23? -Sx' +'...) + B{4x^ + x' + 437= - 8a;^ + ...) + C(8a.-3 + 12a;^+ ...) -\-D{mx' +...)+.... 342 COLLEGE ALGEBRA. That is, a; = 2yla;+ A + 45 x'-2A x'- SA + AB - 7B + 80 + 12C + 16D x^ + .... Equating the coefficients of like powers of x. 2A=1 A+4:B=0 -2A + 4.B-hSC=0 -3A-7B + 12C+16D = 0; etc. Solving these equations, A = -, B=--, (7 = -^, i) = -il, etc. 2 8 16 128 Substituting in (1), we have 1 , , 3 . 13 2^ 8^ 16^ 128 y* + If the even powers of x are wanting in the given series, the operation may be abridged by assuming x equal to a series containing only the odd powers of y. Thus, to revert the series y = x — x^ + x" — x^-^---, we should assume x = Ay + Bf + Cf + Df + ■'• • If the odd powers of x are wanting in the given series, the reversion of the series cannot be effected by the method previously given. But by substituting another letter, say t, for X-, we may revert the series and express t in terms of y ; and by taking the square root of the result, x itself may be expressed in terms of y. If the first term of the given series is independent of x, it is impossible, by the method previously given, to express X in the form of a series proceeding in ascending powers of y; but it is ])()ssible to ex])ress it in the form of a series in Avhich y is the uiily uukuowu quantity. UNDETERMINED COEFFICIENTS. 343 Let it be required, for example, to revert the series y = 2-\-2x + x^-2x^-3x^+--- • Tlie series may be written y-2 = 2x + x'-2x'-SxU We then assume X = A(y - 2) +B(y - 2y+C(y - 2y+D{y - 2)^+ ••. • Proceeding as before, we find x = ^{y-2)-^(y-2Y+ — (y-2Y-—(y-2Y + '-'' EXAMPLES. 481. Revert each of the following to four terms : f l/l y = X -{■ '£- -\- X? -\- x^ -\- '• ' • ^ 2. ?/ = 3x--2x2+3af'-4a;''+.... ^2 4 6 8 ^ i i_3 li - 5. y = x — 7?-\-x' — '3?-\--"' 6. y =■ u ... . ^ 2 3 4 5^ 7. 2/ = 3a; + 5a^ + 7af' + lla;^ + o.oo /y»3 /y*5 /y,7 344 COLLEGE ALGEBRA. XXXIII. THE BINOMIAL THEOREM. FRACTIONAL AND NEGATIVE EXPONENTS. ,482. It was proved in Art. 445 that, when n is a positive integer, /^t . \n 1 , , n(n — l) o , n(n — l)(n — 2) n , (1 + 0;)"= 1 +nx + -^-— — ^-x^ + ^ f;^ ^ar^ + .... PROOF OF THE THEOREM FOR FRACTIONAL OR NEGATIVE EXPONENTS. Note. We shall use the expression "Fractional or Negative Ex- ponent," in the present chapter, to signify a rational exponent which is not a positive integer. 483. I. Wheyi the exponent is a positive fraction. Let the exponent be -, where p and q are positive integers. By Art. 282, (1 + x)^ = ^{1 + xy = ^l+px + -- (Art. 482). In Art. 263, we gave a rule for extracting the ?ith root of a polynomial which is a perfect power of the nth degree. It is evident, therefore, as in Art. 449, that -y/l+jix -\ can be expanded in a series proceeding in ascending powers of X ; thus, px l+pa; + ... 1+'— + ••• q \ 2^x-\- ••• p vx That is, (i + a;)» = l+Y + -... (1) THE BINOMIAL THEOREM. 345 II. When the exponent is a negative integer or a negative fraction. Let the exponent be — s, where s is a positive integer or positive fraction. By Art. 284, (l + x)-' {i + xy 1 1 + sx-Jr -, by Art. 482 or Case I. It is evident, as in Art. 449, that can be ex- ' 1+SX+--- paneled, by actual division in a series proceeding in ascending powers of x ; thus, l + sx + ---)l{l—sx-\ l + sx-\ — sx— ••• That is, ' (1 + x)-' = 1 - saj H (2) From (1) and (2) we observe that, when n is fractional or negative, the form of the expansion is (1 4- xy ^l + nx + A^- + Bx^ ^ (3) Writing - in place of x, we obtain a aj a, a- a'^ Multiplying both members by a", (a + a-) " = a" + na"'^ x + Aa''~^ ar + Ba"'^ x^ -\ • (4) This result is in accordance with the second, third, and fourth laws of Art. 443 ; hence these three laws hold for fractional or negative values of the exponent. We will now prove that the ffth law of Art. 443 holds for fractional or negative values of the exponent. Let P and Q denote the coefficients of x'' and a;''+^, respec- tively, in the second member of (3). 346 COLLEGE ALGEBRA. Then (3) and (4) may be written (l-\-xy = l+nx-] \-Px''-{- Qx''+'' + ..., f5) and {a + x)" = a" -i-na"^x-^ ••• + Pa"-'-a;'- + Qa"-'-'a;'-+^H (6) In (6) put a = l + y and x = z; then, (l+2/ + 2:)»=(l + 2/)"+...+P(l + 2/)"-'-2'- + .... (7) Again, in (5) put x = z + y; thus, (l + z-^yr=l + -+P{z+yy+Q{z + 7jy+' + .^.. Expanding the powers of z + yhy aid of (6), we have (1 + z + 2/)" = 1 + ... + P[z^ + rz^-'y + .-] + Q[,^+i+(r+l),^y + . ..]+.... (8) The first members of (7) and (8) being identical, their second members must be equal for every value of z which makes both series convergent (Art. 452) ; and by the Theorem of Undetermined Coefficients, the coefficients of z'' in the two series are equal ; that is, P(l + ?/)"-' = P+Q(r +1)2/ + terms in y% f, etc. Expanding the first member by aid of (5), this becomes P[l+(,i_r)r/ + ...] = P+Q(r + l)i/+.... This equation is satisfied by every value of y which makes both members convergent, and hence the coefficients of y in the two series are equal ; that is, P(n - r) =Q{r + 1), or Q = P>^zJl. r+ 1 But in the second member of (G), n — r is the exponent of a in the texm whose coefficient is P, and r + 1 is the ex- ponent of X in that term increased by 1. Therefore the fifth law of Art. 443 is proved to hold for fractional or negative values of the exponent. THE BINOMIAL THEOREM. 347 By aid of the fifth law, the coefficients of the successive terms after the second, in the second member of (6), may be readily found as in Art. 444; thus, (a + xy = a" + na^-'x + "^^^—^a"-':^ , n(n — l)(n — 2) _ , ,, ^f^\ The second member of (9) is an infinite series ; for if n is fractional or negative, no one of the quantities n — 1, n — 2, etc., can become equal to zero. The result expresses the value of (a + a;)" only for such values of a and x as make the series convergent (Art. 452). Note. Divicung both members of (9) by a", we have (l I ^V=l 1 n^ 1 »(»-l) ^^ I n(n-l)(n-2) x^ ^ \ aj a \2 a'^ [3 a^ numerically less than 1 ; hence the series (9) is convergent when x is numerically less than a. EXAMPLES. 484. In expanding expressions by the Binomial Theorem when the exponent is fractional or negative, the exponents and coefiicients of the terms may be obtained by aid of the laws of 'Art. 443, which have been proved to hold for all rational values of the exponent. Notes 3 and 4, Art. 446, apply with equal force to the examples in the present article. 1. Expand (a + a;)^ to four terms. 2 The exponent of a in the first term is -, and decreases by 1 in each succeeding term. 348 COLLEGE ALGEBRA. The exponent of x in the second term is 1, and increases by 1 in each succeeding term. The coefficient of the first term is 1 ; of the second term, 2 2 1 - ; multiplying - the coefficient of the second term, by — -, 3 3 o the exponent of a in that term, and dividing the product by the exponent of x increased by 1, or 2, we have — t: as the coefficient of the third term ; and so on. Hence, (^a + x)'^ = J + la-ix-la-^oc^ + ^a-'^x' o y oi 2. Expand (1 — 2 a; ^)-no five terms. (l-2i»-^)-2 = [l+(-2x-^)]-2 = l-2_ 2.1-3. (_2a;-*) + 3.1-^.(-2a;-')2 -4:.l-''{-2x-^y+5-l-'-{-2x-fy- = l-\-4:X~^ + 12x-^ + 32x--^ -\-80x~^-i 3. Expand ^ — to five terms. Va-i + Sx* {a-' + 3xs)^ [(«-) + (3x0] hi-i -f^ia-r'^'isxh^ =ai-Jx^+2a^x^-^a^x+^a'^'x^+... o o THE BINOMIAL THEOREM. 349 Expand each, of the following to five terms : V 4. (a + x) I - 12. {a'-2x-')- 1 13. 5. il + x)-\ i 6. (i-x)-i 7. proximaiely by the Binomial Theorem. 1. Find the approximate value of ^25 to five places of decimals. ■^25 = 25'^ = (27 - 2) 3 = (3^ - 2) J. THE BINOMIAL THEOREM. 351 Expanding by the Binomial Theorem, we have [(3'^) + (-2)]^ = (3^)^ + 3(3TH-2)-^(3T^(-2)^ + A(33)-I(_2r-... = 3. ^ 4 40 3.32 9.3^ 81-3« Expressing the value of each fraction approximately to the nearest fifth decimal place, we have ^/25 = 3 - .07407 - .00183 - .00008 = 2.92402. Separate the given number into two parts, the first of which is the nearest 2')erfect power of the same degree as the required root. Expand the result by the Binomial Theorem. Note. If the second term of the binomial is small compared with the first, the terms of the expansion diminish rapidly ; but if the second term is large compared with the first, it requires a great many terms to ensure any degree of accuracy. EXAMPLES. Find the approximate value of each of the following to five places of decimals : 2. VlO. 4. 1, a""'' = -4 = - = (Art. 210). a cc Whence, by Art. 488, log„0 = - c». Note. Fo literal meaning can be attached to such a result as logaO = — co; it must be interpreted as follows: If, in any system whose base is greater than unity, a number ji]ii)ro.T,ches tlic limit 0, its logarithm is negative, and increases without limit ill ab.S(jiutf value. (^Cmiiiiare Note to Ait. 212.) LOGARITHMS. 355 497. In any system whose base is less than unity, the logo- rithm of zero is infinity. For if a is < 1, a°° = ; whence, log^O = oo. 498. In any system, the logarithm of a i^oduct is equal to the sum of the logarithms of its factors. Assume the equations "*' = ^ I ; whence by Art. 488, | ^ = ^°^«^' a^ = n ) (y = loga^i. Multiplying, a"" x a" = inn, or a'^+^ = mn. Whence, log^ ??i« = x-\-y = log„m + log„7t. In like manner, the theorem may be proved for the product of three or more factors. 499. By aid of the theorem of Art. 498, the logarithm of any composite number may be found when the logarithms of its factors are known. 1. Given log2 = .3010, and logS = .4771 ; find log 72. log72 = log(2x 2x2x3x3) = log2 +log2 + log2 +log3 + log3 = 3xlog2 + 2xlog3 = .9030 + .9542 = 1.8572. EXAMPLES. Given log2 = .3010, log3 = .4771, log5 = .G990, and log 7 = .8451; find: 2. log42. — 6. logll2. 10. logl47. 14. log514.5. 3. log 45. 7. log 144. 11. log 375. 15. log 6048. 4. log63. 8. log216. 12. logGSG. 16. Iogl2005. 5. Iogl05. 9. logl35. 13. logll34. 17. logl5876. 356 COLLEGE ALGEBRA. 500. In any system, the logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator. Assume the equations «^=^n; whence, j-'^^J^-^' a^ = n ) yy = log„7i. Dividing, we have — = — , or a''^^ = • a" n n Whence, log" — = x — v n = logo m — log„n. 501. 1. Given log2 = .3010 ; find log 5. Iog5 = log^=logl0-log2 = 1 - .3010 = .6990. EXAMPLES. Given log2 =.3010, h)g 3 =.4771, and log 7=. 8451 ; find: 2. log^. 5. log35. ^. logi^. 11. log57f. 3. log^'. --' 6. log||. 9. logCf. ^12. log^. 4. logllli. 7. log225.~^ 10. log245. 13. log21x. 502. In any system, the logarithm of any x>oiver of a quan- tity is equal to the logarithm of the quantity multiplied by the exponent of the jiower. Assume the equation a^ = m; whence, x = log„ m. Raising both members to the pth power, a''-' = m'' ; whence, log„ m'' = px —p log,///*. LOGARITHMS. 357 503. In any system, the logarithm of any root of a quantity is equal to the logarithm of the quantity divided by the index of the root. - i 1 For, \og^-{/m = \og^(m'') = -\og^m (Art. 502). 504. 1. Given log 2 = .3010; find log2l log23 = f X log2 = ^ X .3010 = .5017. o o Note. To multiply a logarithm by a fraction, multiply first by the numerator, and divide the result by the denominator. 2. Given log 3 = .4771; find log ^/^ log ^3-=^ = --^=.0596. . EXAMPLES. Given log 2 = .3010, log 3 =.4771, and log 7= .8451 ; find. 3. logSl "-e. logl4^ 9. log253-. 12. log a/5. 4. log7«. 7. log 12.1.;' 10. logV7. 13. logv^. 5. log5i 8.. logl5i 11. log a/2. 14. log a/126. 15. Find log (23 X 30- By Art. 498, log (2^ x 3^) = log 2' + log3T ^log2 + -log3=.6967. "3 ° ' 4 ^^^ Find the logarithms of the following : ^(fj- "^«>S- %4 ... ^f . 17. 3^^x21 19. 3i. Hence, if two numbers have the same sequence of figures, the mantissse of their logarithms are equal. The reason will now be seen for the statement made in Art. 491, that only the mantissas are given in a table of logarithms. For, to find the logarithm of any number, Ave have only to take from the table the mantissa corresponding to its sequence of figures, and the characteristic may then be prefixed in accordance with the rules of Arts. 492 and 493. Thus, if log 3.053 = .4847, then log 30. 53 = 1.4847, log .3053 = 9.4847 - 10, logS05.3 = 2.4847, log .03053 =8.4847-10, log 3053. = 3.4847, log .003053 = 7.4847 - 10, etc. ^ This property is only enjoyed by the common system of logarithms, and constitutes its superiority over others for the purposes of numerical computation. 506. 1. Given log 2 = .3010, log 3 = .4771 ; find log. 00432. log 432 = lug(2' X 3') = 1 log 2 + 3 log 3 ^ 2.0353. LOGARITHMS. 35? Then by Art. 505, the mantissa of the result is .6353. Whence by Art. 493, log .00432 = 7.6353 - 10. EXAMPLES. Given log2 = .3010, log3 = .4771, and log7 = .8451 ; find : 2. log 19.6. ^^ 6. log 7350. 10. log. 06174. 3. log4.8. 7. Iog4.05. 11. log(8.1)^ 4. log. 384. 8. log. 000448. ■ 12. log ^9600. 5. log.00315. 9. log 302.4. 13. log (22.4) «. 507. To prove the relation T log.m logjm = — ^"— log,. Assume the equations cC = m) T (a; = log„m, - ; whence, < t^a ) b'' = m} (i/ = log^m. From the assumed equations, a"^ = b^, or a" = b. Therefore, log„& = -, or y = y ' log,, 6 That IS, logjm By aid of this relation, if the logarithm of a quantity m to a certain base a is known, its logarithm to any other base b may be found by dividing by the logarithm of b to the base a. A- 508. To prove the relation log^a X log„6 = l. Putting m = a in the result of Art. 507, we have log.a = |^ = -J-(Art.495). log,& log„& Whence, log^a x log„^ = 1. 5G0 COLLEGE ALGEBRA. So. 1 2 3 4 5 6 7 8 9 lO 0000 0043 00S6 0128 0170 0212 0253 0294 0334 0374 II 0414 0453 0492 PS3I 0569 0607 0645 06S2 0719 0755 12 0792 0828 0S64 0899 0934 0969 1004 1038 1072 1 106 13 "39 "73 1206 1239 1271 1303 1335 1367 1399 1430 14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 17 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 18 2553 2577 2601 2625 264S 2672 2695 2718 2742 2765 19 2788 2810 2833 2856 287S 2900 2923 2945 2967 2989 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 23 3617 3636 3655 3674 3692 37" 3729 3747 3766 3784 24 3802 3820 3838 3856 3S74 3892 3909 3927 3945 3962 25 3979 3997 4014 4031 4048 4065 4082 4099 41 16 4133 26 4150 4166 4183 4200 4216 4232 4249 4265 42S1 4298 27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 31 4914 4928 4942 4955 4969 4983 4997 501 1 5024 5038 32 5051 5065 5079 5092 5105 5"9 5^32 5145 5159 5172 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 35 5441 5453 5465 5478 5490 55*52 5514 5527 5539 5551 36 5563 5575 5587 5599 5611 5623 5635 5647 5<358 5670 37 56S2 5694 5705 5717 5729 5740 5752 5763 5775 57S6 38 5798 5S09 5821 5832 5843 5855 5866 5877 5888 5899 39 59" 5922 5933 5944 5955 5966 5977 5988 5999 6010 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6S93 49 6902 691 1 6920 6928 6937 6946 6955 6964 6972 6981 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 52 7160 7168 7177 7.8s 7193 7202 7210 7218 7226 7235 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 54 7324 7332 7340 7348 7356 7364 7372 73S0 7388 7396 No. 1 2 3 4 5 6 7 8 9 LOGARITPIMS. 361 No. 1 2 3 4 5 6 7 8 9 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 55 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 57 7559 7566 7574 7582 75S9 7597 7604 7612 7619 7627 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 60 7782 7789 7796 7S03 7S10 7818 7825 7832 7839 7846 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 62 7924 7931 7938 7945 7952 7959 7966 7973 79S0 7987 63 7993 8007 8014 8028 8035 8041 8048 TM 64 8062 S069 8075 8082 80S9 8096 8102 8109 8116 65 S129 8136 8142 8149 8156 8162 8169 8176 81S2 81S9 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 S254 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 68 S325 8331 8338 8344 8351 8357 8363 8370 8376 8382 69 838S 8395 8401 8407 8414 8420 8426 8432 8439 8445 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 72 8573 S579 85S5 8591 8597 8603 8609 8615 8621 l^/ol 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 76 8S0S 8S14 8820 8825 8831 8837 8842 8848 8854 8859 77 8865 8S71 8876 8S82 8S87 8S93 8S99 8904 8910 8915 78 8921 8927 8932 S938 8943 8949 8954 8960 8965 8971 79 8976 8982 8987 8993 8998 9004 9UJ9 9015 9020 9025 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 8i 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 91S6 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 86 9345 9350 9355 9360 9365 937^ 9375 9380 9385 9390 87 9395 9400 9405 9410 9415 94CO 9425 9430 9435 9440 83 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 92 9638 9643 9647 9652 9^57 9661 9666 9671 9675 9680 93 9689 9694 9699 9703 9708 9713 9717 97-2 9727 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 9S23 9S27 9S32 9836 9841 9845 9S50 9854 9859 9S63 C7 9S68 9872 9S77 9S81 9S86 9890 9894 9899 9903 9908 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 8 9996 2I0. 1 1 2 3 4 5 6 7 9 362 COLLEGE ALGEBRA. USE OF THE TABLE. 509. The table, (i)ages 300 and 361) gives the mantissa? of tlie logarithms of all integers from 100 to 1000, calcu- lated to four places of decimals. 510. To find the logarithm of a number of three figures. Look in the column headed "No." for the first tAvo sig- nificant figures of the given number. Then the mantissa required will be found in the corre- sponding horizontal line, in the vertical column headed by the third figure of the number. Finally, prefix the characteristic in accordance with the rules of Arts. 492 and 493. For example, log 1G8 = 2.2253 ; log .344 = 9.5366 -10; etc. 511. For a number consisting of one or two significant figures, the column headed may be used. Thus, let it be required to find log 83 and log 9. By Art. 505, log 83 has the same mantissa as log 830, and log 9 the same mantissa as log 900. Hence, log 83 = 1.9191, and log 9 = 0.9542. 512. To find the logarithm of a number of more thlin three figures. 1. Required the logarithm of 327.6. We find from the table, log 327 = 2.5145, log 328 = 2.5159. That is, an increase of one unit in the number produces an increase of .0014 in the logarithm. Therefore an increase of .6 of a unit in the number will produce an increase of .6 x .0014 in the logarithm, or .0008 to the nearest fourth decimal place. Hence, log 327.6 = 2.5145 -f .0008 = 2.5153. LOGARITHMS. 363 Note. The difference between any mantissa in tlie table and tlie mantissa of tlie next higher number of three figures, is called the tabu- lar difference. The subtraction may be performed mentally. The following rule is derived from the above : Fmd from the table the mantissa of the first three significant figures, and the tabular difference. Multiply the latter by the remaining figures of the number, with a decimal point before them. Add the residt to the mantissa of the first three figures, and prefix the p>roper characteristic. EXAMPLES. 2. Find the logarithm of .021508. Tabular difference = 21 Mantissa of 215 = 3324 ■08 2 Correction = 1.G8 = 2, nearly. 3326 Eesult, 8.3326 - 10. Find the logarithms of the following : 3. 80. 7. .7723. 11. 20.08. 15. 5.1809. 4. 6.3. 8. 1056. 12. 92461. >^ 16. 1036.5. ' 5. 298. 9. 3.294. 13. .40322. - 17. .086676. 6. ,902. 10. .05205. 14. .007178. 18. .11507. 513. To find the number corresijonding to a logarithm. 1. Required the number whose logarithm is 1.6571. Find in the table the mantissa 6571. In the corresponding liiie, in the column headed ''No./' we find 45, the first two figures of the required number, and at the head of the column we find 4, the third figure. Since the characteristic is 1, there must be two places to the left of the decimal point (Art. 492). Hence, the number corresponding to 1.6571 is 45.4. 364 COLLEGE ALGEDRA. 2. Eeqiiired the number whose logarithm is 2.3934. We find in the table the mantissse 3927 and 3945, whose corresponding numbers are 247 and 248, respectively. That is, an increase of 18 in the mantissa produces an increase of one unit in the number corresponding. Therefore, an increase of 7 in the mantissa will produce an increase of -^^ of a unit in the number, or .39, nearly. Hence, the number corresponding is 247 + .39, or 247.39. The following rule is derived from the above : Find from the table the next less mantissa, the three figures corresponding, and the tabular difference. Subtract the next less from the given mantissa, and divide the remainder by the tabular difference. Annex the quotient to the first three figures of the number, and x>oint off the residt. Note. The rules for pointing off are the reverse of those of Arts. 492 and 493 : I. If — 10 is not written after the mantissa, add 1 to the character- istic, giving the number of places to the left of the decimal point. II. If — 10 is written after the mantissa, subtract the positive part of the characteristic from 9, giving the number of ciphers between the decimal point and first significant figure. EXAMPLES. 3. Find the number whose logarithm is 8.52G4 — 10. 5264 Next less mantissa, 5263 ; three figures corresponding, 336. Tabular difference, 13) 1.000 (.077 = .08, nearly. 91 90 According to the above rule, there will be one cipher between the decimal point and first significant figure. Hence, number corresponding = .033608. LOGARITHMS. 365 Find the numbers corresponding to tlie following loga- rithms : -r 4. 1.8055. 9. 8.1648-10. 14. 1.6482. 5. 9.4487 - -10. + 10. 7.5209-10. ■h 15. 7.0450 - -10. - 6. 0.21G5. 11. 4.0095. 16. 4.8016. 7. 3.9487. 12. 0.9774. ■^7. 8.1144 - -10. ^8. 2.7364. -f-lS. 9.3178-10. APPLICATIONS. 18. 2.7015. 514. The approximate value of an arithmetical quantity, in which the operations indicated involve only multiplica- tion, division, involution, or evolution, may he conveniently found by logarithms. The utility of the process consists in the fact that addition takes the place of multiplication, subtraction of division, multiplication of involution, and division of evolution. Note. In computations with four- place logarithms, the results can- not usually be depended upon to more than four significant figures. 515. 1. rind the value of .0631 x 7.208 x .51272. By Art. 498, log (.0631 x 7.208 x .51272) = log .0631 + log 7.208 + log. 51272, log .0631= 8.8000-10 log 7.208= 0.8578 log .51272= 9.7099-10 Adding, log of result = 19.3677 - 20 = 9.3677 - 10 (See Note 1.) Number corresponding to 9.3677 — 10 = .2332. Note 1. If the sum is a negative logarithm, it should be written in such a form that the negative portion of the characteristic may be — 10. Thus, 19.3G77 -20 is written in the form 9.3677-10. 866 COLLEGE ALGEBRA. 2. Find the value of ?^^. 7984 By Art. 500, log ?p| = log 33G.8 - log 7984. log 336.8 = 12.5273 -10 (See Note 2.) log 7984 = 3.9022 Subtracting, log of result = 8.6251 - 10 Number corresponding = .04218. I^ote 2; To subtract a greater logarithm from a less, or to subtract. a negative logarithm from a positive, increase the characteristic of the mmuend by 10, wi-iting — 10 after the mantissa to compensate. Thus, to subtract 3.9022 from 2.5273, write the minuend in the form J2.5273 — 10 ; subtracting 3.9022 from this, the result is 8.6251 - 10. 3. Find the value of (.07396)^. By Art. 502, log (.07396)^ = 5 x log .07396. log .07396 = 8.8690 - 10 5 44.3450 - 50 = 4.3450-10 (See Note 1.) = log. 000002213. 4. Find the value of V. 035063. By Art. 503, log ^.035063 = - log .035063. o log .035063 = 8.5449 - 10 3 )28.5449 - 30 (See Note 3.) 9.5150 - 10 = log .3274. Note 3. To divide a negative logarithm, write it in such a form that the negative portion of the characteristic may be exactly divisible by the divisor, with — 10 as the quotient. Thus, to divide 8.5449 — 10 by 3, we write the logarithm in the foiTQ 28.5449 — 30. Dividmg this by 3, the quotient is 9.5150 - 10. LOGARITHMS. 367 ARITHMETICAL COMPLEMENT. 516. The Arithmetical Complement of the logarithm of a number, or, briefly, the Cologarithm of the number, is the logarithm of the reciprocal of that number. Thus, colog 409 = log -^ = log 1 - log 409. log 1 = 10. - 10 (Note 2, Art. 515.) log 409= 2.6117 .-. colog 409= 7.3883-10.. Again, colog .067 = log -^ = log 1 - log .067. log 1 = 10. - 10 . log .067= 8.8261-10 .-. colog .067 = 1.1739. Tlie following rule is evident from the above : To find the cologarithm of a number, subtract its logarithm from 10 - 10. Note. The cologarithm may be obtamed by subtracting the last significant figure of the logarithm from 10 and each of the others from 9, — 10 being written after the result in the case of a positive logarithm. 517. Example. Find the value of — -" r09 X .0946 log -5138^ = log r.51384 x ^ X -^\ '^ 8.709 X .0946 ^V ^-^^^ .0946; = log .51384 + log -^ + log- ^ 8.709 °.0946 = log .51384 + cotog 8.709 + colog .0946. log .51384 = 9.7109 -10 colog 8.709 = 9.0601 -10 colog .0946 = 1.0241 9.7951 - 10 = log .6239. 368 COLLEGE ALGEBRA. It is evident from the above example that the logaritLiu of a fraction is equal to the logarithm of the numerator plus the cologarithm of the denominator. Or in general, to find the logarithm of a fraction whose terms are composed of factors, Add together the logarithms of the factors of the numerator, and the cologarithms of the factors of the denominator. Note. The value of the above fraction may be found without using cologarithms, by the following formula : log 1^1384 ^ , .51384 - log (8.709 X .0946) ^ 8. 709 X. 0946 ° o^ ^ ; = log .51384 - (log 8. 709 + log .0940). The advantage in the use of cologarithms is that the written work of computation is exliibited in a more compact form. EXAMPLES. 518. Note. A negative quantity has no common logarithm (Art. 488, ]s'ote). If such quantities occur in computation, they may be treated as if they were positive, and the sign of the result determined irrespective of the logarithmic work. Thus, in Ex. 2, Art. 518, the value of 721.3 X (-3.0528) maybe obtained by finding the value of 721.3 X 3.0528, and putting a negative sign before the result. See also Ex. 24. Find by logarithms the values of the following : 1. 130.36 X .08237. 3. (- 4.32G4) x (-.050377). 2. 721.3 X (-3.0528). 4. .27031 x .042809. g 401.8 ^ -.3384 g 15.008 X (-.0843) 52.37" ' .08659 " " .06376 x 4.248 6 '^•^^^1 8 ^-^'^^^ 10 (-2563)x .03442 10.813' ■ .64327' ' 714.8 x (-.511) * J J 121.6 x (-9.0 25) ( - 48.3 ) X 3602 X ( - .< >856) ' LOGARITHMS. 3G9 12. (23.86)1 15. lot. 18. ^3. 13. (-1.0246)'. 16. (.8)T. 19. v'ioo. 14. (.09323)1 17. ( -.003186) i 20. V.4294. 21. V- .02305. 22. V- .00005173. 23. Find the n.„2^5_ 1 . 2\/5 value of ■ — —• 3^ - Ir^n- 9 _J_ Ino- ^VK -U Pnlncr ^^ >.' (\vt. mi^ 1 ^ = log 2 + - log 5 + - colog 3. 3 6 log 2= .3010 log 5 = .6990 ; divide by 3 = .2330 colog 3 = 9.5229 - 10 ; multiply by ^ := 9.6024 - 10 .1364 = log 1.369. 24. Find the value of « - -03296^ \ 7.962 "" \ 7.962 3 7.962 3 ' log .03296= 8.5180- -10 log 7.962= 0.9010 3)27.6170- -30 9.2057 - -10 = = log .1606. Hesult, -.160 Find the values of the following : 25. 2'x3l 26. ^. 4* - (fi) • 370 COLLEGE ALGEBRA 3 28. / .08726 ^3 34. V.0001289 _ V .1321 y * A/.0008276' 41 29. ^r. 35. (-■7469)« - (.2345)^ 30.^|.^l-|. 36. ^^3 (.08291)^ 31. a/2 X v'lO X A/:oi. 32. 37. el 3258 \49309' V298.54 49309 38. (18.9503)" x(-.l)''. 31.63V7 33. f^^^m 429 J 39. V3734.9 X .00001108. 40. (2.6317)^ X (.71272)1 , , ., A/-.008193X (.06285)^ -.98342 42. VA)35 X a/:02667 x -v/.0072163. EXPONENTIAL EQUATIONS. 519. An Exponential Equation is an equation of the form a' = b, where x is the unknown quantity, and a and b are positive real numbers. To solve an equation of this form, take the logarithms of l)oth members ; the result will be an equation which can be solved by ordinary algebraic methods. 1. Given 31' = 23 ; find the value of x. Taking the logarithms of both members, log (310 = log 23. Whence, x log 31 = log 23 (Art. 502) . m f log 23 1.3(117 (..„..o J lieretore, ^ = ; ~ — r = ^r^ r: = -91300. lu<,'31 1.1914 LOGARITHMS. 371 2. Given .2'^ = 3 ; find the value of x. Taking the logarithms of both members, X log .2 = log 3. Whence x = ^-^^= -^^^^ = '^^^^ = - .6825. wnence, x ^^^^^ 9.3010-10 -.6990 EXAMPLES. Solve the following equations : 3. 11^ = 7. 5. 13.18^ = .0281. 7. a^ = &^'c«. 4. .3^ = .8. 6. .7034^ = 1.096. 8. ma'' = 7i\ 9. 21^'-"^ = 9260. 10. .0513^+^ = 384.4. 11. Given a, r, and I ; derive the formula for n (Art. 428). 12. Given a, r, and S ; derive the formula for n. 13. Given a, I, and S; derive the formula for n. 14. Given r, I, and S ; derive the formula for n. 520. 1. Find the logarithm of .3 to the base 7. By Art. 507, W 3 = l^^Si^ = M^I^^IlIO = _ :5229 ^ _ _g^g^^ . ^' logio7 .8451 .8451 EXAMPLES. Find the values of the following : 2. log2l3. 4. log.365. 6. log,3 56.31. 3. logs .9. 5. log.8.0823. 7. logis .007228. Examples like the above may be solved by inspection, if the number can be expressed as an exact power of the base. 372 COLLEGE ALGEBRA. 8. Find the value of logiB 128. Let logic 128 = X ; tlien 16=^ = 128 (Art. 488), That is, (2^)^ = 2", or 2^^ = 2^ 7 Then, by inspection, 4.x* = 7, or x = -' Therefore, logi6l28 = ^- 9. Find the logarithm of 243 to the base 3. 10. Find the logarithm of 7776 to the base 36. 11. Find the logarithm of i to the base 27. 12. Find the logarithm of Jj to the base ^. EXPONENTIAL AND LOGARITHMIC SERIES. 521. Let 71 be greater than unity. By Art. 288, [(l + 1)"]'= (l + i)" Expanding both members by the Binomial Theorem, we have fl-l-u-Vl ""(''-^^ 1 I n(n-l)(n-2) -l 7 L^ 71 |2 n'^ • [3 n'^ J ■ ^ , 1 , nxCnx — 1) 1 = l + nx---\ ^— ^ • - n [2 ?r "•" [3 '71- ^ Since, by hypothesis, n is greater than 1, - is numerically less than 1, and by Art. 466 the series in both members of ( 1 ) are convergent. LOGARITHMS. 373 We may write equation (1) in the form 1-1 A-l'ui (2) J + 1 + — + ^ K ' ^-J ■ x(x ) x( X — -A( x — ~] which holds however great n may be. Now let n be indefinitely increased. 1 2 Then since the limit of each of the terms -, -, etc., is 71 n (Art. 210), the limiting value of the first member of (2) is and the limiting value of the second member is By the Theorem of Limits (Art. 213) these limits are equal ; that is, The series in the second member is convergent for every value of X (Art. 465) ; and the series in brackets is also convergent, for it is obtained from the series in the second member by putting 1 in place of x. Denoting the series in brackets by e, we have which holds for every value of x. 374 COLLEGE ALGEBRA. 522. Putting mx in place of x in (3), Art. 521, e-=l + mcc + — - + —- + .... (4) Let m = logs a, where a is any positive real number. Then e™ = a (Art. 488), and e""^ = a^ Substituting these values in (4), we obtain a== = l+(log,a)a^ + (log,a)^|+(log,a)3| + ...; (5) which holds for all values of x, and all positive real values of a. The result (5) is called the Exponential Series. 523. The system of logarithms which has e for its base is called the Napierian System, from Napier, the inventor of logarithms. The approximate value of e may be readily calculated from the series of Art. 521, and will be found to equal 2.7182818... . 524. To expand log<,(l + x) in ascending piowers of x. Substituting in (5), Art. 522, 1 + a; in place of a, and y in place of X, we have {l-\-xy = l-\- [log<,(l + x) ] ?/ + terms in y^-, y^, etc. ; which holds for all values of y, and all real values of x alge- braically greater than —1. Expanding the first member by the Binomial Theorem, [2 |3_ = 1 + [log,(l -{-x)^y+ terms in y-, f, etc. (6) LOGARITHMS. 