UC-NRLF ^B 531 fll5 V \i, c-.t )}v(i'- I '■■) )■)'■■■ - ;.i ■,(li'.i5KA LIBRARY OF THE University of California. GIFT OF Class I I APPLETOJ^S' MATHEMATICAL SERIES NUMBERS UNIYERSALIZED AN ADVANCED ALGEBRA BY DAVID M. SENSENia, M.S. PROFESSOR OF MATHEMATICS, STATE NORMAL SCHOOL, WEST CHESTER, PA. PART SECOND NEW YORK, BOSTON, AND CHICAGO D. APPLETON AND COMPANY 1890 Copyright, 1890, By D. APPLETON AND COMPANY. PEEFAOE NuMBEKS Universalized is believed to embrace all algebraic subjects usually taught in the preparatory and scientific schools and colleges of this country. For con- venience, it is divided into two parts, which are bound sepa- rately and together, to accommodate all kinds and grades of schools sufficiently advanced to adopt its use. Part Second is treated in five chapters, as follows : One embracing serial functions, including development of func- tions into series, convergency and divergency of infinite series, the binomial formula, the binomial theorem, the exponential and logarithmic series, summation of series, reversion of series, recurring series, and decomposition of rational fractional functions ; one treating of complex num- bers, graphically and analytically, including fundamental operations with complex numbers, general principles of modulii and norms, and the development and representa- tion of sine, cosine, and tangent ; one embodying a discus- sion on the theory of functions, including graphical repre- sentations of the meaning of the terms independent and dependent variables, continuous and discontinuous func- tions, increasing and decreasing functions, and turning values and limits of functions, and also a treatment of differentials and derivatives, and maxima and minima val- ues of functions ; one treating of the theory of equations, including a discussion of the properties of the roots, real and imaginary, of an equation, methods of determining the commensurable roots of a numerical equation, Sturm's theorem for detecting the number and situation of real roots, Horner's method of root extension. Cardan's for- 183681 iv PREFACE. mnla for solving cubic equations, and a short treatment of reciprocal and binomial equations ; one treating of de- terminants and probabilities, so far as these subjects are of interest and value to the general student. The volume closes with a supplementary discussion of continued frac- tions and theory of numbers. The aim of the author in preparing this part of his work has not been so much to give completeness to the various subjects treated as to lead the student to a comprehension of the fundamentals of a wider range of subjects, and to cultivate in him a taste for mathematical investigation. It is believed that the plan adopted will give the general student a broader and more practical knowledge of algebra, and will lead to better results in a preparatory course of study for the university than would a completer treatment of fewer subjects requiring an equal amount of space in their development and more time in their mastery. While a sufficient number of examples have been placed under each head to offer opportunity for the application of the principles and laws developed, there will not be found an unnecessary multiplicity of them to retard the progress of the pupil in his onward course. In conclusion, the author desires to acknowledge his in- debtedness to the English authors, Hall and Knight, Chr^s- tal, Aldis, Whitworth, and 0. S. Smith, whose works he frequently consulted, and from which he obtained many new and valuable ideas. David M. Sensei^ig. Normal School, West Chester, Pa., \ December 2, 1889, \ • OOIfTElsrTS, CHAPTER IX. PAGE Serial Functions 315-353 Definitions, 315. Development of Functions into Series 816-319 Theorem of indeterminate coefficients, 316. Expansion of rational fractions, 316-318. Expansion of irrational functions, 318. Convergency and Divergency of Infinite Series . . . 319-323 General definitions, 319. Fundamental principles, 320. Theorems, 321, 322. Exercises, 323. The Binomial Formula 324-326 Development of, 324. Applications, 325, 336. The Binomial Theorem 326-332 For positive exponents, 326, 327. For any rational exponents, 328- ) = nr«— I, 328. General demonstration of binomial theorem, 328-330. Numerous corollaries and inferences, 327, 330-332. Exercises, 332. The Exponential Theorem 333 The Logarithmic Series 334-337 Development of, 334, 335. Computation of logarithms, 335, 336. Principles, 336, 337. Summation of Series 337-344 Method by indeterminate coefficients, 338. Method by decompo- sition, 339. The differential method, 339-342. To find the (n + l)th term of a series, 340. To find the sum of n terms, 341. To inter- polate terms, 342. Exercises, 343. Reversion of Series . . 344 Recurring Series 345-349 Definitions, 345. To determine the scale of coefficients, 346. To find the snm of n terms, 347. Exercises, 349. Decomposition of Rational Fractional Functions . . . 350-353 Definition, 350. Principles, 350-352. Exercises, 353. vi CONTENTS. CHAPTER X. PAGE Complex Numbers 354-365 Graphical Treatment 354-361 Definitions, 354-358. To add complex numbers graphically, 359. To multiply a complex number by a rational number, graphically, 360. By a simple imaginary number, 360. By a complex number, 361. General Principles of Complex Numbers ..... 361-362 Problem. To find the value of e^+y* 363 Graphical Representation of sin. y and cos. y , . . . 364-365 CHAPTER XI. Theory of Functions 366-388 Definitions, 366-370. Graphical Representation of Functions 370-374 Exercises, 374. Differentials and Derivatives of Functions .... 374-382 Definitions, 374. Principles, 375-379. Exercises, 380-383. Concrete applications, 380. Successive derivatives, 381. Factorization of Polynomials containing Equal Factors . . 382-383 Maxima and Minima of Functions 385-388 CHAPTER XII. Theory of Equations 389-422 Introduction, 389. Normal forms, 390, 391. Divisibility of equations, 393. Number of roots, 393. Relation of roots to coefllcients, 394. Imaginary roots, 395. Fractional roots, 396. Relation of roots to signs of equations, 396. Descartes' rule of signs, 398. Limits of ■ roots, 399. Equal roots, 400. Commensurable roots, 400-403. In- commensurable roots, 403-409. Sturm's series of functions and fundamental principles, 404-406. Sturm's theorem, 406-409. Hor- ner's method of root extension, 409-415. Cubic equations, 415-419. Cardan's formula, 416-419. Recurring equations, 419-422. Bino- mial equations, 422. CHAPTER XIII. Determinants 423-441 Introduction, 423-425. Properties of, 425-430. Development of, 430- 432. Additional properties of, 432-435. Multiplication of deter- minants, 435-437. Solution of simultaneous equations by determi- nants, 438-440. Conditions of simultaneity, 440, 441. Sylvester's method of elimination, 441. CONTENTS. vii PAGE Probabilities 443-453 Definitions and fundamental principles, 442-444. Exclusive events, 444 - 44 6. Expectation, 446. Independent events, 447, 448. Inverse probability, 448-450. Probability of testimony, 450, 451. Exer- cises, 451-453. SUPPLEMENT. Continued Fractions 454-463 Definitions, 454, 455. Formative law of successive convergents, 455- 457. Properties of convergents, 457-460. Keduction of common fractions to continued fractions, 460, 461. Reduction of quadratic surds to continued fractions, 461, 462. Reduction of periodic con- tinued fractions to simple fractions, 462. Approximation to the ratio of two numbers, 462. Exercises, 463. Theory of Numbers 464-482 Systems of notation, 464^i67. Divisibility of numbers and their digits, 468-470. Even and odd numbers, 470-473. Prime and com- posite numbers, 473-477. Perfect squares, 477^79. Perfect cubes, 479, 480. Exercises, 480-482 PAET SECOND. CHAPTER IX. SERIAL FUJVCTIOJTS. I. Definitions. 662. Any expression containing a variable is called a function of the variable. Thus, ax -{-!?, a x~\ Va-\-a^, a", log. {a -\- x), and a-{-l)x-\-cx^-{-doi?-\- etc. , are functions of x. 663. Any series containing variable terms is called a serial function, 664. The expression f{x) represents any function of x, and is read function x, 666. When two or more functions of the same variable are used in a discussion, modified forms are used for dis- tinction ; as, 1. f{x), F{x), (x); read, / minor, f major, phi functions of x. ^' f (^). /" i^\ f" (^) ; read, / prime, f second, f third functions of x. 3. /i (x), /2 (x), fs (x) ; read, / one, f two, f three functions of x. 316 ADVANCED ALGEBRA, Development of Functions into Series. Theorem of Indeterminate Coefficients. 566. If A-\-Bx-{-Ca^-\-D^etc. := A^-^- B^x-{-C^x^ -\- DiO^-\- etc., for any assigned value of x from — cx) to + 00, atid A, At, B, B^, C, C^, D, D^, etc., are independ- ent of X, then will A = A^, B = B^, C=^Ci, D~ D^ etc. Demonstration: Given A->c Bx^- Cx^ -{-Dq? + etc. = A^ + B^x + CiX^ + D^a^ + etc., (A) for any assigned value of x. Let a; = ; then A = Ai. Therefore, A = Ai for every value of x. (1) Subtract (1) from (A), Bx + Gx^ ^Da?-^ etc. = BiX ^t CxX^ + D^a? -\- etc. (B) Divide (B) by cc, B ^Cx + Dxy^ ■\- etc. = ^1 + (7i a; + Di a;2 + etc. (C) Let a; = 0, 5 = B^. Therefore, B = Bi for every value of x. (2) etc., etc., etc. 567. Corollary l. — If A + Bx -^ Cx^ + Dx^ -{- etc. = 0, for any assigned value of x, then will A=0, B = 0, (7=0, D = 0, etc. 568. Cor. 2. — A function of a single variaUe can he de- veloped into a series of the ascending powers of the variable in only one way. For, if possible, let f{x) = a-\-'hx-\-co? -\- etc. ; and f{x) = «i + Jj a; + Ci x^ + etc. ; then will a-^-lx-^cx^ -\- etc. = ax-^-lyX-^-c^o? -\- etc. ; whence a = ai, J = Ji, c = c^, etc., and the two develop- ments will be identical. - 2. Applications. 