375 The first member of (6) is convergent when x is numeri- cally less than 1 (Art. 466). Hence, equation (6) holds for all values of y, and for all real values of x numerically less than 1. Then, by the Theorem of Undetermined Coefficients, the coefficients of y in the two series are equal ; that is, ^ -!4^' + i ^-^ - [t •'^' + - = ^''^^ (^ + ^')- l£ iZ II Or, log:(l+a;) = x-| + f-f + |--; (7) which holds for all real values of x numerically less than 1. (Compare Art. 464.) This result is called the LogaritJimic Series. Putting — a; in place of x in (7), we obtain log,(l-x) = -._f-f-f-f-.... (8) Formula (7) can be used for the calculation of Napierian logarithms, provided x is taken numerically less than 1 ; but unless x is small, it requires the sum of a great number of terms to ensure any degree of accuracy. 525. To derive a more convenient formula for the calcida- tion of Napierian logarithms. Subtracting equation (8), Art. 524, from (7), we ha.ve log, (1 + «) - log,(l -x)=^2x-{-~^^~ + -- o o ■ Whence, by Art. 600, 1 — x V ^ y m. 1 + Y , m-n .. 1 + X. m + n 2 m _ m Let x = : then -— ^ — = = --- = — m + ?i 1 — X- i_'!!L:lVi ^'*' '>^ m + n 376 COLLEGE ALGEBRA. Substituting tliese values iu (9), we obtain n m + n 1 I'm — n\^ 1 I'm — nV ~| 3 \iii + nj 5 \vi + nj J But by Art. 498, log^— = log.m — log,, 71 ; whence, log.m : log,n + 2 — - + — +i •— 7- +••• • [_m -\-n 6 \m + nj 5 \m + nJ J 526. Let it be required, for example, to calculate the Napierian logarithm of 2 to six places of decimals. Putting, m = 2 and n=l in the result of Art. 525, we have log.2 = log.l+2ri + lflY+^riY+-T :^ . Or since log.l = (Art. 494), ' 'C ' jf , " /. -g T 6 '^' '7 ^^ log,2 = 2 (.3333333 + .0123457 + .0008230 + .0000653 + .0000056 + .0000005 +••••) = 2 X .3465734 = .6931468 = .693147, to the nearest sixth decimal place. Having found log^2, we may calculate loge3 by putting m = 2) and n = 2 in the result of Art. 525. Proceeding in this way, we shall find log^lO = 2.302585... . 527. To calculate the common logarithm of a mimber, hav- ing given its Napierian logarithm. Putting 6 = 10 and a = e in the result of Art. 507, 1 log, m 1 T ,n, o< 1 I r 1 logio «i = , ^%,, = ^ onorcT ^ ^^S. ^n = .4o42945 X log, VI. logelO 2.o02i^b5 For example, logi,j2 = .4342945 x .693147 = .301030. 528. Tlie multiplier by which logarithms of any system are derived from Napierian logarithms, is called the modulus of that system. LOGARITHMS. 377 Thus, .4342945 is the modulus of the common system. Conversely, to find the jSTapierigin logarithm of a number when its common logarithm is given, we may either divide the common logarithm by the modulus .4342945, or multi- ply it by 2.302585, the reciprocal of .4342945. Note. Napierian logarithms are sometimes called hyperbolic loga- rithms, from having been originally derived from the hyperbola. They are also sometimes called natural logarithms, from being those which occur first m the investigation of a method of calculating logarithms. MISCELLANEOUS EXAMPLES. 529. Using the table of common logarithms, find the Napierian logarithm of each of the following to foiu- signi- ficant figures : 1. 100. 3.^ 88.2. 5. .3437. 2. .00001. 4. 1325. 6. .085623. (7y What is the characteristic of log3400 (Art. 488) ? ^' ■ 8. AVhat is the characteristic of logo 1523? (9, If log 2 = .3010, how many digits are there in 2'*^ ? (\Q, If log 6 = .7782, how many digits are there in 6-^ ? 11. If log 7 = .8451, how many digits are there in the integral part of 7"8 ? 378 COLLEGE ALGEBRA. XXXV. COMPOUND INTEREST AND ANNUITIES. 530. The principles of logarithms may be applied to tlie solution of problems in Compound Interest. Let P = the principal in dollars ; n — the number of years ; t = the ratio to one year of the time during which simple interest is calculated ; for instance, if the interest is compounded semi-annually, t = \\ R = the amount of one dollar for the time t ; A = the amount of P dollars for n years. 1 . Given P, n, t, R ; to find A. Since the amount of one dollar for the time t is R, the amount of P dollars for the same period will be PR. That is, the amount at the end of the 1st interval is PR. In like manner, the amount at the end of the 2d interval = PR x R = PR" ; 3d interval = PR' x R = PR^ ; etc. Since the whole number of intervals is -, the amount at t the end of the last one, in accordance with the law observed above, will be PR'. That is, A = PR''. (1) By logarithms, log A = log P + " log R. (2) Example. What will be the amount of $ 7326 for 3 years and 9 months at 7 per cent compound interest, the interest being compounded quarterly ? COMPOUND INTEREST AND ANNUITIES. 379 111 this case, P= 7326, n = 3f, t = \, R = 1.0175, -=15. log P= 3.8649 log R = 0.0075; multiply by 15 = 0.1125 log ^ = 3.9774 .-. .4 = 19492. 2. Given n, t, B, A; to find P. Prom (2), log P = log A-- log R. Example. What sum of money will amount to $ 1763.50 in 3 years at 5 per cent compound interest, the interest being compounded semi-annually ? In this case, n = 3,t = ^,R = 1.025, A = 1763.5, "" = 6. log .4 = 3.2464 log R = 0.0107 ; multiply by 6 = 0.0642 log P= 3.1822 .-. P= $1521.40. 3. Given P, t, R, A; to find n. From (2), - log i? = log ^1 - log P. Whence, ^^ ^ ^log^ - log P). logP Exam'ple. In how many years will $300 amount to $ 396.90 at 6 per cent compound interest, the interest being compounded quarterly ? Here, P = 300, t = \, P = 1.015, ^ = 396.9. -,^ - lo^^ 396-9 - log 300 ^ 2.5987 - 2.4771 ^ .1216 4 log 1.015 4 X. 0064 .0256 = 4.75 years. 380 COLLEGE ALGEBRA. 4. Given P, n, t, A; to find R. From (2), log 72 = —1:: 2 — 7 Example. At what rate per cent per annum will f 500 amount to $ 688.83 in 6 years and 6 months, the interest being compounded semi-annually ? Here, P=500, « = 61 t = \, ^ = 688.83, - = 13. log^= 2.8381 lotr P = 2.6990 13)0.1391 log R = 0.0107 .: R=^ 1.025. That is, the interest on one dollar for months is $ .025, and the rate is 5 per cent per annum. 531. To find the present worth of A dollars due at the end of n years, the interest being compounded annually. I'utting t — 1 in (1), Art. 530, we have A = PR" ; whence, P = ^• ANNUITIES. 532, An Annuity is a fixed sum of money payable at equal intervals of time. In the present chapter we shall consider those cases only in which the payments are annual ; in finding th'e present worth of such an annuity, it is customary to compound the interest annually; and when we speak of the annuity as beginning at a certain epoch, it is understood that the first payment becomes due one year from that time. COMPOUND INTEREST AND ANNUITIES. 381 533. To find the present worth of an annuity to continue for n successive years, alloiving comjJound interest. Let A = the annuity in dollars ; Ji = the amount of one dollar for one year ; P = the present worth of the annuity. By Art. 531, the present worth of the 1st payment = — ; R 2d payment = — ; nth payment = Hence the sum of the present worths of the separate payments, or the present worth of the annuity, is R- ^ i2»-i ^ B' R Thatis, P=^^l + ^^ + ...+-l+lj. The expression in brackets is the sum of the terms of a ometrical progression, in w Whence by (II.), Art. 426, geometrical progression, in which a = — , r = R, and I = -- i-i Example. Find the present worth of an annuity of $ 150 to continue for 20 years, allowing 4 per cent compound interest. Here, ^ = 150, ?i ^ 20, ^ = 1.04, i? - 1 = .04. Whence, P = IJi' [l - -_^,]. 382 COLLEGE ALGEBRA. colog 1.04 = 9.9830 20 9.6600 Number correspondiug = .4571. Therefore, ^= i?? (1-.4571) =3750 x .5429. .04 log 3750 = 3.5740 log .5429 = 9.7347 log P= 3.3087 .-. P= $2035.70. 534. We have from (3), Art. 533, P(R-l) PB'^(R-l ) J2" (4) whieh is a formula for finding the annuity to continue for n successive years, when the present worth and the amount of one dollar for one year are given. Note. Formula (4) may also be used to find what fixed annual payment must be made to cancel a note of P dollars due n years hence, E being the amount of one dollar for oiie year. 535. If in (3), Art. 533, n is indefinitely increased, the limiting value of the second member is ^ (Art. 210). E-1 That is, theiiresent loorth of a perpeUml anmiity is equal to the amount of the annuity divided by the interest on one dollar for one year. COMPOUND INTEREST AND ANNUITIES. 383 636. To find the present worth of an annuity to begin after m years and then continue for n years, alloiving compound interest. "Witli the notation of Art. 533, the value of the annuity one year before the first payment becomes due, is B-1' B''(R-1) By Art. 531, the present worth of the above amount, due m years hence, is A(R'>~1) . ^, Therefore, P = M^"-'^) . 537. By Art. 535, the present worth of a perpetual annu- ity to begin after m years, is given by the formula p^—A R'"{E-l) EXAMPLES. 538. 1. What will be the amount of $1000 for 18 years at 6 per cent compound interest, the interest being com- pounded annually ? 2. What sum of money will amount to f 870.50 in 7 years and 3 months at 3 per cent compound interest, the interest being compounded quarterly ? 3. In how many years will $968 amount to $1209.40 at 5 per cent compound interest, the interest being compounded semi-annually ? 4. What is the present worth of a note for f 514.23 due 11 years hence, allowing 41 per cent compound interest, the interest being compounded annually ? 384 COLLECxE ALGEBRA. 5. At what rate per cent per annum will f 2600 gain $416.40 in 3 years and 9 months, the interest being com- pounded quarterly ? 6. In how many years will a sum of money double itself at 5 per cent compound interest, the interest being com- pounded annually ? 7. In how many years will a sum of money treble itself at 7 per cent compound interest, the interest being compounded semi-annually ? 8. What sum of money will amount to $1000 in 11 years and 8 months at 3f per cent compound interest, the interest being compounded every four months ? 9. What is the present worth of an annuity of $200 to continue 15 years, allowing 5 per cent compound interest ? 10. What is the present worth of an annuity of $ 1127 to continue 3 years, allowing 7 per cent compound interest ? 11. What is the present worth of an annuity of $1570 to begin after 11 years and continue for 6 years, allowing 4 per cent compound interest ? 12. What fixed annual payment must be made in order to cancel a note for $ 2000 in 7 years, allowing 3^ per cent compound interest ? 13. What is the present worth of a perpetual annuity of $ 186.25, to begin after 7 years, allowing 3i per cent com- pound interest ? 14. What annuity to continue 10 j'ears can be purchased for $2038, allowing 6 per cent compound interest ? 15. A person borrows $ 525,4 ; how much must he pay in annual instalments in order that the whole debt may be dis- charged in 12 years, allowing 41- per cent compound interest? PERMUTATIONS AND COMBINATIONS. 385 XXXVI. PERMUTATIONS AND COM- BINATIONS. 539. The different orders in which things can be arranged are called their Permutation?. >u ^ ^^ Thus, the perniutatioliS of the letters a, b, c, taken two at a time, are ab, ac, ba, be, ca, cb; and their permutations taken three at a time, are abc, acb, bac, bca, cab, cba. 540. The Combinations of things are the different collec- tions which can be formed from them, without regard to. the order in which they are placed. Thus, the combinations of the letters a, b, c, taken two at a time, are ab, be, ca ; for though ab and ba are different permutations, they form the same combination. 541. To find the number of permutations of n different things taken two at a time. Consider the letters a^, a^, O3, a^, ..., a„. The permutations of the letters taken two at a time, hav- ing tti as the first element, are a^a^, ctiOg, a^tti, ..., aia^\ the number of which is n — 1. In like manner, there are n — 1 different permutations of the letters taken two at a time, having ag as the first element ; and similarly for each of the remaining letters, «3, a^, ..., a„. Therefore the whole number of permiitations of the let- ters taken two at a time is equal to n{n — 1). 386 COLLEGE ALGEBRA. 542. We will now discuss the general case. To find the number of permutations of n different things taken r at a time. Consider the letters ffli, a.2, ttg, ..., a„ a,+i, a,+2, •••? «„■ Take any permutation containing r letters, for example the one consisting of the first r letters in their order ; that is, Oittoag ... a,. Placing after this the remaining n — r letters one at a time, in the following manner, ajajC^ • • • ttr^r+l aia^ag . . . a^a^+2 OiOotta . . . a^a„ there are formed n — r different permutations, each contain- ing r -f 1 letters. We may proceed in a similar manner with the other per- mutations containing r letters, and in each case obtain n — r different permutations, each containing r + 1 letters. It is evident that the permutations of r + 1 letters formed in the above manner are all different ; also, that we obtain in this way all the permutations containing r + 1 letters. Hence, the number of permutations of the letters taken r at a time, multiplied by w — r, is equal to the number of permutations of the letters taken r + 1 at a time. But the number of permutations of the letters taken two At a time, is equal to n{n — 1) (Art. 541). Hence the number of permutations of the letters taken three at a time, is equal to the number taken two at a time, multiplied by n — 2, or n(n — 1) (n — 2). PERMUTATIONS AND COMBINATIONS. 387 The number of permvitations of the letters taken four at a time, is equal to the number taken three at a time, multi- plied by n — 3, or ?i(h — 1) (?i — 2) {n — 3) ; and so on. It will be observed that the last factor in the number of permutations is n, minus a number one less than the num- ber of letters taken at a time. Hence the number of permutations of the letters taken r at a time is n{ii — 1) {n — 2)"-[n — (?• — 1)] ; that is, „P, = n{n - 1) (n - 2)--{n - r + 1). (l^) Note. The number of permutations of n different tilings taken r at a time, is usually denoted by the symbol „Pr. 543. If all the letters are taken together, r = n, and formula (1) becomes „P„ = n{n - 1) (w - 2) ... 3 . 2 • 1 =[n. (2) That is, tlie number of penmitations of n different thuigs taken n at a time is equal to the product of the natural num- bers from 1 to n inclusive. (See Note, Art. 444.) 544. To find the number of combinations of n different things taken r at a time. The number of permxdations of n different things taken r at a time, is n{n-l){n-2)--{n-r + l) (Art. 542). But by Art. 543, each combination of r different things may have [r permutations. Hence the number of combinations of n different things taken r at a time is equal to the number of permutations, divided by [r; that is, (7, == n(n-l){n-2):.{n-r + l) ^^^ Note. The number of combinations of n different things taken r at at a time, is usually denoted by tlie symbol ^C,. 388 COLLEGE ALGEBRA. 545. Multiplying both terms of the second member of (3) by the product of the natural numbers from 1 to n — r inclusive, we have ,a = 7i{n-l) ••• {n-r + 1)- (?t - r) ••• 2 • 1 _ |r X l-2...(n-;-) \r \n-r ' which is another form of the result. 546. By Art. 545, the number of combinations of n differ- ent things taken n — r at a time, is \n \n — , or — \n — r\n — {n — r) | n — r \ r But this is the same as the number of combinations of n different things taken r at a time. Hence, the number of combinations of n different things taken r at a time is equal to the number of combinations of n different things taken n — r at a time. EXAMPLES. 547. 1- HoAV many changes can be rung with 10 bells, taking 7 at a time ? Putting n — 10 and r = 7 in (1), Art. 542, loPr = 10 . 9 • 8 • 7 • G • 5 • 4 = G04800. 2. How many different combinations can be formed with 16 letters, taking 12 at a time ? By Art. 546, the number of combinations of 16 different things taken 12 at a time is equal to the number of com- l)inations of 16 different things taken 4 at a time. I'utting n = 16 and r = 4 in (3), Art. 544, we have ^_16.15.14.13_ PERMUTATIONS AND COMBINATIONS. 389 Find the values of the following : 3. lyPg. 5. 31 P,. 7. I8C12. 4. gXg. b. 1465. O. 22^15' 9. How many permutations can be formed from the 26 letters of the alphabet, taken 5 at a time ? 10. How many different words of seven letters each can be formed from the letters in the word forming ? 11. How many different numbers, of 6 different figures each, can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 ? 12. From a company of 50 soldiers, how many different pickets of 6 men can be takgn ? 13. How many different words of 4 letters each can be made with 7 letters ? How many of 3 letters each ? How many of 6 letters each ? How many in all possible ways ? 14. How many different committees of 12 persons each can be formed out of a corporation of 20 persons ? 15. There are 12 points in a plane, of which no three are in the same straight line. How many different triangles can be formed, having three of the points for vertices ? 16. How many different numbers of 5 different figures each, can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, ? 17. How many different throws can be made with two dice ? 18. How many different throws can be made with three dice ? 19. How many different words, each consisting of 4 consonants and 2 vowels, can be formed from 8 consonants and 4 vowels ? The number of combinations of the 8 consonants, taken 4 . 8 • 7 • 6 • 5 at a time, is - — ; — -; — , or 70. 1 • 2 • 3 • 4 390 collegp: algebra. Tlie number of combinations of the 4 vowels, taken 2 at a time, IS - — -, or 6. Any one of the 70 sets of consonants may be associated with any one of the 6 sets of vowels ; hence, there are in all 70 X 6, or 420 sets, each containing 4 consonants and 2 vowels. But each of the sets of 6 letters may have [6, or 720 dif- ferent permutations (Art. 543). Therefore the whole number of different words is 420 X 720, or 302400. 20. How many different words, each consisting of 3 con- sonants and 1 vowel, can be formed from 12 consonants and 3 vowels? 21. How many different committees, each consisting of 2 Kepublicans and 3 Democrats, can be formed from 14 Kepublicans and 21 Democrats ? 22. Out of 9 red balls, 4 Avhite balls, and 6 black balls, how many different combinations can be formed, each con- sisting of 5 red balls, 1 white ball, and 3 black balls ? 23. How many different words, each consisting of 4 con- sonants and 3 vowels, can be formed from 10 consonants and 5 vowels ? 24. How many different words, each consisting of 4 con- sonants and 1 vowel, can be formed from 10 consonants and 3 vowels, the vowel being the middle letter of each word ? 25. How many different words of 8 letters each can be formed from 4 consonants and 4 vowels, the vowels always occupying the even places ? 26. Out of 11 physicians, 13 teachers, and 8 lawyers, how many different committees can be formed, each consist- ing of 3 physicians, 4 teachers, and 2 lawyers? PERMUTATIONS AND COMBINATIONS. 391 27 o How many different words of 7 letters each can be formed from the letters a, b, c, d, e, f, g, each word being such that the letters a, b, c are never separated ? 28. How many different words of 5 letters each can be formed from the letters in the word Cambridge, each word beginning with a vowel, and ending with a consonant, and having a consonant for its middle letter ? 29. In how many different ways can 52 cards be arranged in four sets, each set containing 13 cards ? 548. To find the number of permutations of n things loliich are not all different, taken all together. Let there be w letters, of which p are a's, q are 6's, and r are c's, the rest being all different. Let JV denote the number of permutations of these letters taken all together. If, in any assigned permutation of the 7i letters, the p a's were replaced by p new letters, differing from each other and also from the remaining n—p letters, then by simply altering the order of these p letters among themselves, with- out changing the positions of any of the other letters, we could from the original permutation form \_p different per- mutations (Art. 543). If this were done in the case of each of the N original permutations, the whole number of permutations would be Again, if in any one of the latter the q 6's were replaced by q new letters, differing from each other and from the remaining n—q letters, then by altering the order of these q letters among themselves, we could from the original per- mutation form [g different permutations ; and if this were done in the case of each of the N x\_p original permutations, the whole number, of permutations would he N x[px[2- 392 COLLEGE ALCxEBRA. In like manner, if in each of the latter the r c's were replaced by r new letters, differing from each other and from the remaining oi — r letters, and these r letters were permuted among themselves, the whole number of permu- tations would heNx[px[qx\r. But the number of permutations on the liypothesis that the n letters are all different, is [n (Art. 543). Therefore, N x\px\q x\r = \n ; or, iV= — — — Any other case may be treated in a similar manner. EXAMPLES. 1 . How many different permutations can be formed from the letters of the word Tennessee, taken all together ? 2. How many different words of twelve letters each can be formed from the letters in the word independence 9 3. In how many ways can 6 dimes, 4 quarter-dollars, and 3 half-dollars be distributed among 13 boys, so that each may receive a coin ? 4. In how many ways can 15 balls be arranged in a row, if 7 of the balls are white, 5 black, and 3 red? 5. How many different words of seven letters each can be formed from the letters in the word Algebra, the first, fourth, and last letters being vowels ? 6. How many different numbers of seven figures each can be formed from the digits 1, 2, 3, 4, 3, 2, 1, the first, third, fifth, and last digits being odd numbers ? PROBABILITY. XXXVII. PROBABILITY (CHANCE). 549. Definition. If an event can happen in a ways, and fail to happen in h ways, and all these ways are equally likely to occnr, the probability or chcuice of the happening of the event is , and the probability of its failing is a+b a-j-b Or, we say that the odds are a to b in favor of the event, if a is greater than b, and a to & against the event, if a is less than b. 550. It follows that if the probability of the happening of an event is p, the probability of its failing is 1 —p. 551. If an event is certain to happen, b is equal to zero, and the probability of the happening of the event is -, or 1. 552. Example .1. A bag contains 5 red > balls, 4 white balls, and 3 black balls. (a.) If one ball is drawn, what is the probability that it is white ? The drawing of a white ball can happen in 4 ways, since either one of the 4 white balls may be drawn. It can fail to happen in 8 ways, since either one of the red or black balls may be drawn. Hence, the probability of drawing a white ball is -, or -• 3 (6.) If 3 balls are drawn, what is the probability that they are all red ? The number of combinations of the 5 red balls, taken 3 K 4 o at a time, is ' ' /] (Art. 544), or 10; that is, the drawing of 3 red balls can happen in 10 ways. 394 COLLEGE ALGEBRA. time, is 12jjl^^ ^^. goQ ; that is, the drawing of 3 balls 1 • U • 3 The number of combinations of the 12 balls, taken 3 at ,. . 12.11.10 a time, is , oi 1-2.3 can occur in 220 ways. Hence, the ijrobability of drawing 3 red balls is , or — 220 22 (c.) If 6 balls are drawn, what is the probability that 2 are red, 3 white, and 1 black ? The number of combinations of the 5 red balls, taken 2 at 5-4 a time, is , or 10 ; the number of combinations of the 4 1*2 4.3.2 white balls, taken 3 at a time, is , or 4. 1.2.3 We may associate together any one of the 10 combina- tions of red balls, any one of the 4 combinations of white balls, and any one of the 3 black balls ; hence there are in all 10 X 4 X 3, or 120 combinations, each consisting of 2 red balls, 3 white balls, and 1 black ball. Also, the number of combinations of the 12 balls, taken 6 ^ ^. . 12.11.10.9.8.7 no< at a time, is , or 924. 1.2.3.4.5.6 ' Hence, the required probability is -^^, or — . Exawx)le 2. A bag contains 30 tickets numbered 1, 2, 3, ..., 30. (a.) If four tickets are drawn, what is the chance that hoth 1 and 2 will be among them ? Tlie number of combinations of the 28 tickets numbered 98 . 97 3, 4, ..., 30, taken 2 at a time, is - — j^; that is, there are 00 ^ 97 . . — '-^^ ways of drawing four tickets, two of -^vhich are num- 1.2 -^ bered 1 and 2. The number of combinations of the 30 tickets, taken 4 at .30.29.28.27 a time, is ^^ ^ ^ ^ — PROBABILITY. 395 Hence the probability tliat, if four tickets are drawn, t^ro of them will be^l and 2, is 28 • 27 ^ .30 • 29 • 28 • 27 _ 3.4 _ _^ 1.2 ■ 1.2.3-4 ~30.29"~145' {b.) If four tickets are drawn, what is the chance that eitlier 1 or 2 will be among them ? Either 1 or 2 will be among the tickets drawn, unless each of the four bears a number from 3 to 30 inclusive. The number of combinations of the 28 tickets numbered 90 97 9fj OK 3, 4, ..., 30, taken 4 at a time, is ^ ^^^ -^o-^^ . The number of combinations of the 30 tickets, taken 4 at 30.29.28.27 a time, is 1-2.3.4 Hence the probability that each of the 4 tickets drawn . 1 . o^ on- 1 • • 28.27-26.25 65 bears a number from o to 30 inckisive, is , or — ' 30 . 29 • 28 • 27 87 Therefore the probability that each of the 4 tickets drawn 65 does not bear a number from 3 to 30 inclusive, is 1 99 87 (Art. 550), or ^. This then is the probability that either 1 or 3 will be among the tickets drawn. EXAMPLES. 553. 1. A bag contains 6 red balls, 5 white balls, and 4 black balls ; find the probability of drawing : (a.) One red ball. (6.) Two white balls, (c.) Four red balls. (cZ.) One ball of each color, (e.) Two red and three white balls. (/) One red, four white, and two black balls. {g.) Three balls of each color. 396 COLLEGE ALGEBRA. 2. If out of every 1584 persons living at tlie age of 14 years, 1512 reacli the age of 21, what is the probability that a person aged 14 years will not reach the age of 21 ? 3. What is the chance of throwing doublets in a single throw with two dice ? 4. What is the chance of throwing at least one ace in a single throw with tAvo dice ? 5. A bag contains 25 tickets numbered 1,2, 3, ..., 25; if six tickets are drawn, find the probability : (a.) That 4, 15, and 21 will be among them. (&.) That either 4, 15, or 21 Avill be among them. 6. If four cards are drawn from a pack, what is the probability that there will be one of each suit ? 7. If four cards .are drawn from a pack, what is the prob- ability that they will be the ace, king, queen, and knave of the same suit ? 8. A man has 3 shares in a lottery, in which there are 3 prizes and 7 blanks ; find his chance of drawing a prize. 554. If an event can hap^Jen in ttco or more independent loayjs ivhose respective jprohahilities are known, the j^'^'ohahility of the haj)pening of the event is equal to the sum ofthepiroha- hilities of its hai^pening in the separate ways. Suppose that a certain event can happen in a particular way a times out of b, and fail to happen in this particular way b — a times ; suppose also that the same event can happen in another way a' times out of b, and fail to happen in this way b — a' times ; all these ways being equally likely to occur. Also, suppose that the different ways in which the event can happen are independent; that is, if the event happens in tlie first way, it cannot at the same time happen in the second way. PROBABILITY. 397 Then the event happens a + a' times out of b, and the probability of its happening is (Art. 549), or - +— • b b b T>ut - is the probabilitv that tlie event happens in the ^ a' ■ first way, and — is the probability that it happens in the second way. Hence, the probability that it happens is equal to the sum of the probabilities of its happening in the separate ways. In like manner, the theorem may be proved to hold when there are more than two independent ways in which the event can happen. 555. Exam2')le 1. What is the probability of throwing 4 in a single throw with a pair of dice ? The event can happen in two ways ; either by throwing 3 and 1, or by throwing double twos ; and these ways are independent, because it is impossible to throw 3 and 1, and double twos at the same time. Each die can come up in 6 ways ; and hence the pair can be thrown in 6 X 6, or 36 different ways. 3 and 1 can be thrown in two ways, for the first die may come up 3 and the second 1, or the first die may come up 1 and the second 3 ; hence the probability of throwing 3 and lisA. 36 Again, double twos can be thrown in only one way ; hence the probability of throwing double twos is — 36 ..21 1 Therefore the probability of throwing 4 is •— H , or -— • 36 36 12 Example 2. A bag contains four $ 10 gold pieces and six silver dollars. If a person is entitled to draw two coins at random, what is the value of his expectation ? 398 COLLEGE ALGEBRA. Note. If a person has a chance of winning a certain sum of money, the product of the sum by his chance of winning is called his expecta- tion. The number of combinations of the four gold pieces, taken 2 at a time, is — ^, and the number of combinations of the 10-9 ten coins, taken 2 at a time, is — - ; hence the probability 4-3 2 of drawinsr two erold coins is , or — 10-9' 15 Then the value of the expectation, so far as it depends on 9 8 the drawing of two gold coins, is ^^ x 20, or - dollars. 15 3 In like manner, the chance of drawing two silver coins is G-5 1 -r^ — , or - ; and the value of the corresponding expectation 10 • 9 3 1 2 is - X 2, or - dollars. 3 3 Also, the chance of drawing a gold coin and a silver coin is (6 • 4) -^ — , or —^ ; and the value of the corresponding expectation is — x 11, or — dollars. lo 15 Hence, the value of the expectation is ( - + ^ + — J dollars, or $9.20. ^^ ^ ^^^ EXAMPLES. 556. 1 . What is the probability of throwing 6 in a single throw with two dice ? 2. What is the probability of throwing at least 5 in a single throw with two dice ? 3. A bag contains three $ 5 gold pieces and five silver dollars. If a person is entitled to draw one coin at random, what is the value of his expectation ? PROBABILITY. 399 4. A lottery has 34 prizes ; four of $ 500, ten of $ 250, and twenty of f 25. If the whole number of tickets is 100, what is the value of each ? 5. What is the probability of throwing 15 in a single throw with three dice ? 6. What is the probability of throwing 11 in a single throw with three dice ? 7. A bag contains six $5 gold pieces, and four other coins which have all the same value. If the expectation of drawing two coins at random is worth $8.40, what is the value of each of the unknown coins ? 8. A bag contains six half-dollars, six quarter-dollars, and six dimes. If a person is entitled to draw three coins at random, what is the value of his expectation ? COMPOUND EVENTS. 557. If there are two independent events ivJiose respective prohahilities are known, the prohability that both ivill happen is equal to the product of their separate probabilities. • Note. Two events are said to be independent when the occurrence of one is not affected by the occurrence of the other. Let a be the number of ways in which the first event can happen, and b the number of ways in which it can fail ; all these ways being equally likely to occur. Also, let a' be the number of ways in which the second event can happen, and h' the number of ways in which it can fail ; all these ways being equally likely to occur. We may associate together any one of the a + b cases in which the first event happens or fails, and any one of the a' -\- b' cases in which the second happens or fails ; hence there are (a -f b) (a' -f b') cases, equally likely to occur. In au' of these cases both events happen. 400 COLLEGE ALGEBRA. Therefore the probability that both events hapjoen is aa' (a + h) («' + &') But IS the i^robability that the first event happens, a + h and '■ — - is the probability that the second happens. Hence, the probability that both events happen is equal to the product of their separate probabilities. And in general, if pi, po, Ps, . . ., are the respective proba- bilities of any number of independent events, the probability that all the events happen is Piihlh • • • • 558. Example 1. Find the probability of throwing an aee in the first only of two successive throws with a single die. The probability of throwing an ace at the first trial is — 5 The probability of not throwing one at the second trial is -• G Hence, the probability of throwing an ace in the first only of two successive throws is - x -, or • — ^ 6 G' 3G Example 2. Required the probability of throwing an ace at least once in three throws with a single die. There will be an ace unless there are three failures. 5 The probability of failing at the first trial is - ; and this 6 is also the probability of failing at each of the other trials. Hence the probability that there will be three failures is 5^5 5 125 - X - X -, or — — G G G 216 Then the probability that there will not be three failures IS 1 ; (Art. 550), or — ^• 21G 21G PROBABILITY. 401 Example 3. A bag contains 5 red balls, 4 white balls, and 3 black balls. Three balls are drawn in succession, each being replaced before the next is drawn. What is the probability that the balls drawn are one of each color ? The probability that the first ball is red is — ; the proba- oi 1 bility that the second is white is — , or - ; and the r)roba- ^ 12 3' ^ bility that the third is black is — , or - •> 12' 4 Hence the probability of drawing a red ball, a white ball, 5 11 5 and a black ball, in this assinned order, is -— x - X -, or -- — ■^ ' 12 3 4 144 But a red ball, a white ball, and a black ball may be drawn in [3, or G different orders (Art. 543) ; and in each 5 case the probability is — — Then by Art. 554, the probability of drawing a red ball, a white ball, and a black ball, without regard to the order 5 5 in which they are drawn, is x G, or ■ — ^ ' 144 ' 24 559. The probability of the concurrent happ)ening of tivo dependent events is equal, to the probability of the first, multi- plied by the probability that ivhen the first has happened the second loill folloiv. Let a and b have the same meanings as in Art. 557. Also, suppose that, after the first event has happened, a' represents the number of ways in which the second will follow, and b' the number of ways in Avhich it will not fol- low ; all these ways being equally likely to occur. Then there are in all (o + b) («'+ b') cases, ecpially likely to occur, and in aa' of these both events happen. Therefore the probability that both events happen is (a + b){u'+b') 402 COLLEGE ALGEBRA. Hence the probability that both events happen is equal to the i^robability of the first, multiplied by the probability that when it has happened the second will follow. And in general, if there are any number of dependent events such that 2>i is the probability of the first, 2'>2 the probability that when the first has happened the second will follow, pg the probability that when the first and sec- ond have hai)pened the third will follow, and so on, then the probability that all the events happen is j^iP-ilh ■• • 560. Example 1. Let it be required to solve Ex. 3, Art. 558, when the balls are not replaced after being drawn. The probability that the first ball is red is -^ ; the prob- ability that the second is white is — ; and the probability 3 that the third is black is — 10 Hence the probability of drawing a red ball, a white ball, and a black ball, in this assigned order, is -^ x — X — 12 11 10 But the balls may be drawn in [3, or G different orders. Therefore the probability of drawing a rod ball, a white ball, and a black ball, without regard to lli'.' order in which they are drawn, is ^ x — X — X u, or — ^ 12 11 10 ' 11 Example 2. An urn contains 5 white balls and 3 black balls ; another urn contains 4 white balls and 7 black balls. AYhat is the probability of obtaining a white ball by a single drawing from one of the urns taken at random ? Since the urns are ('iiiially lik(>ly to be taken, the prol)a- bilily ol' t;il ])r()b;il)ility of ilini dniuin- :i. while ImU in,m it is -• PROBABILITY. 403 Hence the probability of obtaining a white ball from the nrst urn is - X -, or — • 2 8 16 In like manner, the probability of obtaining a white ball 14 2 from the second urn is - x — , or — • A 11 11 5 2 87 Hence the required probability is 1 , or 16 11 176 561. Given the probability of the hapjnnitig of an event in one trial, to find the probability of its happening exactly r times in n trials. Let p be the probability of the happening of the event in one trial. Then 1 — j) is the probability of its failing (Art. 550). The probability that the event will happen in each of the first r trials, and fail in each of the remaining n — r trials, is jp'-(l-p)"-^. But the number of ways in which the event may happen exactly r times in n trials is equal to the number of com- binations of n things taken r at a time, or n{n-l):.{n-r + l) ^^^^_ ^^^ Hence the probability that the event will happen exactly r times in n trials is n(n-l)...(H-r + l) ^.^^_^^„_.^ ^^^ [r For example, putting r = 1, the probability that the event will happen exactly once in n trials is np{l —pY~^ ; putting r = 2, the probability that the event will happen exactly twice in n trials is ^^^^~ ' P^O- —pY~^'i and so on. 404 COLLEGE ALGEBRA. In like manner, tlie probability that tlie event will fail exactly r times in n trials is \l 562. Given the probability of tlie happening of an event in one trial, to find the probability of its hapjyening at least r times in n trials. The event happens at least r times if it happens exactly n times, or fails exactly once, twice, . . ., n — r times. Therefore the probability that it happens at least r times is equal to the sum of the probabilities of its happening exactly n times, or failing exactly once, twice, . . ., n — r times ; which, by Art. 561, is 2r+npf\l -p)+... -f-^A ^ '^ ^ -' jj'(l-j))" ••. \ n — r 563. ExampAe. A bag contains five tickets numbered 1, 2, 3, 4, 5. Five tickets are drawn at random, each being replaced before the next is drawn. Find the probability of drawing the ticket marked 1 exactly three times, and at least three times. In this case, p = --, r = 3, n =5. 5 Then by Art. 561, (1), the probability of drawing the ticket marked 1 exactly three times is l-2-3\5j\5j' 625 And by Art. 562, the probability of drawing it at least three times is (sj^^m l-'Ar, {u • 313S PROBABILITY. 495 EXAMPLES. 564. 1. If eight coins are tossed up, what is the chance that one and onjy one will turn up head ? 2. A purse contains one dollar and three dimes ; another contains two dollars and four dimes ; a third three dollars and one dime. What is the chance of obtaining a dollar by draw- ing a single coin from one of the purses taken at random ? 3. What is the probability of throwing exactly three aces in five throws with a single die ? 4. What is the probability of throwing at least three aces in five throws with a single die ? 5. If three cards are drawn from a pack, what is the chance that they will consist of a king, queen, and knave ? 6. The probability that A can solve a certain problem is f, and the probability that B can solve it is f. Find the probability that the problem will be solved if both try. 7. If a coin is tossed up ten times, what is the chance that the head will present itself exactly five times ? 8. A bag contains ten tickets numbered 0, 1, 2, ..., 9. If three tickets are drawn at random, what is the probability that their sum is 22 ? 9. Two bags contain each 4 black and 3 white balls. A ball is drawn at raiidom from the first, and if it is white, it is put into the second bag, and a ball drawn at random from that bag. Find the chance of drawing two white balls. 10. If the odds are 5 to 3 against a person who is now 40 living till he is 65, and 11 to 6 against a person who is now 45 living till he is 70, what is the chance that at least one of these persons will be alive 25 years hence ? 11. What is the probability of throwing aces with a paii of dice at least three times in four trials ? 406 COLLEGE ALGEBRA. 12. A bag contains 5 v/liite and 8 black balls. Two drawings, each of 3 balls, are made, tlie balls first drawn not being replaced before the second trial ; what is the chance that the first drawing will give 3 white, and the second 3 black balls ? 13. A bag contains ten tickets, five numbered 1, 2, 3, 4, 5, and the rest blank. Three tickets are drawn at random, each being replaced before the next is drawn ; what is the probability that their sum is 10 ? 14. A's skill at a game is two-thirds of B's. What is the chance that A wins at least two games out of five ? 16. A bag contains 4 red, 3 white, and 2 black balls. A ball is drawn and not replaced. Another ball is then drawn. Find the chance that the two balls are of the same color. 16. A's skill at a game is double B's. What is the prob- ability that A wins four games before B wins two ? 17. A bag contains 6 red balls, 5 white balls, and 4 black Ijalls. Four balls are drawn in succession, and are not replaced after being drawn. What is the chance that tAvo of them are red, one white, and one black ? 18. If one vessel out of every ten is wrecked, what is the chance that, out of five vessels expected, at least four will arrive safely ? 19. A and B draw in succession, in the order named, from a purse containing three sovereigns and four shillings. Find their respective chances of first drawing a sovereign, the coins when drawn not being replaced. 20. A, B, and C draw m succession, in the order named, from a bag containing three white balls and five black balls. Find their respective chances of first drawing a white ball, tlie balls Avhen drawn not being replaced. CONTINUED FRACTIONS. 407 XXXVIII. CONTINUED FRACTIONS. 565. A continued fraction is an expression of the form h a + , • d e+ .-. or, as it is usually written in practice, , 6 d c+e + -" We shall limit ourselves in the present work to continued fractions of the form , 1 1 a + - ; 6+C + .-- where each numerator is unity, a any positive integer or 0, and each of the quantities h, c, ..., a positive integer. 566. A terminating continued fraction is one in which the number of denominators is finite ; as, a-{ b+ c+d It may be reduced to an ordinary fraction by the process of Art. 1G9. An infinite continued fraction is one in which the number of denominators is indefinitely great. 567. In the continued fraction ,11 1 a +ao+a,+ Qi is called the Jirst convergent ; ai -) is called the second convergent; ttj H is called the third convergent ; and so on. a.,+ ((■■ 408 COLLt:GE ALGEBRA. Note. If «! = 0, as in the continued fraction J ^ 1 then is considered tlie first convergent. 568. Ayiy ordinary fraction in its loivest terms may he coih- uerted into a terminating continued fraction. Let the given fraction be -, where a and 6 are prime to each other. Divide a by h, and let a^ denote the quotient and hi the remainder ; then, a , &, ,1 - = 0,+ — = a, H — h h ' 6_ Divide h by 6i, and let aeriodic continued fraction may alivays be expressed as the root of a certain quadratic equation. Example. Express — as the root of a certain ^ ^ 1 + 3 + 1 + 3+.. . quadratic equation. Let X denote the vahie of the fraction ; then, = J_ 1 ^ S + x ^ S + x 1 + o + x 3 + x + l 4 + X Clearing of fractions, 4:X -]- X- = 3 + X, or X- + ."5 x = 3. CONTINUED FRACTIONS. 411 Solving the equation, 2 2 Note. The + sign is taken before the radical, since x must evi- dently be a positive quantity. PROPERTIES OF CONVERGENTS. 571. Let the continued fraction be 111 1 + and let p^ denote the numerator, and q^ the denominator, of the rth convergent (Art. 567) when expressed in its sim- plest form. 572. To determine the law of formation of the successive convergents. The first convergent is aj. The second is cti -| = ^ '- • a., a.2 rri 4-1 • 1 • ,11 , ^'". a,aoa.,-^-ai-\-ao The third is Ui -\ = f'l H — = ^ ' '^ — — — ^• tto+ag a/<3+l ttoGg+l The third convergent may be written in the form (aia2 + 1)0'^ + cii . a^ttg + 1 ' in which we observe that : 1. The numerator is equal to the numerator of the preced- ing convergent, mnltipUed by the last denominator taken, plus the numerator of the convergent next but one preceding. . 2. Tlie denominator is equcd to the denominator of the pre- ceding convergent, multiplied by the last denominator taken, plus the denominator of the convergent next but one preceding. We will now prove by Induction (Note, p. 46) that the above laws hold for all convergents after the second, when expressed in their simplest forms. 412 COLLEGE ALGEBRA. Assume that the laws hold for all convergents as far as the ?ith. The nth convergent is &=„.+^^...i. q^ a2+a3+ a„ Then since the last denominator is a„, we have Pn = anPn-l+Pn~2, and qn = Ct^Qn^l + Qn-H- (1) Whence, Pn ^CinPn-^+Pn-,^ (2) q. Cl„q„__,+ q,_2 The {n 4-l)st convergent is J_ J. 1 1 which differs from the ?ith only in having a„ -\ — , or ^'''^"+1 + ^ , in place of a,. Substituting ^^"^^"+1 + ^ for a„ in (2), we have l+i>n-2 ?«+l «,/<„+! + 1 ^^ 1 + g»-2 a»+i(««9«-i + n-l^ by (1). (3) It is evident that the second member of (3) is the sim- plest form of the (n + l)st convergent, and therefore Pn+i = an+iPn+p„-i, aud qn+i = a,ni(]„ + (In-i- These results are in accordance witli the laws stated on the preceding page. Hence, if the laws hold for all convergents as far as the 7ith, they also hold as far as the (n + l)st. CONTINUED FRACTIONS. 413 But we know that they hold as far as the third convergent, and hence they also hold as far as the fourth ; and since they hold as far as the fourth, they also hold as far as the fifth ; and so on. Therefore the laws hold for all convergents after the second. Examx>le. Find the first five convergents of 1 + 2 + 3+4+.. . The first convergent is 1, and the second is 1 + 1, or 2. Then by aid of the laws just proved, ^1 ^T 1 ■ 2.2 + 1 5 the third is — • the fourth is the fifth is 1. 2 + 1 3' 5. • 3 + 2 17 3. .3 + 1 10' 17 . 4 + 5 73 10-4 + 3 43 573. The difference between two consecutive convergents £1' and -L^ is equal to •. g« qn+i Quqn+i The difference between the first and second convergents is «i + - - tti = -. a.,/ c(2 Thus the theorem holds for the first and second conver- gents. Assume that it holds for the nth. and (n + l)st convergents ; that is, f^ ~f^' = ^, or p„q,^_, ~,.„,,g,. = 1. (1) Qn y-. + l UnQn+l Then, 414 COLLEGE ALGEBRA. yn+iyn+2 yn+l5H+2 Hence if the theorem holds for any pair of consecutive convergents, it also holds for the next pair. But we know that it holds for the first and second conver- gents, and hence it also holds for the second and third ; and since it holds for the second and third, it also holds for the third and fourth ; and so on. Therefore the theorem holds universally. 574. It follows from Art. 573 that j9„ and g„ can have no common divisor except unity ; for if they had, it would be a divisor of p„ g„+i ~ p„+i g„, or unity, which is impossible. Therefore all convergents formed in accordance with the laws of Art. 572 are in their lowest terms. 575. The even convergents are greater, and the odd con- vergents less, than the fraction itself. I. The first convergent, a^, is less than the fraction itself, 1 is omitted. ao+' II. The second, a^ -{ , is greater, because its denomina- tor cia is less than a, -\ , the denominator of the fraction. III. The third, a. H , is less, because, by II., the I ch+a-I 11 denominator a., -\ is greater than a, -\ , the Ug ^ a^+a^-i denominator of the fraction ; and so on. Hence the first, third, ..., convergents are less, and the second, fourth, ..., convergents greater than the fraction itself. CONTINUED FRACTIONS. 415 576. Any convergent is nearer than the 2'>rececUng convergent to the value of the fraction itself. By Art. 572, Pn+2^<^n+2Pn+l+ 2^.^ qn+2 an+oQu+l-hgn The fraction itself is obtained from its (?i-|-2)d con- 1 vergent by putting a„+2 in place of a„+2- «n+3 + Hence, denoting the vahie of the fraction itself by a;, we have i>„4-l +Pn q»+i + ?« mPn+\-\-Pn where m stands for a,.+2 + ISTow, tMu+i + qn qn ^{Pn+iqn-Pnqn+l) qn{mq„^, + qn) qn{^Mn+l + qn) (Art. 573). (1) Also, X Since Pn+l^ '>nPn+l+Pn _Pn+\ «ig,-+i + qn ~ qn+i Pnqn+l-^Pn+iqn qn+i qn+1 (mg„+i + g„) g„+i (mg„+i + q„ ) 1 is a positive integer, a,^_^.2 + CW3+' (2) is>l; that is, m is > 1. And since q„^i = a„_^_lq„ + q„_]^ (Art. 572), g„^i is > q„. Therefore the fraction (2) is less than the fraction (1), for it has a smaller numerator and a greater denominator. Hence the (n + l)st convergent is nearer than the ?ith to the value of the fraction itself. 416 COLLEGE ALGEBRA. 577. By Art. 576, the difference between the fraction itself and its nth. convergent is m 1 gn(m^n+i + g,.) 9«(^^+i + | (1) Since m is > 1 (Art. 576), the denominator 5„f g„+i + ^] is q„q„+i- Hence the fraction (1) is > -, and < qniqu+i + qn)- qnqn+i That is, the erroj- made in taking the nth. convergent for the fraction itself lies between the limits 1 and 1 qn{qn+i + qn) q.qn+1 EXAMPLES. 578. Convert each of the following into a continued frac- tion, and find in each case the first five convergents : 1. ^. 39 3. 3.61. 5. J^. 326 7 436 ■ 345' 2. ^. 91 4. 11-2. 153 6. Ml. 89 g 3015 6961 Convert each of the follovv^ing into a continued fraction, find in each case the first four convergents, and determine limits to the error made in taking the third convergent for the fraction itself : 9. V5. 12. V3. 15. 2V5. 18. Vi4. 10. ViO. 13. VfJ- 16. V7. 19. V33. 11. V17. 14. yj^. 17. ^'^^ + ^ - 20. 2V35. CONTINUED FRACTIONS. 417 Express each of the following in the form of a surd : 21. 1111 2 + 0+2 + 0+--- 23. 2 + ^-^—- 1+1 + ... 22. 1111 1 + 4+1 + 4+-" 24. 3+111 1 ^ 1 + 8+1 + 8+.. . 25. The ratio of the circumference of a circle to its diame- ter is approximately equal to 3.14159; express this decimal as a continued fraction, and find the first four convergents. 26. The modulus of the common system of logarithms is approximately equal to .43429; express this decimal as a continued fraction, find its seventh convergent, and deter- mine limits to the error made in taking this convergent for the fraction itself. 27/ The base of the Napierian system of logarithms is 2.7183 approximately ; express this decimal as a continued fraction, find its eighth convergent, and determine limits to the error made in taking this convergent for the fraction itself. 28. Express the positive root of the equation ^.2 _ 3, _ ;L]^ ^ as a continued fraction, and find the first five convergents. Convert each of the following into a continued fraction, and find in each case the first four convergents : 29. Vl3. 30. -^. 31. V9cr + 3. 32. V22. V33 33. Exi)ress « -| in the form of a surd. 418 COLLEGE ALGEBRA. XXXIX. SUMMATION OF SERIES. 579. The Summation of an infinite literal series is the process of finding a finite expression from which the series may be developed. The result represents the series only for such values of the letters involved as make the series convergent. A method has already been given (Art. 429) for finding the sum of an infinite geometrical series. RECURRING SERIES. 580. A Recurriiig Series is an infinite series of the form a-o + a^x + agO^ + • • •? where any r + 1 consecutive terms are connected by a rela- tion of the form p, q, ,.., s being constants. The above recurring series is said to be of the rth order, and the expression 1 4-^.13 + gx^ +■•• + sx" is called its scale of relation. For example, in the infinite series any three consecutive terms are so related that the last term, plus —2x times the preceding term, plus x- times the next but one preceding, is equal to 0. Hence the series is a recurring series of the second order, and its scale of relation is 1 —2x -{-a^. Note. An infinite geometrical series is a recurring series of the first order. SUMMATION OF SERIES. ' 419 581. To find the scale of relation of a recurring series. If the series is of the first order, the scale of relation may be found by dividing any term by the preceding term, and subtracting the result from 1. If the series is of the second order, and ao, ctj, a2, a^, ..., are its consecutive coefficients, and 1 -\-px + qa? its scale of relation, we shall have 1 a. +pa2 -t- gcti = ; from which p and q may be determined. If the series is of the third order, and ao, aj, a^, a-s, ^4, tts, ..., are its consecutive coefficients, and 1 + ^j.t + g;z;^ + ?-ar' its scale of relation, we shall have r ttg +pa2 + qai + ra^ = 0, \ a^ +pa^ + qa^ -f ra^ = 0, [ «5 + 2^«4 + qa.^ + ra.^ = ; from which p, q, and r may be determined. It is evident from the above that the scale of relation of a recurring series of the rth order may be determined when any 2r consecutive terms are given. To ascertain the order of a series, we may first make trial of a scale of relation of three terms ; if the result does not correspond with the series, try a scale of four terms, five terms, and so on until the true scale of relation is found. If .the series is assumed to be of too high an order, the equations corresponding to the assumed scale will not be independent. (Compare Art. 215.) 582. To find the sum of a recurring series iclien its scale of relation is Tcnoion. Let l-\-px + qx? be the scale of relation of the series 420 COLLEGE ALGEBRA. Denoting the sum of the first n terms by S,„ we have >S„ = Oo + «i x-\- a.^x- -\- ■•• + a„_ix" ^ Whence, and qx^Sn = gaoX^ + ••• +ga„_3ic''^^ + qa,^^2^'' + 5a„_ia;"+\ Adding these equations, and remembering that, by virtue of the scale of relation, «2 -\-p(h + Q^O =0, •••, a„_i -}-J9a„_2 + (?«n-3 = 0, we have = tto + («! +i:'«o)a: -f (pa„-i + 5«»-2)^" + 5«n-ia;"+^ Whence, which is a formula for the sum of the first n terms of a recurring Series of the second order. If X is so taken that the given series is convergent, the expression (i5o„_i + qcin-2)^'^ + ga„_ia;''+^ approaches the limit when n is indefinitely increased, and (1) becomes l+px+qaf ' ^^^ which is a formula for the sum (Art. 579) of a recurring series of the second order. li q = 0, the series is of the first order, and therefore «i +iJao = ; whence, s = -^^ ; (3) I+JIX which is a formula for the sum of a recurring series of the first order. (Compare Art. 129.) SUMMATION OF SERIES. 421 In like manner, we shall find the formula ^ ^ a,j + (cti +pao)x + (cio +P% + gfto)ar^ r^\ 1 +px + ga;^ + rar' for the sum of a recurring series of the third order. Note. It will be observed in each case that the denominator of the fraction is the scale of relation. 583. A recurring series is formed by the expansion in an infinite series of a fraction, called the generating fraction. The operation of summation reproduces the fraction; the process being the reverse of that of Art. 471. 584. Example. Find the sum of the series 1 + Ox - 15ar^ + 57x3 - ISOa;" + ••• . To determine the scale of relation, we first assume the series to be of the second order (Art. 581). Substituting a^ = 1, a^ — 9, a.^ = — 15, and a^ = 57 in the equations of Art. 581, we have ■15+ 9p+ g = 0, 57-15j9 + 9r^ = 0. Solving, we find p = 2 and g = — 3. To ascertain if 1 + 2 x — 3 a;- is the true scale of relation, consider the fifth term. Since -159x*+(2a;) (57ar'^) + (-3x2) (-15^2) is equal to 0, it follows that l + 2x — 3a:^is the true scale. Substituting the values of a^, a^ p, and q in (2), Art. 582, ^^ l+(9 + 2)a; ^ l + llx l+2a;-3ar^ l + 2a;-3a;2' 585. It is possible, by aid of Art. 474, to find an expres- sion for the rth term of the series of Art. 584. . 1 + llx A , B Assume, ~ = l + 2a; — 3a;- 1-x 1+ox 422 COLLEGE ALGEBRA. Then, 1 + llx = A(l + 3x) + B (1 - x). Putting a; =1, 12 = 4, A; whence ^1 = 3. Putting x = — -, — ^ = -iJ; whence B = — 2. o o o Then, 1 + 11^ _ 3 2 l+2a;-3a"' 1 - a; l+Sic = 3{l-j-x + x^ + x"+.:) -2ll+{-3x) + (-3xy+...]. Therefore the Hh term of the given series is 3x^-1 - 2 (- 3xy-\ or [3 - 2 (- 3)'-']x'-\ EXAMPLES. 586. In each of the following, find the" generating frac- tion, and the expression for the Hh term : 1. l + 5x + 19x' + 65x'+211x*-\--'. 2. 2-x-|-5.^--7.r' + 17a;^ 3. l-4.x-2x'-10o^-Ux* 4. 2 - 5x +^ 17x^-650^ + 257 x' 6. 3 + 5x-5a^-115a.-3_845.'c^ 6. 5 + 8x + 56x^-\-176x'-i-800x'+-" 7. i-x + 61x'-319x' + 2U9x* 8. l-13a; + 89a^-517a^ + 2801a;* In each of the following, find the generating fraction, and continue the series to two more terms : 9. i+3a;_ar_5ar^-7a;*-ar' + lla;'' + 10. 1+ix- 7af - 2x' + 9.x-' + lO.x'' - 51 a;" + ••• • 11. 2 - 7.« + iDx- - 54 .*.-' + UGa;^ - 397.c-^'+ \i%7x''^ SUMMATION OF SERIES. 428 THE DIFFERENTIAL METHOD. 587. If the first term of any series is subtracted from the second, the second from the third, and so on, a series is formed which is called the first order of differences of the given series. The first order of differences of this new series is called the second order of differences of the given series ; and so on. Thus, in the series 1, 8,« 27, 64, 125, 216, ..., the successive orders of differences are as follows : 1st order, 7, 19, 37, 61, 91, .... 2d order, 12, 18, 24, 30, .... 3d order, 6, 6, 6, .... 4th order, 0, 0, 588. The Differential Method is a method for finding any term, or the sum of any number of terms of a series, by means of its successive orders of differences. 589. To find any term of the series tti, tta, ttg, a^, ..., a„, ftn+i} ••• • The successive orders of differences are as follows : 1st order, a^ — a^, a^ — a.^, a^ — ar^, •••, a„+i — «„, ••• • 2d orderj ag — 2a2 + ai, a4 — 2a3 + a,, ••• • 3d order, a^ — 3 a^ -\- ^ a.^ — a^, •••; etc. Denoting the first terms of the 1st, 2d, 3d, ..., orders of differences by dj, d., d^, ..., respectively, we have di = a2 — cti', whence, ctj = aj + c^i- ^2 = ag — 2 ao + «! ; whence, a^ = - a,+ 2ao+ d. = - a^+ 2ai+ 2di+ d. = «!+ 2d,+ d^ cZg = a4 — 3 ag + 3 tta — tti ; whence, a^ = cii - 3a2 + Sag + dg t= a, + 3cZi + ScZa + dg ; etc. 424 COLLEGE ALGEBRA. It will be observed, in tlie values of ag, a^, and Ui, that the numerical coefficients of the terms are the same as the coefficients of the terms in the expansion by the Binomial Theorem of a + x to the Jirsty second, and third powers, respectively. We will now prove by induction that this law holds for any term of the given series. Assume the law to hold for the nth term, a„ ; then the coefficients of the terms will be the same as the coefficients of the terms in the expansion by the IJinomial Theorem of a + X to the (n — l)st power. That is, a„ = a, + (n-l)cZ,+ (^^-^)(^^-^> d, \1 \1 If the law holds for the nth term of the given series, it must also hold for the nth. term of the first order of differ- ences ; whence, a„^i-a„ = d, + (n-l)d,+ ^''~^|^''~^^ d3+-' (2) Adding (Ij and (2), we have a„+i = ai + [(n-l) + l]di + ^[(n-2) + 2]d, ^ (n-l)(.-2) ^^^_3^^3^^^^_^^,, , , , n(n — l\ , , 7i(n — l)(n — 2) , , ,o\ = ai + ndi + V ^ ^d.,-\-^ ^^ ^^3-1 (3) This result is in accordance with the above law. Hence, if the law holds for the ?ith terra of the given series, it also holds for the (/i + l)st term; but we know tint it holds for the fourth term, and hence it also holds for the fifth term ; and so on. SUMMATION OF SERIES. 425 Therefore (1) holds for any term of the given series. Note. If the differences finally become zero, the value of «„ can be obtained exactly. 590. To find the sum of the first n terms of the series tti, a,, a,, a^, as, ... . (1) Let S denote the sum of the first n terms. Then S is the (w + l)st term of the series 0, ai, ■ Gi + a.,, ai + a2 + ttg, ••• • (2) The first order of differences of (2) is the same as (1) ; whence it follows that the rth order of differences of (2) is the same as the (r — l)st order of differences of (1). If, therefore, cl^, dj, ..., represent the first terms of the 1st, 2d, ..., orders of differences of (1), a^ d^, dj, ..., will be the first terms of the 1st, 2d, 3d, ..., orders of differences of (2). Putting tti = 0, di = Oi, do = d^, etc., in (3), Art. 589, we have ^^^a, + ^^<^d, + ^^^^-V.^^^"^^ ^.+ -- (3) \A \^ 591. Example. Find the twelfth terin and the sum of the first twelve terms, of the series 1, 8, 27, 64, 125, .... Here, n = 12, and aj = 1. Also, di = 7, ^2 = 12, c?3 = 6, and d, = Q) (Art. 587) Substituting in (1)," Art. 589, the twelfth term = 1 + 11.7+^:^^.12 + ^1-11^.6 = 1728. ^ ^ 1-2 1.2-3 Substituting in (3), Art. 590, the sum of the first twelve -12 + 12ill. 7 + 12.11-10 . i2 + l^-ll-lQ-9. 6 = 6084. ~ ^ 1-2 ^ 1.2-3 1-2.3.4 426 COLLEGE ALGEBRA. 592. Piles of Shot. Example. li shot are piled in the shape of a pyramid with a triangular base, each side of which exhibits 9 shot, find the number in the pile. The number of shot in the first five courses are 1, 3, 6, 10, and 15, respectively ; we have then to find the sum of the first nine terms of the series 1, 3, 6, 10, 15, .... The successive orders of differences are as follows : 1st order, 2, 3, 4, 5, .... 2d order, 1, 1, 1, .... 3d order, 0, 0, .... Putting Ti = 9, tti = 1, f?i = 2, and ch = 1 in (3), Art. 599, ^ = 9 + 1^.2 + 1^.1 = 165. EXAMPLES. 593. 1 . Pind the first term of the sixth order of differ- ences of the series 1, 3, 8, 20, 48, 112, 256, .... 2. Find the eleventh term, and the sum of the first eleven terms of the series 1, 8, 21, 40, 65, .... 3. Find the ninth term, and the sum of the first nine terms of the series 7, 14, 19, 22, 23, ... . 4. Find the thirteenth term, and the sum of the first thirteen terms of the series 4, 14, 30, 52, 80, .... 5. Find the sum of the first n natural numbers. 6. If shot are piled in the shape of a pyramid with a square base, each side of which exhibits 31 shot, find the number contained in the pile. 7. Find the fourteenth term, and the sum of the first fourteen terms of the series 8, 16, 0, -64, -200, -432, .... SUMMATION OF SERIES. 427 8. Find the sum of the first ten terms of the series 1, 16, 81, 256, 625, 1296, 2401, .... 9. If shot are «piled in the shape of a pyramid with a triangular base, each side of which exhibits n shot, find the number contained in the pile. 10. Find the nth term, and the sum of the first n terms of the series 1, 5, 12, 22, 35, ... . 11. How many shot are contained in a pile of ten courses whose base is a rectangle, if the number of shot in the upper course is 15 ? 12. How many shot are contained in a pile of n courses whose base is a rectangle, if the number of shot in the upper course is m ? 13. Find the eighth term, and the sum of the first eight terms of the series 30, 144, 420, 960, 1890, 3360, .... 14. Find the sum of the squares of the numbers 1, 2, . . ., n. 15. Find the sum of the cubes of the numbers 1, 2, ..., n. 16. Find the 7ith term, and the sum of the first n terms of the series 1, 4, 10, 20, 35, 56, ... . 