1. Expansion of Rational Fractions. 569. A rational fraction of a single variable may gen- erally be developed into a series by dividing the numerator EXPANSION OF RATIONAL FRACTIONS. 317 by the denominator, but a more expeditious method con- sists in the application of the principle of indeterminate coefficients. 1 — x Ulustrations. — 1. Develop ^ into a series of the ascending powers of x. Let \^ = A + Bx + Cx^ + Da^ + etc. (A) 1. •{• X Clear of fractions, and arrange the coefficients of the like powers of X into columns. 1-x = A + B + A x + C + B x^ + D I x^+ etc. (B) + C Equate the coefficients of the like powers of x [566], A = l', A + B = -1, B + G = 0; C + D = 0, etc. /. A = 1, B = -2, C = 2, D = -2, etc. Substitute these values of the coefficients in (A), 2-^ = 1 - 2a; + 2a;2 - 2a;8 + etc. 1 + a; Let the student divide the numerator by the denominator, and show that the same result will follow. 570. The first term of the expansion may be obtained by dividing the first term of the numerator by the first term of the denominator, and the remaining terms by indeterminate coefficients. 2. Develop , „ in the ascending powers of x. X —J~ Xr ^^^ A 2 = »^~* + B^ + Cic + Dx^ + etc. (A) X "T X Clear of fractions and column coefficients, x^+ D + Cb ic« + etc. (B) a = a + B I x+ C + ab \ + Bb Equate coefficients, (!) B + ab = 0. (2) G + Bb = 0. (3) i) + C& = 0, etc. B = — ab, G = ab\ D = - ab\ etc. Substitute these values in (A), J— ^ = aa;-* — aft + a6'a; — a62a;' + etc. a; + &a;* 318 ADVANCED ALGEBRA, EXERCISE 87. Develop to four terms : 2x — S x-\-x^ + l 2. 3. 1-x 1 2x 1 l-x-\-x^ 5. 9. a-\-x 6x + 2x^ l-\-x-{-a^ 2. Expansion of Irrational Functions. Elnstrations. — 1. Expand to four terms Vl—x-\- x^, ^- 1 + Bx + Cx^ + Do? + etc. Put Vl — a: + a;' Square both members and column the coefficients, Equate the coefficients, + 3(7 a;8+ 2i) + 3^(7 7? + etc. (A) (B) (1)3^=- B^- (2) ^ + 3(7= 1. 3 (3) 32> + 3^C = 0. ^-8 D = W etc. Substitute these values in (A) 1 2a; + |a;«+^a:«+etc. -s/l - a; + ^2 = 1 2. Expand to three terms v8 — ^. Put V8-a;2 = 2 + ^a; + Cx^ + Z>a:3 + ^a;* + etc. Cube both members and column coefficients. 8-a;8 = 8 + 13^ I a; + 13 C ' + 6^ a; 2 + J53 + 13i> + 13^(7 a:3 + 13^ + 3J52(7 + 6(7« + 13^i> Equate the coefficients, (1) 135 = 0. (3) 13 C + 6-B2 = - 1. (3) 135C+13i> + 53 = 0. . (4) 12jE^+3J52C7+6C8 + 135i> = 0. ... 5 = 0, C=-l, i> Substitute these values in (A), 0, ^= -So5» etc. V8 ^ = ^-lV-38-8^-^^- (A) xl^ + etc. CONVERGENCY OF INFINITE SERIES. 319 EXERCISE 88. Expand to four terms : 1. a/4 — a; 4. Vl + a; 7. Va-{-x 2. ^l^X'-x' 5. a/27 + 0,-2 8. Vo^^ 3. V9 + a;-3a;2 6. Vs + M^ 9. VoH^ Convergency and Divergency of Infinite Series. General Definitions. 671. The limit of a series is the limit of the sum of n terms of the series, when n is indefinitely increased ; that is, when lim. n= co, 672. A series is convergent when its limit is a finite constant, including zero, 673. A series is divergent when its limit is infinity. 674. A series is indeterminate when the sum of n terms is finite hut does not approach any definite yalue as n is indefinitely increased. Thus, 1 — 1 + 1 — 1 + 1 — 1 + is indeterminate, since, when n is even the sum is 0, and when n is odd the sum is 1, however great n he taken. 676. For convenience of discussion, the following nota- tion will be adopted : 1. The terms of a series will be represented in order by Wi, Uz, Us w„, Un+i 2. The sum of n terms will be represented by Z7„, so that Un = Ui-{-U2-\-Us-{- + w«. . 3. The limit of the series will be represented by U, so that [7= wi + t^2 + % + .... + «^n + w«+i + 320 ADVANCED ALGEBRA. Fundamental Principles. 576. X No series whose terms are all of the same sign can he indeterminate. For either the sum of n terms increases numerically without limit as n is increased indefinitely, or else it can never exceed some fixed value which it approaches as a limit. Such a series is, therefore, either convergent or divergent. 577. 2, A series of finite terfns whose signs are all alike is divergent. For, if we let a represent the numerical value of the smallest term, then, numerically, U>na, whose limit is c» , when lim. n = co and a is a finite quantity. Thus, the series 1 + 2 + 44-8 + 16 + .... is divergent. 678. 3. If a series is convergent it will remain con- vergent, and if divergent it will remain divergent, if any finite number of terms he added to or subtracted from the series. For, the sum of any finite number of terms is finite, and, therefore, can not change the nature of the limit of the series when combined with the series by addition or subtraction. 579. Ji.. If a series is convergent when its terms are all positive, it is also convergent when its terms are all negative, or some positive and some negative. For its limit will have the same numerical value when its terms are all negative as when they are all positive, and will be numerically less when the terms do not all have the same sign as when they do. It must not be inferred from this principle that a series THEOREMS. 321 is necessarily divergent when its terms are not all of the same sign, if it is divergent wh«n they are alike in sign. Such may or may not be the case. Theorems. 580. I. In order that a series may he convergent, the limit of the {n + l)th term, and the limit of the sum of . any numder of terms beginning with the {n + ^)th term must he zero, and conversely. DemonBtration : If a series is convergent, then ultimately, if ti is indefinitely increased, (1) U-Un = o [4981 (2) U- Un+l = o (3) U- Un+^ = o (4) U- Un+z = o Subtract (1) from (2) ; (1) from (3) ; (1) from (4), etc. ; then, (a) Un — Un + \ = o ; or, Un+\ = o ; whence, lim. w^^ i = (6) Un — Un+i = o ; or, Un+\ + w„4.8 = o ; whence, lim. (Wn+l + Wn+j) = (c) Un — Un + 3 = o ; or, Un + 1 + Un + i + Un+z = o ; whencc, lim. (Un + l + Un + 1 + Un + z) = etc., etc., etc., etc. 581. II. If each term of a series whose terms are alter- nately positive and negative is numerically greater than the following term, the series is convergent. Demonstration : Let U=ui — Ui + U9 — Ui+ ± w« T Un+i. . . ., in which -Mi > 'Wa > Ws > '^4. . . ., be the given series. (1) U — {Ui — Wa) + (Ws — Ui) + (We — w«) + etc. (2) U = Ui — (Ui — Ws) — {Ui — -Mb) — etc. From (1) it is evident that U is positive. From (2) it is evident that, since U is positive, U<,Ui. .'. U approaches Wi or some quantity less than Ui as a limit, and the series is, therefore, convergent. 582. III. A series is convergent if after some particu- lar term the ratio of each term to the preceding term is less than unity. 322 ADVANCED ALGEBRA. Demonstration: The most unfavorable case to convergency sup- posable, under the conditions given, is evidently the one in which all the terms have the same sign (say plus) and all the ratios described are equal and each equal to the greatest of them. This is, therefore, the only case that needs proof. Let r be the greatest ratio after the nth term, but < 1 ; then, Un + ^n + l + t^n + 2 + ^n + S + etC. = W„ + W» 7* + Wn r« + CtC. = q— ^?— 1 — r [499, P.] = a finite quantity. Therefore, the whole series is con- vergent [578J. 683. IV. A series of all positive or all negative terms is divergent, if after some particidar term the ratio of each term to the preceding term is equal to or greater than unity. Demonstration : The most unfavorable case to divergency, and the only one that needs investigation, is the one in which all the ratios described are equal and each equal to the least of them. Let r be the least ratio after the nth term, but = or > 1 ; then, «*n + w«+i + -Wn+s + 'Wn+s + ctc. is divergent [577] ; and hence the whole series is divergent [578]. 584. V. A series of positive terms is convergent if each term is less than the corresponding term of a given convergent series of positive terms. Demonstration : Let CT = Wi + w, +....+ w„ + u^^i + be a given convergent series ; and T = Vi + Va + + Vn + Vn+\ + . . . . a series in which Vi < Wi , iJa < "Wa, . . . . t^n < w„ , VnJ^\ < w„+i , From the nature of addition, it is evident that Vn < Cn ; and hence, too, lim. F„ < lim. C/« , or V l. 3. When x = l. 3. Testtheseries:i-^ + ^-^+.... for convergency. when a; < 1 ? 6. Testtheseries:i + ^ + ^ + ^ + etc. for convergency. 6. Test the series : . 1 rr^ 1 3 ^5 1 . 3 . 5 ^.7 ^+2-3+2T4-5+2TT:^-T + '*'- for convergency 1. When xl. 3. When ic = 1. Suggestion.— Lim. jj " = -j. Why! 7. Test the series : l + aj-f-a^^ + a^^H- etc. for con- vergency 1. When x = l. 2. When x< 1. 3. When x>l, 8. Test 1 + 1 + ^ + ^^^+.... for con- vergency. 324 ADVANCED ALGEBRA. The Binomial Formula. 587. The binomial formula is used to find the prod- uct of any number of binomial functions of the form of Development. {x + a){x + 'b) = x^ + (a+b)x+ab Multiply both members hj x + c, {x + a){x+b){x + c) = x^+(a + b)x^ + abx + cx^ + (ac + bc)x+abc = x^ + {a+b + c)x^ + {ab + ac + bc)x +abc Multiply both members hj x + d, {x + a) (x + b) {x+ c) (x + d) = a^+{a+b + c)x^ + {ab + ac + bc)x^ + abcx + dx^ + {ad + bd+cd)x'^ + {abd + acd + bcd)x+abcd = a^+{a + b + c + d)a^ + {ab + ac + ad + bc + bd+cd)x^ + {abc + abd + acd + bcd)x + abcd Observe the following laws in these products : i. The number of terms is one greater than the num- ber of binomial factors, 2. The exponent of x in the first term equals the num- ber of binomial factors, and decreases by unity in each succeeding term, S, The coefficient in the first term is unity ; in the second term the sum of the second terms of the binomial factors ; in the third term the sum of the products of the second terms taken two together j in the fourth term the sum of the products of the second terms taken three to- gether, etc, 4' The last term equals the product of all the second terms. Are these laws true for any number of factors ? Assume them true for r factors, so that {x-{-a){x-\-b) {x-\-m) = x^-^pi of-^-\-pz af-*+ .... Pr-\ x-\-p^, in which THE BINOMIAL FORMULA. 325 p^ = a-\-h-{- +m p^ = al)-\-ac-{- am-\-hc + bd-{- + Jm + etc. ^3 = abc-\-aI)d-\- + «5m + .... p, = abc m. (A) Multiply by {x + n), the (r + l)th factor, then {x-\-a){x-{-b)..,.(x + n) = X' + ^+PlX'-\- ^2^-^ + ....+ Pr^ -\- nx' + npi iC"- ^ + .... + npr-i x + npr =:af+-'-^{pi-\-n)3f + (p2-{-np^)af-''-\-.... + npr Laws 1 and 2 are evidently still true. Pi-\-n = a -\- h -\- c -{- n, Vi + ^ A = {d^ -\- d -\- am-\-'bc-\-hd-\-.,,. •\-lm-\- etc.) -\-{an-\-ln-{- -^-mn), which is still the product of the second terms taken two and two. • ••••• np, = alcd n. Therefore, all the laws still hold true. Hence, if they are true for r factors, they are true for r + 1 factors. But we found them true for four fac- tors by multiplication ; hence, they are true for five fac- tors ; and, if so, for six factors ; and so on. Therefore, formula (A) is general. Uote. — The number of products that enter into each coeflBcient may be determined by the principles of combination. Applications. Ulnstrations. — 1. Expand {x + 1) (^ + 2) {x - 3) (a; + 4). Solution : pi = 1 + 2-3 + 4 = 4 i?a = (1 X 2) + (1 X -3) + (1 X 4) + (2 X -3) + (2 x 4) + (-3 x 4) = -7 i)3 = (lx2x-3) + (lx2x4) + (lx-3x4) + (2x-3x4) = -34 i>4 = lx2x— 3x4= —24 /. (x+l)(a;+2)(a;-3)(a;+4) = ic4 + 4ic»-7a;»-34a;-24. 326 ADVANCED ALGEBRA. 2. Factor a;* + 14 a;^ _|. ^^ ^2 _|_ I54 ^ _l_ ^go, if possible. Let {x + a){x + h){x + c){x+d) = o^ + l^a? + 1l\x^ + l^^x + \20. Then, 1. a-\-h + c + d = +14: 2. ab + ac + ad+bc + bd + cd = +71 3. a&c + a6c^ + «C(^ + 6ccZ = +154 4. aJc6? = +120 Resolve if possible +120 into four factors whose sum is +14. These we find to be 2, 3, 4, 5. .*. a = 2, b = d, c = 4, and d = 5. Will these values satisfy 2 and 3 ? ab + ac + ad+bc + bd + cd = 6 + 8 + 10 + 12 + 15 + 20 = 71, correct. abc + abd + acd + bcd = 24:+S0 + 40 + 60 = 154, correct. .-. x^ + Ux^ + 71x^ + 154:X + 120 = (x+2)(x+d)(x + 4)ix+n). EXERCISE 90. 1. Expand (x -{-2){x-\- 3) (^ + 1) 2. Expand (x -j-3)(x — 2) (x — 3) 3. Expand {x + 2) (ir + 3) {x - 1) {x - 2) 4. Expand (:r + 3) (:r + 5) (a; - 2) (:c - 6) 5. Expand {x + 2) (rr + 2) {x -\-%){x-\- 2) 6. Expand {x — h){x — 5) (ic — 5) (a; — 5) 7. Expand (2a; + 1) (2a; + 3) (2:r - 5) (2 a;- 1) Suggestion. — Put y for 2 a:. 8. Factor a;^ _j_ 9 ^^^s _{_ 26 a; + 24 9. Factor x^ -%x^ -%^x-\-m 10. Factor a:* + 5 a;^ + 5 a;^ — 5 a; — 6 11. Factor it-* - 2 a;^ - 25 a;^ + 36 a; + 120 12. Factor a;^ + 4 a;* - 13 a;^ - 52 ar^ + 36 a; + 144 The Binomial Theorem. I . For Positive Exponents. 588. If, in the binomial formula [587, A], we assume a = h=:c = dy etc., and r = w, then will TEE BINOMIAL THEOREM. 327 1. {x^a){x + h){x-{-c).... ={x^ay. 2. a;'" = cc" ; x'-^ = aj~-^ ; a:'- ^ = x""-^ ; etc. 3. p^z=a-\-a-\-a-{- to n terms = na. 4. j92 = «^ + «^ + ^^ + = ^^ taken as many times as there are combinations of 2 in 7i ; or, n(n — l) 2 5. p^ = fl^aa + aaa + «a« + = a^ taken as many times as there are combinations of 3 in w ; or, Ps = -^ i ^' • •••••• 6. p'=aXaXaXa to n factors = a\ 589. Cor, 1, — If a and x le interchanged, {a + xy = ar^na--^x-{-'^^^^a^-^x^-^...,-\-x\ (C) From (B) and (C) it will be seen that the coefficients of any two terms equidistant from the first and last terms are numerically equal. 590. Cor, 2, — If X le made negative in (C), (a — xf = a- -na''-^x-\- ^^^"^^ a'^-^x^ -,...±x\ 591. Cor. 3, — The sum of the coefficients in (C) equals zero. For, put a = 1 and a: = 1 ; then Li Li ADVANCED ALGEBRA. 2. For any Rational Exponents. — — — J = Tir*"^ for any Demonstration : I. Let n = any positive integer. Now, — ^-^ = ic"-^ + ra;«-2 + r^a;"-^ + + r«-J [134]. I =: lim. a:"— ^ + lim. roc:^—^ + x-r Jx = r + lim. r»-i [401, 413] = r^-^ + r»-> + r«-i + . . . . to n terms = nr^—'^. If) II. Ze^ w, = i- , a positive fraction. -T a?^ — r"^ Xq — rq .. Now, = — -. (1) ' a; — r x — r ^ ' Put Xq =y, or x = y9 '^ and r? = s, or r = s? ; then a;» — r** yp — sp yp — sp y^ — s? X — r ~ yt — si ~ y — s y — 8 ' Since x-=yi and r = s^, lim. y = s when lim. ic = r. ., Lim. (^^^:r.\ = ita. -S ^^^1^' ^ ^1^' i = \aJ — r/x = r i y — s y — s)y = 8 pyP-i ^ qyq--^ [I, 416] = ^ yp- = i>w; etc. A = n, B = — g— , C= ^ n{n D= ''' ^''^ ^,etc. Substituting the values in (C), {1 + zY = l + nz+ -^ — '-z^ + r£ 2» + n(n-l)(n-2)(n-S)^,_^^^^^ ^^^ 330 ADVANCED ALGEBRA. Substitute — for z (B), and multiply by o*» (A), Cb n(7i-l)(w — 2)a"-3 , w(7i — l)(w — 2)(n— 3) , ^ ' ^t. —5^ '-j^ x'+— '^ '^ ^-a^-*x*+ (N) 594. Cor. 1. {X + i/)" = a- ^1 + ^= = X" + «a?-' y + ^i^^af-^/ + "('^-^>("-^> x-V+etc. (P) This is the most general form of the binomial theorem, inasmuch as X and y may be both variables. 595. Cor, 2, — By inspection it will le seen that 1, The rth term of the development of {x -\- yY = n(n-l){n-^)..,.{n-r-^%) ^«_,+, ,_i |r — 1 * ^ 2, The {r + l)th term = n{n^l){n-^2).,.,{n-r + 2)(n-r + l) |r-l xr = |r '^ ^ 3, The ratio of the (r -f- l)th term to the rth term = n^r-\-l y r x' ' { r ) X 596. Cor, 3, — 1. If n is a positive integer equal to r — 1, the coefficient of the {r + l)th term, which is also the coefficient of the (n + 2)th term, will reduce to zero. Therefore, the series will terminate with the {n-\-l)th term, which will he y\ 2. If n is negative or fractional no factor of the rth THE BINOMIAL THEOREM. 331 term (r being a positive integer) will reduce to zero, how- ever great r he taken. Therefore, the series will he infinite. 597. Cm-. 4,— Since Urn, \ I —^ -'^ll' = —1 ,^ , it follows : 1. That the coefficients of all terms in the hinomial theorem are finite hoioever far the theorem he expanded. 2. That if y X, the literal part {not coefficient) of the {r + Vjth term will increase indefinitely as r increases and will ultimately become infinitely great, and as the coefficient remains finite the whole term will become infinitely great. Therefore the expanded form will be divergent [580]. Jf.. If y ■=. ±x the expansion will be indeterminate ; hut (x + y)'' = (2 a:)" or (0)" = 2" a;* or 0. 5. The expansions of {x-\-yY and {y-\-xY can not both be convergent for particular values of x and y ; only the one that has the greater first term. 598. Cor, 5, — 1. The coefficient of the rth term will evi- n "^ r I 1 dently be greatest when ^^— is first < 1 ; or when n — r-\-l is first w + 1, or when r is first > T" . 2. The rth term when the expansion is convergent, or n ~~ r I 1 1/ when xy y, is evidently greatest when ^^^— - . — is r X first < 1 ; or when {n — r-\-\)y is first <,rx ; or when {n-\-l) y is first < {x-\-y) r ; or when r is first > ( -^— | y. Illustration. — In the expansion of (8 + -^) , the 332 ADVANCED ALGEBRA. greatest coefficient belongs to the term whose number is — 4 + 1 1 first greater than — ^ — > ^^ TKy which is the first term. The greatest term in the expansion of I 8 + -^ j is the one which immediately follows in number, — \7~ x -5- , 1 . . ^t ^ or — , which is again the first term. 00 EXERCISE 91. Expand : 1. (a - 3 T'Y 3. (a; - 3 af 5. {x^ - 5)8 2. (2 + 5 xY 4. (2a^-\- 5y 6. (3 x^ + a^y Expand to four terms : 7. (1 - x)i 9. (a^- 1)^ 11. («rr + b)-^ 8.{a—x)^ 10. (a;^ + «)~^ 12. {x^ — a^~^ 13. Extract the cube root of 126 to six decimal places. Suggestion : Vl26 = Vi25Tr = (125 + l)i = j53(l + ^)P = 5(l + ji^)* -^r 3 125+ \2_ "" V125; + [3_ (^y+ etc. [ = 5(1 + 0026666 - -0000071 + -0000001) = 5-0132975. 14. Find to 5 decimal places : V65, VSO, V344, V3128 15. Find the 7th term of {2x-}-SY^ 16. Find the 5th term of V4+^ a 17. Find the 6th term of Va-\-x 18. Find the rth term of {a — x)~^ 19. Find the greatest coefficient of (2 + x)^ 20. Find the coefficient of the 5th term of {a — a^)~^. 21. Find the numerical value of the 10th term of (7 — 5 3/^)", when y = 27 and n = S TEE EXPONENTIAL THEOREM. 333 The Exponential Theorem. 599. The exponential theorem is the expansion of a" in ascending powers of x, and is derived as follows : T> ^ /^ lA"" - 1 nx(nx — l) 1 And (l . 1)'= 1 . 1 + ^ . ^-— I^^^ etc. (C) Substitute (B) and (C) in (A), x{x-l) x(x-^)(x-l) 1 + ^+ — \r^*- \r — -^'"'- Suppose lim. n = ao, then Put e for 1 + 1 + i-ft- + fo" + iT" + 6tc. ; then, [£_ l£_ l± e» = l+a: + j^ + -- + g-+ etc. (F) Put ca; for a;, then e"" = 1 + ex + -rjr- + -r^- + etc. (G) Let e" = a, and assume e as the base of a system of logarithms [465], then c = loge a, read logarithm a to the base e. Substitute these values in (G), »- = l + ^log.a+^ = l + ,log. a+ t^^ + t^^^ ,t,. p99,H]. (A) Transpose 1 and divide by y, -y- = logea + 2/f-^ + '-^"-^ + etcj. (B) Let lim. y = 0, then loge a = lim. I I Put 1 + a; for a, then loge (1 + a;) = lim. - ^1 + xy -l\-^^^ y r 12. |3_ ij/=o = a; — -^r^ + -o — etc. Therefore, lO o logo (l+a:) = aj-^a;2+ -^x^— jX^ + etc. (C) This is known as the Logarithmic Series. 