17. How many shot are contained in a truncated pile of seven courses whose bases are rectangles, if the numbers of shot in the length and breadth of the upper course are 10 and 6, respectively ? 18. How many shot are contained in a truncated pile of n courses whose bases are squares, if the number of shot in each side of the upper base is m ? IISTTERPOLATION. 594. Interpolation is the process of introducing between the terms of a series other terms conforming to the law of the series. Its usual application is in finding intermediate numbers between those given in Mathematical Tables. 428 COLLEGE ALGEBRA, The operation may be effected by giving fractional values to n in equation (1), Art. 589. 595. 1. Given V5 = 2^361, VG = 2.4495, V7 = 2.6458, V8 = 2.8284, ... ; find V6.3. In this case the successive orders of differences are : .2134, .1963, .1826, .... -.0171, -.0137, .... .0034, .... Whence, d^ = .2134, da = - .0171, d^ = .0034, .... Since the required term is distant 1.3 intervals from V5, we have n = 2.3. Substituting in (1), Art. 589, we have, approximately, VaB = 2.2361 + 1.3 X .2134 + ^f^f (-M71) 1x2 1.3X.3X--.7 Q(j3^ 1x2x3 = 2.2361 + .2774 - .0033 - .0002 = 2.5100. EXAMPLES. 2. Given log 22 = 1.3424, log 23 = 1.3617, log 24 = 1.3802, log 25 = 1.3979, ... ; find log 24.5. 3. Give n a/TO = 4.12129, -^tI = 4.14082, \/72 = 4.16017, ...; find a/70.12. 4. The reciprocal of 22 is .04545 ; of 23, .04348 ; of 24, .04167 ; etc. What is the reciprocal of 22.8 ? 5. Given log 109 = 2.03743, log 110 = 2.04139, log 111 = 2.04532, ... ; find log 110.7. 6. Given V37 = 6.08276, V38 = 6.16441, V39 = 6.24500, ... ; find V37.48. 7. Given log 11 = 1.04139, log 12 = 1.07918, log 13 = 1.11394, log 14 = L14613, ... ; find log 13.28. DETEEMI^TANTS. 429 XL. DETERMINANTS. 596. Consider the equations a^x + biy = Ci, cioX + boy = C2. Solving, we obtain ^ _ boCi — &1C2 _ CaCTi — Ciag The common denominator may be written in the form a„b,\^ W which is understood as signifying the product of the upper left-hand and lower -right-hand quantities, minus the prod- uct of the lower left-hand and upper right-hand. The expression (1) is called a Determinant of the Second Order. 597. The numerators of the fractions in the preceding article can also be expressed as determinants ; thus. 6.c,-6,C2= ll ^^ ,andc2a,- — Cjtta = "2, C2 598. Consider the equations a^x + b^y -f CiZ = ■ch a^ ■\- boy -\- c^z = :d2, a^x + 63?/ + CqZ = ■■d,. Solving, we obtain ^ _ diboCs — di^sCo + dobgCi — chb 1C3 + d^h^ Cg — ■ dAci 0'lb2Cs — C(,lbsC2 + (i^^sCl — tta^l^S + «3&lC2 — a3&2Ci with results of similar form for y and z. (1) 430 COLLEGE ALGEBRA. The denominator of (1) may be written in tlie form ttl, K Ci «2, b,, Co ttg, h, Cg (2) which is understood as signifying the sum of the prod- ucts of the quantities connected by lines parallel to a line joining the upper left-hand corner to the lower right-hand, in the following diagram, minus the sum of the products of the quantities connected by lines parallel to a line join- ing the lower left-hand corner to the upper right-hand. The expression (2) is called a Determinant of the Third Order. 599. The numerator of the value of x can also be expressed as a determinant, as follows : f?l, K Ci ch, b,, Ca ds, bs, Cs as may be verified by expanding it by the rule of Art. 598. EXAMPLES. 600. Evaluate the following determinants 2, 3, 5 10, 2, 8 -2, -3, 4 7, 1, 4 2. n, 4, 3. 5, -0, 7 6, 2, 3 3, 1, 7 -8, 9, 1 DETERMINANTS. 431 1, a, b a, h; g 1, b, c 5. h, b, f 1, c, a 9, f, c 1, c, -b — c, 1, a b, -a, 1 Verify the following by expanding the determinants 7. 8. ag, 63, C3 Ct'25 0^5 Co "3; ^3J C3 - n I ^2, Co I , ; J C2, 0^2 I , . I CU, b., I '^SJ *^3 I I ^^S) ""3 I I "3? ^^3 «1, ao, «3 bu b,, h Cl, C2, C3 61, ai, Cl b.2, 0,-2) C2 ^35 <^3> ^^3 10, «1, tti, &I a.,, «2, &., ^3, «3, &3 0. 11. mai, 61, Cl ma2, &2, C2 = m mas, h, C3 «i, K Cl ao, h, C2 as, h, C3 601. General Definition of a Determinant. If, in any permutation of the numbers 1, 2, 3, ..., n, a greater number precedes a less, there is said to be an inversion. Thus, in the case of five numbers, the permutation 4, 3, 1, 5, 2, has six inversions ; 4 before 1, 3 before 1, 4 before 2, 3 before 2, 5 before 2, and 4 before 3. Consider, now, the n^ quantities ^n, I, ^n^ 2, a„^ 2 a„ (1) Note 1. The notation in regard to suffixes, in tlie above, is tliat the first suffix denotes tlie liorizontal row, and tlie second the vertical column, in which the quantity is situated. Thus, aii,r is the quantity in the klh row and rth column. 432 COLLEGE ALGEBRA. Let all possible products of the quantities taken n at a time be formed, subject to the restriction tliat each product shall contain one and only one quantity from each row, and one and only one from each column, and write them so that the second suffixes shall occur in the order 1, 2, ..., n. Note 2. This is equivalent to wi'iting all the permutations of the order 1, 2, ..., n in the first suffixes. Give to each product the sign + or — according as the number of inversions in the first suffixes is even or odd. The expression (1) is called a Determinant of the nth Order. 602. The expanded form may also be obtained by writing the Jirst suffixes in the order 1, 2, ..., n, and giving to each product the sign + or — according as the number of inver- sions in the second suffixes is even or odd. For let the absolute value of any product, obtained as in Art. 601, be ap,ia,,2 -"^'r.n; (1) where jy, q, ..., r is a permutation of 1, 2, ..., n. Writing the first suffixes in the order 1, 2, ..., n, we have «1,8 «2,« ••• C*K,t;) (2) where s, i, ..., ^; is a permutation of 1, 2, ..., n. ■ It is evident that there are just as many inversions in the first suffixes of (1) as in the second suffixes of (2) ; and hence the products (1) and (2) will have the same sign. 603. The quantities aj^i, Oj^s, etc., are called the constitu- ents of the determinant, and the products aj,] aj^j ••• ^n^M etc., occurring in the expanded form, are called its elements. The constituents lying in the diagonal joining the upper left-hand corner to the lower right-hand, are said to be in the principal diagonal; the element whose factors are the constituents in the principal diagonal is always positive. Note. By Art. .54.3, the number of elements in the expanded form of a determinant of the nth order is | n. , DETERMINANTS. 433 604. It may be shown that the definition of Art. 601 agrees with those of Arts. 596 and 598 ; for consider the determinant ^i,i> %,25 <^i,a 0^2, U <^2,2> C(2 3 ^S, 1} %, 2? '^3, a The only possible products of the quantities taken 3 at a time, subject to the restriction that each product shall con- tain one and only one constituent from each row, and one and only one from each column, the second sufl&xes being written in the order 1, 2, 3, are %, 1 <^2, 2 <^3, 3) ^1,1^,2^2,35 ^2, 1 <^1, 2 <^3, 3? <^2, 1 %, 2 <^1, 3) %1<^1,2^2,3J ^^^ %, 1 f'2, 2 %, 3- In the first of these there are no inversions in the first suffixes ; in the second there is one, 3 before 2 ; in the third there is one ; in the fourth, two ; in the fifth, two ; in the sixth, three. Then according to the rule of Art. 601, the first, fourth, and fifth products are positive, and the second, third, and sixth are negative ; and the expanded form is «1, 1 «2, 2 ^3, 3 — «1, 1 ^3, 2 <^2, 3 — «2, 1 «1, 2 «3, 3 + ^2, 1 %, 2 «1, 3 + «3, 1 Ctl, 2 «2, 3 — «3, 1 «2, 2 «1, 3) which agrees with Art. 598. PROPERTIES OF DETERMINANTS. 605. A determinant is not altered in value if its roios are changed to columns, and its columns to rows. (Compare Ex. 8, Art. 600.) Consider the determinants <-"!, 1> "'1,2) •^'2,15 .'^'2, 2j and ''M,l; '-''2,1) a^i 2, ^^2 2) 434 COLLEGE ALGEBRA. Since the first suffixes of the first determinant are the same as the second suffixes of the second, if the first deter- minant is expanded by the rule of Art. 601, and the second by the rule of Art. 602, the results will be the same. Therefore the determinants are equal. 606. A determinant is changed in sign if any ttvo consecu- tive columns, or any tivo consecutive rows, are interchanged. (Compare Ex. 9, Art. 600.) Consider the determinants ■n,«5 a«,r, a. and Cf2,lJ '••) ^2,r1 <^2, }J »•■> tt2,; the gth and rth columns of the first being, respectively, the rth and gth columns of the second. Let the absolute value of one of the elements of the first determinant be a,,i...a(,a„,,...a,,„; (1) where s, ..., t, u, ..., v is a permutation of 1, 2, ..., n. Since the constituent in the ith row and gth column of the second determinant is a<_^, and the constituent in the ?*th row and rth column a„^ „ the absolute value of the cor- responding element of the second determinant may be derived from (1) by replacing a,,, and a,,,^ l^y «(,r and a„,,, respectively ; that is, a^^ 1 . . . a<_ ^ a„_ , . . . a„_ „. The latter expression is also the absolute value of one of the elements of the first determinant, since it has one and only one constituent from each row, and one and only one from each column ; and writing it so that the second suf- fixes shall occur in the order 1, 2, ..., ?;, wc have . (2) DETERMINANTS. 435 Now whatever the number of inversions in the first suf- fixes of (1), s, ..., t, u, ..., V, tlie number of inversions in the first suffixes of (2), s, ..., u, t, ..., v, differs from it by unity ; for in the first case t precedes u, and in the second u precedes t. Hence the elements (1) and (2) of the first determinant are of opposite sign (Art. 601). That is, any two elements of the given determinants of equal absolute value are of opposite sign; and hence the determinants themselves are of equal absolute value and opposite sign. It follows from Arts. 605 and 606 that if two consecu- tive rows are interchanged, the sign of the determinant is changed. 607. A determinant is changed in sign if any two roics, or any two columns, are interchanged. Consider the m letters a, b, c, .. ., e, f, g. By interchanging a with b, then a with c, and so on in succession with each of the m — 1 letters to the right of a, a may be brought to the right of g. Then, by interchanging g with f, then g with e, and so on in succession with each of the m — 2 letters to the left of g, g may be brought to the left of b. It is evident from this that a and g may be interchanged by (m — 1) -t- (m — 2), or 2m — 3, interchanges of consecu- tive letters ; that is, by an odd number of interchanges of consecutive letters. It follows from the above that any two rows, or any two columns, of a determinant may be interchanged by an odd number of interchanges of consecutive rows or columns. But every interchange of two consecutive rows or columns changes the sign of the determinant (Art. 606). Therefore the sign of the determinant is changed if any two rows, or any two columns, are interchanged. 436 COLLEGE ALGEBRA. 608. If two rows, or two columns, of a determinant are identical, the value of the determinant is zero. (Compare Ex. 10, Art. 600.) Let D be the value of a determinant having two rows, or two columns, identical. If these rows, or cohimns, are interchanged, the value of the resulting determinant is — 7) (Art. 607). But since the rows, or columns, which are interchanged are identical, the two determinants are of equal value. Hence, D = — D ; and therefore D = 0. 609. Cyclical Interchange of Rows or Columns. By 7i — 1 successive interchanges of two consecutive rows, the upper row of a determinant of the nth order may be brought to the bottom. Thus, by Art. 606, the determinant is equal to (— 1)" ' as, &2J •• a„, 5„, The above is called a cyclical interchange of rows. In like manner, by n — 1 successive interchanges of two consecutive columns, the left-hand column of a determinant of the ?ith order may be brought to the end. 610. If each constituent in one roiv, or in one column, is the sum of m terms, the determinant can be expressed as the sum ofm determinants. Consider the determinant ^"1, 15 «2,1> «■« n (1) DETERMINANTS. 437 Let each constituent in the rth column be the sum of 7n terms, as follows : ai,r =fh fci-i \-fi, C('2,r = &2 + C2 H +/2, &„ + c„ + •••+/„. Let ap,i'"a,,^---a^^„ be the absolute value of one of the elements of (1) ; then, = (a^,,...6,...a,,„)+... + (a,.i.../,...a,,„). It is evident from this that the determinant (1) can be expressed as the sum of the determinants ttl 1} • , b„ . ; «l,n tti 1) • ; fv . ., tti n a2 1) • V b,, . •, «2,n + • • + a„ I' .* •, J2, . ., fn, . n a, , 1' • ., K . •, «„,„ n 611. If all the constihients in one row, or in one column, are multiplied by the same quantity, the determinant is multi- plied by this quantity. Consider the determinant •, ai,r, ., a„ , (1) Multiplying each constituent in the rth column by m, we have ti,i, ..., maj,r, ma. 2,r) '••■> ^'2,1 6„,„ ..., ma^^,, (2) Let Op 1 . . . a, ,. . . . a,, „ be the absolute value of one of the elements of (1). 438 COLLEGE ALGEBRA. Eeplacing a^^^ by ma^^^, the absolute value of the corre- sponding element of (2) is wia^_ i . . . a,_ ^ . . . o,_ „. It is evident from this that the determinant (2) is equal to the determinant (1) multiplied by m. 612. If the constituents in any row, or column, are multi- plied by the same quantity, and either added to, or subtracted from the corresponding constituents of another row, or column, the value of the determinant is not changed. Let the constituents of the rth column of the following determinant be multiplied by m, and added to the corre- sponding constituents of the gth column. , "-!> •••> "'gi '••) "-rj •••5 "^11 We then obtain the determinant (1) a, + ma^, b^ + 7nb„ a„ .. h,, .. ki, ..., Jc^ + mk^, ..., k^, ..., kn rhich, by Arts. 610 and Gil, is equal to (2) bu ..., b„ ., a^. ki, •1 l^rj k„ + m b„ ..., K, ki, ..., k^, ., a,, ., br. But the coefficient of m is equal to zero (Art. 608). Hence the determinant (2) is equal to (1). ., a„ 613. Minors. If the constituents in any m rows and any m columns of a determinant of the nth order are erased, the remaining constituents form a determinant of the (n— m)th order. DETERMINANTS. 439 This determinant is called an mth Minor of the given determinant 5 thus, a„ d„ 6] %, d„ es a« ds, Go is a second minor of «!, 6i, Ci, c?i, ei <*2) O2, C2, tto, 62 0^3) "3j ^3? ^3j 63 (X45 0^ C4J Ct4, 64 %, &s, Co, d„ 65 obtained by erasing the second and fourth rows, and the second and third columns. 614. To find the coefficient of a^^-^ in the determinant «2,1J «2,2> •• ; «1, ', f2, «n.b «n.2) • ., a,. (1) By Art. 601, the absolute values of the elements which involve a^^i are obtained by forming all possible products of the constituents taken 71 at a time, subject to the restric- tions that the first constituent shall be aj^i, and that each product shall contain one and only one constituent from each row except the first, and one and only one from each column except the first. It is evident from this that the coefficient of a^^i in (1) may be obtained by forming all possible products of the following constituents taken n — 1 at a time, %25 f'3,3> •••? 0.3, n a«,2, «„,3J •••, ««,» subject to the restriction that each product shall contain one and only one constituent from each row, and one and only one from each column, writing the second suffixes in the order 2, 3, ..., n, and giving to each product the sign + or — according as the number of inversions in the first suffixes is even or odd. 440 COLLEGE ALGEBRA. Then by Art. 601, the coefficient of aj, i is a,,,, a,, 3, ..., a.„ that is, the minor obtained by erasing the first row and the first column of the given determinant. 615. By aid of the theorem of Art. 614, a determinant of any order may be expressed as a determinant of any higher order ; thus, 1, 0, 0, 0, 1' ^' 0' 0, 0, ai, b„ Ci 0, a.2, b.2, c, 0, as, 63, Cg oi, bi, ci as, &2, C2 = as, &3; C3 0, 1, 0, 0, 0, 0, ai, bi, ci 0, 0, a.2, 62? ^2 0, 0, aa, 63, C3 etc. 616. We will now consider the general case. To find the coefficient of a^^^ in the determinant ai.b ••; a,,, ..., aj (1) By fc — 1 successive interchanges of consecutive rows, and r—1 successive interchanges of consecutive columns, the constituent a^ ,. may be brought to the upper left-hand corner; thus, by Art. 606, the determinant is equal to (-iy\-iy-^ ai,r; ai,b a* „ Then by Art. 614, the coefficient of a^^^ is «!,!, •••, a, (-1)''+'-^ But (-1)^+'- 2 = ( - l)*"-( - 1) ^ = ( - 1 )*+'•. DETERMmANTS. 441 Hence the coefficient of a^^^ is equal to (—1)*+', multi- plied by tliat minor of (1) Avhich is obtained by erasing the ^th row and ?-th column, 617. By aid of Art. 616, a determinant of any order may be expressed in terms of determinants of any lower order. Thus, since every element of a determinant contains one and only one element from the first row, we have, tti, &i, Ci, di a.,, bo, Co, cZa <^3; ^g, Cg, ttg a^, &4, C4, di = «! b.,, c.2, d.2 b-i, Cg, dg &4, C4, di -by a.2, C2, d.2 Cls, Cg, dg (li, C4, d, + Ci a.2, &2, di ttg, 63, dg a^, bi, di - dl ttg, 63, Cg 5 cii, bi, C4 and each of the latter determinants may in turn be expressed in terms of determinants of the second order. 618. Evaluation of Determinants. The method of Art. 617 may be used to express a deter- minant of any order higher than the third in terms of determinants of the third order, which may then be evalu- ated by the rule of Art. 598. The theorem of Art. 612 may often be advantageously employed to shorten the process, as shown in Ex. 1. 5, 7, 8, 6 11, 16, 13, 11 14, 24, 20, 23 7, 13, 12, 2 1 . Evaluate Subtracting the first row from the last, twice the first row from the second, and three times the first row from the third (Art. 612), the determinant becomes , by Art. 61L 5, 7, ^, 6 5. 7, 8, 6 1, 2, 1, 3, -3, -4, -1 5 = 2 1, 2, -1, 3, -3, -4, -1 5 2, 6. 4, -4 1, 3, 9 — 2 442 COLLEGE ALGEBRA. 3, 23, 11 5, -7, 4 , by Art. 616. 1, 5, -1 Subtracting five times the second row from the first, add- ing the second row to the third, and subtracting the second row from the last, we have 0, -3, 23, 11 1, 2, -3, -1 0, 5, -7, 4 0, 1, 5, -1 The object of the above process is to put the given deter- minant in such a form that all but one of the constituents in one column shall be equal to zero ; the determinant can then be expressed as a determinant of the third order by Art. 616. The last determinant may be evaluated by Art. 598 ; but it is better to subtract five times the first column from the second, and then add the first column to the last : thus, 38, 8 ■32, 9 0, = -2 ■32 9!= -2(342+256) = -1196. EXAMPLES. Evaluate the following 3. 5, 15, 10 6, 21, 13 7, 25, 16 6. 11, 12, 13 14, 15, 16 17, 18, 19 • 7. 30, 15, 17 29, 18, 23 20, 19, 22 . 8. 1, a, a- 1, b, h- 9. 1, c, c- 1, 2, 3, 4 1, 3, 6, 10 10. 1, 4, 10, 20 1, 5, 15, 35 9, 13, 17, 4 18, 28, 33, 8 30, 40, 54, 13 24, 37, 46, 11 13. 1, 15, 14, 4 12, 6, 7, 9 8, 10, 11, 5 13, 3, 2, 16 7, 13, 10, 6 5, 9, 7, 4 11. 8, 12, 11, 7 4, 10, 6, 3 a, 1, 1, 1 1, h, 1, 1 1, 1, c, 1 1, 1, 1, d 3, 2, 1, 4 15, 29, 2, 14 • 12. 16, 19, 3, 17 33, 39, 8, 38 0, a, b, c a, 0, c, b b, c, ^, a c, b> a, a, b, c, d b, a, d, c c, d, a, b d, c, b, a DETERMINANTS. 443 14. 15. 18. X — 4:y, x — y, x + 2y x — oy, X, X + 'dy x-2y, x + y, a; + 4?/ x-i- y, z — y, z — X x — y, y + z, x — z y — x, y-z, z + x p, q, r, s ~p, q, r, s -P, -Q, ^» « — p, —q, —r, s 19. 16. 7,' —2 5, -2, 0, 5 6, -2, 2 -2, 5, 3 2, 3, 4 0, a, b, c 17. — a, 0, 11, VI -n, 0, I -c, -m, - I, a-{-x, a, a, a, ^ b+x, b, b, c, d c, d c+x, d c, d+x 619. Let A^^r denote the coefficient of a^^^ in tlie deter- minant a, 1 «! .., ..., a, a^ (1> Then since every element of the determinant involve? one and only one constituent from the rth column, the valu^ of the determinant is It follows from the above that is the value of a determinant which differs from (1) only in having b^ b.,, ..., b„ instead of ai_^, 03,^? •••; ««,rj as the con- stituents in the rth column. Hence, if q is any number of the series 1, 2, ..., w except r, the expression for it is the value of a determinant whose qth. and rth col- umns are identical (Art. 608). 444 COLLEGE ALGEBRA; 620. Multiplication of Determinants. Consider the determinant ttidi + a,ei + tts/i, bidi + 62^1 + hfi, Cid^ + c^ei + cj^ a^do + a^e^ + 03/2, 61^2 + 6262 + hU c^d^ + 0263 + C3/2 ttjcZg + 02^3 + a 3/3, bids + 6263 + &3/3J Cjdg + Cgeg + C3/3 ■ By Art. 610, this can be expressed as the sum of twenty- seven determinants, of which the following are types : aA, he^, C3/1 a^, b-^So, Csf, , aA, b^es, C3/3 ttidi, bA, Cidi aA, ^Aj, CA2 aA, bA} cA That is, by Art. 611, aib2C3 d„ ei, /i do, 62, /a , aAc2 ch, es, f-i (h, di, e, C?2, d.2, 6.2 , ciAci ds, cl„ e.. d„ d„ di d,, d,, d. ds, d„ d, The eighteen determinants of the second type, and the three determinants of the third type, are all equal to zero by Art. 608. Hence the given determinant is equal to aAcs H-aa&sCj di, 61, /i d,, e,, /a d^j €3, fs ■j-aAc2 di, fi, d^ ^2, fi, d^ +a3^iC2 63? fi, di di, fi, 61 d,, f, e, d^, f, 63 +(hbjC3 A, di, e, f, d,, 62 f, d^, 63 4-a3&2Ci eij d„ f 62, d,, f ^3, d3. f f. Ci, di f, e,, d. f, es, d3 By Art. 606, the above is equal to {ciACs — aAc2 + <^^A<^i - aa&iCs + cta^iCs - ctAci) or, ay, &1, Ci «2, b,, C-2 X (hi h, Cy di, e.,, f, dz, Co, f d„ Ci, /i d„ 62, /2 d3. 63, f That is, the product of two determinants of the third order can be expressed as a determinant of the third order. DETERMINANTS. 445 621. We will now discuss the general case. Consider the determinants , and Q ■■ , K R = (1) (2) Let a third determinant be formed from P and Q by aid of the relation Ck, r = tti, 1 &r, 1 + «i, 2 ^r, 2 H h «l, „ ^r, ; To prove that R = Px Q. One of the elements of i2 is ± Cp_i...Cg,„, where j9, ..., q is a permutation of 1, 2, ..., 7i ; the sign being + or — accord- ing as the number of inversions in j?, ..., g is even or odd. By (2), the value of this element is ± (%, 1 ^1, 1 + • • • + «>, nKn) — (S. 1 ^«, !+•••+ «*, « ^«, ») ■ Expanding, we obtain a series of terms of the type ±(cv,...a„,)x (&,,.•..&.,.); (3) where s, ..., t are numbers of the series 1, 2, ..., n. Then R is equal to the sum of all the terms of which (3) is the type ; the sign of each being + or — according as the number of inversions in p, ..., q is even or odd. Now since p, ..., g is a permutation of 1, 2, ..., n, by Art. 601 the sum of all terms involving 6i, , . . . &„, , is X &],.•••&„,«! (4) and this expression vanishes identically unless s, permutation of \, 2, ..., n (Art. 60S). 446 COLLEGE ALGEBRA. But if s, ..., t is a, permutation of 1, 2, ..., n, (4) may be written o, 1, .... o, a„ X &],, (5) the sign being + or — according as the number of inver- sions in s, ..., t is even or odd (Art. 607). Hence the sum of all the terms of which (5) is the type is «n,l5 •••5 «n Therefore E=Px Q. 622. It was shown in Art. 621 that the product of two determinants of the nth order can be expressed as a deter- minant of the ?ith order. But a determinant of any order can be expressed as a determinant of any higher order (Art. 615). Hence, the product of any two determinants can be expressed as a determinant of the same order as that of the factor of highest order. 623. Application to the Solution of Equations. Let it be required to solve the following set of n indepen- dent, simultaneous simple equations, involving n unknown quantities : «2, 1 ^1 + Cf L', 2 ^'2 + • • • + «2, n^n = ^2) Let Ai,^r denote the coefficient of a^.^^ in the determinant Z) = "1,1? "1,2? •• a..^, «„ 2, ■• DETERMINANTS. 447 Multiplying the given equations by A-^^^, A^^^ respectively, and adding the results, we have + X^ (tti, ,. ^1, r + «2, r^2,r-\ + «„, r -4„, r) + = 6l A,,- + &2 Ar H h K^n,r- a) By Art. 619, the coefficient of each of the unknown quan- tities except XV is equal to zero ; also, the coefficient of x^ is D, and the second member is a determinant which differs from D only in having &i, 62) •••? ^n i^ place of aj^^, aa,^, ..., a„,r as the constituents of the ?-th column. Denoting the latter by D^, we have x. = ^- D ^ Example. Find the value of .?/ from the equations 3x-5y + 7z= 28, 2x + 6y-9z = -23, 4.x-2y-5z= 9. We have y = 3, 28, 7 2, -23, -9 4, 9, -5 3, -5, 7 2 6, -9 4, ■"5 -5 630 210 = -3. 624. Elimination. Consider the following n homogeneous simple equations, involving n unknown quantities : ttg iXi -\- a2,2-'^'2 + ••• + 02,n^n = 0, <^»t,l^'l + f*n,2^'2 +•••+«« 0. 448 COLLEGE ALGEBRA. Dividing the terms of each equation by x„, and transpos- ing, we have ai,i^ + «i,2^ + ••• + «i,n-i^ «!,, (1) Xi Xo Xn-i a2,l ;^ + «2,2 r^ + ••• + ^.n-l^T" = - ^2.' Xi X2 X„_i By solving the last n — 1 equations, we may obtain the values of the n — 1 quantities — > •••> -^^ ; thus, by Art. 623, — «2,n> «2,2J • •, «2,„-I — a„,n> «n,2, • V «n,H-l C^2,b «2,2, •• •, «2,„-l ««,!, Ct„,2, . V a„,„_i a2,2, ■■} a2,n-l, Hn (_!)«-! Cb,^2i •••) Cln,n-\^ ^n,n %,1) 01.2, 2J •••} <^2,n-l a«,i, a„,2, ..., a„,„_i by Art. 606 ; and results of similar form will be found for X.2 cc„_i xj ' Xn Substituting these values in (1), clearing of fractions, and transposing, we have av,2) •••) Ct2,n-1) ^2,) «l.l(-l)""' ■«1,2(-1)" <*2, 1> <^2,3> •••> <*2, -f ... + ai That is (Art. 616), ^*2,l' '^'2,25 •-•) (^2,1 ^n, 1> tt„_2> •••> tt„ n_i = 0. (2) DETERMINANTS. 449 Equation (2) expresses the relation which must hold between the coefficients of the unknown quantities in the given equations in order that all the equations may be satis- fied by the same sets of values of the unknown quantities. EXAMPLES. 625. Solve the following by determinants i 2x + 3y-\- z= 4. 1. < x-\-2y-\-2z= 6. {5x+ y + 4.z = 21. r 4a; + 5^-72; = -8. 2. J 3a;-42/-2z= 25. [ X + 32/+ 2 = -9.. 4. 2x + 5y-^z= 17. Qx — 2y — 5z=— 3. [ 3a; + 7y-f 4z = -18. 2^4- y+ z+ u= 0. x-2y+3z-4:u=29. 2x+Sy-4:Z-5u= 9. l3x-4:y-\-5z + 6u= 1. 5. Express 3, -2 -8, 7 ^ it 6 --^« 6. Express 2, 3, -5 1, 0, 2 3, -2, -4 X 5, 0, -2 0, -3, -4 -6, 1, 3 as a determinant. 6, 1, 2 7. Express —4, 3, 0, -7, 5 (Compare Art. 622.) 8. Express the square of 0, c, b c, 0, a b, a, as a determinant. as a determinant. 450 COLLEGE ALGEBRA. XLI. THEORY OF EQUATIONS. 626. Every equation of the wth degree (Art. 179) involv- ing one unknown quantity, can be written in the form cc" + PiX"~'^ + p.^x""^ + ••• -Fi>„_iX+^„ = 0; (1) where the coefficients p^, p2, •■-, Pn may be positive or nega- tive, integral or fractional, rational or irrational, real or imaginary, or zero. If none of the coefficients 2h, P2, ••) Pn are zero, the equa- tion is said to be Complete; if one or more of them are zero, it is said to be Incomplete. We shall hereafter speak of (1) as the General Form of the equation of the 71th degree. 627. It will be proved in Appendix II., that every equation of the above form has at least one root, real or imaginary. 628. A function of x (Art. 213) is often represented by the symbol /(cc), or F{x). If, in any investigation, a certain function of x is repre-- sented by /(a;), then, whatever a may be, f{a) is taken to represent the result obtained by substituting in the given function a in place of x. Thus, if f{x) = o? + ^x-2, then /(3) = 32 + 3-3-2 = 9-f-9-2 = 16; /(-3) = (-3)-4-3(-3)-2 = 9-9-2=-2; etc. 629. If a is a root of the equation X" +pia;"~^ -I + 2)„-iX -\- Pn = 0, then the first member is divisible by x — a. The division of the first member hj x — a may be carried out until a remainder is obtained which does not contain x. Let Q denote the quotient, and R the remainder. THEORY OF EQUATIONS. 451 Then the given equation may be made to take the form {x-a)Q + E = 0. (1) Since a is a root of the given equation, equation (1) must be satisfied when x is put equal to a. Putting x = a,we have, since M does not contain x, B = 0. Therefore aj — a is a factor of the first member of the given equation, for it is contained in it without a remainder. 630. Conversely, if the first member of ic" -fpiX"-^ -I \-2^n~i^ -\-Pn = is divisible by x — a, then a is a root of the equation. For since the first member of the given equation is divisi- ble by cc — a, the equation may be made to take the form (x-a)Q = 0; and it follows from Art. 350 that a is a root of this equation, 631. It follows from Art. 630 that if the first member of PqX" + piX"~^ -\ +P„_ia;+p„ = is divisible by ax + b, then is a root of the equation. a EXAMPLES. 632. Prove by the method of Arts. 630 or 631 : 1. That 5 is a root of a;^- 2^2 -19a; + 20 = 0. 2. That -3isarootof 2ar' + 3a;2_2a; + 21 = 0. 3. That - is a root of 33;" - 8 a;^ + 13 a;- - 9 a; + 2 = 0. o 4. That - 4 is not a root of a;" - a;^ + 7 x - 12 = 0. 5. That - - is a root of 8 a;' + 6 a;" - 15 ar - 16 a; -3 = 0. 4 6. That - is nit a root of 15 ar^ + a;^ -f- 14 a; - 3 = 0. 452 COLLEGE ALGEBRA. 633. Number of Roots. All equation, of the nth degree cannot have more than n dif- ferent roots. Let tne equation be X" +pia;""^ -t-2hx"-^" -\ -i-Pn-i^+Pn = 0; (1) By Art, 627, thia equation must have at least one root. Let a be this root ; then by Art. 629, the first member is divisible by cc — a, and the equation may be put in the form (X — a) (X"-^ +^23?""^ -I h Qn-l^ + Qn) = 0. Then by Art. 350, the equation may be solved by placing X — a = 0, and «"-^ + (h x"^^ H 1- g«-i a; + ^„ = 0. (2) Equation (2) must also have at least one root. Let b be this root; then (2) may be written (cc - 6) (a;"-2 + rgcc^s -\ h r„_i a; + r„) = 0, and the equation may be solved by placing x-b = 0, and ic""^ + Vgcc"^^ + • • • + ''n-i x + r„ = 0. Continuing the process until n — 1 binomial factors have been divided out, we shall arrive finally at an equation of the first degree, X — k = ; whence, x = 7c. Therefore the given equation has the n roots a, b, ..., k. Note. It should be observed that the roots are not necessarily unequal ; thus, the equation a;^ — 3 x'^ + 4 = can be written in the form {x + 1) (x — 2) (x — 2) = 0, and its three roots are — 1, 2, and 2. 634. It is customary to enunciate the principle demon- strated in Art. 633 in the following form : An equation of the nth degree has n roots; by which we mean that it may have n, different roots, but cannot have more than n. THEORY OF EQUATIONS. 453 635. It follows from Art. 633 that if m roots of an equa- tion of the 7ith degree are known, the equation may be depressed to another of the {n — m)th degree which shall contain the remaining roots. Hence, if all the roots of an equation are known except two, these two may be obtained from the depressed equation by the rules for quadratics. 1. Two roots of the equation 9 a;'' -37 o.-^- 8 a; + 20 = are 2 and — - ; what are the others ? 3 Dividing 9a;V37a;2 - 8a; + 20 by (x- 2)(3a; + 5), or 3 a;^ — a; — 10, the depressed equation is 3 a;- + a; — 2 = 0. 2 Solving by the rules for quadratics, we have a; = - or —1. o EXAMPLES. 2. One root of af' — 37 a; + 84 — is 3; what ?re the others ? 3. One root of 2a;3_^5a^.2_43^_ 90 ^ is - 2 ; what are the others ? 4 One root of 24 x^ - 46 .«- + 29 a; — 6 = is - ; what are the others ? ^ 5. One root of 32a;3 _ 32^52 _ 94^; + 39 = is - ^ ; what are the others ? 6. Two roots of 6 a;* - af' - 42 a;^ + 15 a; + 50 = are 2 and — 1 ; wiiat are the others ? 7. Two roots of 36a;'' - 445a;2 + 49 = are I and -^; what are the others ? 8. Two ro<5ts of a;* - 10 aar' + 35 aV - 50 a^a; + 24 a'' = are a and 3 a ; what are the others ? 9. One root of the equation a:? — (m + 2)x^ — (m^+ 4m + 5)a; + m^ + 6 ?7i- + 11 w -f- 6 = 9 is m -\-l; what arc the others ? 454 COLLEGE ALGEBRA. 636. Formation of Equations. It follows from Art. 633 that if the roots of ic" -\-piX"-^^ + ••• +2^n-i^ +2K = ^ are a, h, ..., k, the equation may be written in the form {x-a){x-h)--{x-k) = Q. Hence, to form an equation which shall have any required roots, Subtract each of the roots from x, and jJlace the product of the resulting expressions equal to zero. (Compare Art, 354.) 1. Form the equation whose roots shall be 1, — 6, - and _5 ^ 4" By the rule, {x - 1) (x + 6) fx - ^\ fx + ^\ = 0. Multiplying the terms of the third and fourth factors by 2 and 4, respectively, we have (x -l)(x+6){2x- 1) (4x + 5) = 0. That is, 8a;^ + 46ar^ - 23 ar - 61 x + 30 = 0. EXAMPLES. Form the equations whose roots shall be : 2. 2, 3, 5. 6. -2, -2, 1|,1|. 3. - 2, - 3, - 4, 9. 4. 1, - 2, 3, 0. 7 4 r, 11 g . 1 2 3 ^- ^'-2'-3'~4' 6. 6,-l,H,-|. 9. 2±V3, -,2±V3. 10. i±-^^, 2 -2±V5 2 jj V-i±V-3 -V-i±V-3 THEORY OF EQUATIONS. 455 637. Composition of Coefficients. By Art. 636, the equation of the nth degree whose roots are a, b, c, d, ..., k, I, m, is {x - a) {x - b) {x - c) {x - d) ■■■ (x - m) = 0. (1) By actual multiplication, we obtain (x — a) (x—b)=x^— (a + b)x-{-ab; (x — a){x—b) {x — c) = x^—{a-\-b-^c)x^-\-{ab-\-bc-{-ca)x — abc; etc. When all the factors of the first member of (1) have been multiplied together, we shall have a result of the form cc" H-Pix"-! + 2hx"~^ +^3^""^ -\ \-Pn = ; where p^ = — (a + 6 + c + • • • + A: + Z + ?)i) ; p,^ — ab -\- ac -^ be -\- ••• + Im ; Ps = — {abc + abd + acd + • • • + klm) ; p„= ± abed '••klm, according as n is even or odd. Hence, if an equation of the nth degree is in the general form. The coefficient of the second term is equal to minus the sum of all the roots. The coefficient of the third term is equcd to the sum of their products, taken two at a time. The coefficient of the fourth term is equcd to minus the sum of their jyi'oducts, taken three at a time; etc. The last term is equal to plus or minus the product of cdl the roots, accord'ng as n is even or odd. 638. It follows from Art. 637 that, if an equation of the nth. degree is in the general form, If the second term is wanting, the sum of the roots is 0. If the last term is wanting, at least one root is 0. If the last term is integral, it is divisible by every inte- gral root. 456 COLLEGE ALGEBRA. 639. If all but one of the roots of an equation of the «th degree in the general form are known, the remaining root may be found either by adding the sum of the known roots to the coefficient of the second term of the given equation and changing the sign of the result, or by dividing the last term of the given equation by plus or minus the product of the known roots according as n is even or odd. If all but two are known, the coefficient of the second term of the depressed equation (Art. 635) may be found by adding the sum of the known roots to the coefficient of the second term of the given equation; and the last term of the depressed equation may be found by dividing the last term of the given equation by plus or minus the product of the known roots according as n is even or odd. EXAMPLES. In each of the following obtain the required roots by the above method : 1 . Two roots of ar^ — 4 x^ — 17 ic + 60 = are — 4 and 5 ; what is the other ? 2. Three roots pf a;* - 45x2 -f 40 a; + 84 = ^re 2, 6, and — 7 ; what is the other ? 3. Four roots of ar^ — 4a;* — 5aT'' + 203:^ + 4.'» — 16 = are 1, — 1, — 2, and 4 ; what is the other ? 4. Two roots of a;* + 2a;=' - 13 a;^ - 38 .t - 24 = are - 1 and 4 ; what are the others ? 5. One root of loa.-' + x*^ — 31 a; + 15 = is — - ; what are the others ? 6. Three roots of a;* - 74 ar^ - 24 a;- + 937 a; - 840 = are 1,-7, and 8 ; what are the others ? 7. Two roots of 2 a;" - 13 x" - 91 a;^ + 390 a; + 216 = are 4 and : what arc the others ? 2' THEORY OF EQUATIONS. 457 640. Fractional Roots. An equation in the general form whose coefficients are inte- gral, cannot have as a root a rational fraction (Art. 154) in its lowest terms. Let the equation be where pi, p2, •••■> Pn are integral. If possible, let -, a rational fraction in its lowest terms, h be a root of the equation ; then. Multiplying each term by 6""*, and transposing, 6 By hypothesis, a and h have no common divisor. We then have a rational fraction in its lowest terms eqiial to an integral expression, which is impossible. Therefore the equation cannot have as a root a rational fraction in its lowest terms. 641. Imaginary Roots. If a pure imaginary or complex number (Art. 327) is a root of an equation in the general form with real coefficients, its conjugate (Art. 337) is also a root. Let the equation be X" + pi x"-^ H f- p„_i x-\-p„ = 0, (1 ) where p^, ..., p,^ are real numbers. Let a + ftV— 1, where a and & have the same meanings as in Art. 327, be a root of the equation ; then, (a + bV^iy +Pi(a -f 6V^=l)"-i + ... + P„ i(a + ^V-^) -i-Pn = 0. 458 COLLEGE ALGEBRA. Expanding by the Binomial Theorem, we have by Art. 333, » 17 / ^ n(n — 1) „_o,o n(w-l)(/i-2) 3,3 / — ^ ^^ + Pi[a''-^+(n-l)a"-^6V^- (^'~^H^^-2) ^,.-3^2 -] l£ + - +P„-i (a + W- 1) +1\ = 0. (2) Collecting the real and imaginary terras, we shall have a result of the form where P and Q are real. In order that this equation may hold, we must have P = 0, and Q = 0. Now substituting a — &V— 1 for x in the first nxember of (1), it becomes (a - 6 V^l)" -\-lh{ci - 6 V^T)"-^ +-' + P«-i(a-&V^^)+i\. (3) Expanding the powers of a — 6V— 1, we shall have a result which differs from the first member of (2) only in having the odd terms in each expansion, or those involving V — 1 as a factor, changed in sign. Then, collecting the real and imaginary terms, the expres- sion (3) is equal to where P and Q have the same meanings as before. But since P = and Q = 0, P - Q V^^ = 0. Therefore a — 6V— 1 is a root of (1). THEORY OF EQUATIONS. 459 Note. The product of the factors of the first member of (1), Art. 641, coiTesponding to the conjugate imaginary roots a+ 6\/^^ and a — bV— 1, is [X - (a + i) V^n:)] [x - (a - & V^ ] = (a; - a)2 - (6 V^)- = (x - «)2 + 62 ; and is therefore positive for every real value of x. 642. It follows from Arts. G33 and 641 that every equa- tion of odd degree has at least one real root ; for an equation cannot have an odd number of imaginary roots. TRANSFORMATION OF EQUATIONS. 643. To transform an equation into another which shall have the same roots loith contrary signs. Let the equation be a;" + xh •'^"^ + Ih a^*"~^ H h l>«-i x+p,^ = 0. ( 1 ) Substituting — y for x, we have {-yy+Pi{-yY-^+ih{-yY''-\- ••• +i^«-i(-2/) +p„= o. That is, or, 2/" - Pi 2/""^ + P-'.V""^ ± Pn-i yTPn = Q; (2) the upper or lower signs being taken according as n is odd or even. It follows from (1) and (2) that the desired transforma- tion may be effected by simply changing the signs of the alternate terms beginning with the second. Note. If the equation is incomplete, any missing term must be supplied with the coefficient zero before applying the rule. 460 COLLEGE ALGEBRA. Example. Transform the equation r»^ — 10 ic + 4 = into another which shall have the same roots with contrary signs. The equation may be written a;3 + 0-a;--10x + 4=0. Then by the rule, the transformed equation is jc3_0.a^-10a;-4 = 0, or .t'' - lOx- 4 = 0. -, 644. To transform an equation into another whose roots shall he m times those of the first. Let the equation be x^ +j)ix""^ +^2-'^""^ + ••• + Pn~\^ +Pn = ^- Substituting ^ for x, whence y = mx, we have m (i:)"+^ffrr negative root^. Hence, since the equation has three roots (Art. 633), one of them must be negative and the other two imaginary (Art. 642). Note. If two or more successive terms of an equation are wanting, it follows by Descartes' Rule that the equation must have imaginary roots. EXAMPLES. The roots of the following equations being all real, de- termine their signs : 2. 2a;^-3a.-2- 17a; + 30 = 0. 3. 3ar^- liar- 19a; -5 = 0. 4. a;''-8ar^ + 17a.-2 + ^a;-24 = 0. 5. a;" - 53 ar' + 441 = 0. 6. 4a;^ + 28a;'^ + 39ar'-7a;-10 = 0. 7. ar'-41ar'' + 12a;2 + 292a; + 240 = 0. 8. 3a.-5 - 2x^ - 45ar' + 92a; - 48 = 0. Determine the nature of the roots of the following : 9. a;3_8a;2-12 = 0. 12. a;^ - 4x^- 5 = 0. 10. a;* + 3ar + l = 0. 13. a;-' + 2x'' + 3a;2 + 1 = 0. 11. a;^ + l = 0. 14. a;'' + 2;c--l = 0. (Put ar = ?/.) DIFFERENTIATION. 657. We shall first demonstrate three propositions in rocrard to limits. THEORY OF EQUATIONS. 471 I. The limit of the sum of any number of variables is the sum of their Hunts. Let x', y\ z', ..., be the limits of the variables x, y, z, ... . Then x'—x, y'—y, z'—z, ..., are variables which can be made less than any assigned quantity, however small (Art. 209, Note). Therefore, {x'- x) + {y'- y) + (z'- z')+ ■■■ , or {x'+y'+z'+---)-{x + y+z+---), can be made less than any assigned quantity, however small. Hence, x'+y'+z'-\ is the limit of x + y + z-\ II. The limit of the product of a constant and a variable is the constant midtiplied by the limit of the variable. Let a be a constant, and let x' and x have the same mean- ings as above. Then a{x'—x), or ax'— ax, can be made less than any assigned quantity, however small. Whence, ax' is the limit of ax. III. The limit of the product of any number of variables is the product of their limits. Let x', y', z', ..., and x, y, z, ..., have the same meanings as above, and let x'—x= I, y' — y = 7n, z'—z = n, .... Then I, m, n, ..., are variables which can be made less than any assigned quantity, however small. Now, x'y'z'--- = {x -h I) (y + m){z + n)--- = xyz--- + terms involving I, m, n, ... . Whence, x'y'z' '•• -xyz--- = terms involving I, m, n, ... . (1) The second member of (1) can be made less than any assigned quantity, however small. Therefore, x'y'z' ... is the limit of xyz... . 472 COLLEGE ALGEBRA. 658. Derivatives. Ill any function of x (Art. 213), let x + 7i be substituted for x; subtract from the result the given function, and divide the remainder by h. The limiting value of the result as h approaches the limit zero, is called the derivative of the function luith respect to x. Let it be required, for example, to find the derivative of with respect to x. Substituting x + h for x, and subtracting from the result the given function, we have (x + hy- 2{x + hy+ 5 - (ar* - 2ar^ + 5) = 3 xVi + 3xh^ + h^ - 4 xJi - 2 1i\ Dividing this result by h, we have 3x^ + 3xh + h''-4.x-2h. (1) The limiting value of (1) as h approaches 0, is 3 x^ — 4 a;. Hence the derivative of ar* — 2 x^ + 5 with respect to x is 3 ar^ — 4 a;. . The process exemplified above is called Differentiation. 659. In general, let u represent any function of x ; and suppose that, when x is changed to x + h, u becomes u -\- h'. Then the derivative of u with respect to a; is lim h = nu + h')-u where ^™ is used as an abbreviation for " the limit as h h = approaches zero of." It follows from the above that d lira r/i'l /-i\ where —u stands for the derivative of u with respect to x. dx THEORY OF EQUATIONS 473 660. The process of (iifferentiation is much facilitated by means of the following formulae, in which a and n repre- sent constants, and u, v, w, ..., any functions of x: I. ^x = l. dx TT f^ / . \ d II. — (u-\-a) = — u. dx ' dx TTT d , . d III. — (au) = a — u. dx dx TTT d , , , , . d , d , d IV. — {a + v + w -\- •••) = — w-l v-\ w -\- ■•• • dx dx dx dx V. — iuvvj ■•■) = (vw • • • ) -^ w -f- (uw •••) — V -\ • dx dx dx VI. — (U") = 71U"- ^ M. dx dx VII. — (aa;") = wax"~^ dx 661. In proving the formulae of Art. 660, we shall sup- pose that, when x-\-h is substituted for x, u, v, w, ..., are changed in consequence to m -f h', v -f- h", w + h'", — Proof of I. By Art. 658, f^ o. - lini [ {x + li) — x '^ _ ^ dx h = (^l h \~ e derivative with respect to : [I. — (u + a)= ^™ \ {u-lr^l'-[-a)-{u + a) '^ ^ lim f/i^l ^ ^ ^^ dx^ ' /i = 0|_ /i J /t = o|_/iJ dx ' by Art. 6.59, (1). That is, the derivative with respect to x of a function of x plus a constant is equal to the. derivative of the function of x. For example, -^ {3x- ~5)= — (?^x"). dx dx That is, the derivative with respect to x of x itself is unity. Proof of II. 474 COLLEGE ALGEBRA. Proof of III. d , . lim To (« + /i') — awl lim VaV'~\ d^^'"^=/.^oL — I — j=/.^o[tJ = a X ^^'|!^'_ T-lcArt. 657, II.) = a — u (Art. 659, (1) ). That is, tlie derivative ivith resjiect to x of a constant times a function of x is equal to the constant times the derivative of the function of x. For example, — (3x-) = 3— (x'). dx dx Proof of lY. — (?i + v + z«H ) dx ^ lim r {u + h' + v + h" + iv + h"' + ..-)-{u + v + io + '--) l h=o\_ h J im r h'+h"+h"'+... l RH , lim p"1 , lim R"n, /A,.t h lim 657, I.) d , d , d = — u-\ v-\ w + dx dx dx That is, the derivative with respect to x of the sum of any number of functions of x is equal to the sum of their derivatives. Proof of V. Consider first the case of two factors. (I („„^ _ lim r {u + h'){v + h")-uv '] dx ^''"^ -h = 1 I J ^ lim r uh"+{v + h")h '~\ - h = o\_ h J „^ lim [/*""]. lim r, , /jH^ lira RH by Art. 657, I. and III. THEORY OF EQUATIONS. 475 As h approaches the limit 0, h" also approaches 0, and therefore the limiting value of v ■\- 7i" is v. Whence, — (uv)= u —v +v — u. (1) clx dx clx Consider next the case of three factors. — hivio) = - r (uv) ■ ?«] = IV ^^ (uv) +110-^ 10, by (1) dx dx"- dx dx d -f- uv — w dx f d , d \ \ dx dx J d . d , d = vw — u-\- uio — V + uv -— w. dx dx dx In like manner the theorem may be proved for any num- ber of factors. That is, the derivative with respect to x of the product of any number of functions of x is equal to the sum of the results obtained by multix)lying the derivative of each factor by all the other factors. For example, —[(•x + l)^-] = (.x- + l)-^(arO+ar^-^(aJ+l). dx dx ax Proof of YI. d /...x lim [ {n + h'Y-u- '\ dx^''^--h^^\_ h J Expanding by the Binomial Theorem, and cancelling %", r n 17/ , n(n—l) „_2 7f2 I 1 nu"-^h'-\ 5^-^^ — Lu"- ^h'^+ ••• d , „x lim i- d^.(^) = .^o h Vnn pl!"!^ lim [,,,,.-!+ !L(!iZ^ ,,-2;,,+ lim nn lim r^^,^,. ^Olh] h=0\_ by Art. G57, III 476 COLLEGE ALGEBRA. As ^ approaches the limit 0, h' also approaches 0, and the limiting value of the second expression in brackets is nit""\- Whence, — (li") = nu'^~^ — ii. dx dx For example, ~\_{x^ + 1)^] = 3 (x- + ly^ (x" + 1). dx dx Proof of VII. By III., — (ax") = a— (x«) = anx"-'-^x, by VI., ' dx^ ^ dx^ ^ dx ' ^ ' = a?ia;"~\ by I. That is, the derivative with respect to x of a constant times any power of x is equal to the constant, times the exponent of the power, times x raised to a power whose exponent is less For example, — (3 x') = 12 x;\ dx EXAMPLES. 662. 1. Find the derivative with respect to x of 2x^-5x^ + 7x-6. By II. and IV., Art. 661, — (2a;^ - 5x^ + 7a; -6) = — (2x') - ^- {5x') + — (7a-) dx^ ^ dx^ ^ dx^ ' dx^ ^ = 6x--10a; + 7, by VII. Find the derivatives with respect to x of the following : 2. 3^2-5. 6. 6.'c'' + ar^-12a;2 + 8x-2. 3. 2x'-x' + x. 7. 5x''-2.t'-.9x-'-6x2. • 4. x' -lx'' + 2x' + l. 8. x" -\-8x' + 10x^ -2X + 1. 5. 3.r^ + 5.x-''-llx\ 9. ?,x''-Sx' + x'-iryx-12. THEORY OF EQUATIONS. 477 10. Find the derivative witli respect to x of (.^•- + l)'l ByVI.,f-[(a)^+l)«] = 3(x^+l)^|-(^*^+l) ax nx = 3(a;-+l)-- 2x, by II. and VII., = 6x{x^+iy. 11 . Find tlie derivative with respect to it' of (.c + 1 ) (ar — 2) . = (a; + l)2.^ + (a;2-2) =2af+2x + x--2 = 3a-2+2x-2. Find the derivatives with respect to x of the following : 12. (3a^-l)^ 17. {l-x){2-3x'){l + x'). 13. {ax'+bx + cy. 18. {x + 'i)\x;'-2x + 3). 14. x\x'-3). '19. (x'-2Y(x-^iy. 15. (2x-3)(3x + 4). 20. (4a; + 5)^(4 - 5x)'- 16. {x-\-2){x + 3)(x-5). 21. (aj-+x-l)-(ar-x+l)l 663. Successive Differentiation. If u is any function of x, the derivative of the derivative of u is called the Second Derivative of u ivith resjject to x, and is represented by -— u. dx^ The derivative of the second derivative of u is called the Third Derivative of u loith respect to x, and is represented by . — w; etc. dx" ' 1. Find the successive derivatives with respect to x of 3x«-9a;--12x + 2. We have, — (3 a;^ - 9 a;^ - 12 a; + 2) = 9 x' - 18 x - 12. dx 478 COLLEGE ALGEBRA. . — , (3x^ - Qx" - 12x + 2) = 18x - 18. —ASx"- dx' - 12x + 2) = 18. ^(3af'-9a;2-12a; + 2) = 0; etc. Note. It will be understood hereafter that when we speak of the derivative of a function of x, the first derivative is meant. EXAMPLES, rind the successive derivatives with respect to 03 of : 2. 2a:2_,_^_,_i^ 5^ x'-x^- 3x^+7. 3. of-Sx' + ix. 6. 2a;^ + 9a;^-21x. 4. 3a;^ + 8a.'5-12ar^. 7. 5x^ - Ax' ^Sx" -2. MULTIPLE ROOTS. 664. If an equation has two or more roots equal to a (Art. 633, Note), a is said to be a Multiple Root of the equation. In the above case, a is called a double root, tri2)le root, quadnqyle root, etc., according as the equation has two roots, three roots, four roots, etc., equal to a. 665. Let the equation i^o X" + Pi x"~'^ -\ \-li„-iX+ p„ = (1 ) have m roots equal to a. By Art. 636, the first member can be put in the form (^x-arf{x) (Art. 628); (2) where f{x) is the product of the factors corresponding to the remaining roots of (1), and is therefore an integral expression of the {n — m)th degree with respect to x. THEORY OF EQUATIONS. 479 By Art. 660, V., the derivative of (2) with respect to x is or, {x - a)%{x) -\-m{x- ay-'f{x), by Art. 660, VI. (3) Note. The derivative of f(x) witli respect to x is usually denoted by /i(x). It is evident that the expression (3) is divisible by (x — a)"*"^ ; and therefore the equation formed by equating it to zero will have m — 1 roots equal to a. Hence, if any equation of the form (1) has m roots equal to a, the equation formed by equating to zero the derivative of its first viemher ivill have m — 1 roots equal to a. 666. It follows from Art. 665 that, to determine the existence of multiple roots in an equation of the form . 2hx''+Pi^"'^-\ \-Pn-i^+Pn = 0, we proceed as follows : Find the H.C.F. of the first member and its derivative. If there is no H.C.F., there can be no multiple roots. If there is a H.C.F, by equating it to zero and solving the resulting equation, the required roots may be obtained. The number of times that each root occurs in the given equation exceeds by one the number of times that it occurs in the equation formed from the H.C.F. 1. Find all the roots of af + »"- Gar''- 5ar + 16x4- 12 = 0. (1) The derivative of the first member is 5x^ + 4.^3 -21 X-- lOx + 16. The H.C.F. of this and the first member of (1) is Qi?-x-2. Solving the equation cc-— ic — 2 = 0, we have a; = 2 or —1. 480 COLLEGE ALGEBRA. Hence the multiple roots of the given equation are 2, 2, — 1, and — 1. Dividing 12 by minus the product of 2, 2, — 1, and — 1, the remaining root is — 3 (Art. 639). Therefore the roots of (1) are 2, 2, - 1, - 1, and - 3. EXAMPLES. Find all the roots of each of the following : 2. af5 + 3 0^2-24 3^ + 28 = 0. 3. x''-4:x'-llx-C^ = 0. 4. 8c(^ + 4x'-66x + 63 = 0. 5. a;4 + 6a^ + a^- 24a; + 16 = 0. 6. x* + 7x^ + 9a^-27x-54: = 0. 7. x'-7x'-i-2a^-\-12x-8 = 0. 8. a^''-6r^-28»2 + 120.'c + 288 = 0. ' 667. An equation of the form a;" — a = can have no multiple roots ; for the derivative of x" — a is wa;""^, and a;" — a and nx"'^ have no common factor except unity. Therefore the n roots of a;" = a are all different. It follows from the above that every expression has two different square roots, three different cube roots, and in general n different wth roots. LOCATION OF THE ROOTS. 668. To find a siqjerior limit to the positive roots of the eqaalion f{x) = a;" + 27] .-c"-^ -\ f-p„-i x + p„ = 0. (1) Let p^ be the absolute value of the negative coefficient of greatest absolute value, and x"~' the highest power of x which has a negative coefficient. Then none of the first s terms have negative coefficients. THEORY OF EQUATIONS. 481 Now f{x) will be positive when x is positive, provided X" — j>,.a;""^ — j:»r^"^''~^ — ••• —PrX —p^ (2) is positive ; for f{x) is equal to (2) plus 2JiX^~^ H \- 2)^-1 x"-'+^ + (p, -\-p,) X"" -\ h (Pn +Pr), a positive quantity. We may write (2) in the form .-c" — Pr (a;""' + x"-'-^ H + cc + 1), or, X- _^ x--^+^ - 1 ^^^^ -^^^^ ^3^ cc — 1 Then /(cc) will be positive when x is positive, provided (3) is positive. But if x is greater than unity, (3) is greater than a»-p, -• (4) x — 1 Therefore for values of x greater than unity, f{x) will be positive if (4) is positive; or, if (cc — l)£c"— 2>,.a;"^'+' is positive ; or, if {x — 1) x'-'^ —p, is positive. But if X is greater than unity, {x — 1) x'~'^ —p^ is greater than {x — 1) (cc — 1)'"^ — pr, or (a; — 1) ' — p^ ; hence for values of X greater than unity, f{x) will be positive if {x — iy—p^ is positive or zero; or, if (aj — 1)* is greater than or equal to Pr ; or, if a? — 1 is greater than or equal to '^Pr- That is, for all values of x greater than or equal to 1 + -\fpri /(^') is positive. Hence, no root of (1) can equal or exceed 1 + -{/Pr\ that is, 1 + -^r is a superior limit to the positive roots. Example. Eind a superior limit to the positive roots of CF*- 19x2-460;+ 120 = 0. Here, p^ = 46, and w — s = 2 ; whence, s = 2. Then a superior limit to the positive roots is 1 + V46, or 7.78 approximately. 482 COLLEGE ALGEBRA. Note. In applying the principles of Art. 668, the term independent of X, in the equation must be considered as the coefficient of x^. 669. By changing the signs of the alternate terms of an equation beginning with the second, an equation is formed Avhose roots are those of the first with contrary signs (Art. 643). The superior limit to the positive roots of the trans- formed equation, obtained as in Art. 668, with its sign changed, will evidently be an inferior limit to the negative roots of the given equation. Example. Find an inferior limit to the negative roots of cc^ + S.T^ + lire -13 = 0. (1) Changing the signs of the alternate terms beginning with the second, we have a;* + 5^2-11.^-13 = 0. (2) Here, p^ = 13, and ?i — s = 1 ; whence, s = 3. Then a superior limit to the positive roots of (2) is 1 + Vl3, and therefore an inferior limit to the negative roots of (1) is - (1 + V\3>^. EXAMPLES. 670. Find a superior limit to tlio positive roots, and an inferior limit to the negative roots, of : 1. ic3_2ar'-3a; + 7 = 0. 4. x' -1 x^ -?,^ + ^ = Q). 2. af'- 5x2^.2.^4.9 = 0. 5. u;^ + 6;c-- 3a;- 11 = 0. 3. a;^ + 3.'B3-4a;-10 = 0. 6. a;'* + 2 .x-^ - 5 a.-^ + 7 = 0. 671. Jf two real numbers, a and b, not roots of the equation f(x) = a;" + 2h ^""^ H H Pn^i x+Pn = 0, when substitided for x in f(x), give results of opposite sign, an odd number of roots off{x) — lie between a arid b. THEOKY OF EQUATIONS. 483 Let a be algebraically greater than b. Let d, ..., g be the real roots of f{x) = lying between a and b, and h, ...,lc the remaimng real roots. Then by Art. G36, f(x) = {x-d)...{x-g).{x-h)...ix-k).F{x), (1) where F{x) is the product of the factors corresponding to the imaginary roots of /(.«;) = 0. Substituting a and b for x in (1), we have (Art. 628), f{a) = {a-d)...{a-g)-{a-h):.{a-k).F{a), and f{b) ^(b-d)--{b-g)-{b-h)---{b-k)-F{b). Since each of the quantities d, ..., g is less than (( and greater than b, each of the factors a — d, ..., a — gin posi- tive, and each of the factors b — d, ..., b — g i's, negative. Again, since none of the quantities A, ..., k lie between a and b, the expression {a— h)---(a — k) lias tlie same sign as {b-h)--{b-k). Also, F{a) and F{b) are positive ; for the product of the factors corresponding to a pair of conjugate imaginary roots is positive for every real value of x (Art. 641, Note). But by hypothesis, /(a) and/(&) are of opposite sign. Hence the number of factors b — d, ...,b — g must be odd; that is, an odd number of roots lie between a and b. If the numbers substituted differ by unity, it is evident that the integral part of at least one root is known. 672. 1. Locate the roots of x'^ + ^' — 6x — 7 — 0. By Descartes' Rule (Art. 653), the equation cannot have more than one positive, nor more than two negative roots. The values of the first member for the values 0, 1, 2, 3, — 1, — 2, and — 3 of .x are as follows : a; = 0; -7. .x' = 2; -7. a; = -l; -L a; = -3; -7 x=1; -11. x = 3; IL x = -2; 1. 484 COLLEGE ALGEBRA. Since the sign of the first member is — when x = 2, and -f- when X = 3, one root lies between 2 and 3. The others lie between — 1 and — 2, and — 2 and — 3, respectively. The integral parts of the roots are 2, — 1, and — 2. Note. In locating roots by the above method, first make trial of the numbers 0, 1, 2, etc., continumg the process until the number of positive roots determined is the same as has been previously indi- cated by Descartes' Rule. Thus, in Ex.' 1, the equation cannot have moi'e than one positive root ; and when one has been found to lie between 2 and 3, there is no need of trying 4, or any greater positive number. The work may sometimes be abridged by finding a superior limit to the positive roots, and an inferior limit to the negative roots of the given equation (Arts. 068, 669) ; for no number need be tried which does not fall between these limits. EXAMPLES. Locate the roots of the following equations : ', 2. x^- 5x^+3 = 0. » 4. .'B3^8.'c2-9a;-12 = 0. 3. x'* — 8a;- + 15 = 0. 5. x* — 5x^ + x' -j-13x — 7 = 0^ 6. Prove that the equation x'^ — x- + 2x — l = has at least one root between and 1. 7. Prove that the equation x* — 2x^ — 3 a;- -+- a; — 2 = has a root between — 1 and — 2, and at least one between 2 and 3. 8. Prove that the equation x* + x^-i-2x- — x — 1 = has a root between and 1, and at least one between and — 1. 673. Location of Roots by Synthetic Division (Art. 650). Let Q denote the quotient, and R the remainder obtained by dividing the expression a;" +27i^'" ' H \-P„-iX +Pn (1) hy X — a ; then, X" +2h X" ' + •••+ ;>„ , X + p„ = Q (x - a ) (- 11. (2) THEORY OF EQUATIONS. 485 Putting X = a in (2), we obtaiu a" + 2h rt" "' + ••• +Pn~i « + pH = -K- That is, the value of (1) with a written in place of x is equal to the remainder obtained by dividing (1) by a; — a. It follows from the above, in connection with Art. 671, that if a and b are real numbers, not roots of the equation / (.^•) = X" + /)i x"-^-{ f- p„ _ 1 .1- + p, = 0, and, when f{x) is divided by x — a and x — b, the re- mainders are of opposite sign, an odd number of roots of /(.t) = lie between a and b. The remainders may be obtained by Synthetic Division. 1. Locate the roots oioif-{-x- — ox — A = 0. By Descartes' Rule, the equation cannot have more than one •i^)ositive, nor more than two negative roots. Dividing x'^-j-ay^—5x—4: by x, the remainder is — 4. (o) Dividing the first member successively by x — 1, x — 2, x — S, x+1, ic + 2, and x + 3, we have 1 +1 -5 -4 1-1 (7) - 1 5 -0 ^ 1 +1 _5 _4 1-2 (8) — 2 2 () ^ ^ ~2 1 +1 -5 -4 1_--^ (9) — 3 6 — 3 In (5) and (6), the remainders are —2 and +17, re- spectively ; hence one root lies between 2 and 3. In (3) and (7), the remainders are — 4 and + 1, respec- tively ; hence a root lies between and — 1. In like manner, a root lies between — 2 and — 3. 1 + 1 — 5 -4 LI (4) 1 o -3 2 ^ — 7 1 + 1 — 5 -^[2 (5) 2 6 '> 3 1 « 1 + 1 ^5 -4L3 (G) 3 12 21 4 7 17 486 COLLEGE ALGEBRA. Note 1. The above process is nothing more than a convenient way of applying the test of Art. 671. It, has moreover the advantage over the method of direct substitution that, when the integral part of a root has been found, the work performed is identical with the first part of Horner's method (Art. 718) for determining additional root- figures ; thus, in Ex. 1, the work m (5) is identical with the first three lines of the determination by Horner's method of the root of the given equation lying between 2 and 3. Note 2. The Note to Art. 672 applies with equal force to tne method of Art. 673. EXAMPLES. Locate the roots of the following equations : 2. ar^-5ar + 2a; + 6 = 0. 4. x' -15ar -{-3x + U = 0. 3. ar' + 2ar'-x-l = 0. 5. x* + 6 ar'' - 42 .x - 44 = 0. 6. Prove that the equation a;^ + 3.x — 5 = has one root between 1 and 2. 7. Prove that the equation x* — 3x^ -{- 6x- -{-x — 1 = has one root between and — 1, and at least one root between and 1. 674. The methods of Arts. 672 and 673, though simple in principle and easy of apj^lication, are not sulticient to deal with every problem in location of roots. Let it be ret^uired, for example, to locate the roots of x!^ + 3x^-\-2x + l = 0. We know by Art. 642 that the *^uation has at least one real root. By Descartes' Eule, the equation has no positive root. By Art. 669, — 4 is an inferior limit to the negative roots. Putting X equal to 0, — 1, — 2, and — 3, the correspond- ing values of the first member are 1, 1, 1, and — 5. Therefore the equation has either one root or three roots between —2 and —3; but the methods already given are iusuincicut to determine which. THEORY OF EQUATIOXS. 487 Sturm's Theorem (Art. 675) affords a method for deter- mining completely the number and situation of the real roots of an equation. It is more difficult of application than the methods of Arts. 672 and 673, and should be used only in cases which the latter cannot resolve. 675. Sturm's Theorem. Let /(.^•) = x" -j-2),x''~' + ••• +Pn-i3^+Pn = (1) be an equation from which the multiple roots have been removed (Art. 666). Let/i(x) denote the derivative oi f(x) with respect to x (Art. 658). Dividing /(cc) by /i(a;), we shall obtain a quotient Qi, with a remainder of a degree lower than that of /(.i-). Denote this remainder, with the sign of each of its terms changed, ^J fXx), and divide f{x) hy f{x), and so on; the operation being precisely the same as that of finding the H.C.F. of /(a;) and/i(aj), except that the signs of the terms of each remainder are to be changed, while no other changes of sign are permissible. Since /(a) = has no multiple roots, /(cc) and/i(a;) have no common divisor except unity (Art. Q>QQ)) ; and we shall finally obtain a remainder /^ (a?) independent of x. The expressions f{x), fi{x), fix), ..., fn(x) are called Sturm's Functions. The successive operations are represented as follows : f{^)=QiM^)-f2{x), (2) f{x)=QMx)-f{x), (3) f2(x)=QJsix)-f{x), (4) fnMx)=Qjj„^,(x)-fXx). We may now enunciate Sturm's Theorem. 488 COLLEGE ALGEBRA. If two real numbers, a and b, are substituted in place of x in Sturm/s Functions, and the signs noted, the difference be- tween the number of variations of sign (Art. 653, Note 1) in the first case and that in the second is equal to the number of real roots off{x) = lying between a and b. The demonstration of the theorem depends upon the fol^ lowmg prmciples : I. Tivo consecutive functions cannot both become for the same value of x. For if, for any value of x, f{x) = and fo{x) = 0, then by (3), fsix) = 0; and since f{x) = and f{x) = 0, by (4), f[x) = 0; continuing in this way, we shall finally have f,,{x) = 0. But by hypothesis, /„ (a;) is independent of x, and conse- quently cannot become for any value of x. Hence no two consecutive functions can become for the same value of x. II. If any function, except f{x) and fn(x), becomes. for anji value of x, the adjacent functions have opposite signs for this value of x. For if, for any value of a;, f{o:) = 0, then, by (3), we must have/i(a;) = —fi{x) for this value of x. Therefore fx{x) and f{x) have opposite signs for this value of X ; for, by I., neither of them can equal zero. III. Let c be a root of the equation fr{x) = 0, where /,(.r ) is any function except /(x) and/„(a;). By 11., fr-i{x) and/.+j(£c) have opposite signs when x=c. Now let h be a positive quantity, so taken that no root of fr^i{x) = or /,.