602. The ratio of the (w + l)th term to the nth. term x'^'^'^ x"" ^ n _ 1 IS I 7 ! — j — . X — r- . X . n-{-l n n-\-l i_4__ Now, lim. / .x\ =x. Therefore, if a; < 1, numerically li+i'7„=» the series is convergent. It is, therefore, convergent for all values of X between — 1 and + 1. When x = l, loge 2 = 1— — + -^ — -j + etc., which is converg- ent [5831. When x= — l, log, ^=-'^('^+2+'o+T + ^^^-j' which is divergent, since lim. ( -^-^ ) = lim.-! —1 ( r | >■ = — 1, and all the terms have the same sign [583]. THE LOGARITHMIC SERIES. 335 603. Eesume Put — X for oCf then \og.{l-x)^-x-\x^-\7?-\^-\o?-.... (3) Subtract (3) from (1), then log. (1 + *) - log. (1 - X), or log, ([±|) [467, P. 3] = 2(.+ J + |'+....) (3) 171 — n ^ ' Put — I — for X, then. m-{-n Put m = n-{-l, then iog,(„+i)-iog.«=2J^+i(^^y+ iog.(«+i)=iog.»+3J^-^+j(^^y+ + ....[ (D) 5 \2 ^ + 1/ As this formula converges very rapidly for all values of n, it may be used to find the Napierian logarithm of any number from that of the next preceding number, n being regarded an integer. Computation of Logarithms. 604. The logarithms of composite numbers may be readily found, when the logarithms of primes are known, by Art. 467, P. 2. / OF THE 336 ADVANCED ALGEBRA. The logarithms of prime numbers are found by formula D, Art. 603. niustrations. — loge 1 = [466, P.] loge 2 = loge (1 + 1) = ^ "*" ^ \ 3 + d^iW + 5l<^ + 7x~3^ + • • • • j = 0-69314718 (by actual reduction). loge 3 = loge (2 + 1) = =1-09861228.... loge 4 = 2 loge 2 = 1 -38629436 .... loge 5= loge (4 + 1) = loge4 + 2(i + 3^3 + 5^ + ....) = 1-60943791.... loge 6 = loge 3 + loge 2 = 1 '79175946 .... loge 7 = loge (6 + 1) = loge 6 + 2(1 + 3^ + ^^ + ....) = 1-94591.... loge 8 = 31oge2 = 2-07944 loge 9 = 21oge3 = 2-19722 loge 10 = loge 5 + loge 3 = 2-30258509 etc., etc., etc. 605. Let a and i represent the bases of two systems of logarithms and n any number. Let logb n •= Xf then l' = 7i (1) Let loga l — m, then oT =.1) (2) dT" z=: If =: n, or log. n = mx (3) loff. n mx .*. , ^ = — = m : or logb n X log, n = m logb n. Therefore, COMPUTATION OF LOGARITHMS. 337 Principle. — Multiplying the log^ of a number hy the loQa of b gives the log^ of the number. 606. Loge n = logio n X loge 10 [605, P.] .-. log.o^ = loge^ X j^ = log.^ X ^^30^ = loge /J X 0-4342944.... 607. The number 0-4342944 is called the modulus of the common system, and is represented by m. Therefore, JPrin, 2, — Log^Q n = m log, n. By means of this principle the Briggean or common logarithms may be derived from the Napierian or natural logarithms. I. Since loge (^ + 1) = logio (» + !) = ]og.„»t + 3mj^^ + i(^^y+....| (H) By means of this formula the common logarithms may be computed directly. Summation of Series. 609. No general method of summing series can be given. Series of special types may sometimes be summed by special methods. The student has already learned how- to sum an arithmetical progression [486], a geometrical progression [493], an infinite series of the geometrical type [499], an arithmetico-geometrical progression [«500], a series of square numbers [506], a series of cubic num- 338 ADVANCED ALGEBRA. bers [507], and series dependent upon or resolvable into these. A few additional methods will be given here. 610. I. Method by Indeterminate Coefficients. This method is applicable when the nih. term is a rational integi-al function of n. niustration. — Find the sum Sn of n terms of the series : lX22 + 2x32 + 3x4:2 + ..., + w(w + 1)2. Solution : Put Sn= 1 X22 + 2x32 + 3x42 + 4x52 +....+ 7i(W+l)2 == ^ + ^n + (7n2+i>w3+^7i4+ and, Sn+\ = 1x22 + 2x32 + 3x42 + 4x52 +....+ (?i + l)(n + 2)2 = ^ + ^(71 + 1) + (7(?i + l)2 +Z>(n + l)3 + ^(n + l)4+.... Then, by subtraction, (w+l)(w + 2)2 = ^ + (2w + l)(7+(3n2 + 3n + l)i) + (4n3 + 6n2 + 4?i + l)^ + , or w3 + 5n2 + 8w + 4 = {B+G+D+E) + {2G+^D + 4.E)n + {^D+QE)'n? + 4:En\ since all coefficients after E are zero, there being no more than four terms in the expansion. Equating the coefficients of the like powers of ti, 1.4^=1, 2. 3i> + 6^=5, 3. 2C+3i) + 4^=8, 4. 5 + C+Z) + ^ = 4; 17 7 5 whence, E = -^, 7) = -^, C = -r, and B = -^» 5 7 7 1 Sn= A+ -^n+ -^'n? + -^n^ + -jn^ To find A, put n = 1, then Sn = Si — the first term = 1 x 22. ...(lx2)2 = A+| + | + |- + i = ^ + 4; whence A = 0; and 5 7 7 1 = i^(3n* + 14 n8 + 21^2 + lOn) la = ^(3n»+147i»+21»+10)=^(w+l)(n+2)(3w+5) SUMMATION OF SERIES. 339 611. 2. Method by Decomposition. This method is sometimes applicable when the wth term is a rational fractional function of n, and is resolv- able into the algebraic sum of the nth. terms of two or more other series of the same nature. niustration. — Find the sum S,, of n terms of the series : ^r — ^ — 7 + 7 , 10 , , 37^ + l 3x4x5 ' 4X5X6 ' •**• ' (/i + 1) (/^ + 2) (/^ + 3) * Solution : 3/1 + 1 _ A B C ^^^ (w + l)(w + 2)(n + 3) - n + 1 "^ n + 2 "^ 71 + 3' ^^®" A = —ly B =z 5, and C = — 4 ; whence 3n + l _ / 1_ 5 4_\ (n + l){n + 2)(n + S)~ \ n + l'^n + 2 n + dj' •• " ](7i + i)(n + 2)(w + 3)f ^V n + lj 2 3 4 \ n + 2) 4 n + 1 5 5 5 ■o +^ + 3 4 n + 1 n + 2 2/ 4_\ _4_ 4 4 ^ V ^ + 3/ 4 71 + 1 n + 2 n + S Adding the last three series, we have "~V 2"^3"^w4-2 n + s)~6'^n + 2 n + 3 If lim. n = ao, then /Soo = 77 . The Differential Method. 612. If the first term of any series be taken from the second, the second from the third, the third from the fourth, and so on, a new series will be formed which is 340 ADVANCED ALGEBRA. called the first order of differences. If the first order of differences be treated in the same manner as the original series, a second order of differences will be formed, and so on. Thus, if we let a, I, c, d, e, , , . , be any series, then h — a, c — h, d— Cy e — d, .... will be the first order of differences ; c — %'b-\-a, d — 2c-}-b, e — 2d-^c, the second order of differences ; d — dc-\-3b-a, e — Zd^^c-h, the third order of differences, and so on. 613. If we let «!, Ji, Ci, fZi, represent the first order of differences ; «2? ^2? Cgj ^2* • • • • the second order of differences ; ^3 J ^3> ^3? ^3^ •••• the third order of differences; and so on, we have the following scheme : Series : a, l, c, dy e, 1st Differences: ^i, Jj, Ci, (?i, 2d Differences : ct^, hi c^y 3d Differences : ^3? ^3? 4th Differences : ^4, , and so on. (hi ci2f (hy <^if are the^r^^ terms of the succes- sive order of differences. 614. Problem 1. To find the {n + l)th term of a series. Solution : Take the series a, 6, c, d, e, then from the above scheme, 1. b —a =ai, whence b = a + ai (1) bi — ai = a%, " bi = ai + a^ (2) &a — aa = Os , " 63 = O9 + as (3) &3 — as = a4 , " ba = a3 + at (4) 2. c =b +bi = a +2ai + tti, from (1 and 2) (5) Ci = &i + 6a = ai + 2 as + as , from (2 and 3) (6) Ca = Ja + 6s = as + 2 as + a4 , from (3 and 4) (7) SUMMATION OF SERIES. 341 3. S„ = «o + aiX-{- a^x^ -\- + «„_! x'"'^ —pxSn= —pa^x—pa^T? — — ^«„_2a;"~* — qx^S^ = —qa^x^ — — qa^^^af-^ — q a^-z^"" — q cin-\^'^'^ 348 ADVANCED ALGEBRA. Adding and remembering that the sum of the co- efficients of each term from the third to the nth. in- clusive is zero, (1 —px — q 0^) /S'« = «o + («i — i? «o) ic - (P «n_i + q a^-z) x^-q a^-i a;"+i ; whence, S. = ^o + {a.-pa,)x 1 —px — qar {pan-i-{-qan-2)x''-{-qan-i3f^'^ 1 —px — qx^ 626. Cor, — If X <1 and Urn. n = co, then ^ ___ ao-\-{ai—pao)x 1 ^px — qoir Example.— Sum l^^x^bx^-\-l^ a;3+48 rr^+145 x^-{- .... to 7 terms and to infinity, when x <1, Solution: 1. 5jt? + Sg- + r = 18 3. 18^9 + 5g + 3r = 48 3. 48p + 18^ + 5r = 145 .*. p = 2, g = 3, and r = — 1. S-, = l + 3a:+5a;2 + 18a;8 + 48a;4+1453:5^.4i6a;6 -2 a; /S't = -2a;-6a;2-10a;8-86a;4- 96a;«-290a;«-832a;'' -3a;2,S'T= -3a;2- 9a;8-15a;4_ 54a;*-144a;«-435a;''-1248ic8 + x3>S'7= + x^+ 3a;4+ 5^^+ iga^e^ 48a;7+ 145a:« + 416a;9 .-. (l-2a;-3a;2 + a;3)>S'7 = l+a;-4a;2-1219a;''-1103a;8 + 416a:» l + a;_4a:2_1219a;'-1103a:8 + 416a;9 /St 1+X — 4:X^ l-2a;-3a;2+a;» 627. If /S'ao is developed into a series by the method of indeterminate coefficients, the original series may be reproduced to any number of terms desired. Therefore, /Soo is often called the generatrix of the series. 628. If the generatrix can be decomposed into partial fractions, the general, or ni\i term, of the series may easily be obtained, and hence, too, the sum of n terms. RECURRING SERIES. 349 niustration. — Let it be required to find the nth term and the sum of n terms of the series Solution : It may readily be determined that i? = 2, g = 3, and the l + 3a; 11 3 1 generatrix = ^_^^_.^^, = " 2 ' iT^ + 3 ' T^Ts^' 11 1.11, ^^^'-2 •n:T-=-2 + 2^-■2^+•••• r _ l^n ^ _ (-1)"^"-^ r493 b1 • 3 1 3 9 27 , ^^^ 2T335=2 + 2^+-2^ +•••• _ (-l)na;n_l 3n+la;n_3 '^''~ 2a; + 2 "^ 6a;- 2 ' 2^—1 3»i nth term = (— 1)** . -g— + -g- . a?*-*. EXERCISE 94. Sum to infinity : 1. l + 3ic + 5a;2 + 13a;3 + .... 3. 2 + 2a; + 4a;2 + 14a;34-.... 4. 3 + 2a;-7a;2- 38:^3-.... 5. l + 2a; + 3a;2+ll:r3^35^_j_121a;5 + .... 6. l-3:zj + 5a;2_^5^^13^_|.61^5_|.__ 7. Sum l + 2ir + 9a;2 + 33iz^+ to 6 terms. 8. Sum 1 — 2a;— 7a;2 — 8a;3 4- to 7 terms. 9. Sum 2 — a; + 6a;2 — 14a;3 + to 8 terms. 10. Sum l — ^x-{-^x^^Qo(?^x^ — ^x^-\- to 9 terms. Find the nth. term and the sum of n terms of 11. l + 2a;-82;2 + 20a;3- 12. 1 + 5 a; -f 9 a;2 4- 13 a;3 _^ 350 ADVANCED ALGEBRA. Decomposition of Rational Fractional Functions. 629. To decompose a rational fraction is to find two or more other fractions whose sum equals the rational fraction. 630. It will be only necessary to show how to decom- pose proper fractions, as all improper fractions may be reduced to mixed numbers, which process will already lead to a partial decomposition. Principles. 