^i(.r) = lies between c — h and c + h. Then as x changes from c — h to c + h, no change of sign takes place in fr.i{x) or f+i{x) ; while f{x) reduces to zero, and changes or retains its sign according as the root c occurs an odd or even number of times in/,.(a;)=0. TFIEORY OF EQUATIONS. 489 Therefore, for values of x between c — h and c, and also for values of x between c and c + h, the three functions. fr-i{x), f,.{x), and fr+i{x) present one permanence and one variation. Hence, as x increases from c — k to c -\- h, no change occurs in the number of variations in the functions fr-i{^-), f,i^'), and./^^.i(a;) ; that is, no change occurs in the number of variations as x increases through a root of /.(x) = 0. IV. Let c be a root of the'equation /(.!■) = ; and let /« be a positive quantity, so taken that no root oij\{x) = lies between c — h and c + h. Then as x increases from c — /i to c -f h, no change of sign takes place in fi{x) ; while f{x) reduces to zero, and changes sign. Putting x = c — h in (1), we obtain /(c - k) = (c - hY+p,{c - hy ' + ••• + p„_i(c - h)+p^. Expanding by the Binomial Theorem, and collecting the terms involving like powers of h, we have /(c - h) = c" +;?i c"-^ + • • • + p„ -1 c -h 21. — h[nc" '^ + (n — 1 )29, c"~- + ■■■ + p„_i] + terms involving /i^, /i'', ..., h". But since c is a root of /(«) = 0, we have by (1), c^+iJ^c" ^-] \-Pn-iC+P„ = 0. Also, it is evident that the coefficient of — /* is the value of /i(x) when c is substituted in place of x; therefore, /(c — /i) = — /i/i(c) + terms involving h-, h^, ..., h'\ (5) In like manner it may be shown that /(c + 7i) = -{- /i/j|(c) + terms involving /r, h^, ..., h". (6) Now if h is taken sufficiently small, the signs of the sec- ond members of (5) and (6) will be the same as the signs of their first terms, —hfi(c) and +/'/i(c), respectively. 490 COLLEGE ALGEBRA. Hence, if h is taken sufficiently small, the sign of /(c — h) will be contrary to the sign of /,(c), and the sign of /(c + h) will be the same as the sign of /i(c). Therefore, for values of x between c — h and c, the func- tions f{x) and fi{x) present a variation, and for values of X between c and c-\-h they present a permanence. Hence a variation is lost as x increases through a root of/(a;) = 0. We may now demonstrate Sturm's Theorem ; for as x increases from h to a, supposing a algebraically greater than b, a variation is lost each time that x passes through a root of f{x) = 0, and only then ; for when x passes through a root of f^(x) = 0, where fr{x) is any function except f(x) and/„ (x), no change occurs in the number of variations. Hence the number of variations lost as x increases from & to a is equal to the number of real roots of f{x) = in- cluded between a and b. 676. It is customary, in applications of Sturm's Theorem, to speak of the substitution of an indefinitely great number for X, in an expression, as substituting /y:! for x. The substitution of + co and — oo for x in Sturm's Func- tions determines the number of real roots of f(x) = 0. The substitution of -f oo and for x determines the num- ber of positive real roots, and the substitution of — co .and determines the number of negative real roots. 677. Since Sturm's Theorem determines the number of real roots of an equation, the number of imaginary roots also becomes known (Art. 633). 678. If a sufficiently great number is substituted in place of X in the expression F{x) = 2hx"-\-PiX^-'^-\---- +Pn^iX+p„, the sign of the result will be the same as the sign of its first term, poX". THEORY OF EQUATIONS. 491 It follows from tlie above that : 7/" + c» is substituted in place of x in F{x), the sign of the result is the same as the sign of its first term. Jf —yD is substituted in x)lace of x in F{x), the sign of the result is the same as, or contrary to, the sign of its first term, according as the degree of F{x) is even or odd. 679. In the process of finding /^(a;), fi{x), etc., any positive numerical factor may be omitted or introduced at pleasure ; for the sign of the result is not affected thereby. In this way fractions may be avoided. 680. 1. Determine the number and situation of the real roots of /(x) = a;3 - 63.-2 + 5a; + 13 = 0. Here, f {x) = 3 a;^ - 12 a; + 5. Multiplying f{x) by 3 in order to make its first term divisible by 33;^, we have 3a;2 - 12a; + 5)3ar^ - 18a;2 + 15.a; + ,39(a; - 2 3a;" — 12a;2+ 5a; - 6a;2 + 10a; + 39 - 6a;- + 24a; -10 7) -14a; + 49 - 2a;+ 7 3a;2-12a;+ 5 2 . /,(a;) = 2a;-7. 2a;-7)6ar^-24a; + 10(3a; Qx'-llx - 3a; + 10 2 - 6a; + 20(-3 - 6a; + 21 - 1 .-. /3(a;) = l. 492 collegp: algebra. Substituting -co for x in f{x), fi(x), /^{x), and fs{x), the signs are — , -\-, — , and -\-, respectively (Art. 678) ; sub- stituting for X, the signs are +, +, — , +, respectively; and substituting + a> for x, the signs are all + . Hence the roots of the equation are all real, and two of them are positive and the other negative. We now substitute various numbers to determine the situation of the roots : A^) M^) Mx) fz{^) a; = — CO, — + — + 3 variations. x = -2, — + — + 3 variations. x = -l, + + — + 2 variations. X=Q, + + — -f 2 variations. x = l, + — — + 2 variations. x = 2, + — — + 2 variations. X = 3, + - - + 2 variations. a;=4, + + + + no variation. a = co, + + + + no variation. We then know that the equation has one root between — 1 and — 2, and two roots between 3 and 4. Note 1. In substituting the numbers, it is best to work from in either direction, stopping when the number of variations is the same as has been previously found for -f-co or —go, as the case may be. 2. Determine the number and situation of the real roots of /(.T) = 4«''-2a;-5 = 0. Here, fiix) = 12.'c^ — 2 ; or, Ga;- — 1, omitting the factor 2. ^x^-2x- 5 3 6c^-l)12x--()x-l5{2x 12a;'' -2a; -4«-15 .-. /,(a;) = 4x + 15. THEORY OF EQUATIONS. 493 6x^-1 2 4a; + 15) 12 a;-- 2(30; 12a;- + 45£C -45x- 2 4 -180x- 8(-45 - 180 X- 675 667 .-. /3(a;) = -667. Note 2. The last step in the division may be omitted ; for wc only need to know the sig7i of /j (x) ; and it is evident by inspection, when tne remainder — 45 a; — 2 is obtained, that the sign of /j (x) will be — . /(^) M^) f2{x) f,{x) x = — 00, — + — — 2 variations. cc = 0, — — + — 2 variations. x = l, — + + — 2 variations, a; = 2, + + + — 1 variation. x = oo, + + + — 1 variation. Therefore the equation has a real root between 1 and 2, and two imaginary roots. EXAMPLES. Determine the number and situation of the real roots of ; 3. ar^_4a;2-4a;+12 = 0. 7. a;*- 12a;2 + 12cc- 3 = 0. 4. a;^ + 5a; + 2 = 0. 8. 2a;*- 3a;- + 3.r - 1 = 0. 5. a;3 + 3a;2-9a;-4 = 0. 9. a;*+2ar^-6a;2-8a;+9=0. 6. ar''-4a;2- 10a; + 41 = 0. 10. a;'' + 4a;3 + 3a; + 27 = 0. 681. As X increases from — oo to +go, f{x) and /i(a;) change signs alternately, for they are ahvays unlike in sign just before /(a;) changes sign (Art. 675, IV.); hence, if the roots of /(x-) = and/i(a;) = are all real, a root of /i(a;) =0 lies between every two adjacent roots of /(a;) = 0. P,(-a,h) Y P,(a,h) h\ i „' 1 o a j . ^ \n m\ ' h\ Z, P,{'a,-b) Y' 1 Pi {.a,-h 494 COLLEGE A1>GEBRA. GRAPHICAL REPRESENTATION OF FUNCTIONS. 682, Rectangular Co-ordinates. Let Fi be any point in the plane of tlie lines XX' and YY', intersecting at right angles at 0. Draw Pi 31 perpendicular to XX'. Then 03f and A M are called the rectangular co-ordinates of -b) PjI OM is called the abscissa, and Pi M the ordinate. The lines of reference, XX' and YY', are called the axes of X and Y, respectively, and is called the origin. It is customary to express the fact that the abscissa of a point is a, and its ordinate b, by saying that, for the point in question, x = a and y = b; or, more concisely still, we may refer to the point as "the point (a, 6)," where the first term in the parenthesis is understood to be the abscissa, and the second term the ordinate. 683. If, in the figure of Art. 682, OM=ON=a, and P, 7^4 and P2P3 are drawn perpendicular to XX' so that PiM=P.,N=P^N=PiM=b, the points Pi, Po, P3, and P, will have the same co-ordinates, {a, b). To avoid this ambiguity, abscissas measured to the right of are considered positive, and to the left, negative; and ordinates measured above XX' are considered positive, and beloio, negative. Then the co-ordinates of the points will be as follows : Pi, (a, h) ; P, (-a, h) ; P,^{-a,-h); P„ (a, - b). If a point lies upon XX', its ordinate is zero ; and if it lies upon YY', its abscissa is zero. The co-ordinates of the origin are (0, 0). THEORY OF EQUATIONS. 495 To plot a point when its co-ordinates are given, lay off the abscissa to the right or left of 0, according as it is positive or negative, and then draw a perpendicular, equal in length to the ordinate, above or below XX' according as the ordinate is positive or negative. Thus, to plot the point (—3, 2), lay off 3 units to the left of upon XX', and then erect a perpendicular 2 units in length above XX'. 684. Graph of a Function of x (Art. 213). Consider the function x- — 2x — 3. and put y = x- — 2x — o. If we give any numerical value to x, we may, by aid of the relation y = x' — 2x — 3, calculate a corresponding value for y. • The following are the values of y corresponding to the values 0, 1, 2, 3, 4, - 1, - 2, and - 3 of a: : ^; x = 0, y = -o. E] x= 4, y= 5. B; x = l, 2/ = -4. F; x = -l, y= 0. C; x = 2, y = -3. G; x=-2, y^ 5. D; X = 3, y= 0. H; X = — 3, 2/ = 12. Now let the above be regarded as the co-ordinates of points, and let these points be plotted, as explained in Art. 683. The points will be found to lie on a certain curve, GBE, which is called the Graph of the given func- tion. The point H falls without the limits of the figure. By taking other values for x, the curve may be traced beyond E and G ; and since y increases indefi- nitely with X, the gra])h extends in either direction to an indefinitely great distance from 0. 496 COLLEGE ALGEBRA. By taking values of x between those assumed above, the curve may be located to any desired degree of precision between E and G. 685. Let it be required to construct the graph of Putting y = 2x'^ — ox, we have the following; results : x = (), y= .=1 —4' x = l, y = -i -1 - ! -1- 5 4' -i,y = 1. 3 "2' ^^' 9 4° The graph of the function is the curve AOB. This graph also extends indefinitely beyond A and B. 686. To determine the points where the graph of any function of x intersects XX', we require to know what values of x will make y equal to zero. This may be effected by placing the given function equal to zero, and solving the resulting equation. It follows from this that the graph of the first member of PoX"" -\- pix"-'^ -\ l-i5„-iic+P„ = intersects XX' as many times as the equation has real unequal roots, and that the abscissas of the points of inter- section are the values of the roots. Thus in Art. 684, the graph of cc^ — 2 a; — 3 intersects XX' twice ; at cc = 3 and a; = — 1. > Hence, the equation x^ — 2x — S = has two real roots ; 3 and - 1. THEORY OF EQUATIOXS. • 407 Again, in Art. 685, the graph of 2 x^ — 3 a; intersects XX' '3 three times ; once between x = l and x=-, once between 3 x = — l and x = , and once at x = 0. Hence, the equation 2cc^ — 3^ = has three real roots; one between 1 and -, one between — 1 and — '-, and one 2 2 equal to zero. If the graph does not intersect XX' at all, the equation formed by placing the function equal to zero has only imaginary roots. 687. The principles of Art. 686 may be used to locate the roots of an equation. For if, in the graph of f{x), the points Avhose abscissas are a and b, respectively, lie on opposite sides of XX', an odd number of roots of /(x) = lie between a and b (Art. 671). The method is simply a graphical representation of the process of Art. 672, and is subject to the limitations stated in Art. 674. EXAMPLES. 688. Plot the graphs of the following functions : 1. 3a; -5. 3. x^-1. 5. a;'' -7a; + 4. 2. a;- + 4a; + 4. 4. x'-4:x' + 2. 6. a.-^ - 9 a;'^ + 23 a; -13. Locate the roots of the following equations : 7. a;3 + 4x--7 = 0. 9. a;'' + 9a;2 + 23a; + 17 = 0. 8. x^-6x' + 7x + 3 = 0. 10. x' + x'-Sx'-x + l^O. 498 COLLEGE ALGEBRA. XLII. SOLUTION OP HIGHER EQUATIONS. COMMENSURABLE ROOTS. Note. We shall use the term commensurable root, in Chap. XLII., to signify a rational root (Art. 209) expressed ni Arabic numerals. 689. By Art. 040, an equation of the nt\\ degree in its general form (Art. 626), with integral numerical coeffi- cients, cannot have as a root a rational fraction in its lowest terms. Therefore, to find all the commensurable roots of such an equation, we have only to find all its integral roots. Again, by Art. 638, the last term of an equation of the above form is divisible by every integral root. Hence, to find alj the commensurable roots, we have only to ascertain by trial which integral divisors of the last term are roots of the equation. I The trial may be made in three ways : V^ I^ By actual substitution of the supposed root. II. By dividing the first member of the equation by the unknown quantity minus the supposed root (Art. 629) ; in this case, the operation may be conveniently performed by Synthetic Division (Art. 650). III. By the Method of Divisors (Art. 091). In the case of small numbers, such as ±1, the first method may be the most convenient. The second method has the advantage that, when a root has been found, the process gives at once' the depressed equation (Art. 635) for obtain- ing the remaining roots. If the number of divisors is large, the third method will be found to involve the least work. Considerable work may sometimes be saved by finding a superior liinit to the positive roots, and an inferior limit to the negative roots (Arts. 668, 669) ; for no number need bo tricid which does not fall between these limits. SOLUTION OF HIGHER EQUATIONS. 499 Descartes' Eule of Signs (Art. 653) may also be advan- tageously employed to shorten the process. Any multiple root should be removed (Art. 666) before applying either method. Example. Find all the roots of a;* — 15 a;- -|- 10 a: -|- 24 = 0. By Descartes' Rule, the equation cannot have more than two positive, nor more than two negative roots. By Arts. 668 and 669, 1 + Vl5 is a superior limit to the positive roots, and —(1 +Vl5) is an inferior limit to the negative roots. The integral divisors of 24 lying between 1 -|- Vl5 and — (1 + Vi5) are ±1, ±2, ±3, and ± 4. By actual substitution, we find that 1 is not, and that — 1 is, a root of the equation. Dividing the first member by a; — 2, a; — 3, etc. (Art. 650), we have 1-+0 -15 +10 +24 [2 1 +0 -15 +10 +24 |_3 2 4-22-24 3 9-18-24 2-11-12, Rem. ~~3 ^^ ^I~8, Rem. The work shows that 2 and 3 are roots of the given equation ; and since the equation cannot have more than two positive roots, these are the only positive roots. The remaining root may be found by dividing 24 by the product of — 1, 2, and 3 (Art. 639), or by the same process as above. Dividing the first member by a; + 2, a; + 3, etc., we have 1+0-15+10+24 1^-2 1+0-15+10+24 \^-^ -2 4 _22 -64 - 4 16-4-24 -2^11~~32l^ 1^4 i 6 1+0-15+10+24 1 -3 -3 9 18 - 84 Zs ^^ ^8 ^^ 600 COLLEGE ALGEBRA. The work shows that the remaining root is — 4. Thus, the four roots of the given equation are — 1, 2, 3, and — 4. 690. By Art. 645, an equation of the nth degree in its general form, with fractional coefficients, may be trans- formed into another whose coefficients are integral, that of the first term being unity. , The commensurable roots of the transformed equation may then be found as in the preceding article. Examj)le. Find all the roots of 4ar^ - 12. t^ + 27a; - 19 = 0. Dividing through by the coefficient of x', we have x^ — 3 X- H = 0. 4 4 Proceeding as in Art. 645, it is evident by inspection that the multiplier 2 will remove the fractional coefficients ; thus the transformed equation is a:3_2.3a;2 + 22.?^-23.— = 0, 4 4 or, ar"^- 6x^4- 27a;- 38 = 0; (1) whose roots are those of the given equation multiplied by 2. By Descartes' Rule, equation (1) has no negative root. The positive integral divisors of 38 are 1, 2, 19, and 38. Dividing the first member by a; — 1, a; — 2, etc., we have 1 _ 6 + 27 - 38 |JL_ 1 - 6 + 27 - 38 |_2^ 1 - 5 22 2-8 ^8 ^ "^ ^16 _4 19 The work shows that 2 is a root of (1). The remaining roots may now be found by depressing the equation; it is evident from the right-hand operation above that the depressed equation is x^ — 4 x + 19 = 0. SOLUTION OF HIGHER EQUx\TTONS. 501 Solving this by the rules for quadratics, we have a; = 2 ± V4 - 19 = 2 ± V-lo. Thus the three roots of (1) are 2 and 2 ±V- 15. Dividing by 2, the roots of the given equation are 1 and l±j V^^IS. 691. Newton's Method of Divisors. If a is an integral root of the equation x" + PiX"-^ -\ h ;?„_^ x^ + p^ J ic + p„ = 0, where 2?!, ...,p„ are integers, then a" +pia"-' H \-Pn-2a- +i>,.-i« +Pn = 0. Transposing, and dividing by a, Ph - =-p„-i-i>„-2« Pja"-2-a»-i; (1) Pi from which it is seen that -" must be an integer. We may write (1) in the form Pn Pi a'' Representing -^ +Pn-i by q„ j, and dividing by a, Qn-i from which it is seen that -^ must be an integer. Proceeding in this way, it is evident that, if a is a root of -^+Pn~2 „ the equation, each of the quantities or ^^^, .... — +lh or - , must be an integer, and — + 1 must equal 0. 502 COLLEGE ALGEBRA. We then have the following rule : Divide the last term of the equation by one of its integral divisors, and to the quotient add the coefficient of x. Divide the result by the same divisor, and, if the quotient is an integer, add to it the coefficient ofa^. Proceed in this manner with each coefficient in succession ; then, if the divisor is a root of the equation, each quotient will be integral, and the last quotient added to unity will equal zero. If a fractional quotient is obtained at any stage, the corresponding divisor is not a root of the equation. Example. Find all the roots of aj* — or^ — 7 x^ + a; + 6 = 0. By Descartes' Rule, the equation cannot have more than two positive, nor more than two negative roots. The integral divisors of 6 are ±1, ±2, ±3, and ± 6. By actual substitution, we find that 1 and — 1 are roots. We will next ascertain if 2 is a root; a convenient arrangement of the work is shown below : . 1 _1 _7 +1 +6 \2_ 2 3 -5 4 The operation is carried out as follo^\^s : Dividing 6 by 2, gives 3 ; adding 1, gives 4. Dividing 4 by 2, gives 2 ; adding — 7, gives — 5. Dividing — 5 by 2, the quotient is fractional ; therefore 2 is not a root. l-l-7+l-f-G|_3 1_1_7+14-G |_-2 --1 -2 1 2 -13 1-3 ~~0 -3 -6 3 ~0 2 -6 -2 In these cases, each quotient is integral, and the last quotient added to unity gives 0; therefore 3 and —2 are roots. SOLUTION OF HIGHER EQUATIONS. 503 Thus, tlie four roots of the given equation are 1, —1, 3, and - 2. Note. There is no necessity for trying + 6, in the above example, for we know that the equation cannot have more than two positive roots. EXAMPLES. 692. Find all the commensurable roots of each of the following equations, and the remaining roots when possible by methods already given: 1. a;^-8x-2 + 19i(;-12 = 0. 4. 2a.-3 + a;2_ 23x + 20 = 0. 2. a;3-31a;-30 = 0. 5. x" -7x- ~Ux + 'i8 = 0. 3. x'' + 5x'-6x-24: = 0. 6. 3a;3 + 2a;- - 3x - 2 = 0. 7. a;'' + 2.r^-7cc--8a: + 12 = 0. 8. x' + 6x^ + x--2ix-20 = 0. 9. 4:X*-12x' + Sx^-\-13x~6 = 0. 10. x* + llx^-\-4=lx''-\-61x + 30 = 0. 11 . x* + x^ - 31 a-- + 71 x - 42 = 0. 12. 4=x* - 31 x' + 21 a; -f 18 = 0. 13. a;*-lla;« + 35a;--13a;-60 = 0. 14. x^ + 14 af^ - 6 .T- + 45 x - 54 = 0. 15. 9x*-16x^-3x + 4: = 0. 16. a;''-7x'^ + 15x2-a;-24 = 0. RECIPROCAL OR RECURRING EQUATIONS. 693. A Reciprocal Equation is one such that if any quan- tity is a root of the equation, its reciprocal is also a root. It follows from the above that, if - is substituted for x X in a reciprocal equation, the transformed equation will have the same roots as the given equation. 504 COLLEGE ALGEBRA. 694. Let be a reciprocal equation. Putting - in place of x, the equation becomes X aj" x"~^ x"-^ ^ X '■ ' or, by clearing of fractions, and reversing the order of the terms, _?)„a;" +l)„-ix"-^ +P„-2-'c""^ ^ VV-i^ -Vlh^ +Po = 0. (2) By Art. 693, this equation has the same roots as (1), an& hence the following relations must hold between the coefficients of (1) and (2): Po = ± P„, i\ = ± i>,, ^1, P2 = ± Pn-1^ etc. ; or, in general, Pr = ± Pn-r ; all the upper signs, or all the lower signs, being taken together. We may then have four varieties of reciprocal equations : 1. Degree odd, and coefficients of terms equally distant from the extremes of the first member equal in absolute value and of like sign ; as, a^ — 2a;^ — 2 a; + 1 = 0. 2. Degree odd, and coefficients of terms equally distant from the extremes of the first member equal in absolute value and of opposite sign ; as, 3 a^ + 2 cc* — cc'^ + x- — 2 cc — 3 = 0. 3. Degree even, and coefficients of terms equally distant from the extremes of the first member equal in absolute value and of like sign ; as, x^ — 5 x^ + 6 a;^ — 5 a; + 1 = 0. 4. Degree even, and coefficients of terms equally distant from the extremes of the first member equal in absolute value and of opposite sign, and middle term wanting; as, 2x'' + 3a;'^ - 7.1;^ + 7a;^ - 3a; - 2 = 0. On account of the pro]ierties stated above, reciprocal ('r|uations are also ml led RRcurring Efjtiations. SOLUTION OF HIGHER EQUATIONS. 505 695. Every reciprocal equation of the first variety may be written in the form Po^^" +Pl.^*""^ +P2^""^ H \-lh^'+PiX + Po = 0, or, i5o(^" + t) +i^i^(x''-' + l) +ij,x\x" ' + l)+- = ; . (1) the number of terms being even. By Art. 113, since n is odd, each of the expressions x" + 1, X" - + 1, etc., is divisible by a; + 1. Therefore — 1 is a root of the equation. Dividing the first member of (1) by x + 1, the depressed equation is Po(^" ' ' — cc" - + a" '^ +x- — x -{-1) + p,x (x"-^ - a;"-^ 4- .^■""^ {-x' — x + l) -\- 2h^-{x''-^ — cc"-" + x"^'' — • • • + a;- — X + 1) + • • • = 0, or, po^^""' + {Pi -i^o) a;"-- + (p. -Pi -i-Po) x""-'^ + ••• + ( Vi - Pi + i^o) X- + (Pi- ih) x + po = ; which is a reciprocal equation of the third variety. 696. Every reciprocal equation of the second va,riety may be written in the form l^oX" +2hx''-' +2hx''-^ -\ (••• +P2X^ +PiX +po) = 0, or, 2,^{x"-l)+2hx{x^-'-l)+P2x\x'^-*-l)-\---'=0. (1) Since each of the expressions x" — 1, x"''- — 1, etc., is divisible by ic — 1, + 1 is a root of the equation. Dividing the first member of (1) by a; — 1, the depressed equation is PoX''^^ + {Pi + Po) x"-- + (P2+ Pi +Po) x"-^ + ... + {Pi +Pi +Po) X- + (p, +Po) X +po = 0; which is a reciprocal equation of the third variety. 506 COLLEGE ALGEBRA. 697. Every reciprocal equation of the fourth variety may be written in the form PoX^+PiX''-^+2hx"^^-\ (■'•+P2X^+PyX+Po) = 0, or, Po{x--l)+PiX{x'^-'-l)+p,x''(x--'-l)-\-... = 0; (1) the number of terms being even (Art. 694). Since each of the expressions x" — 1, x"~^ — 1, etc., is divisible by x^— 1, both 1 and — 1 are roots of the equation. Dividing the first member of (1) by x^ — 1, the depressed equation is Po{x'^~' + x"-" + ... -f a;* + a;- + 1) +PiX {x--* + x--' + ... +x' + x' + 1) +P2x'{x- ' + x--' Jr-+x' + x' + 1) -^ ... = 0, or, Pq a;"-2 + j^i x"'^ + (p. + Po) ^" " + • • • + (P2 +Po) ^" + Pi a^ + Po = ; which is a reciprocal equation of the third variety, 698. Every reciprocal equation of the third variety may be reduced to an equation of half its degree. Let the equation be Poa^^+Pia^""^-! +i'm^"'H +Pia;+po = 0. Dividing through by x" the equation may be written Put .+1 = X = y- Then, --1- -h ^J- 2 = 2/— 2; --1- =(- ^)e '^^-i: 4) = ui>/' -2)- -y = f-^ ?/; SOLUTION OF HIGHER EQUATIONS. 507 The general law is expressed by an expression of the rth degree with respect to y. Substituting these values in (1), the equation becomes go 2/"* + (iiv"'^'^ + Q^y'"'- + ••• = 0. 699. It follows from Arts. 695 to 698 that any reciprocal equation of the degree 2 m + 1, and any reciprocal equation of the fourth variety of the degree 2 m + 2, can always be reduced to an equation of the mth degree. 700. 1. Solve 2.x'^- 5a;*- 13ar' + 13iK2 + 5x- 2 = 0. The equation being of the second variety, one root is 1 (Art. 696). Dividing by a; — 1, the depressed equation is 2a;* - 3x3 - 16x- - 3x + 2 = ; a, reciprocal equation of the third variety. Dividing by x", 2 fx' -\- ^}j - sfx + -\ - 16 = 0. Putting x-\-- = y, and x^ + — = ?/2 _ 2 (Art. 698), we have 2(/-2)-3^-16 = 0. Solving this equation, y = 4 or — -• Taking the first value, x + - = 4, ox x- — 4x = — 1. X Whence, x- = 2 ± Vo. 608 COLLEGE ALGEBRA. 1 5 Taking the second value, cc + - = , ot 2x^ + 5x = — 2. X 2 Whence, a; = — 2 or 2 Thus the roots of the given equation are 1, — 2, — -, and 2 ± V3. ^' Note. That 2 + VS and 2 — V3 are reciprocals may be shown by multiplymg them together ; thus, (2 + VS) (2 — VS) = 4 — 3 = 1. EXAMPLES. Solve the following equations : 2. C)x^-7x''-7x-\-6 = 0. 4. 5 a;^ -|- 26 .^•3 - 26 .t - 5 = 0. •3. x^ + ox^ — ox — l=0. 5. ar* — aar + «x — 1 = 0. 6. 45a;^-48ar'-250ar'-48a; + 45 = 0. 7. .^'^-29a,•3 + 29a;^-l = 0. 8. x^ + 7x* + x'+x^ + 7x + l = 0. 9. 24 a.-^ - 34 a;* - 67 ar^ + 67 a^ + 34 X- 24 = 0. 10. 3ar' + 16a;'' + 29ar'' + 29a;2+16a; + 3 = 0. 11. 4a;«-29a:^ + 55a;''-55.'c2 + 29a;-4 = 0. 701. Binomial Equations. A Binomial Equation is an equation of the form a;" = a. Binomial equations are also reciprocal equations, and, in certain cases, may be solved by the method of the preced- ing article. 702. Putting a; = ay, the equation x" = ± a" becomes 2/" = ± 1 ; Avhich is a form to which every binomial equa- tion may be reduced. In Arts. 351 and 358, methods were given for the solution of the V)in()iuial equations x^ = ± 1, a;* = ± 1, and a:" = ± 1. SOLUTION OF HIGHER EQUATIONS. 509 The forms a^ = ± 1 are readily solved by the method of Art. 700. Binomial equations of any degree may be solved by a method involving Trigonometry, EXAMPLES. Solve the following equations : 1. x' = l. 2. af = -l. 3.03^ = 32. (Putx = 2^.) 703. The Cube Roots of TJnity. By Ex. 3, Art. 351, the roots of the equation a;'' = 1 are 1, -1±VE?, and -^-^^ . 2 ' 2 The third root is the square of the second ; for 14.V_3 Y_ 1-2V-3-3 ^ -l-V-3 4 2 (-^¥-^J Hence, if the second root is denoted by w, the three cube roots of unity are 1, w, and or. 3 1 Or, since w^ = 1, they are 1, w, and — or -• (O to 704. If the second root is denoted by a, the three roots are a, aw, and aw- ; for these are respectively equal to w, w^, and co'' or 1. In like manner, if the third root is denoted by a, the three roots are a, aw, and awl Hence, if either of the cube roots of a quantity is denoted by a, the other two roots are aw and aw-. CUBIC EQUATIONS, 705. A Cubic Equation is an equation of the third degree (Art. 179), containing but one unknown quantity. 510 COLLEGE ALGEBRA. 706. By Art. 652, the cubic equation x^ + pi X- + P2 ^ -\- Pz = 0) where pi is not zero, may be transformed into another whose second term shall be wanting, by substituting y —■— in place of x. Therefore, every cubic equation caii be reduced to the form ar* + rtx + & = 0. 707. Cardan's Method for the Solution of Cubics. Let it be required to solve the equation x' + aa; + & = 0. Putting x = y + z, the equation becomes f + oyz{y + 2) + r + a(2/ + 2) + & = 0, or, f + z' + {^yz-\-a){y + z) + h = 0. We may take y and z in such a way that 3^2 + a shall be equal to zero ; whence, Then, 2/^ + 2^ + 5 = 0. (2) Substituting in this the value of z from (1), we have 3 _ _a^ + & = 0, or y^J^hf=^—- This is an equation in the quadratic form (Art. 3G3) ; solving by the rules for quadratics, we have Then by (2), ^ = - f - h = -^^-^ yj^^- (4) Since x = y + z, the values of x corresponding to the upper and lower signs in (3) and (4) will evidently be the same. SOLUTION OF HIGHER EQUATIONS. 511 Therefore, ^•=xI(-IWM)s'(-2-nR> ^^> The remaining roots may be found by depressing the equa- tion (Art. 635), or by the method explained in Art. 710. 708. From (1), Art. 707, we derive the following rule: To solve a cubic equation of the form x^ -\-ax + b = 0, substitute y fo7- x. 709. 1 . Solve the equation ar' + 3 a;- — 6 x + 20 = 0. We first transform the equation into another whose second term shall be wanting ; putting x = y — 1 (Art. 706), we have f - 3y2 + 3y - 1 ^ 3f' - Gy + 3 - 6y -[- 6 -\- 20 = 0, or, y'-9y + 2S = 0. The latter equation may now be solved by putting 3 y = z + - (Art. 708); or, by substituting a = — 9 and 6 = 28 in (5), Art. 707. Using the second method, we have y = ^/_U-hVi96-27 + ^-14-Vl96-27 = ^^ri+i/Zr27 = -l-3 = -A. Therefore, x = y — 1 = — 5. Dividing the first member of the given equation by x -f- 5, the depressed equation is x--2x + 4: = 0. Solving, we have x =1 ± V— 3. Thus the roots of the given equation are — 5 and 512 COLLEGE ALGEBRA. EXAMPLES. Solve the following equations : 2. X-3 + 15a; + 124 = 0. 7. ar'^ -f- a;^ - 33 a; + 63 = 0. 3. ar^- 27a;- 54 = 0. 8. a;'^ + 12a;-' + 57a; + 74 = 0, 4. a;'5 + 105a;-218 = 0. 9. ar^ - 4x2- 11a;- 6 = 0. 5. ar^_ 6x2- 33a; -70=0. ^q. .r^- 2.t2 + 3 = 0. 6. a;3-9a;2 + 63a; + 73 = 0. 11. .^3 + a.-^- 7a; - 52 = 0. 12. Find one root of .r'' — .x + 1 = 0. Note. A cubic equation having a commensurable root is solved more easily by the method of Art. 689 than by Cardan's rule. 710. If h is any one of the cube roots of \- a/— + — , and Ti any one of the cube roots of \\ — |- — , the ^ 2 \4 27 three cube roots of the first expression are h, hw, and hw,% and the three cube roots of the second are k, kw, and kw^ (Art. 704). This would apparently indicate that x has nine different values. But by (1), Art. 707, yz = - - ; that is, the product of o Hence the only possible values of x are h + k, ho) + kw^, and har + yfcw ; for in each of these the product of the terms is hk ; that is, 3 IP P oF a \ , or — ; while in any other case the product \4 4 27 3' -^ ^ is either — -la or — - w. 3 o SOLUTION OF HIGHER EQUATIONS. 513 Putting for w and w' their values (Art. 703), tlie second and third values of x become and hl ~'^~y~''^ ]+k 1^-). Hence the three values of x are li + k, _^r_ + ___V-3, and ^ Y^^~^- Thus in Ex. 1, Art. 709, 7i = - 1 and k = - 3. Then the values of y are - 4, 2 + V^^, and 2 - V^3. EXAMPLES. Solve the following equations : 1. a^ + 6a; + 2 = 0. 2. af' + 9rc-6 = 0. 711. Discussion of the Roots. «-^(-2W!-l)-"=^'(-|-VM/ the roots of af^ + aa; -f- 6 = are fe + fc, and -^*±'l^V^^ (Art. 710). 2 ^ 1. If a is positive, or if a is ne-;ative and — numerically i.'ss than — , /t and k are real and unequal. Therefore one root is real, and the other two imaginary. 2. If a is negative, and ^ numerically equal to p li and fc are real and equal, and Ji — k is zero. Hence the roots are all real, and two of them are equal. 514 COI.LEGE ALGEBRA. 3. If a is negative, and — numerically greater than — , Ji i 4 the values of h and A; involve imaginary expressions. In this case, h must have some value of the form h'-\-hW^-i, where A' a^d k' are real (Art. 627) ; that is, Raising both members to the third power, Equating the real and imaginary parts (Art. 338), ^. = h''-3h'k'\ 9 and J^ + 1^ = oh'-k'V^l - k'W- 1. 4 27 Subtracting, _ ^ _^1' + |- = /i'^-37i'2fc' V^- 3/i'A;'2+ fc'^V^. Extracting the cube root, k = h'-k'V^l. (2) From (1) and (2), h + k = 2h', and h-k = 2 k'V^^. Then the three roots are 27i' and — h'± k'V— iV— 3. That is, 27i' and -h'^ k'VS. Therefore the roots are real and unequal. In the above case, Cardan's method is of no practical value ; for since there is no method in Algebra for finding the cube root of an expression which is in .the form of a rational expression plus a quadratic surd (Art. 291), the values of h and k cannot be found. In this case, which is called the Irreducible Case, Cardan's method is said to fail. SOLUTION OF HIGHER EQUATIONS. 515 It is possible, in cases wliere Cardan's method fails, to find the roots by Trigonometry ; but in practice it is easier to lind them by Art. 689, or by Horner's method (Art, 718), accord- ing as the equation has or has not a commensurable root. 712. Consider the equation x' + ax- + bx + c = 0. Putting x = y — -, the equation becomes o , , a-y a^ , .-, 2a-y , a* , , ab , ^ ^ Multiplying the roots by 3 (Art. 644), the equation becomes f + 3{3b~a-)y-\-2cc'-9ab-{-27c = 0. Then it follows from Art. 711 that * 1. If 3b — a^ is positive, or if 36 — cr is negative and 4(3& — a2) 3 numerically less than {2a^ — 9 ab -{-27 c)-, the given cubic has one real and two imaginary roots. 2. If 3& — a^ is negative, and 4(3& — a-y is numerically equal to {2a^ — 9ab + 27c)^, the roots are all real, and two of them are equal. 3. If 3b — a- is negative and 4(3& — a-)' numerically greater than (2a^ — 9ab + 27c)-, the roots are all real and itnequal. BIQUADRATIC EQUATIONS. 713. A Biquadratic Equation is an equation of the fourth degree (Art. 179), containing but one unknoAvn quantity. 714. Euler's Method for the Solution of Biquadratics. By Art. 652, every biquadratic equation can be reduced to the form x^+ox- + b.v-\-c = 0. (1) 516 COLLEGE ALGEBKA. Let xi=u + y-\-z; then, x'==ir + t/ + z' + 2uy + 2yz + 2zu, or, x' - (u- + y' + 2') = 2 {uy + yz + zu). Squaring both members, we have x^ — 2ar{v? + y-+ z-) -\- (u^ + 2/^ + zy= 4:(uy + yz -{- zuY = 4(rty + y-z'^ + zhi?) + 8 uyz{u + y+z). Substitiiting x for it.+ ?/ + ^, and transposing, x'^ — 2af{u^ + ?/2 + 2;-) — 8 uyzx + (m2+ ?/2+ 2-)2 - 4(My + 2/V + zhv") = 0. This equation will be identical with (1) provided a = -2{it' + y' + z'), b=-8mjz, (2) and c = ( u- + r +*") " - H^y- + y-z^ + 2^%^) . Therefore, w" + 2/' + 2' = ~ «> "^ 64' and uY~ + 2/^.^ + .V = (lL±l±fr^ 4 a - — 4c 4 16 If, now, we form the cubic equation ^3 _ (,^2 _|. 2/2 + z-)t- + (?ty + y-z- + ^-u2)« - ir^/V = 0, the values of t will be u% /, and z- (Art. 637). Hence, if the roots of the cubic equation are I, m, and n, -we shall have u=± Vl >l ^ ± Vm, and z = ± Vn. SOLUTION OF HIGHER EQUATIONS. 517 NoAv x=: u + y -\-z; and since each of the quantities u, y, and z has two vakies, apparently x has eight vakies. But by (2), the product of the three terms whose sum is a vakie of x must be equal to — -• o Hence the only values of x are, when h is positive, — V^ — Vm — V/*, — V^ + V«t 4- V?i, V^ — Vm + Vrt, and VZ -f V»i — Vh; and when h is negative, V^ + Vm + Vn, vT — Vwi — V^i, — V^ + V»i — V», and — v'/ — V»i + V». Equation (3) is called the auxiliary cubic of (1). 715. 1. Solve the equation x^ - 46 aj- - 24 x + 21 = 0. Here, a = — 46, 6 = — 24, and c = 21 ; whence, ^^'-^^=127, and ^ = 9. 16 64 Then the auxiliary cubic is f — 23 «- + 127 « - 9 = 0. By the method of Art.. 689, one value of t is 9. Dividing the first member by ^ — 9, the depressed equii/- tion is «- — 14^ + 1 =0. Solving, we obtain t = l ± V49 — 1 = 7 ± 4 V3. Proceeding as in Art. 313, we have V(7 ± 4 V3) = V(4 ± 2 Vl2 + 3) = 2 ± V3. Then since h is negative, the four values of x are 3 + 2+V3 + 2-V3, 3-2-V3-2+V3, -3 + 2+V3-2+V3, and _ 3 - 2- V3 + 2- Va That is, 7, — 1, - 3 + 2 V3, and - 3 - 2 V3. 518 COLLEGE ALGEBRA. EXAMPLES. Solve the following equations : 2. a;4-42a;2 + 64aj+105 = 0. 3. a;" - 54 tc-- 24 a; + 77=0. 4. a;^-76a^- 16a; + 896 = 0. 5. .r^-10a;2 + 20.x -16 = 0. 6. x'- 36 x^ + 16 X- + 195 = 0. 7. a-" + 4x3 + 3a;- -44a; -84 = 0. 716. Discussion of the Roots. The auxiliary cubic of x^ + ax- + &x + c = is ^3 a^2 o^^-4c 6^^^ (Art. 714). 2 16 64 ^ ^ Since the last term is essentially negative, the equation must have either three positive, one positive and two nega- tive, or one positive and two imaginary roots (Art. 637). Multiplying the roots by 4 (Art. 644), the equation becomes e-\-2at- + {a--4:c)t-b^ = 0. Denoting 2a, a^ — 4c, and — h^ by a', h', mid c', we have 36'- a'- = 3(a- - 4c) - 4a- = - (a- + 12c), and 2a"^- 9a'b'+ 27c'= 16 a" - 18 a (a^ - 4c) - 276- = - (2 a'^ - 72 ac + 27 6-). Then it follows from Art. 712 that : 1. If a^ + 12c is negative, or if a^ + 12c is positive and 4(a- + 12c)'^ less than (2a»- 72 ac +276^)2, the auxiliary cubic has one positive and two imaginary roots. If V^ = p, Vm = g + rV— 1, and Vh = Q — rV— 1, the roots of the biquadratic are — p ± 2q and p ± 2r V— 1, or p±2q and — p ±2rV — 1, according as b is positive or negative. SOLUTION OF HIGHER EQUATIONS. 51? That is, the biquadratic has two real and two imagioary roots. 2. If a- + 12c is positive, and ^{0^ -\-12cy is equal to (2a^ — 72 ac + 27 b^y, the cubic has two roots equal. If Vw = V»i, the roots of the biquadratic are — Vl ±2 Vm, VT, and V^ or ■\/T± 2 Vm, — VT, and — V^ according as b is positive or negative. That is, the biquadratic has two roots equal. 3. If a^+12c is positive and 4(a^ + 12 c)'^ greater than (2a^— 72 ac-\- 27 b^y, the cubic has either three positive, or one positive and two negative roots. In the first case, the roots of the biquadratic are all real; in the second case, they are all imaginary. 4. If a2+12c=0 and 2a^-72ac+27b''=0, then c = -^. Substituting from the third equation in the second, 8 a^ + 27b^ = 0, or a = - J 6^ ; whence, a--4c = — = 3 b\ In this case, the auxiliary cubic becomes 16 64 ' V 4y b^ 4 and each of its roots is equal to The roots of the biquadratic are —- — , — , — , and — , or 2 2 2 2 3&^ b^ b^ 1 b^^ T , . , , , and — — , according as is positive or negative ; that is, the biquadratic has three roots equal. 5. If a^ — 4c = and & = 0, the biquadratic becomes x'^ + ax- + ^' = 0, or (x" + "Y = 0, and its roots are ±-v/— - and ± a/— -• That is, the biquadratic has two pairs of equal roots. 520 COELEGE ALGEBRA. INCOMMENSURABLE ROOTS. 717. We will now show how to find the approximate numerical values of those roots of an equation which are not commensurable (Art. 689). 718. Horner's Method of Approximation. ' 'w- u q, Let it be required to find the approximate value of the root between 3 and 4 of the equation {^^-^ }^ - ar^_3x'2-2a;-f 5 = 0. Diminishing the roots of the given equation by 3, by the method explained in Art. 650, we have 1 _3 -2 +5 [3_ 3 0-6 -2 -1 3 9 3 7 3 6 The transformed equation is ^/'^ + 6?/^ + 7?/ — 1 = 0. (1) This equation is known to have a root between and 1 ; if, then, we neglect the terms involving ?/^ and y^, we may obtain an approximate value of y by solving the equation 7?/ — 1 = ; thus, approximately, y = .\ and x = 3.1. Diminishing the roots of (1) by .1, we have ■6 +7 .1 .61 -1 L .761 6.1 7.61 .1 .02 - .239 6.2 8.23 .1 6.3 The transformed equation is z'' + 6.3 2^ + 8.23 z — .239 = 0. SOLUTION OF HIGHER EQUATIONS. 521 Neglecting the z" and 7} terms, we have, approximately, 8.23 Thus the value of x to two places of decimals is 3.12. The process may be continued until the value of the root has been found to any desired degree of precision. The work is usually arranged in the following form, the coefficients of the successive transformed equations being denoted by (1), (2), (3), etc.: -3 _ -2 + 5 1 ai28 3 -6 - -2 (1) -1 3 _9 .761 3 (1)' 7 (2) - .239 3 ,61 .167128 (1) 6 7.61 (-) - .071872 . .1 .62 6.1 (2) 8.23 .1 .1264 6.2 8.3564 .1 .126<^ (2) 6.3 (-/ 8.4832 .02 6.32 .02 6.34 .02 (3) 6.36 Dividing .071872 by 8.4832, we have .008 suggested as the fourth figure of the root. Thus the value of x to three places of .decimals is 3.128. 522 COLLEGE ALGEBRA. We derive from the above the following rule for, finding the approximate value of a positive incommensurable root : Find hy Arts. G71, 073, or 687, or by Stur7n^s Theorem, the integral part of the root. (Compare Art. 674.) Transform the given equation into another ivhose roots shall be less by this integral part. Divide the absolute value of the last term of the transformed equation hy the absolute vahie of the coefficient of the first poiver of the unJcnoivn quantity, and ivrite the approximate value of the result as the next figure of the root. Transform the last equation into another ichose roots shall be less by the figure of the root last obtained, and divide as before for the next figure of the root; and so on. Note. In practice, the work may be contracted by dropping sucli decimal figures from the right of each column as are not needed for the required degree of accuracy. 719. To find the approximate value of a negative incom- mensurable root, change the signs of the alternate terms of the equation beginning with the second (Art. 643), and find the corresponding positive incommensurable root of the transformed equation. The result with its sign changed will be the required negative root. 720. In finding any particular root-figure by the method . exi)lained in Art. 718, we are liable, especially in the first part of the process, to get too great a result ; the same thing occasionally happens when extracting square or cube roots of numbers. Such an error may be discovered by observing the signs of the last two terms of the next transformed equation ; for since each root-figure obtained as in Art. 718 must be posi- tive, the last two terms of the transformed equation must be of opposite sign. SOLUTION OF HIGHER EQUATIONS. 52^ If this is not the case, the last root-figure must be diminished ui*fcil a result is obtained which satisfies this condition. Let it be required, for example, to find the root between and — 1 of the equation or^ -f 4 a;- — 9 a; — 5 = 0. Changing the signs of the alternate terms beginning with the second (Art. 719), we have to find the root between and 1 of the equation a^ — 4a^ — 9cc + 5 = 0. Dividing 5 by 9, we have .5 suggested as the first root- figure ; but it will be found that in this case the last two terms of the second transformed equation are — 12.25 and — .375. This shows that .5 is too great ; we then try .4, and find that the last two terms of the second transformed equation are of opjjosite sign. The work of finding the first three root-figures is shown below : .469 1 -4 .4 - 9 - 1.44 + 5 -4.176 -3.6 .4 - 10.44 - 1.28 (1) .824 - .713064 (1) -3.2 .4 -2.8 .06 (1) -11.72 - .1644 - 11.8844 - .1608 (2) .110936 -2.74 .06 (2) - 12.0452 -2.68 .06 (2) -2.62 The required root is therefore —.469, to three i^laces oE decimals. o24 COLLEGE ALGEBRA. In any case, the root-figure to be taken is the greatest number luhich will ensure that the last two terms of the next transformed equation shall be of opposite sign. Note. In some cases, the first trausforined equation gives very little information in regard to the first decimal root-figure. Thus, in the equation x* — 7a;2 — 5 a; — 1 =0, which has a root be- tween 2 and 3, the first transformed equation is yi + 8?/3 + 17y2_y_23 = 0, the last two terms being negative. The rule directs us to take the greatest number, less than unity, which will ensure that the last two terms of the second transformed equation shall be of opposite sign ; and in the present case this will be found to be .9. 721. If tKe coefficient of the first power of the unknown quantity in any transformed equation is zero, the next fig- ure of the root may be obtained by dividing the last term by the coefficient of the square of the unknown quantity, and tak- ing the square root of the result. For if the transformed equation is y^ + ay- + & = 0, it is b evident that, approximately, ay- -\-b = 0, or y = We proceed in a similar manner if any number of consec- utive terms immediately preceding the last term are zero. 722. Horner's method may be used to find any root of a number approximately ; for to find the 7ith root of a is the same thing as to solve the equation a;" — a = 0. 723. If an equation has two or more roots which have the same integral part, the first decimal root-figure of each must be obtained by the method of Art. 671 or 673, or by Sturm's Theorem, If two or more roots have the same integral part, and also the same first decimal root-figure, the second decimal root- figure of each must be obtained by the method of Art. 671 or 673, or by Sturm's Theorem ; and so on. SOLUTION OF HIGHER EQUATIONS. 525 Note 1. Horner's method may be used without change to deter- mine successive figures in the integral,, as well as in the decimal, portion of the root. Note 2. If all but one of the roots of an equation are known, the remaining root may be found by adding the sum of the known roots to the coefficient of the second term, and changing the sign of the result (Art. 639). EXAMPLES. 724. 1. Find the roots between 1 and 2, and —1 and —2, of 2. Find the root between 5 and 6 of a^_(- 20)2 -23a; -70 = 0. 3. Find the root between — 2 and — 3 of — _- -4- -t- 4. Find the root between and 1 of ar' + 6iK- + 10a;-l = 0. 5. Find the root between — 5 and — 6 of x3-a^-25x + 81 = 0. 6. Find the root between 3 and 4 of a;*-10ar^-4a; + 8 = 0. 7. Find the root between — 2 and — 3 of «* + 6 a;3 + 12.^2 - 11 .T - 41 = 0. 8. Find the root between and 1 of x' -f 3x^ -3x' + 19x - 12 = 0. Find the real roots of the following : 9. a;2- 2.3a; -1.29 = 0. 10. ar'-2>K--a; + l = 0. ■ 11„ a;^_3.^._l = 0. 12. a;3 + 3a;2 + 4a;-f 5 = 0. Find the approximate valu( 17. i^2. 18 is a units above O in the line Y. Again, with the notation of Art. 28, —ai may be represented by, the line OB', where B' is a units below in the line 01''. The imaginary numbers +i and —i are represented by the lines DC and 00', where C and C are, respectively, one unit above, and one unit below 0, in the line YY'. Note. It will be understood hereafter that, in any figure where the lines XX' and YY' occur, they are fixed straight lines which are perpendicular to each other and intersect at ; that all positive or negative real numbers are represented by lines laid off to the right or left of 0, respectively, in the line A'A'' ; and that all positive or nega- tive pure imaginary numbers are represented by lines laid off above or below 0, respectively, in the line YY'. Y ■B c- +ai + i ^ , -i c'. 0-i-a A -ai ■B' y' B hi 731. We will now show how to represent any complex number (Art. 327). Let the number be a-\-hi; and let the real number a be repre- sented by the line OA, and the pure imaginary number hi by the hue OB. Draw ^C equal and parallel to OB, on the same side of XX' as OB, and join OC. -X Then the complex number a ■\- hi is represented by the line OG. With the notation of Art. 28, the complex number — (a -f hi) may be represented by the line OG ', where OG' is equal in length to OG, and is drawn in the opposite direction from 0. In like manner, any complex number whatever may be represented by a straight line drawn from 0. APPENDIX I. 531 It follows from Arts. 730 and 731 that we may regard —1, in the product of any real, pure imaginary, or complex number by —1, as an operator which turns the line which represents the first factor through two right angles, in a direction opposite to the motion of the hands of a clock. (Compare Art. 729.) 732. In the figure of Art. 731, let C'A' be drawn peqiendicular to OX' ; then the right triangles OA'C and OAC are equal, having the hypotenuse and an acute angle of one equal to the hypotenuse and an acute angle of the other, respectively. Then OA' and A'C are equal to OA and AC, -respectively ; that is, OA' represents the real number —a, and A'C is equal and parallel to OB', where OB' represents the imaginary number —hi. Therefore OC represents the complex number — ({ — bi. But OC also represents — (a + hi) (Art. 731). Whence, —{a + bi) = — a — bi. 733. The modulus of a real, pure imaginary, or complex number is the length of the line which represents the number. The amplitude is the angle between the line which represents the number and OX, measured from OX in a direction opposite to the motion of the hands of a clock. If, for example, in the figure of Art. 731, the angle XOC is 30°, the amplitude of the complex number represented by OCis 30^', and the amplitude of the complex number represented by OC is 210°. The modulus is always taken positive, and the amplitude may have any value between 0° and 3(30°. The pure imaginary numbers +ai and —ai have the modulus a, and the amplitudes 90° and 270°, respectively ; and the real numbers + a and —a have the modulus a, and the amplitudes 0° and 180°, respectively. We have, in the figure of Art. 731, 0C= V 07l'^ + AC' ^=Va^ + b- ; that is, the modulus of the complex number a + bi is Va- + b'^ I and this is also the modulus of each of the complex numbers ±a±bi. Whatever number is represented by a, the amplitude of —a is always equal to the amplitude of a increased by 180°. Note. We may regard zero as having the modulus zero. Addition and Subtraction of Imaginary Numbers. 734. The representation of a complex number, as explained in Art, 731, shows that the result of adding a pure imaginary to a real num- ber may be represented by a straight line drawn from 0. 532 COLLEGE ALGEBRA. We will now show how to represent the result of adding h to a, where a and h represent any two real, pure imaginary, or complex numbers. Let a be represented by OA, and h by OB. Draw AC equal and parallel to OB., in such a way that C shall be in the same direction from A that B is from 0. Then the result of adding 6 to « is represented by the line OC. That is (Art. -5), a + hi?, represented by OC. Note 1. The above construction holds equally when OA and OB lie in the same direction, or in opposite directions, from 0. Note 2. The form of addition exemplified in the above construc- tion is known as Geometric Addition. In like manner, the result of adding any number of real, pure imaginary, or complex numbers may be represented by a straight line drawn from 0. 735. In the figure of Art. 734, draw BC. By Geometry, OACB is a parallelogram, and therefore BC is equal and parallel to OA. Then OC represents the result of adding a to h. But OC also represents the result of adding h to a. Whence, a-\-b=b + a. (Compare Art. 36.) The above result holds if either of the letters a and h repre- sents the sum of any number of real, pure imaginary, or complex numbers. 736. . We shall define the subtraction of b from a, where a and b represent any two real, pure imaginary, or complex numbers, as the process of finding a number such that, when h is added to it, the sum shall be equal to a. (Compare Art. 41.) Let a be represented by OA , and b by OB ; and complete the parallelogram OB AC. By Art. 734, OA represents the result of adding the number represented by OB to the number represented by OC; that is, if b is added to the number represented by OC, the sum is equal to a. b is represented by the line OC. APPEXDIX I. 533 737. In the figure of Art. 73G, let OB be produced to B', making OB' equal to OB. Then since J.O is equal and parallel to OB'^ 00 represents the result of adding the number represented by OB' to the number repre- sented by OA. J3ut by Art. 731, the number —6 is represented by OB'. Whence, a — 6 = a + (— 6). (Compare Art. 42.) Again, the result of adding — 6 to fe is represented by a line joining O to the extremity of a line drawn from B towards 0, equal and parallel to OB' . That is, , & + (- h) = 0. (Compare Art. 39.) 738. Let a, 6, and c be any three real, pure imaginary, or complex numbers, represented by the lines 0^4, OB., and OC, respectively. ^ ^ Complete the parallelograms OADB and /^^-^^ ,^T^\ OB EC ; then OD represents a + b (Art. 734) / /^"t-'-Os and in like manner OE represents b + c. / / ''^--^ ^ Complete the parallelogram OBFC ; then / ^'^^.^"-'''''''^^^ ' a + b + c, being the result obtained by add- O ^C^i^ Hc'^ i ing c to ffl+6 (Art. 5), is represented by \ "~^s<-^ \ / the Ime OF. . V^^-lljl-V' Join A F and EF. ^ D Since, by construction, DF and BE are equal and parallel to OC, they are equal and parallel to each other, and BDFE is a parallelogram. Therefore EF is equal and parallel to BD, and consequently to OA ; and hence OAFE is a parallelogram. Then OF represents the result of adding the number represented by OE to the number represented by OA ; that is, OF represents a+(& + c). But OF also represents a + ft + c. Hence, a + (5 + c) = a + 6 + c. (Compare Art. 37. ) The above result holds if any or all of the letters a, &, and c repre- sent the sum of any number of real, pure imaginary, or complex numbers ; and hence the Associative Law for Addition (Art. 37) holds when any or all of the numbers involved are pure imaginary or complex. 739. "We will now prove that the Commutative Law for Addition (Art. 36) holds when any or all of the numbers involved are pure imaginary or complex. 534 COLLEGE ALGEBRA. Consider the expression a + b + - + c + d + e+f+- + g, where a, h, c, etc., are any real, pure imaginary, or complex numbers. By Art. 738, a + b + --- + c+ cl+e +f+ — + g = a+b + - + c+id + e)+f-\-- + g =ra+b + --- + c+(e + d)+f+-- + g (Art. 735) = a + 6 + - + c+ e + d +f+- + g (Art. 738). Tliat is, any two consecutive terms of an expression may be inter- changed without altering the value of the expression. Now by successive interchanges of consecutive terms, the terms of an expression may be written in any order whatever. Hence the Commutative Law for Addition holds when any or all of the numbers involved are pure imaginary or complex. Note. It follows from what has already been proved, that the results in Arts. 43 to 49 inclusive hold for any pure imaginary or com- plex values of the letters involved. Multiplication of Imaginary Numbers. 740. Since -f ai may be written (-f 1) x (+ aO, the product of -f 1 by -I- ai is represented by turning the line OA, which represents the number -f 1, through one S " right angle, in a direction opposite to the motion +«-* of the hands of a clock, and multiplying the X- ^^ result by a — a-. > ■X And since — ai may be -^ATitten (+ 1) X(— ai), the product of -F 1 by — ai is represented by a -ai line equal in length to that which represents .j3' the product of -1- 1 by -t-ai, but drawn in tho opposite direction from 0. •*• This suggests the following : The product of any real, pure imaginary, or complex number by -\-ai may be represented by turning the line which represents the number through one right angle, in a direction opposite to the motion of the hands of a clock, and multiplying the result by a. The product of. any real, pure imaginary, or complex number by — ai may be represented by aline equal in length to the line which represents its product by -^ai, but drawn in the opposite direction from 0. APPENDIX I. 535 ^ , y\ , r^'y /^ +1.1 741. Since a + 6i may be written (+ 1) X (a + hi), the product oi + 1 by a-\rhi is represented by turning the line OA, which represents the number + 1, through an angle equal to the amplitude of rt+6t (Art. 733), in a direction opposite to the motion of the hands of a clock, and mul- ' --m/ i ■ ^l, tiplying the result by the modulus of a + hi. And since — (a + hi) may be written (+ 1) X (—a — 60 (Art. 732), the product of + 1 by — (a + fei) is represented by a line equal in length to that which represents the product of +1 by a-\-hi, but drawn in the opposite direction from 0. This suggests the following : If a and h are any positive or negative real numbers, the product of any real, pure imaginary, or complex number by a-^hi may be represented by turning the line which represents the number through an angle equal to the amplitude of a+6i, in a direction opposite to the motion of the hands of a clock, and multiplying the result by the modulus of a + hi. The product of any real, pure imaginaiy, or complex number by — {a-\-hi) may be represented by a line equal in length to the line which represents its product by a + hi, but drawn in the opposite direction from 0. 742. Let a and h be any two real, pure imaginary, or complex numbers, represented by the lines OA and Oi?, respectively. Then the result of multiplying a by & is represented by OC, where the angle XOC is the sum of the angles XOA and XOB, and OC is equal to OA x OB. That is (Art. 7), ah is represented by OC. In like manner, the product of any number of real, pure imaginary, or complex numbers may be represented by a straight line drawn from 0. It is evident from the above that the modulus of the product of two or more numbers is the product of their moduli, and that its amplitude is the sum of their amplitudes. Note. The form of multiplication exemplified in the above con- .struction is known as Geometric Multiplication. 636 COLLEGE ALGEBRA. 743. With the figure and notation of Art. 742, bxa is represented by turning OB through an angle equal to XOA, in a direction opposite to the motion of the hand.s of a clock, and multiplying the result by OA. That is, 6 X a is represented by OC. It is evident from this that axb^bxa. (Compare Art. 58.) The result holds if either or both of the letters a and b represent the product of any number of real, pure imaginary, or complex numbers. 744. In the figure of Art. 742, let OC be produced to C", making OC" equal to OC. Then by Arts. 740 and 741, a X (— fc) is represented by OC. But OC also represents —ah. Therefore, aX(-b) = -ab. (1) Again, let OA be produced to A', making OA' equal to OA. Then — a is represented by OA' ; and consequently (— a) X & is represented by turning OA' through an angle equal to XOB, in a direction opposite to the motion of the hands of a clock, and multiply- ing the result by the modulus of b ; that is, (—a)xb is represented by OC. Therefore, (— a) X & = — ab. (2) By (2), (_a)x(-6) = -aX(-6) = _(_a6),by (1) = + ab. Hence the results in Art. 56 hold if either or both of the letters involved represent pure imaginary or complex numbers. 745. Let «, b, and c be any three real, pure imaginary, or complex numbers. Then ax(,bxc) is represented by turning the line which repre- sents a through an angle equal to the sum of the amplitudes of b and c (Art. 742), in a direction opposite to the motion of the hands of a clock, and multiplying the result by the product of the moduli of b and c. Again, axbxc, being the result of multiplying axb by c (Art. 7), is represented by turning the line which represents a through an angle equal to the amplitude of b, in a direction opposite to the motion of the hands of a clock, and multiplying the result by the modulus of b ; and then turning the resulting line in the same direction through an angle equal to the amplitude of c, and multiplying the result by the modulus of c. APPENDIX I. 537 It is evident from this that axibxc) = axbxc. (Compare Art. 59.) The above result holds if either of the letters a, b, and c represents the product of any number of real, pure imaginary, or complex num- bers ; and hence the Associative Law for Multiplication (Art. 59) holds when any or all of the numbers involved are pure imagmary oi complex. 746. We will now prove that the Commutative Law for Multipli- cation (Art. 58) holds when any or all of the numbers involved are pure imaginary or complex. Considei the expression axbx--- XcXdXeXfX--- Xg, where «, b, c, etc., are any real, pure imaginary, or complex numbers. By Art. 745, axbx--- XcX dxe Xf<---Xg = axbx--- XcX(dXe) xfX ■■■ X (J = axbx-- Xcx(ex d) X/X ••• X g (Art. 74-3) = axbx — XCX exd XfX---Xg (Art. 745). That is, any two consecutive factors of a product may be inter~ changed without altering the value of the expression. Now by successive interchanges of consecutive factors, tlie factor.s of a product may be written in any order whatever. Hence the Commutative Law for Multiplication holds when any or all of the numbers involved are pure imaginary or complex. 747. Let a, 6, and c be any three ^^ real, pure imaginary, or complex num- bers, represented by the lines OA, OB, and OC, respectively. Complete the parallelogram OBDC; then by Art. 734, 6 -f c is represented by OD. Hence, by Art. 742, aX{b-\- c) is represented by OE, where the angle XOE is the sum of the angles XOA and XOD, and OE is equal to OAxOD. Again, ab is reiiresented by OF,wliere the angle XOF is the sum of the angles XOA and XOB, and P is equal to OA X OB ; and ac is represented by OG, where the angle XOG is the sum, of the angles XOA and .YOC, and OCr is equal to OAxOC. iX'Axr Y Y \ CI \\ 1 h ^-^•1 X' y' A' 538 COLLEGE ALGEBRA. Join EF, EG, BF, and CG. . ,. OE OF OG r>A By construction, ^ = ^ = ^-^ = 0A Whence, by alternation (Art. 385), 9^ = ^ and^ = — OF OB OG OC' Again, the angles EOD, BOF, and COG are equal, since, by con- struction, each is equal to the angle XOA. Therefore the angles EOF and BOD are equal ; for EOF is the sum of BOF and EOD, and BOD is the sum of DOF and BOF, which is equal to EOD. In hke mamier, the angles EOG and COD are equal. Therefore the triangles EOF and BOD are similar, as also are the triangles EOG and COD ; for, by Geometry, two triangles are similar when they have an angle of one equal to an angle of the other, and the including sides proportional. Then the figure OFEG is similar to OB DC, and hence OFEG is a parallelogram. Therefore OE represents the sum of the numbers represented by OF and OG ; that is, OE represents ab + ac. But OE also represents « x (6 + c). Therefore, a x (b + c) =^ ab + ac. Hence the Distributive Law for Multiplication (Art. 60) holds when any or all of the numbers involved are pure imaginary or complex. Division of Imar/inary Numbers. 748. We shall define the quotient of a divided by h, where a and h are any two real, pure imaginary, or complex numbers, as the process of finding a number such that, when it is multiplied by b, the product shall be equal to a. (Compare Art. 67.) y Let o be represented by OA, and b by OB. Draw OC so .that the angle XOC shall li be equal to the angle XOA minus the -X angle XOB, and OC equal to — • ^^ Then the angle XOA is equal to the angle XOC plus the angle XOB, and OA is equal to OC x OB ; and hciice, by Art. 742, OA represents the product of the number repre- Rcntt'd by Or,';uid the number rrpri'seiitod by 07?. APPENDIX I. 539 That is, if the number represented by OC is multiplied by 6, the product is equal to a. Therefore, - is represented by C. b It is evident from the above that the modulus of the quotient of two numbers is equal to the modulus of the dividend divided by the modu- lus of the divisor, and its amplitude equal to the amplitude of the dividend minus the amplitude of the divisor. Note. It follows from what has already been proved, that the results in Arts. 68 to 72 inclusive hold for any pure imaginary or com- plex values of the letters involved. 749. We define y/a, where a is any real, pure imaginary, or com- plex number, as a number whose ?tth power is equal to a (Art. 121). Let a be represented by OA. Draw OB so that the angle XOB shall be equal to one-nth of the angle XOA, and the length of OH equal to the nth root of the modulus of a. Then the angle XOA is ?i times the angle XOB, and the modulus of a is the nth power of the length of OB ; and hence, by Art. 742, OA represents the nth power of the number repre- sented by OB. Therefore, \^a is represented by OB. 750. We have proved that every result in Chapter II. holds when any or all of the numbers involved are pure imaginary or complex ; and therefore every statement or rule, in Chapters III. to XVI. inclu- sive, in regard to expressions where any letter involved represents any real number whatever, holds equally when this letter represents any pure imaginary or complex number. (Compare Art. 321.) 