631. 1. Any rational fraction of the form of p , ■ ., _.-,. j—j- — r may he decomposed so that [X -f- a) \x -\- 0) .... {x-\-n) (x-]-a){x-\-I)) (x-{-n) x-\-a x-\-b x-{-n Illustration. — Put 3^24. 14:^-29 _ A . B . C .,. {x-l){x-\-%)(x-^) x-1 ' x-{-2 ' x-3 Clear of fractions and arrange the terms according to the descend- ing powers of a;, 3x^ + Ux -29 = (A + B + C)x^ - {A + 4:B - C)x - {6A-SB + 2C). Equating the coefficients [566], we have (1) A +B+C=d (2) A + 4B-C= -U (S) 6 A-BB + 2C= 29 Finding the values of A, B, and C by elimination, and substi- tuting them in (A), we obtain dx'^ + Ux-29 _ 2 d_ 4 {x — l)(x + 2){x-S)~ x-l x + 2'^ x-S' 632. In a similar manner it may be shown that (a X -{- b) (c X -\- d) {mx-\-n) ax-\-l) cx-\-d ' mx-\-n RATIONAL FRACTIONAL FUNCTIONS. 351 633. 2, Any rational fraction of the form of P {p(^ -{- a X -{- h) {x^ -\- c X -\- d) {a^ -\- m x -\- n) may he decomposed so that P {x" ^ ax ^h) {7^ -\- ex ^ d) . . . . {x^ -\- mx ^ n) ^ Ax + B Cx-\-D Mx-^JV ~r ^2 I ^ ^ I ,7 "T • • • • ~r x^-\-ax-{-bx^-\-cx-\-d x^-\-mx-{-n' „, , ^. T, i. 4:X^-8x^-63^-Ux-{-3 mustration.-Put ( ^^^^i)(^_^^i)(^.^^^^) Ax + B Ox-^D Bx-\-F .. ^ x^-\-x-\-l ' x^ — x-\-l ' x^-\-x^% Clear of fractions and arrange the terms according to the descend- ing powers of ic, 4a;4 - 8a:3 - 5a;2 - 15a; + 3 = {A + C + E)0!^ + {B + 2 C + D + F)a^ + {2A-\-4.C + 2D-[-E)x'-^{-A + 2B + ^C+4:D + F)x^ + (2A-^ + 2(7+3i) + ^)a; + (25 + 2i> + i^) (1) Equating the coefficients [566], (1) A + C+E = (2) ^ + 2C+Z) + ^=4 (3) 2A + 4(7+2/> + J^= -8 (4)^-25-3(7-4i9-i^=5 (5)2A-^ + 2(7 + 3i; + ^=-15 (6) 2^ + 2Z) + ^=3 Finding by elimination the values of J., B, C, D, E, and F^ and substituting them in (A), we have 4a4_8a^_5a;2_i5a;4.3 _ (a;2 + ic + 1) {x^-x+ 1) (ic* + a; + 2) ~ 3 4 5 x^ + X + 1 x^ — X + 1 x^ + X + 2' 634. In a similar manner it may be shown that P {a x^ -^ b X -\- c) {dx^-\-ex-\-f) {m x^ -\- n x -{■ p) Ax-\-B Cx-\-D Mx-\-N ax^-\-bx-{-c d:x^ -\-ex +/ mx^ -\-nx -{-p * 352 ADVANCED ALGEBRA, P 635. S, A rational fraction of the form of -, — ; — r- -' J J J {x-\-aY may he decomposed so that p A B jsr t /^ I ^\2 ~r • • • • "r (x-^aY x-\-a (x-\-aY {x-\-ay Illustration. — Put i^-^r _ A B G D " x-%'^ {x-%)^^ {x-^f'^ {x- 2)* ^^^ Clear of fractions and arrange the terms according to the descend- ing powers of x ; then, 3a:3 _ i4a;2 + 19a. _ 5 _ ^^3 _ (6 ^ _ ^)a;2 + (12J-4^+ C)x-{%A-4:B + ^G-D). Equating the coefficients, we have 1. ^ = 3 2. 6.4-5 = 14 3. 12A-4:B + C= 19 4. 8^ -45 + 2 C- 2) = 5 Solving these equations, and substituting the values of A, B, C, and D in (A), we obtain Sa^-Ux^ + 19x-5 3 4 1 1 {x-2)* x-2 (x- 2)2 (x - 2f {X — 2)4 636. In a similar manner it may be shown that 1 _p _^_+ -g + + ^ (ax-i-by ax + b ' (ax-^-bf ' ' (ax-^by 2 P ^ Ax-]-B (x^ + ax-\-by a^-\-ax-\-b^ Cx + D Mx-^-N {^j^ax^bf ' •••• • {x'-^-ax-irby {aT^^bx-^cy a:^ + bx-^c'^ Cx^D Mx-{-JSr {as^ + bx^cf ' •••* ' {a3^-\-bx-\-cy 4. Any rational fraction whose denominator may be resolved into linear and quadratic factors may be decom- posed by a combination of the above methods. RATIONAL FRACTIONAL FUNCTIONS. 353 Ulustration. — To decompose ^^^_ a)^^-iY i^^+p^ + ^y ' P^* {x — a) {x — iy {x^ -\- p X -\- qY x — a x — b , M Px^Q P'x-\-Q' {x — df af-\-px-{-q {x^ -\- p x -{- q)* EXERCISE 93. Decompose into partial fractions : 5cc + 2 2a;-5 _ A B ^' A ^2 1 O ^ I -I I x(x-{-l){x-\-2) X ' ic + 1 ' x-^2 3x-2 x^-x-{-l ^ 3x^-Ux-{-25 (a;4-lf (^ + 3)3 (a;-3) (a;2-a:+6) aJ + (a — ^)a; — ic^ ■ {px-{-qy ^ ^ 10. X^-{-X^ + l X(1-4:X^) 3a:3_8a:2 + 10 2a; + l 11< 7 TTT 12* (a; - 1)* (a; - 1) (a;^ + 1) 13. -n r^ 14. a^^6x-{-6 (x-l)(x^-i- ly X5. .4t^+l^ 16. ^ (a; + 1) (x^ + 1) -"• a^ _|. :r7 _ 2^ _ ^ 2a;g~lla; + 5 5 ^^3 _p g ^ _j_ 5 ^. ^ (^~3)(a;2_|.2a;-5) ' (ic^ _ j) (^ _|_ 1)3 1 x^-{-x-\-l 10 on ■ ^^- a^^x'-x^-a^ (a;+l)(^' + l) 21. ^+^t^ . 22. ^' + ^ l_a;-a.'*-|-a:« "'•• {x-\-lf{x-2){x-{-3) CHAPTER X. COMPLEX J{ UMBERS, Graphical Treatment. 637. If a straight line of any assumed length be taken to represent the number one, then will a straight line twice as long represent the number two, one three times as long the number three, and so on. Thus, we see that any number may be represented by a line. 638. A line representing a number is called a graph number, or a vector. The point where a vector is sup- posed to begin is called the origin, and the point where it ends, the extremity. 639. A vector is fully determined when both its length and direction are given. In a system of graphical repre- sentation of numbers, a vector running rightward from its origin represents a positive number and is positive, and one running leftward from its origin represents a nega- tive number and is negative, 640. If the vector +« be made to revolve about its origin A, through an angle of 180°, or ir, it will become the vector — a, or will be multiplied by — 1 ; and if j^, ~ ^ i "'" " r^ the vector — a be revolved about its origin A through an angle of 180°, or tt, it will become the vector -|- a, or will be multiplied by — 1. GRAPHICAL TREATMENT. 355 Therefore, 1. Revolving a vector through an angle of 180°, or tt, is equivalent to multiplying it hy — 1. 2. Revolving a vector about its origin through an angle of 360°, or 2 tt, is equivalent to multiplying it twice hy — 1, or once ly + 1^ which does not affect its length or direction. 3. Since — 1 = V— 1 X V— 1, revolving a vector about its origin through an angle of 90°, or \ tt, is equiva- lent to multiplying it by V— 1, or i [v. P. I. 299]. 641. Motion about the origin of a vector in the direc- tion the hands of a clock go is considered negative, and counter-motion positive. The factor -\-i may, therefore, be taken to represent circular motion about an origin through an angle of 90°, or \ it, counter-clock-wise ; and — i through an angle of 90°, or \ tt, cloclc-wise. 642. Since (+ i) X (+ i) X (+ i), or (+ if = - i, the factor symbol — i may also denote circular motion about an origin through an angle of 270°, or | tt, in a positive direction. 643. Since {± i) X (± i), or (± if = - 1, the factor symbol — 1 may denote circular motion about an origin through an angle of 180°, or TT, in either direc- tion. Illustrations. — 1. The B- ^ vector -\-a multiplied by -{-i = AB revolved about A in the positive direction through an angle of 90° = AB\ B' +a —at 356 ADVANCED ALGEBRA. 2. The vector + a multiplied by —l^AB revolved about A in the positive or the negative direction through an angle of 180° = A B", 3. The vector + a multiplied by —i = AB revolved about A in the negative direction through an angle of 90°, or in the positive direction through an angle of 270° = A B'". 4. In a similar manner it may be shown that (— a) X (+^) =AB"'', (-«)x(-l) =AB, and {- a) X {- i) = AB\ 644. From what has been explained thus far it will be seen that, if a vector a units long running rigJitward from its origin represents + «^ running leftward from its origin, it will represent — a ; running upward from its origin, -\-ai'y and running doiunward, — ai. 646. One vector is added to another by placing its ori- gin to the extremity of the other and giving it the direc- tion indicated by its factor symbal. The vector of their sum is the length and direction of the line joining the origin of the vector to which addition is made with the extremity of the vector added. ^ +« ? +5 ^ Illustrations. — 1. The vector B C ^ ±^ (=+&) added to the G vector A B {= -\-a) —h ■^^r-B gives the vector A C ^ A ~^^ (= + (« + 5)). 2. The vector B C (= - b) added to the vector A B (= -f a) gives the vector A C {= -{- {a — h)] , when a>h. 3. The vector BO (= — !)) added to the vector AB (= -f a) gives the vector A C {= — (b — a)}, when a to E, making DE = d times A C, and revolve D E about D through an angle of 90° clock-wise, then is DE' the vector of —di times —a + bi constructed at the extremity of A D. Join A and E'. A E' is the vector of the sum of c times —a + bi and —di times —a + bi. Exercise.— Multiply graphically : 1. a-\-bihyc-\-di 4. — a-^-bi hj c-\-di 2. a — hi hj c-\-di b, — a — lihj— c-\-di 3. a — bihjc — di 6. — a — bi hy — c — di General Principles. 656. 1. The sum, the difference, the product, and the quotient of two complex numbers are, in general, complex numbers. For, 1. {a^bi)-\r(c^-di) = {a^c)-\-{b-\-d)i. 2. {a-^bi)-(c + di) = {a-c)-{-{b-d)i. 3. {a-{-b i) {c + di) = ac-}-bci-{-adi — bd = {ac — bd)-\-{bc-i-ad)i, a-{-bi _ {a-\- bi) (c — di) c-\-di ~ (c-\- d i) {c — d i) 4. {a c -\-b d) -\- {b c — a d) i _ ac-\-bd + (be — ad\. 862 ADVANCED ALOEBRA. 667. 2. The sum and the product of two conjugate complex numbers are real. Eor, 1. {a-\-hi)-^{a — 'bi) z=%a. 2. {a-\-di)(a-bi) =a^ + bK Scholium, a^ + b^ is the square of the modulus of ±a-\rbi and of ± a — bi, and is called the norm of each. Therefore, Cor, — The product of two conjugate complex numbers equals their norm. 668. 8. The norm of the product of two complex num- bers equals the product of their norms. For, norm {a-\-b i) {c + di) = norm {{ac — bd)-\- (ad-\-hc)i] -{ac-bdf-\-{ad-\-b cf [657, Sch.] = a^ c" -\-b^ d^ -^ a^ d^ -^-b^ d" = norm {a-\-b i) multiplied by norm {c-\- d i). Cor, — The modulus of the product of two complex num- bers equals the product of their moduli. 669. Jf. If a-^bi = 0, then a = and b = 0. For, it a -\- b i = 0, bi= —a and —b^ = a^; whence, a^-\-P = 0, which is possible only when a = and b = 0. Cor, — If a complex number vanishes, its modulus van- ishes ; and conversely, if the modulus vanishes, the complex number vanishes. 