751. It is evident from Arts. 734, 736, 742, 748, and 749, that any expression which is the result of any finite number of the following operations performed upon one or more real, pure imaginary, or com- plex numbers, may be represented by a straight line : 1. Addition or Subtraction. 2. Multiplication or Division. 3. Rais- ing to any power whose exponent is a rational number (Art. 209). 4. Extracting any root. That is, any such number can be expressed in the form a+ hi, where a and h are real numbers, either or both of which may be zero. 640 COLLEGE ALGEBRA. APPENDIX 11. CAUOHT'S PROOF THAT EVERY EQUATION HAS k ROOT. 752. To prove that Va + bi, where a and b are real numbers, can be expressed in the form c + di, where c and d are real numbers. Squaring the equation Va + bi = c + di, we have a + bi = c^-\- 2 cdi — d^. Whence (Art. 338), c"- - cV = a, (1) and 2 cd = b. (2) Squaring (1 ) , c* - 2 c'^ cZ^ + d* = a^ (3) Squaring (2), 4c'^d^=b'^. (4) Adding (3) and (4), c* + 2 c^d^ + d^ = a^ + h\ Whence, c'^ + d^= Va^ + b'^. (5) The upper sign must be taken before the radical in equation (5) ; for since by hypothesis c and d are real number^, c- + d- is positive. Adding (1) and (5), 2 c- = « + Va- -r 0-. (0) Subtracting (1) from (5), 2 cZ- = — a + Va- + h'. (7) Equations (6) and (7) show that c- and d- are positive, and there- fore c and d are real. 753. To prove that, if n is a positive integer, each of the equations X" = ± 1 , and X" - rb i has a root of the form a + hi, where a and & are real numbers, either of which may be zero. Case I. a;» = l. It is evident that 1 is a root of this equation. Case II. x" = — 1, where n is odd. It is evident that — 1 is a root of this equation. Case III. %"■= —\, where n is even. Let ra = 2 m, where m is a positive integer ; then, x^" = — 1. Extracting the square root of each member, x'" = ± i. The latter forms are included in the four following cases. APPENDIX II. 541 Case IV. a;" = i, where n is odd. If m is a positive integer, i*"^-*-^ = i (Art. 333) ; hence, if n is of the form 4 j/« + 1, Jis a root of the equation. Ao-ain, (— i)4'n+3= _ j^m+s^ _ (_j) (Art. 333) = i ; hence, if n is of tlie form 4 hi + 3, - i is a root of the equation. Case V. x"- = i, wliere n is even. Let n -= 2'>p, where p is an odd integer ; then, x'^^p = i. Let x^ = y; tlien 2/P = i, and, by Case IV., y = i or —i according as p is of the form 4 ni + 1 or 4 jw + 3 ; that is, a;-' = i or - i. The value of x may be obtained from this equation by q successive extractions of the square root ; and since it has been proved that the square root of a + hi, where a and b are real, can be expressed in the same form, it follows that x can be expressed in the form a + hi. Case VI. x"-= — i, where n is odd. By Art. 333, (— j)''"'+i = — i-*"'+i= — i ; hence, if n is of the form 4 j?t + 1, —i is a root of the equation. Again, ii"'+^= — i ; hence, if n is of the form 4?n + 3, i is a root. Case VII. a;"= — t, where n is even. As in Case V., x may be obtained in the form a + hi. 754. We will now consider the general case. To prove that the general equation of the ?«th degree x» + i)iX"-i+i>,a;«-^+ ••• +p«-ix + p„=0 (1) has a root of the form a + hi, where a and h are real numbers. Substituting a + hi for x in (1) we have (a + &0" + Pi (« + ^0"'^ + ••• + Pn~\ (« + 60 + Pn = 0. Expanding by the Binomial Theorem, and collecting together the real and imaginary terms, we shall have a result of the form C/+Fi = 0, (2) where U and V are real numbers. Transposing Vi in (2), and squaring both members, we have m = - F^ or U^ + V^ = 0. We have then to prove that such real values may be found for a and h as will make U- + F^ = 0. As a and & change in value, U and F also change ; and if U- + F- cannot become zero for any values of a and h, there must be some pos- itive real number which is the least value that U'^+V'^ can assume. 542 COLLEGE A1.GEBKA. Let a and /S be the values of a and b, respectively, for wlilch U^ + V has this minimum value. Let P+ Qi be the value of the first member of (1) when a+ pi is substituted for x ; then P^ + Q^ is the minimum value of U'^ + V'^. Writing o + yOi + h in place of x in (1), we have (a + /Si + hy+l\ (a + Hi + /0«^1 + ••• +P«-1 (a + Hi + /i) + i5„ = 0. Exisanding by the Binomial Theorem, and arranging the result in ascending powers of h, (a+/30"+P^(a+;30«-l+ ... + ;,„ _i (a + ^i) +^„ + h In (a + Hiy-^ + Pi (« - 1) (a + /aO''-^ 4. ... j^ p^^j-j + (terms involving 7t', /i^, ..., A") — . 0. (3) The first line of (3) is equal to P + Qi. The coefficients of some of the powers of h may be zero ; but they cannot all be zero, since the coefficient of ft" is unity. Let /t™ be the lowest power of h whose coefficient is not zero ; and denote its coefficient by /? + -S'j, where R and S are not both zero. Then (3) becomes P + (2» + (P + Si) h'^ 4- (terms involving powers of h higher than the mih.) = 0. Let this be denoted by P' +Q'i=^Q. (4) Now let h = ct, where c is a positive real number, and t a root of the equation i™ = 1 or ("^ = — 1 . By Art. 753, t is in cither case a number of the form a + hi. Then, pi^ qti = p^ Qi ± (p + Si) C" + - . Whence (Art. .338), P' ^ P± lie"' + ••• , and Q'=r Q±Sc-" +■■•■ Therefore, P'2 + Q'^= p-^+ Q^±2 {PR + QS) c" + ••• • That is, P'2 + Q'2 - pi - Q2 = ± 2 (PP + Q^S) C" + (terms involving powers of c higher than the ?7ith). (5) If PR + QS is not zero, c may be taken so small that the sign of the second member will be the same as that of ± 2 {PR + Q*S')c"». Hence, if PR+QS is positive, the sign of P'- + Q'^ — P'^ — Q'^ may be made negative by taking i"* = — 1 ; and if PR + QS is nega- tive, the. sign of P'^ + Q'^ — P^ — Q- may be made negative by taking «- = + L '■J'hus, in either case, P'- + Q'- can, by properly choosing c and t, be madi; less than P- + Q-. APPENDIX II. 54a If PR + QS= 0, let «>" = ± i in (4). By Art. 753, t is in either case a number of tlie form- a + hi. Then, P'+q'i = P+Qi±{R + Si)ic^ + - = P + Qi±{Bi-S)c"'+-' Whence, F' = PT Sc" + ■■■, and ■ Q'=Q±Rc"^+-- Therefore, P'' + Q'' = P' + Q^±2 (QB- PS) c- + •.• » That is, P'' +Q'^-P-'-Q-' = ± 2 {QR-PS) c- + - ■ Now, (PA' + Qsr + (QR-PSy' = P'R' + Q'S-' + q^ R^ + P'^S-' And since, by hypotliesis, P-^ + Q- is not zero, and R and S are not both 0, it follows that {PR+ QSy +(QR- PSy is not zero. But PR + QS= 0, and hence QR - PS is not zero. Therefore, if c is taken sufficiently small, the sign of P'-+ Q'-—P-—Q^ will be the same as the sign of ±2{QR-PS)c'" ; and we can ensure that this sign shall be negative by taking r = - i when QR -PS is positive, and t"'=i when QR — PS is negative. Thus, by properly choosing h, P'^ + Q'^ may be made less than P- + Q^ ; that is, a value of U' + F- may be obtained which is less than P2 + Q^, and the latter is not a minimum value of U^ + V\ Hence, no positive real number can be a minimum value of U- + V'^ ; and therefore values of a and b can be found which will make f/^ + F'^ = 0. We will now prove that the values of a and b which make U- + F- = are finite. The first member of (1) may be written \ X X^ X"J Putting a + bi in place of x, we have ^7+ vi=(a + bi)"[i + -^ + ^fW+ ••• + 7— rhrnT (^^ ^ L a+bi (a + bi)- {a + bi)"J Consider the term 2),- _ pr(a — biy ia + bi)'- [(a + bi){a-bi)y (a^+fe-) (rt — 50'' _P^^{ar-ra-^bi-'^^^;^a^-'b^~+ - a-^ + b-^yl (a-^ + b-^yl \2 = Ar + P,i, say. 544 COLLEGE ALGEBRA. Now, A=^^^wk-'%^«^^^^^+-l ■'■[(#-'"S"(.T!)' ■(?:)■•■■■]■ It is evident from tliis that, when a and b are indefinitely increased, Ar is indefinitely diminished ; and in like manner it may be shown that B, is indefinitely diminished when a and b are indefinitely in- creased. Thus (6) may be written U+ Vi =(a + ?)i)" [1 + ^1' + S'i], (7) where A' and B' are indefinitely diminished when a and b are indefi- nitely increased. If a — bi is substituted for x in (1), we shall have a result which may be obtained from (7) by simply changing the signs of the terms involving i ; thus, U-Vi=(a-biy[l + A>-B'i]. (8) Multiplying (7) and (8), m + V^ = (a^ + b'^)" [(1 + A'^ + B''^]. (9) The second member of (9) increases indefinitely when a and b are indefinitely increased; for the factor (a- + fe"^)" increases indefinitely, and the factor (1 + A')'^ + B'^ approaches the limit 1. Hence, U"'^ + V^ cannot be zero when a and b, or either of them, are indefinitely increased ; and therefore the values of a and b which make U'^ + V^ = are finite. 755. The demonstration of Art. 754 holds equally whether the coefhcients of the terms in equation (1) are real or imaginary. It follows from the above that y/— a, where n is any even integer and a a positive real number, and v'a + bi, where n is any positive integer and a and 6 any real numbers, can be expressed in the form c + di, where c and d are real numbers. That is, any even root of a negative real number, or any root of a pure imaginary or complex number, can be expressed as a pure imaginary or complex number. (^Compare Art. 33G ; also, Appendix I., Art. 751.) ANSWEES. Note. In the following collection of answers, all those are omitted I'hich, if given, would destroy the utility of the example. Art. 84 ; page 28. 1. ^x-^+Sx+2. 2. Sa + Sb+Sc + Sd. 3. 5fl3 -3n62 -46^. 4. a^-ix^. 5. -3x3-5a:2^ + 8.r^2 + 2/. 6. 5x^+Gxy-l y-^-Gx-~ y+6. 7. 3.r5-5;ri — .r^- ll.r2+ 4.r-2. 8. 4x3 + 9.r2(/-4.ry2_3y3. 9. 4«4 _i_ 7^3 _ 2a _ 4. 10. 2x + 4i/. 11. a-c. 12. -3a-l. 13. 4,r-2. 14. Gm + 2. 15. a + 21,. Art. 96 ; pages 34, 35. 1. 12x6 + 7t*+ 5a:3+ 10a:-4. 2. - ?«5 — 37 m2 + 70 ?« - 50. 3. a5-5ai6+10a362-10n263+5a64_i5. 4. _6x5-25a:* + 7x^J4-81x2+3.r-28. 5. 8a7-44a5_40a*+76a3+ 112«2-,32. 6. x^ + y^ + z"" - 3 xyz. 7. a262 4- c^d' - d^c^ - Jf-d'. 8. 9 x^ - 6 .r ' — 50 x^ + 60 x^ + 28 x + 4. 9. x8 + x* + l. 10. x6- 50x1 + 769x2 -3600. 11. «6_ea452 + 9„2;,4_4;,6. 12. ^2 - 4x^ + 4j/2 - 9^2. 13. x^-/. 14. 62-rf2 15 _4^s. 16. 0. 17. 80c. 18. 0. 19. - 9 m* + 82 »i2?)-2 - 9 n*. 20. 8 a^. Art. 106 ; page 40. 1. 3x2 + 6x + 9. 2. x3-2.r2+x-2. 3. 024.306+5^,2. 4. »n3 _ ^2 _ 14 ,„ 4. 24. 5. n^ - 2 an> -6ab'^+7 b^. 6. x8-2x2-x+l. 7. x+2y-Sz. 8. a" - 6'« + C". 9. x3 + 2x2-x+l. 10. 2«2_oft + 262. 11. x+a. 12. (6+.c)o + 6c. 13. (x+^)-3. 14. x + a. 2 collegp: algebra. 15. (m-ny + 2(m-n) + l. 16. x^'+ (a - b)x -ab. IT.x'^-bx + c. 18. (n-3i).r+ (2« + 5 6). Art. 117 ; page 49. 9. a2_2a6 + &2_c.2. 10. .r*-.r2-2a- - 1. 11. a:* — 49.t2+ 84.r-3G. 12. a*-14a2 + 25. Art. 143 ; page 65. 7. 8.T-70. 8. 2x-5. 9. 3a + b. 10. 4a:2-6ar. 11. 3x-l. 12. 2a + 36. Art. 152 ; page 69. 6. x^-Wx^y + 86 x^f - 176 x^^ ^. 105 yi, 7. a-! ^ 2 a5 _ 4 «^ - 7 a^ - 16 a^ + 32 a - 8. 8. 6 .j-s - 31 x3 - 4 ar* + 44 x3 + 7 .r^ _ 10 x. 9. rS + 3 .r* - 23 x3 — 27 2-2 + 166 3; — 120. 10. 24 a6 _ 64 a^Z) — 58 a*62 +114 aSft-' _ 36 a^b*. Art. 170 ; pages 82 to 84. , 0x2 — Or +4 ^ ^ ^ a2 + i2 ^ x + ;/ + z 3x + 2 a ar — 3/ + z 5 r+l g 6m -n ^_ 3x-2 g x2-3x,y + y2 ' x2 + X + 1 ' 5 H( — 7 « X + 3 x2 — xy + 3 j/2 11 2x-^- -^"-^^ - i2..»'-^+12»>— 6m + 6. i3_ x + 5. 4 16x'^+12 42^3 x+i 14 — ?— . 15. -^^. 16. -^^. 17. •^'+^^+^ • . x-2y a + b a2 + 62 (., + 1) (.r? _ 1) 18. 2:^-3^/. 19. --ii!L. 20. x2 + x^. 21. J':i^=l^. 2x + 3j/ 7n + n x + jy — 2 22. 0. 23. ^^'"-")- 24. -2i!£:^. 25. -J^- (m + »)- x^-a* • x2-_y2 „p -ot+ic + ra 2^ 3 ^^j, _2( x2— 1).. _ (a + 6)(/> + c)(c + «) l + 9x x* + x2+l ANSWERS. 3 Art. 186 ; pages 92 to 94. 1. -3. 2. 91. 3. -1. 4. -7. 5. 8. 6. 1. 7. 5. 8. 17. 9. _ii. 10. _?. 11. 31. 12. 3(a-l). 13. ^^- 14. — • ' 5 " a + i a 15. „. 16. 1 17. -n. 18. -^^- 19. .7. 20. -.04. Art. 188 ; pages 95 to 97. 3. 29, 14. 4. 3 dollars, 48 dimes, 6 cents. 5. 82, 31. 6. A, a m(n-\) , -3 a{n-\) ^ IQQ («_;,) „ 17 Hi — 71 m — n pt 23 L. 55 min. 13. A, 12 miles ; B, 14 miles. 14. 283. 9. 27A min. after 5. 10. ?-=^, 2^^- 11. 55 min. 12. $15,000. 15. First, )i -\- up + pm uin — np + pm 12. A, 10; B, 15. 13. ^^"' " "" dollars at ^^^(^-f>) per cent. m — n bm — an 14. A,$117;B,$63; C,.$36. 15. Hind-wheel, 15 ft.; fore-wheel, 10 ft- 16. Eate, 40 miles an hour; distance, 112 miles. 17. A, 8 hours ; B, 9 hours ; C, 12 hours. 18. A, 6 ; B, 5. Art. 227 ; pages 128, 129. 3. x If. 6. x<2a. 7-. x2, y>4. 9. .r < 24, // > 3. 10. :r > 5, a: < 15. 11. 8. 12. 19. 13. 32 or 33. Art. 233 ; page 135. 7. £!!_i? + 6 -—-}--. 9. 9a:6_12.r5-2xt+28.T3-15a-2_8a:+16. b^ b a a^ ■ 10. x8 - Sx^ + 28 -r" - 56x5 + 70x* - 56x3 ^ 28x2 _ 8x + 1. 11 . 8 x9 -F 60 x' + 150 .7-5 + 125 x^. 12. 64 a^xS - 144 a^b^x^y + 108 aVAry'^ - 27 b^yK 14. o^"' + 15 a*"" 6" + 75 a^™ b-"' + 125 b^». 15. xe-3x5-}-6x4-7.r3 + 6.r2_3x-f 1. 16. 8.r6-.36x5+42x«-f9x3-21.r2-9x-l. 18. 8 x9 - 36 x8 + 42 x^ - 39 x^ -f 123 x^ - 69 x* + 23 x^ _ 156 x2 - 48 x - 64. Art. 268 ; pages 162, 163. 7. 2a2-5aZ. + 8Z;2. 8. l-x-f-- 9. 2.r3 - 3x2 -f 4x- 5. 10. 21.12. 11. 900.8. 12. .8253. 13. Sa'^ -'2ab - b^. 14. l-^-'^^ 15. x3-x2 + x-l. 16. 31.7. 17. 10.13. 3 2 18. .0534. 19. .t2-2x-2. 20. a^ - b. 21. r2-2x-2. 22. 21.4. 23. .40. 24. 12.3. ANSWERS. 5 Art. 280 ; page 170. 1.73205. 5. 1.05409. 9. 1.25992. 13. .72112. 2.64575. 6. .44721. 10. 1.81712. 14. 1.07722. .37947. 7. .64550. 11. 1.93098. 15. .63764. .04472. 8. .42492. 12. .31072. 16. .87358. 28. 243 37. Art. 290 ; page 177. 29. -128. 30. ±243. 31. 1296. 32. 6x2 — 7r3 — 19x3 + 5ar + 9a:t — 2xi. 2x-^y-l0 xy-i + 8 x^y-^. 34. 2-4 a~ t xt + 2 a~ I x^. 35. ai-ah^ + bi. 36. a-i6-2 - a-'-^^-s - a-^t"*. x-^-y-i + T^-y-l 38. ah-^-2 + a-h^. 39. x^-3xi + 2xi 40. a-i^V". 41. x«-^ 42. a'^y. 43. x. Art. 317 ; pages 191 to 193. y/2. 2. V3. 3. ^5^. 4. v/^. 5. 4^. 7. da^b^VUFc. 8. 3a6v'2a-56. 9. (m-9n)V'W. V3. 6a72, 1 6 lV30. 11. ^ 9 , 3a6v2a — 56. 12. -^ ^18a-'6c2, 4a-^ m 13. V448. \/25^. 15. '86 37. X 40. V^ /2^9. gl — gibi + a2 bi — a%= + gt 4 — g^ft + ahi - ab^ + gU? - 6? a^ — 6- 7.24263.... 46. .10102.... 47. 1.91245.... 48. Vl + Vb. 5-2V2. 50. \/5-V3. 51. 7 + 3V2. 52. 4V2-3V3. Vm + n — Vm — n. 54. Va + .r + Va. 55. v'3(7+\/5). v^(3V2-2\/3). 57. -3. Art. 335; pages 200, 201. K'^f'-c). 59. 12. g-36 -2\/l5. 2. -V^IT^. 3. 1. 4. 38_14\/^^. 5. 102. 5 -x^+2xy\/xy-y^ 1. - 10 + 9\/^^. 8. 0. 9. 7^2. V3. 11. -4\/:r2. 12. -2y/^Z. 13. ~^^^~^ . rt-2_6 + 2av'-6 15. 10 _ 23 V- 10 12 a2 + 6 70 19. V-^ + V^^. 21. Va - \/^^ 22. \/3 + V- 16. 12 V- 5. 0. 5 + V^. Art. 341 ; pag-e 204. . ±2\/in:. 3. ±1. 4. ± Va + 6. 5. ± \/l9. g + i 2. 7. ±3. 8. ±2. 9 2. 13. ±1. 14 (a -6). 10. ± V2. 11. 15. ±2. 16. ± - VlO, Art. 346 ; page 209. 17 2. 3, - 2. 7. •^.-*. 4 9 , , 3 1 2 1_ 5 5 1 1 2' 3* 12. 4. -^, -^. 7 7 -I+2V5 - 5 ± V- 23 5. 15, 1. 2 4. 3, 2 2± V7 ANSWERS. Art. 347; pages 210, 211. 5. .-1. 10. -2. 4 4 5,2. 2 3 9 12. Il- 13. 3 3 ■ 39± 3\/ir3 Art. 348; pages 212, 213. -1, -4. 4 15 5 5 1 1 4 ' 2 ■ 2' 14" 6.13 1. 7.-1,-?. 4 2 2 -" -'• 9. 5, -3. 10. 4±2V3. 11. 4, - -. - 10 ± V/ ■y. 13. 1,~- 14. 2,1. 36 3 15.4,-1 16.-^,-^. ^■f 18. 4, ~. 19. 1, -• 4 9 20. 1^,-2. 21. 1,-18. 23 I-- 23. 2. 24. 17,-1. 25. -2, — • 26. 3,"^. 65 9 ^'-f 28. 3,-2. 29. 5,- ■ -• 30. 2V3-3, 2-V3. 21±2vT09. 32.1,-10.33.1- 29 „. 8-5V2 13-9\/2 21 ■ 7 ' 7 Art. 349 ; pages 215, 216, 20. 1 _2 lm-1. 4. 04 -.h. 5. a ,-b. 6. - -1, — a . 7. -16, -2c. m ■:\- .3. 9. -^ a ' c 10 . 2p, - 5;j. 11. - h 3n -2 3^' « ' 2' ■ 4'2" n. n 1 + 1 14 ?n n 111 15. a. 1 a 16. a + 6, 4 6. .1^- (a + ;.)^ - -(« -M -. 18. 9m, -I n. 19. a, -2a. a + h c c a + h 21. , ■i + b, 2ab a + h 22. w —2n, n — 2 m. 2 — a -2a-l. 24. — a, -b. 25. a-6,- ■a- -c. 23 6-2« + c_ c + a-26 2 ac 6 3a + 5 6c il'. 38. « a + 6 rt — 6 - 6' a + b 29. 3 n + 4 6 a a ' 2a-\- b 30. a + 6 — c, a — h + c. 31 -2a6±0 2 COLLEGE ALGEBRA. S2 t±±±l, «'-" + l . 33. ^^^ + "^) - 6. a^ — a + l a- + a + l a (2 a + 3) 34. 5a + 6 3rt —46 3a — 26' 2« + 56' 35. a+6+c a—h—c a — b — c a -{-b + c Art. 351 ; pages 218 to 220. 11. 0, -1 . 0, 3, -8. l± V^^ ). 2, -^, -4. 2 10. 0, 3 2 12. 3, 3±3V-3 14. +a,~,-~- 15. -1,2, ±3, ±4. 16. ±1 '22 '2 4 3 13. ±2, ±2^^^. 5 5±5V;^ io 4a 2a±2aV'^ ^^- -y 3 19. 1, ±3. 21. -5, 0, ±2V3. 24. -a, -b. 3 25. 0, =1±:^. 26. 0, - 1, 1 ± V-3 2 2 5 J. ±1, ±2. 0, +5V2. 30. ±2a, ±3a. 31. 1, —4, 5± >/41 0, ± Vl3. 33. 0, ± Va^ + 6-^. 3. II. 36. a, b, IL+i. 34. 1,2,3, 11 ± V-23 37. -1 ± V5, i2V-l. 1 _2 2' 3" Art. 354; page 224. 2. a:2-9r=-20. 3. .r2+2.r = 3. 4. 5r2_12.r=9. 5. 3x2 + 40ar = -133. 6. 12.r2_ 17r = - 0. 7. 21x2 + 44a: = 32. 8. 33-2+11x1^0. 9. G.r2 + 31.r = -35. 10. x2 _ (2 « + 6) X = - «2 - ,t6 + 2 6- 12. a:2-4x=71. 11. .r2 -2 //(.(• = 13. 4 x- _ 4 11, f = H — «j2. ANSWERS. 9 Art. 355; pages 226, 227. 6. (2x-Z){x+5). 7. (5x+l)(x+7). 8. (2:c-3)(4a:-3). 9. (13 + x)(3-a-). 10. (l + 2.r)(2-3a). 11. (x + 2 4-V3)(^ + 2- V3). 12. (3x-H- V5)(3x-1- Vo). 13. (6:r-5)(2x+l). 14. (2 + x)(3 - 2.0- 15. (4.c - l)(2.r + 5). 16. (.r-2)(10z-3). 22. (Sx- 7 7«n)(7x- 3//)?0- 17. (Vl7 + 4 + x)(Vl7-4-.r). 24. (x + Sy -2)(x - 2 y + 3). 18. (6+12.i)(3-2x). 25. (x + y + l)ix + 2y + 2). 19. (2.T-7a)(3z+5a). 26. (2 - 3^ + x)(3 + 2 */ - a). 20. {dx + 4:m)(4x + 5w). 27. (x- ;/ + 2)(2r + j/ - 1). 21. (3x-2(/)(4x + 5(/). 28. (3fl + 6 - 4)(a + 6+ 3). 29. (x-3//-s)(x-2j/-4s). Art. 356; page 228. 5. (x-3)(4x + 3). 6. (2x-3)(8x-5). 7. (3x-2)(x + 3). 8. (3x-2+ V3)(3x-2- V3). 9. (6x- l)(Gx + 5). 10. (4x + 5)(2x+7). 11. (G + 5x)(7-2x). 12. (5x-3)(3x-l). 13. (5x-2+ V'6)(5x-2-\/6). Art. 357 ; pages 229, 230. 4. (x2 + x+l)(x2-x+l). 5. (x2 + 3x+l)(x2_3x+l). 6. (2a2 + 2a6-62)(2a2-2a6-62). 7. (?«2 + 3mn + 5n2)(m2 — 3?nn + 5n2). 8. (l + 3&-262)(l_36-262). 9. (x2 + 4x^ + 2«/2)(x2-4xy + 2!/2). 10. (2a2 + 2a + 3)(2a2_2a + 3). 11. (2 w2 +Sm + 4) (2 m2 _ 3 w + 4) . 12. (a2 + ax V3 - x2) (o2 _ ax VS - x2) . 13. (x2 + 3xV2 + 9)(x2-3xV2 + 9). 14. (2a2 + a& + 462)(2a2-a6 + 462). 15. (4 .r2 + 5 mx — 3 m^) (4 x2 — 5 mx — 3 ni^) . 16. (3.r2 + 3xV2+2)(3x2_3xV2 + 2). 17. (3a2 + 4o,n + 5 Jn2)(3a2 — 4am + 5w2). 18. (2 + 2n-7«2)(2-2H-7n2). 19. (4 x2 + 3 xy - 5 y2) (4 x^ _ 3 xy - 5 ^2) . 10 COLLEGE ALGEBRA. Art. 358; page 230. 1. \/2 ± V^, - V2 ± V=^. 2. 1 ± \^, - 1 ± V2. 5. 1 ± V3, - 1 ± V3. 6 ^^^^^^^ - Vl4±\^r2 Art. 362 ; pages 235 to 238. 3. 19,8; or, -^,-^. 4.3,4,5. 5. 4, 16; or, - 5, 25. 6. 8, 5. 7. 1, 2, 3, 4 ; or, 5, 6, 7, 8. 8. 16 barrels, at $ 6 apiece. 9. § 30. 10. 4, 5, 6 ; or, -4,-6,-6. 11.61 miles an hour. 12. 27 and 36 miles an hour. 13. 9 c/. per dozen. 14. $80 or $20. 15. 20. 16. 9 miles an hour. 17. Area of court, 529 sq. yds. ; width of walk, 4 yds. 18. Hind-wheel, 16 feet ; fore-wheel, 12 feet. 19. Larger, 5 hrs. ; smaller, 7 hrs. 20.24. 21. $2000. 22.8. 23.5. 24. 4. 25. 6. 26. 14400 and 625 sq. ft. ; or, 8464 and 6561 sq. ft. 27. 12. 28. 38 or 266 miles. 29. 4, - 5. 30. 70 miles. 31. A, 18; B, 12. 32. 100 shares at $15 each. 33. 42. Art. 364; page 241. 4. ±4, ±3. 5. ?, -2. 6. ±3 ±2 7 ? -1 8 -^ 1_ 9. ±lV3, ±1. 10. -1, -i- 11. vl2T. 4. 12. 25, ^. ^3 2 '16 13. -243, v'265. ■"■ '• •^^" ' ^^^- u. (..,., {-ff ■". le, (P?- n. I (If- 18^ .l(-r\^. 19. -32.1,^2 4 \'6J 8'\ 74y ■ "' 4 Art. 365; pages 244, 245. 3. 5,-l,±3. 7.1,3,-5,-7. 8. ± 2, + ^30 9 2 ^ 3^a/505 2 2' 4 10.-1,-2,^,^. ll.±3,±3V2. l2.-l,-2,±^. 3 3 2 13. 1, 15 14. 3 17. 2, 3, 1, I- 2 3 ANSWP:ilS. .-21. 15.6 13 18. -Sf-lO, 11 -9. 16. 4, -6, ±1. 1 ± V- 4679 19. 0, -2, -1±2V-1. 22 i, -1, ^^^^ . 23 3 3 50 25. -2,-3,-4,-5. 5 ± V2T -2, -VIS. 21 1 3± V;^503 a±Va2_462 24. 1, 1±2\/15. 27. -1, 39. 7, 1 31. 0, 2, - 7, . 19 3. 0, 2 + 2a±2V\ + 2a. 29. 2, 1, 9W=31. 4 8 3± a/5 rl 7,t 33. a + 65, a + Sn^. 34. 1, 2, 1 o 2 7 _10 3' 3' 35. 3. -1 2± V61 36. t, -2, - 5' 11 37. 5± V2l -5±\/2T ^ V7, ± — v'- 665. Art. 367 ; pages 247, 248. Note. In this and the four following articles, the answers are arranged in the order in wliich they are to be taken ; thus, in Ex. 2, the value a- = 3 is to be taken with y = + 5, and x= — S with y = ± 5. ,77 99 3. a:, or ; w, ± - or ± — '2 2 ^' 2 2 2. a-, 3 or — 3 ; y , ± 5 or ± 5. 4. ar, 2V3or-2\/3; y, ±2\/2 or ±2V2. 5. x,2a — boT — 2a + b;y,±(a + 2b)ov± (a + 26). 6. a-, -or--; y, ±-or +-• 7. ar, 3 or - 3 ; y, ± 2 or ± 2. '2 2'^' 3 3 ' ^' r, 1 or ; w, — 2 or 25 ^ 25 T, 6 or - 9 ; y, - 9 or 6. Art. 368; page 249. 46 4. .r, 8 or — 7 ; y, 7 or — i X, 10 or — 3 ; y, 17 or 4. T, 2 or — 5 ; y, 5, or — 2. COLLEGE ALGEBRA. x,6ot1; y,l or 6. x,a-\-l OT —a ; y,a or — a — 1. X, 4; y, 6. 1 2 12. a;, 3 or — ; y, 2 or - — • 3 -^ 3 10. X, a ±b; y, a + h 14. :r, 2«-3 or 10a 13 13. X, 4 or y, 3a — 2 or 12 12 7 ' ■" 7* 126 a -169 Art. 369 ; pages 251, 252. 4. r, 8 or 6 ; !/, 6 or 8. 10. x, - 1 or - 8 ; ^, - 8 or - 1 5. X, 1 or - 11 ; y,-\\ or 1. 11. x, 2 or - 16 ; y, 16 or - 2. 6. x, — 1 or — 4 ; ?/, 4 or 1. 12. x, 8 or — 2 ; y, — 2 or 8. 7. X, 8 or 3 ; y, - 3 or - 8. 13. x, 17 or 4 ; j/, - 4 or - 17. 8. X, ± 3 or ± 2 ; y, ± 2 or ± 3. 14. x, ± 11 or ± 8 ; y, + 8 or + 11 9. X, 5 or - 3 ; !/, 3 or - 5. 15. x, 2 or - 10 ; y,- 10 or 2. 16. X, a + h ; y, a ^ b. 17. X, 3a + 2 or 2a- 3; y, 2«- 3 or 3a + 2. 18. X, - 6 or - 25 ; y, 25 or 6. 19. X, a 4- 1 or — a ; y, a or — a — 1. Art. 370; page 253. 2. X, ± 4 or ± - V2 ; y, ± 1 or + - y/2. 3. X, ± 3 or ± - \/3 ; v, + 1 or + i Vs. 3 ^ 3 4. X, ± 2 or ± - \/5 ; V, + 3 or ± - Vs. 5 -^ 5 5. X, ± 4 or ±1 ; y, =f 2 or + 3. 6. X, ± 4 or ± ? V? ; !/, ± 3 01 T - v^. 7 ^ 7 7. X, ± 2 or ± A \/-13; y, + 1 or ± A 13 -^ 13 8. -r, ±5or il6; y, + 1 or + -• 1 2 or ± — V3 ; y, ± 5 or + — V3. 15 15 1 o 3 9 ± 1 or ± 2 ; V, ± - or ± — ' •" -1 8 ANSWERS. 13 Art. 371 ; pages 256 to 258. 6. .r, — or - 2 ; y, - or - 5. 11 •^ 11 7. X, 3, 2, or - 3 ± VS ; y, 2, 3, or - 3 + VS 8. :r, ± 2 or ± - a/5 : y, + 3 or + - V'5. 5 ' -^ 5 9. z, ±2 or ± 14; y, +3 or t5. 10. X, 12. X, 11. x,^ OT-3; ij,±-OT ±V3. '2 ' -" 2 4, -, 8, or - -; y, 3, 8, -5, or 16. 13. x, 1 or 1; y, -1 or - 1. 1111 - or - : (/, or — 2 9 -" 9 2 3 14. X, 2, 1, or 1,-2, or 7 4- "' 4 - 3 ± V^^SS 15. X, 3, -6, 1, or --; y, - 1, 1, - -, or -• 2 2^ 4 4 16. X, a — b or b — a; y, a + b or 2 a. 1111 1 1 4 3-^0 4 17. X, 19. X, 2 or 16 ; y, 2 or 18. X, ± a + b; y, ± a — b. ±3 or ±2; y, t 2 or +'3. 21. X, ± 2 or ± i VT ; y, ± 5 or + — V?. 7 ^ 7 22. X, + (2a-6) or +(a-2b); y,±{a-2b) or ±(2 a — 6). 23. X. 2, 1, or 3-AV=X9 ; ,, i. 2, or ^ ^ ^^ . 10 6±V'193 . „, 63±3vi;93 • y, — 5, —21, or ' ' 3 4 - - - 4 343 or - 125 ; y, 125 or - 343. 26. x, 2 or - 1 ; y, - 1 or 2. 27. X, 29. X, 31. X, 2 or ; w, — 1 or 2. 27 27 ^ 9 9 ^ X, — , , or 0: ?/, -, -, or 0. '2 4 -^ 2 4 3 or — 7 ; y, — 1 or — 21. 30. x, 2 a or — a ; y, 2 i or -.- b. 2, 0, or ± \/2 ; y, 2, 0, or 2 + V2. 32. x, 3 ; y, - 2. 33. X, ± XflliJ: or ; y, ± Va'^ + 1 or 0. 34. X, ± 3 or ± V- 7 ; y, 2 m- 6 36. X, — 1 or — 1 ± VlO ; y, - 1 or 9 + 2 VTO, 37. X, 2, 1, 6, or - 3; y, 1, 2, - 3, or 6. 35. X, ± 1 or ± — : v, + - or ± — 59^ 3 31 14 COLLEGE ALGEBRA. 38. a:, 3, - 1, - 1, or - 2 ; y, 1, - 3, 2, or 1. 39. X, 3, 4, or — 6 ± V43 ; y, _ 4, - 3, or 6 ± ViS. 40. X, 8, 2, or 2 ± 4 V^^; y, 2, 8, or 2 + 4\/^^. 41. a:, 2 or 1 ; i^, 1 or 2. 43. ar, 4 or — 5 ; y, — 5 or 4. 42. a:, ± - or ± - ; y, ± 1 or + — '2 7^ 19 44. 1 or — ; V, 2 or - —. 20 -^ 60. x,±^-y/2; y,±--\/2: '6 ' ^' 2 V2. Art. 372 ; pages 259 to 261. 1. ± 9 and ± 5 ; or, ± V53 and ± v'53. 2. 8and±3;-8and±3; 3\/^and±8\/^; or,-3\/^and±8 V^. 3. Length, 10 rods ; breadth, 6 rods. 4. 4, 7. 5. 8, 5. 3 -U5 14 6. Duck, #1.75; turkey, $2.25. 7. 21 or 12. 8.4,9. 9. 10. Length, 150 ft. ; breadth, 100 ft. 11. Length 16 rds., breadth 10 rds. ; or, length 13V rds., breadth 12 rds 12. Rate in still water, 4 miles an liour ; rate of stream, 2 miles an hour. 13. $3. 14. A, 6 hrs.; B, 3 hrs. ; C, 2 hrs. 15. Distance, 504 miles; A, 24 miles a day; B, 18 miles a day. 16. 7 and 5 ; 1.3+V^^^^-1,3-V-71. 2 2 17. 5 and 2 ; ^^^7+V-3O3^^^,7-V-303. 18. 3 and 1 ; - 1 and - 3 ; 1 + V^ and - 1 + V^ ; or, 1 - V^ and — 1 — V— G. 19. 3 miles an hour. 20. -150 miles. ANSWERS. 15 21. A, 8 miles an hour ; B, 6 miles an hour. 22. A, 11 hrs. ; B, 22 hrs. ; C, 33 hrs. Art. 375 ; pages 266, 267. 4. x=9, y = l; x = 6, y = S; or, x = 3, ij = 5. 5. X = 4, y = 13 ; or, X = 8, »/ = 6. 6. x = 3, y = 5. 7. x = 4, y = 122; x = 13, y = 9\; x=22, y = 60; or, x = 31, y = 29. 8. X = 3, 2/ = 50 ; X = 10, y = 26 ; or, X = 17, y = 2. 9. X = 3, y = 2. 10. X = 3, y = 89 ; or, X = 14, y = 43. 11. x=78, !/ = 4; x=59, y=12; x = 40, y = 20 ; x = 21, y = 28; or, X = 2, y = 36. 12. x=r2, 3/=l. 13. x=5, j/ = 2. 14. x = 8, y = 6, 15. x=3, y=ll. 16. x=7,.y=l. 17. x = 9, y = 4. 18. X = 2, y = l, z = S. 19. X = 2, y = 31, 2=3; or, x = 13, y = l, z = 3i. 20. Either 1 and 17, 3 and 12, 5 and 7, or 7 and 2, fifty and twenty cent pieces. 21. Either 6 and 3, or 1 and 7, florins and half-crowns. 22. 1^ and ?; 1^ and I ; or, ^ and !?• 9 5 9 5 9 5 23. Either 1, 18, and 1; 4, 10, and 6; or, 7, 2, and 11, half-dollars, quarter-dollars, and dimes. 24. 5 pigs, 10 sheep, and 15 calves. 25. Either 50, 2, and 33 ; 28, 21, and 50 ; or, 6, 40, and 67, half-crowns, florins, and shillings. Art. 398; pages 276 to 278. 5. 9. 6. 25. 7. 5. 8 a-S^ 9 21. ^^ ^_3 35 27 3 a + 3 2 11. Srtjui. 12. 3, --• 13. 2,-3,^. 14. ±~. 2 11 9 b 15. X, ±a26; 2/, +62a. 16. x, 32; y, 18. 17. 25, 11. 18. 31, 17. 19. 6,8. 20. 9,3. 21. ^^, ^i^- 26. 3:4. 27. «:-6. 28. ^ = -JL^ = ^- 29. - = ^ = i- 30. A,§105; B,$189; C,.$270 17 -23 -27 a b c !«) COLLEGE AL(;EBRA. 31. First, wine : water =1:2; second, wine : water = 2:1. 32. 5 : 4. 33. 6, 4, 12, 8 ; or, 44, -110, -64, 160. Art. 413; pages 283, 284. 3. 6. 4. 3/ = |.2. 5. 70. 6. 14. 7. — • 10 8. J-V,. 9. 788-rVft. 10. 5x, -1 X n. f -¥■ 13. y=S + 5x-4x^. 14. 9. 15. 6 in. 16. 3. 17. 10. 18. 8. 19. 15(\/3-l) in. Art. 418 ; page 286. 2. I, 147 ; ;V, 1425. 3. /, - 131 ; S, - 1496. 4. I, 36 ; S, - 264. 5. l,-^; S,-78. 6. /, 229. ^,2925. 7 / 61 3439_ 4 4 4 ' 2 ' ' 6 8. Z, - — ; S,- 165. 9. /, - 6 ; ,S', - — • 4 5 10. /, 420+196; 6', 198a + 636. 11. / 17// -8a: o 80,y-35x^ 2 ' ' 2 Art. 419; pages 289 to 291, 4. a, 1 ; ,9, 540. 5. a, 7 ; /, -69. 6. d,3; S, 552. 7. f/, -5; 7,-145. 8. n, 35 ; d,-- 9. a,§: (^,_J-. 4 5 15 10. /,~;n, 16. 11. n, 22;^,i. 12. a, -3;/, 5. 13. a, - - ; n, 14. 12 2 5 .. 14. a,l;d,-l. 15. n, 43 ; J, - 1. 16. Z, -12;rf,_l 2 3 3 ' 3 17. a = -l, n=16; or, a = — , n = 25. 18. n, 15 ; /, — • 3 15 .4 21 ^^ 2.9-fm . ^^^ 2(.S-a«) « ' n(n-l) 22. a_ 2^-»(n-l)«? . ^ _ 2 .9+ » (n - 1) t/ 2« ' 2h 23. n^ ^-" + ^ - ^._ (/+«)(/-a + - ^•"■"'^•'•il- '■■'•^S.L- 18 COLLEGE ALGEBRA. 8. ,-5.^-19171 §,^.4039. 2 384 2 384 9. r,l;„,9. 10. ^'~324'"'^' 11. a, 3; n, 7. 12. r, 1 ; n, 10- 13. 1 «+('--l)'^. 14. r-'^-«. 15. a- r S-l rl-(r-l)S. 16. a- '-,8- ^(f-^). n-g-'^C-^);/- Sr'^-Hr-1) r"-i r«-i(r-l) r" - 1 ,-»_l 18 , "-1/7. o /"~-a^ 18. r ^ 6 ^ _^. Art. 429 ; page 300. 2. 9. 3. 'A 4. -^ 5. -5. 6. 14. 7. 12. 8. - 2 5 6 5 55 Art. 430; page 301. 9 Q 64 'io' ''• 147" 2. 8 3 11 4 25 g 581 g 916 y 2284 11 '27 '36 ■ 990 ■ 8326 2475* Art. 433; page 302. 1. 7- = 2. 2. r=-3. 3. r=±2. 4. r=±5. 5. r=-4. 6. r=±?. 7. "'+^^^. 8. 5. 9. 4x2-9(/2. lo. ?. 3 6 Art. 434; pages 303, 304. 2. 3. 3. 5, 10, 20, ^0 ; or, - 15, 30, - 60, 120. 4. 5, 15, 45 ; or, 40, - 20, 10. 5. ± 4. 6. 3100 ft, 7. 2, 4, 8, 16; or, 810 _ 540 300 _240, g_ _81_. 13 13 13' 13 8192 9. 3, 9, 27; or, 25, -35, 49. 10. 8, 4, 2 ; or, - 8, 4, ^2. 11. -3,4, 11; or, 13,4, -5. 12. 4, 2, 1. 13. A, $108; B, §144; C, $192; D, §256. 14. -4, 1,6, 30; or, 8, 1, -6, 36. 15. 3. 16. a,6;r, ±2. 5700 1710 513 ■' 139' 139' 139' 17. 12, 18. 27 ; or. 5^. i^, *^^-^ 18. ""I^/^ ANSWERS. 19 Art. 441 ; pages 307, 308. 3. A. 4. -A. 5. ^. 6. -A. 7. -A. 11. _72. 74 19 169 29 142 5 12. 5?.:^. 13. . ^ -. 14. "^(" + 1) . a2 + 62 na—nb — a + 2b bm + 2a — b ,r XV x// xi/ ,« o 7 ,„ 3 16. 3,--- 17. -— • 18. 24, 2x-y3x-2y ix — Sy 2 19 19. 1 1 1- or 1 i I 20. MzlLziiO. 2 5 8 8 5 2 mq — np Art. 446 ; pages 313, 314. 9. zS"" - 10 x*™ j/2" + 40 x^'" y4™ - 80 x-^"" (/C» + 80 x" ?/8« - 32 ^/W". 12. x-W _ ^ x-8 v3 + - X-6 y6 _ 12 x-4 y^ + — X-2 V^^ - — lA^ 3 ^9 ^ 27 81 •^ 243 -^ 13. J b- ¥ + 7 J b- ¥ + 21 at 6-2 + 35 a5 6- S + 35 a- 2 6t + 21 a"! 62 + 7a-t6¥ + o-2-6¥. 14. o« + 16 a¥ + 96 a V + 256 a '^ + 256 aJ. 15. 32.r5^-t — 40xty-i + 20x^yi -bxiy^ + - x-^yi - — x~^y^. 8 32 16. a-18 - 2 a-15 xi + - a-12 x - — a'^xt + A a-6x2 - A a-3;j.t + — x^. 3 27 27 81 729 17. x3 + 15 x"V- !/- 1 + 90 xt j/" i + 270 xt ?/" t + 405 xt ^- 1 + 243 y-^. 18. 81 a-362 _ 108 a-2 6-2 + 54 a-^-^ - 12 6-w + a6-i*. 19. a36-3 + 12 a26-2 + 60 a6-i + 160 + 240 a-^b + 192 a-262 + 64 a-W. 21. x6 + 3x5+6x* + 7x3+6x2+3x+ 1. 22. 27 ac - 27 a^x - 99 a4x2 + 71 a^x^ + 132 a^x^ - 48 ax^ - 64 x^. 23. 1 +8x + 20x2 + 8x3-26x4-8x5 + 20x6_8x7 + x8. 24. 10 xi2 + 32 xio + 96 x9 + 24 x^ + 144 x'^ + 224 x« + 72 x^ + 217 x* + 228 x3 + 54 x2 + 108 x + 81. 25. 1 _ 5x + 15 x2 - 30x3 + 45x4 _ 51 ^^ + 45 x« - 30 x^ +15x8-5x9 + x"'. 26. .ri" + 10 x^ + 30 xs - 120 x« - 48 x^ + 240 x* - 240 x^ + 100 x - 32. 20 COLLEGE ALGEBRA. Art. 447 ; page 315. 2. 462a5a-6. 3. ^368 vr\ 4. -792^/", 5. SOGOaS. 6. 84 ''^ 6m n +9 7. -5005x " . 8. ^a-%'\ 9. 219648 a;-6(/i 10. - 486486 a-3.r25. H. 126720. 12. -^z"a?\ 13. 110565.riV. 14. -i^.r". Art. 468 ; page 327. 1. 2-ll.r+33.T2-99.r3+-... 2. 1 _ 4.t + 20.r2 - 92.r3 + ... 3. 24^8 16 3 3^9 3 5. 1_.,_1,-2_1,3+.... 6. «_26-3^-3^+.... 2 2 2 a a2 7. ^2:r2 16 .r6 ^ 64 .rio ^ 3 9 ^ 81 ^ 9. 2«2_ll-l-^_Ai!i_.... 11. Convergent. 4 a" 32 aW 768 aie 12. Convergent. 13. Convergent. 14. Convergent. 15. Convergent, 16. Convergent. 17. Divergent. Art. 471; pages 331, 332. 2. l-2.r + 2.r2-2x3 + 2.ri 3. 2 + 11 .T + 33x2 + 99^.3 ^ 297 .r* + ... . 4. 3-19x2 + 95.r*-475.r6 + 2375x8 ..2 , 4 , , 8 5 , 16 - , 32 „ , 3 9 27 81 243 6. l-2x+2x3-2.c* + 2.r6+.... 7. x-x^-2x^-5xi-l2x^ 8. 2-X+ 3x2-x3 + 3.r*+ ■■■ • 9. 1 -2.r + 5.x-2- 16.t3+ 47x1+ ..- 10. 2 - 7 x + 28 x2 - 91 x3 + 322 x* + . • • • 12. l + lx + 1..2_§..3_ll.,5+.... 3 3 9 27 13. 2 4 8 16 ^32 ^ 15. 2 _„ , 8 _i , 32 , 128 , 512 , - X 2 4- _ .c 1 4 ^ x-\ t2 3 9 27 81 2 13 16. x-J + 3 + 2x'-5x2-16.,' ANSWERS. 21 17. a;-2-.r-i-2.r + 2x2-4x3+ ••• • la 3 _3 1 _2 1 _i , 31 , 23 , 2 4 8 16 32 ^ Art. 472 ; page 333. 2. l + .r_ix2+l.r3-^x-i+ ... . 2 2 8 2^ 8"^ IG'' 128 "^ 4. 1- 2 5. l+l:r-^:r2+-:r3-ilx4+ ... 2 8 16 128 6. l-^x-'^x^- — x^- — xi 3 9 81 243 7. 1+1.. + 2,._L3,3.+ A,4+... 3 9 81 243 Art. 474; page 335. 4. ^ i—. 5. ^ 1 1 2.r + 5 2.r-5 .t 3x+5 2x x+S x-S 2 I 3 ^ 2 ^ 3a 2a g 1 , 2 T 3x+l 2.1--5 ■ x+a x-ia ' 3 + ix 3~x -J_+^ ^. 10. -1 ^ ^ + -1-. x+l 2:f+3 2z-3 .r+2 .r-2 x+1 x-l 11. 1+ V2 ^ 1-V2 2X-5+V2 2X-5-V2 Art. 476; page 337. 2 2 x+5 (:» 6 23 3 • + 5)2 1 ,r+l 1 (x+iy (.r+l)3 4 , 14 4 -2 (x-2)2 (x-2)3 2 4 3 3x+2 (3a- + 2)2 (3.r + 2)3 1 1 1 5(5ar-2) 5(5a;-2)3 x + 1 (z + 1)2 (x + 1)3 (r+l)^ 8. -2 i__.+ _3 1_. x-l (x-l)2 (x-l)3 (x-1)* 9. 1 27 27 2(2x-3) 2(2 X- 3)3 (2x-3)* 22 COLLEGE ALGEBRA. Art. 477 ; page 339. „23 5 ,515 X x+2 (x+2)^ X x^ x+l (x +!)■'= -1— 1 + § 5.1 + -^+^- + -^— x-2 2(2x-3) 2(2x-3)2 x x-1 x-2 (x - 2)2 1_2 3^_^L_ 7 5_1,2 5 4 X X2 x3 X + 5' ■ X X2 x3 X+1 (x + 1)2 Art. 478 ; page 340. 2x-5- 17 2 2. 3 + 1 x + 2 5 18 2x-5 ' 2x+l (^+2)2 (a : + 2^3 3. 6.^.+ l-l + 4. X x-' x^ 1 x+1 '■'^ ' x + 1^: 2 (-r+l)2 ^.4i 1 8 . (X-1)2 5. 2x2-7-^ + 1- X x3 5 x-1 Art. 479 ; page 341. 2. 7 5x-3 :c _ 1 x2 + X + 1 3. .r2 + 2 1 x2 + X + 2 3 x + 2 2 x-10 x-2 x2+4 ^■J ,.^ + 6 3x-4 + X + 1 x2 - X + 1 ^3- 2 3x + 4 x-3 x2 + 3 7. - -l^h X X+1 . ^X2+1 (x2+l)2 Art. 481; page 343. 1. x=y-y2 + f-i/+ •••• 2 a; = ^ y J. _ ,,2 _ ^ ,,3 _ ^"^ j,4 I ... 3^ 27-^ 243-^ 2187^ 3. a: = 2(/ + 6/ + ^»/3 + 98/+ .•• • . x=(^-l)-l(^-l)2 + |(^-l)^-^( ■2- (ij) \-Q^ IVI306043 Q/l IS 3 THE UNIVERSITY OF CALIFORNIA LIBRARY <,'* ^' A/ \ y ^