660. S. Ifa-^bi = c-\-di, then a = c and b = d. For, if a-\-bi = c-\-di, (a — c) -{- {b — d)i = ; whence, a — c = and b •— d = [P. 4], and a = c and b = d. COMPLEX NUMBERS. 363 661. Problem. To find the value of e*+^*. Solution : Assuming the exponential law of multiplication [275, P.], and Formula (G), Art. 599, sufficiently general to include imaginary- exponents; then ( 2/H"2 y^i^ y*i* ^ ) e«+y» = e' X ev' = e' . B D will be tan. y. For, from the similar triangles DAB and CAB we have BD'.CE '.'.AB'.AE', or, B D : sin. y ::1 : cos. y ; whence, sin. y BD = tan. y. sin. y, and A L ^ cos. «/. The tri- ces, y If BF=zy, FL angles A FL and B A K will be similar, and B K=^ tan. «/. If B F G = y, then G L =. sin. y, and ^ Z = cos. ?/. The triangles GAL and DAB wiU be similar, and D B =:^ tan. y. If ^ i^ZT = y, then HE = sin. y, and AE = cos. «/. The triangles KAB and -ST^ ^ will be similar, and KB — tan. y. Scholium, — So long as y < \ it [90°], sin. y and cos. y are positive ; hence, tan. y (B D) is positive. When y > i TT but < IT, sin. y is positive and cos. y negative ; he^ice, tan. y (B K) is negative. When y > tt tut < f t, sin. y is negative and cos. y negative ; hence, tan y {D B) is posi- tive. When y > i TT but < 2 v, sin. y is negative and cos. y is positive; hence, tan. y {KB) is negative. CHAPTER XI. THEORY OF FUJVCTIOJVS. Definitions. 669. A quantity whose value changes, or is supposed to change, according to a definable law, is a definite variable, or simply a variable. 670. A variable whose law of change is not dependent upon that of another variable is an independent variable, 671. A variable whose law of change is dependent upon that of another variable is a dependent variable, and is called a function of that variable. Hence it is, that any expression containing a variable is a function of that variable [562]. • 672. Any law of change may be imposed upon an inde- pendent variable ; but, when it is once imposed, the law of change of any function of the variable becomes de- termined. 673. The simplest treatment of functions of a single variable is that in which the variable is supposed to in- crease or decrease uniformly by equal increments, finite or infinitely small. 674. A function is said to be continuous so long as an infinitely small change in the independent variable pro- duces an infinitely small change in the function, and dis- THEORY OF FUNCTIONS. 367 continuous when an infinitely small change in the inde- pendent variable produces a finite or infinitely great change in the function. Ulustration. — Thus, the function - — — assumes all values between + 1 and + oo as a; assumes all values be- tween and + 1, and is, therefore, continuous from to + 00 ; but, as the value of x continues to increase from a quantity infinitesimally less than + 1 to a quantity infini- tesimally greater than + 1, or takes an infinitely small step across + 1, the function takes a leap through the whole gamut of numbers from + "^ to — oo , and is, therefore, discontinuous between these values. 675. So long as a function increases in value as the independent variable increases in value, and hence, too, decreases in value as the independent variable decreases in value, it is an iticreasing function ; but when it decreases in value as the independent variable increases in value, and, hence, increases in value as the independent variable decreases in value, it is a decreasing function. Ulustration. — Let y =f(x) = x^ — 4,x-\-3. Assign values to x and calculate the corresponding values of y by synthetic division [106], you will obtain results as follows : For a; = - 3, - 2, -1, 0, +1, +2, +3, +4, +5 y = +24, +15, +8, +3, 0, -1, 0, +3, +8 Here y decreases from + 24 to — 1 as a; increases from — 3 to +2, and is, therefore, a decreasing function be- tween these values of x ; and it increases from — 1 to +8 as X increases from +2 to +5, and is, therefore, an in- creasing function between these values of x. 676. The maximum value of a function is the value at which the function changes from an increasing to a de- creasing function. 368 ADVANCUn ALGEBRA. 677. The minimum value of a function is the value at which the function changes from a decreasing to an in- creasing function. 678. The maxima and minima values of a function are often called the turning values of the function. 679. A turning value of a function may be a finite constant, zero, or infinity. niustrations.— 1. Take ^^ = /(i») = 3 + (4 - xf. As X increases from to 4, «/ decreases from 19 to 3 ; and as x continues to increase from 4 to oo , y increases from 3 to 00. Therefore, 3 is a turning value (a mini- mum) of y. 2. Take y = {a- xf. As X increases from to a, y decreases from a^ to ; and as x continues to increase from a to co , y increases from to 00 . Therefore, is a turning value of y. As X increases in value from to + 1, (1 — xy decreases from 1 to 0, and y increases from 1 to oo ; and as x con- tinues to increase from 1 to oo , (1 — xY increases from to 00 , and y decreases from oo to 0. Therefore, oo is a turning value (a maximum) of y. 680. The limit of a function is the value of the func- tion at which it ceases to be continuous. Note. — Notice the distinction between the meaning of the word limit as here used and as used in Art. 398. In the latter sense, would be the limit of y in illustration 3, Art. 679, instead" of a maximum. 681. The limit of a function may be a finite constant, zero, or infinity. Illustrations. — 1. Take y =f(x) = 2 — —, 2 As X increases from to oo , ~ decreases from 2 to 0, THEORY OF FUNCTIONS. 369 and y increases from to 2 ; and, as x can n(# be sup- posed greater than oo , y can not become greater than 2, neither can y begin to decrease at 2. Therefore, 2 is the limit of y. 2. Take y z=z f{x) = x^{l^ x^). As X decreases from 1 to 0, y decreases from 2 to ; and as x can not be taken less than (negative) without making y imaginary, y can not become less than 0, neither can y change from an increasing to a decreasing function at 0. Therefore, is the limit of y, 3. We have already seen [674] that y — f(x) = __ increases from 1 to oo as a; increases from to +1, and thereafter becomes discontinuous. Therefore, (» is the limit of y, 682. A function may have two sets of values approach- ing the same or different limits for the same set of values of the independent variable. niustrations.— 1. Take y^ =/(a:) = 16 — a;^ ; then y = ± a/16 - a;^. Here are two values of y for each value of x, numeri- cally equal but opposed in sign. As x increases from to 4 one value of y decreases from 4 to 0, and the other increases from — 4 to 0. It x becomes infinitesimally greater than 4 both values of y become imaginary. There- fore, is the limit of both values of y. 2. Take?/2 =/(^) = 4a:; then «^ = ± 2 Vx. Here, again, are two values of y for each value of x. As X increases from to + oo , one value of y increases from to -f °° and the other decreases from to — oo ; and as x can not be supposed greater than -\- co , + °o ^s the limit of one value of y and — oo the limit of the other value. 683. The limit of an increasing function is a superior 370 ADVANCED ALGEBRA. or maximum limit ; that of a decreasing function an iw- ferior or minimum limit. Graphical Representation of Functions of a Single Variable. 684. Every function of a single variable may be ap- proximately represented by a line, straight or curved, called the graph of the function. Method. — Let y = f{x). Assign successive values to x and calculate the corresponding values of y. Construct two indefinite straight lines intersecting each other at right angles, one running right and left and the other up and down from their intersection. These are the axes of refer- ence. The first is the a;-axis and the second the i^-axis, and their intersection the origin. Regard distance right- ward from the y-axis positive^ and distance leftward nega- tive ; distance upward from the a^-axis positive, and dis- tance downward negative. Assume a fixed length as a unit of scale, and lay off on the a;-axis from the origin the successive values of x based on this scale, and at the extremity of each x value, and on a line parallel to the y-axis, lay off the corresponding values of y. Thus will be located a series of successive points ; draw a continuous line through these points ; it will be the graph of the function, and its accuracy will depend upon the nearness to each other of the successive values of X taken, the relation of the unit of scale to that of x and y, and the correctness of the instruments used in plotting. niustrations.— 1. Take y =f{x) = x^ — 20 x^ -\- QL Assign special values to x and calculate the correspond- ing values of y by synthetic division [106]. You will readily derive the following table of values and make the following plot : FUNCTIONS OF A SINGLE VARIABLE. 371 X y 64 1 44 2 3 -35 4 5 189 00. 00 - 1 44 - 2 - 3 -35 - 4 - 5 189 — 00 00 PLOT. + 2/ Observa- tions. — 1. The curved line on the plot is the graph of the function. 2. The unit of scale used in plotting the graph repre- sents 20 units of y to one unit of X. + 200 + 180 -f- 160 1 + 140 + 120 + 100 + 80 +^ v60 / /. .oN \ / + 20 \ / 4 -3 -2 A A +2 +3 u \ / 20 \ y V / 40 \ / __ 60 -y 3. The graph exhibits three turning values of the function ; two minima at the points (ic = 3, y = — 35) and (— 3, — 35), and one maximum at the point (0, 64). 4. When a:=+oo, 2/r=+oo, and when a;=— oo, y = + co. The graph, like the function, descends from (— oo , +oo) to (—3, —35), then ascends from (— 3, — 35) to (0, 64), then descends from (0, 64) to (3, — 35), then again ascends from (3, — 35) to (+ oo , + oo ). It is a continuous graph from beginning to end. 5. At a? = 2, 4, — 2, and — 4, the graph crosses the ir-axis, exhibit- ing the fact that for these values of x, y = f(x) = x^ — 20 x^ + 64t — 0. The values of x that render f(x) = are, however, the roots of the equation f(x) = 0; therefore, the values of the roots of f(x) = Q may- be approximately found even if incommensurable, by plotting /(a;) = y and determining with a scale of equal parts where the graph crosses the a^^axis. 372 ADVANCED ALGEBRA. 2. Plot y = ± Va^-x = ± Vx(x+l){x -1). The following table of values may readily be obtained -^07 + y 3 PLOT. / + 3.5 / + 2 / + 1.5 / + 1 / / + .5 / -1 /d ■^ +.5 / -fl.5 +2 +2.5 V ^ .5 \ 1 \ 1.5 \ y 2 \ 2.5 \ _ 3 \ \ +3 Observations. — 1. The graph consists of two branches between the points (—1, 0) and (0, 0), symmetrical with respect to the a;-axis. These branches are confluent at the points mentioned. 2. The graph is discontinuous for all values of X antecedent to — 1, counting from a; = — oo , and also for all values of x between and + 1. 3. The graph again consists of two branches, symmetrical with respect to the a;-axis, for all positive values of x greater than + 1. 4. The limits of the first branch are at (—1, 0) and (0, 0) ; the limits of the second branch are also at (—1, 0) and (0, 0) ; the limits of the third branch are at (+ 1, 0) and (+ oo , + oo ) ; and the limits of the fourth branch are at (+ 1, 0) and (+ oo , — oo ). + x X y + < + l 1 1-5 ±1-4 2 ±2-4 2-5 ±3-6 3 ±4-9 -1 -<-l V- -•2 ±•44 -•4 ±•58 -•5 ±•61 -•6 ± ^62 -•8 ±•53 FUNCTIONS OF A SINGLE VARIABLE. 373 5. The first branch has a turning-point (maximum) somewhere between (— '5, + -61) and (— '8, + 'SS). The second branch also has a turning-point (minimum) between (— '5, — '61) and (— '8, — '53). 6. The branches of the graph meet the a;-axis when a: = 0, +1, and — 1. These values are, therefore, the roots of f{x) = x^ — x=.0, 3. Plot 2/2 = a;3_9^^24a;^16^ or y = ± v^ y X y <1 V- 1 1-5 ± 1-76 2 ±3 3 ±1-41 4 5 ±2 6 ±4-47 7 ±7-34 + 00 ±00 PLOT. 16. Questions. — 1. How many branches has this graph ? 2. How many ^/ turning-points? Locate and _ name them. 3. What are the roots of the "~ function x^ — 9x2 -h 24a; -16 _ = 0! 4. Between what limits are "" the branches of the graph con- _ tinuous ? 5. Where is the function an ~~ increasingfunc- tion and where a decreasing? 6 / 5 / 4 / / 3 / 2 / 1 f \, / i 1 2 •\ / 5 6 1 \ / \ 2 V / \ 3 \ 4 \ 5 \ \ 6 \ \ y \ 374 ADVANCED ALGEBRA. EXERCISE 96. Plot and discuss the following functions : (Use paper ruled in squares, called l)lotting-paper.) 1, y =^x-\-Q 6, 1/^ = a;^ 2. y =Sx 1. y^ =:x^{x— 1) 3, y =81a;-3 8. y =a^ -'8x^-\-20x -10 4. y^ = 4:X 9. y =3x-\-lSx^ — 2a^ b. y^ = 16-x^ 10. y^ = 3^-{-ds^-5x-20 Differentials and Derivatives of Functions. Definitions. 685. The limit of the ratio of the increment of a func- tion to the increment of the independent variable pro- ducing the increment of the function, when the limit of the increment of the independent variable is zero, is called the derivative of the function. Thus, if we let y =/(a;), and represent the increment of a; by A a; and the corresponding increment of y hj Ay, then will lim. ( — - ] = the derivative of the function. m 686. The limit of the increment of the independent variable is called the differential of the independent vari- alle, and is represented hy dx\ and the limit of the incre- ment of the function is called the differential of the func- tion, and is represented by dy. dy : -■ (fl) Therefore, x^^x. , , _ , Q ax Notice, dy and dx represent single quantities (differentials) and are not equivalent to d x y and d y. x. DIFFERENTIALS AND DERIVATIVES. 375 Ulustration. — Let y = 7^ (1) then, y-\- /\y = {x -^ /^xf = x^ -{-%x(Ax) -\-{^xf (2) Subtract (1) from (2), ^y = '^xi^x) + {^xf (3) Divide by a x, -^ = 2a; + A a; (4) A X .-. Lim.fA|) ^lim.(2a: + Aa;) [401,P.] (5) .: -r^ = 2x = the derivative of a;', dx and dy = 2xdx = the differential of a;'. 687. The differential of a function equals the derivative of the function multiplied ly the differential of the inde- pendent variable, 688. The derivative of a function equals the differential of the function divided by the differential of the independ- ent variable. 689. // the differential of a function, and hence, too, the derivative of a function, is positive, the function is an increasing one; if negative, a decreasing one. Principles. 690. Let y =f{x) = x% (1) then, y-]- Ay = {x-\- Axy = x''-\-nx''-'^ .Ax-{-A.{AxY (2) in which A = n{n-l) ^_, _^ n(n-l)in-2) ^_3 . ^ ^ + ^tc. [593]. Subtracting (1) from (2), Ay = nx^-^ . Ax + B .{Axf (3) A tJ Dividing by A x, — ~ = wa;*-i + -B . A a; (4) A X Lim. ( ^^ ) = lim. (na;«-i + 5. A a;)^ ^_o .'. ~ = nx^-\ since Km. ^ = a finite constant [582], and lim. A a; = 0. (5) .'. dy = nx^-'^dx. Therefore, 376 ADVANCED ALGEBRA. JPrin, 1. — The differential of a variable with a con- stant exponent equals the continued product of the expo- nent, the variable with its exponent diminished by unity, and the differential of the independent variable. Ulustrations. — 1. d{a^) = 4.a^dx 2. d{x)-^= -^x'^dx 3. dia + bxy^pia + bxy-'^dia + bx) 691. Let y = ax (1) then, y -\- Ay = a(x-\- Ax) =zax-\-a{Ax) (2) Subtracting, Ay = a{Ax) (3) Dividing, — - = a A X Lim. — - = a A X whence, -j^ = a, and dy = adx. Therefore, Prin, 2, — The differential of a constant times a vari- able equals the constant times the differential of the vari- able. Thus, d{dx') = d . d(x') = 3 X 6x^ dx = 15a^dx. 692. Let y = ax-\-b (1) then, y-\- Ay = a{x-{- Ax)-]-b = ax-\-a(Ax)-\-b (2) Subtracting, Ay = a(Ax) Dividing, -^ = a; whence, ^ = «» and dy = adx. Therefore, I*rin, 3, — The diffetential of a constant term is zero. 693. Let V = f{x), w =/' (x), and z =/" {x) ; and let y =z v-\-w — z (1) then, y -\- Ay = v-Yav-^w-\-Aw — {z-]rAz) ■=V-\-W — Z-\-AV-[-AW— AZ (2) DIFFERENTIALS AND DERIVATIVES. 377 Subtracting, Ay = Av + aw— a z (3) T^. .,. , Ay A V AW AZ DmdmgbyA^, — = — + — -— (4) , ,. Ay T AV ,. AW ,. AZ whence, lim. - — ~ = lim. h lim. lim. (5) [413,P.]. ^"^ ^^ ^"^ AX '> dy _dv dw d z dx ~ dx dx dx ^ ' whence, dy = dv + dw — dz. Therefore, Prin, 4, — The differential of a polynomial whose terms are functions of the same independent variable equals the algebraic sum of the differentials of its terms, niustration. ^ (a;^ + 3 a;^ — 2 a; + 5) = d{7^) + d{^x^) ^ d(-%x) -\- d{h) = da^dx-{-6xdx^2dx= {Sa^ -\-6x — 2)dx. 694. Let V =f(x) and z = f {x), and y = vz (1) then, y-\- Ay = {v -\- /\v) {z -\- A z) =iVZ-\-V./\Z-{-Z.l\V-\-llV.AZ (2) Subtracting, 'Ay = v.Az + z.Av-\-Av.Az (3) TV. .,. , Ay A z A V A V Dividing by a x, — - = v . + z . + — - . a z ^ ^ 'ax ax ax ax Lim. — ~ — lim. | v . ) + lim. ( z . ) AX \ AxJ \ AxJ + lim. (-^ . A zj whence, dy = vdz + zdv. Therefore, rrin, 5. — The differential of the product of two con- tinuous functions of the same independent variable equals the swn of the products obtained by multiplying each func- tion by the differential of the other. niustration. — d{--^7^Xhx^) = -dar'xd(6x^)-\-6x^Xd{-Sa^) = {-da^XiX6x-^-{-6xixSxi-3a^)}dx = — 65x^ dx. 378 ADVANCED ALGEBRA. Cor, d{vwz) = v.d{wz)-\-wz.dv [694, P.] ='Vwdz-\-vzdw-\-wzdv. [694, P.] ; etc. 696. Let V =: f{x), and z =f'(x); and ^ -1 y = — = vz ^ : z then dy = v , d {z-^) -\- z-"" d v [694, P.] = —vz~^ dz-\-z~'^dv _dv vdz_zdv — vdz Therefore, Prin» 6, — The differential of a fraction whose terms are continuous functions of the same independent variable equals the denominator into the differential of the numera^ tor minus the numerator into the differential of the de- nominator, all divided by the square of the denominator, \yy y' _2xy^dx-'3x^y^dy " ? • 696. Let y = log, x then y-\- Ay = ]og,{x+Ax) = log, x(l + ^j = log, X + loge (l + -^) [467, P. 2] = log.a;+ — - ^ . L_^ + - . L_l - etc. [601, C] — = - + 5. A2:; in which 5=_-.- + -.--^_ etc.; which for very small values of A a; is convergent [582]. .-. Lhn. (^\ = lim. (^ +B. ax) \AxJi,x = \X J^x-0 whence, -^ = —; and dy = — . Therefore, CuX X X DIFFERENTIALS AND DERIVATIVES, 379 Prin, 7. — The differential of the log, of a quantity equals the differential of the quantity divided hy the quan- tity itself Cor, — Since logio x — m log, x [607, P.] , ., . mdx ^.(logio^) = — ^. Illustration. — — ^ J ^■^^-\- (x-\-a)dx \ __ 1 /2x-\-a\ , "^ x-\-a{ x^ \'~ x-\-a\ X J 697. Let y = «*, in which « is a constant, then, log. y =^ X log. a [468, P.] d (log. y) = log. a . dx or, -^ = log. a . t?a; whence, dy = a" log. a .dx. Therefore, JPrin, 8, — The differential of a constant with a vari- able exponent equals the continued product of the original quantity, the logarithm of the constant, and the differential of the variable exponent. Thus, d{a-\-l)'^=-d{a-^b)^={a^ bf^ log. {a + b) X d (x^) = i x-^ (a + if^ log. (a + J) ^ x. 698. Problem. Find the differential of 05*. Let y — ^ then, log. y = X log. a; and d (log. y) = x . d (log. ic) + log. x , dx dy dx . , J or, — J = a; . 1- log. X . dx X/ X whence, dy — xf{\-\- log. x) dx, G 380 ADVANCED ALGEBRA. EXERCISE 97. Differentiate : 1. y=.hao[^ —'^Ix^-^-'ilcx — d 2. y = 5x^z^ -\-z ^ ' 14. ?/ = 5' y— 2^ jg_ y = {x-\- ay (x - h) = /^ 7. 2^=V5aJ+6 19. 2/ = 3^' z.f^%px 20. y = a;^^ 9.f = Za^ 21. 2^ = ic*(a + ^^)-i •^ ' , 22. «/ = lOge (a + Xf ^ ' ' 23. y = (f -^ d" 13. 2/ = Vx-\-a 25. y = d'^-' Applications. EXERCISE 98. 1. At what rate is the area of a circle increasing when the radius is 6 inches and is increasing at the rate of 3 inches per second ? Solution : Let y = the area, and x = the radius ; then, y z=z V x^ and dyz=2'irxdx. This denotes that at any instant the rate of increase of the area is 2 rr X times as great as the rate of increase of the radius at the same instant. But when the radius is 6 inches, it increases at the rate of 3 inches per second ; or, when x = 6 inches, dx = B inches. .'. dy = 2ir X 6 inches x 3 inches = 36 ir square inches ; that is, the area is increasing at such a rate that, if kept uniform for one sec- ond, the increase would amount to 36 v square inches. DIFFERENTIALS AND DERIVATIVES. 381 2. At what rate is the area of a square increasing when the side of the square is 4 inches and is increasing at the rate of 2 inches per second ? 3. The volume of a sphere increases how many times as fast as its radius ? When its radius is 6 inches and in- creases at the rate of 1 inch per second, at what rate is the volume increasing ? 4. At what rate is the diagonal of a square increasing when the side of the square is 8 inches and is increasing at the rate of 2 inches per second ? 5. The radius of a circle is 4 inches and its circumfer- ence is increasing at the rate of 2 tt inches per second. At what rate is the radius increasing at the same instant ? 6. A boy approaches a tree 90 feet high standing on a level road at the rate of 3 miles an hour. At what rate is he approaching the top of the tree when he is 220 feet from the base ? 7. The diagonal of a cube is increasing at the rate of 36 inches per second, when the side of the cube is 5 inches long. At what rate is the side increasing at the same time ? 8. If X increases at the rate of '5 per instant, at what rate is logio x increasing when a: = 42 ? 9. The logio 42 = 1-62325. What, then, would be the logio 42*5, if the increase were uniform ? How does the result compare with logio 4:2 '5 as found in the table ? Successive Derivatives. 699. If the derivative of f{x) be treated as a new func- tion of X [/i(^)], there may be found from it a second derivative of f(x) [/g {x)"] in the same way as /i {x) was derived from f{x), and so on, until a derivative is found that is independent of x \_f{x^\ 382 ADVANCED ALGEBRA. Illustration. — Let/ (a:) = x^ -]-4:a^ - '^a^ ^%7? - bx^^ -, then, f^{x) - 5a:* + 16a,-3-9a;2 + 4^-5 [693, P. 688] f^{x) = 20a;3^48^2_i3^_j_4 f^{x) = 60a:2_^96^,_i8 /4(a;) = 120a;4-96 f{x,) = 120 EXERCISE 99. Find the first derivative of : 1. ic3_4ic2 + 7a; + 2 4. (a + a:)5 (a _ a:)3 2. (a; + 2)3 (:z; - 2)* 5. {a + a:^) (a - t?) 3. 2:(a; + 2) + a;2(:r + 3) 6. (« + a;)^ ^ (« - ic)*^ Factorization of Polynomials containing Equal Factors. 700. Let f{x) = {x-\- tti) {x + a^) {x-\-a^) . . . . {x-^ a^) = any polynomial composed of binomial factors of the form of x-\-a; then /i (x) = {x-\- ttz) (x + as) (x-\-a^) ....(x + «„) + {x + «i) (a? + %) {x-{-a^) .... (x + aj + {x + «i) (:r + «2) (:r + «4) . . . . (a; + o^J + (^ + «i) (a; + ag) (^ + «3) i«^ + «» H- [694, Cor. 688]. Observations. — 1. If no two factors of f(x) are alike, f{x) atid fx (a;) have no common factor. 2. If two, three, or r factors of f{x) are equal, and all equal to jc + a, then will a; + a, (a; + a)*, or (a; + a)'"^ be a common factor of S{x) and /i (x). POLYNOMIALS CONTAINING EQUAL FACTORS. 383 3. In general, if f{x) contains the factor x -^ a p times, {x + h) q times, {x + c) r times then will {x + a)p-'^ {x + 6)?-* {x + cy-'^ .... be the H. C. D. of f(x) and /i (x). 4. The H. C. D. of f{x) and /i {x) contains one factor less of each kind than does f{x). 701. Theorem, — Every polynomial composed of bino- mial factors of the first degree, some of which are equal, may he decomposed into factors containing no equal bino- mial factors of the first degree. For, let f{x) be a polynomial composed of binomial factors of the first degree, some of which are equal, /i {x) its first derivative, /' {x) the H. C. D. of f{x) and /i {x), and <\>{x) the other factor oi f{x) ; then, 1. <^{x) will be devoid of equal factors of the first degree [700, 4]. 2. If /' {x) still contains equal factors of the first de- gree it may be resolved into two factors, /" {x) and <^' {x), in which {x)=f{x)-^P{x) = x + l .-. /(a;) = (a; + 2)(a; + 2)(a;-2)(a;-2)(a; + l). EXERCISE 100. Factor : 1. x^-^-'^Q^-llx^-VUx-m 2. a;«-5ic5 + a^ + 37tc^-86a;2_j_Y6a;-24 384 ADVANCED ALGEBRA. 3. o^-^x'-^Q x^-'^l ic5_|_2i6 2^+243 a^-^^Q x^-^^ a;-739 4. x^"" _ 30 a:8 + 345 x"^ - 1900 cc* + 5040 3? — 5184 5. a;^«-13a;8 4-42ic6-58ar* + 37a;2_9 Graphical Significance of /i (x). 702. Let m/i be the graph of y =f(x). Let P be a point on the graph whose co-ordinates are x and y. Let GE: = PR = Ax\ then will F'R = Ay. Draw the secant line P'JPS, also the tangent line T'P T. Take SB = 1, and draw BC = sin. /S and /SC = cos. S. Now, the triangles P' PR and jB/SC are similar. .-. ^ = §^=tan.>S'[668] (1) .-. -^ =:tan. /S (2) AX ^ ' Let the point P' approach the point P on the graph* so as to make ax diminish uniformly; then will the secant line P' P S re- volve about P and approach the tangent line T' P T &s its limit, and the angle S will approach the angle T as its limit. Lim. f -^"i = lim. (tan. S)i or, :t- dy^ dx tan. T. Therefore, The first derivative of a function is equivalent to the tangent of the angle which a tangent line to the graph of the function makes with the axis of abscissas. MAXIMA AND MINIMA OF FUNCTIONS. 385 Maxima and Minima of Functions. 703. The maximum or minimum value of a quadratic function may readily be found, as follows : Example 1. — What is the maximum or minimum value of x^-\-^x-\- 6, and what value of x will render it a maxi- mum or minimum ? Solution : Let f{x) = a;* + 8a: + 6 = w Complete the square, ic* + 8 a: + 16 = w + 10 Extract the \/, x + 4 = ± ^/nl + l6 Transpose, x = —4± ^m + 10 Now, w < — 10, else would x be imaginary. m = — 10 is the minimum value of / {x). But when m — — 10, a; = —4; then, a; = — 4 renders f{x) = a;* + 8 a; + 6 = — 10, a minimum. Example 2. — What is the maximum or minimum value of 8 a; — 3 2;^ + 9, and what value of x will render it a maximum or a minimum ? Solution : Let f{x) = 8a;-3a;2 + 9 = m Complete the square, 9 a;* — 24 a; + 16 = 43 — 3 m Extract the V» 3 a; - 4 = ± \/4S-dm Transpose and divide, x = -^ ± -w \^4d — 3m Now, 3 wi > 43, or m > 14 -o , else would x be imaginary. 1 ^ .'. w = 14-^ is the maximum value of f(x). 11 1 But, when m = 14-^ , x = 1-^; therefore, x = l-^ renders f(x) = 8a;— 3a;* + 9 = 14-;3- , a maximum. o Example 3. — Divide 36 into two parts whose product shall be the greatest possible. Solution : Let x and 36 — a; = the two parts, and X (36 — a;), or 36 a; — a;* = m. Then, a; =i 18 ± \/^24: - m. Now, m = 324 is a maximum ; x = 18 and 36 - a; = 18. 386 ADVANCEB ALGEBRA. 704. General Method.— Let mn he the graph ot y = f(x). Y /^ ^^ — pVU /a A X / / P5^ J. 1 V V V Conceive a point, P, to move along the graph, carrying with it a tangent Hne to the graph, in such a manner as to cause the abscissa (ic) of the point to increase uniformly. Let v be the value of the vari- able angle which the tangent line makes with the ic-axis. At P' 1? < 90 ; hence, tan. v, or /i {x\ is positive [668, Sch.]. This is true, however near Pi is to pii. At P° the tangent line is parallel to the rc-axis; hence, v = 0, and tan. v, or fx{x) =0. At P™, t;> 90; hence, tan. v, or /i (a;), is negative [668, Sch.]. This is true, however near P™ is to P°. Again, just before P arrives at P^, v > 90°, and tan. V is negative ; when P is at P^,v = and tan. v = 0; when F has just passed F^, v < 90 and tan. v is positive. Therefore, 706. Frin, 1» /i(a;) = at turning values of f(x), Prin, 2, Immediately before a maximum value of f{x), f\{x) is positive y and immediately after, negative, Prin* 3, Immediately before a minimum value of /(^)> /i(^) *^ negative, and immediately after, positive. 706. Caution 1.— A root of /i (x) = is not necessarily the abscissa of a turning point. For a tangent line to a graph may be parallel to the a;-axis where there is no turning point, as where two branches tangent to the same line coalesce at the point of tangency. (See diagram.) It is only when Prin. 2 or Prin. 3 is satisfied, as well as /i (x) = 0, that a turning point is established. Caution 2. — There may be turning points under peculiar conditions when /i (x) 4= 0. For there may be turning points where the tangent line to the graph is not parallel to the a;-axis; as where two branches coalesce and cease. (See diagram.) MAXIMA AND MINIMA OF FUNCTIONS. 387 707. Observations. — 1. So long as f{x) remains continuous, its maxima and minima values succeed each other alternately. 2. If two successive turning values of f{x) have the same sign, the graph of f{x) between these values can not cross the a;-axis, or f{x) 4= between these values. 3. If two successive turning values of f(^) have opposite signs, the graph of f{x) must cross the ic-axis between these values, or f{x) = somewhere between these values. 4. \i x = a and x = h render f{x) = 0, and a^h, there must be a turning value of f{x) between x = a and a; = 6. Example. — Find the turning values of f{x) = a;3 - 9 a;2 _^ 24 a; + 16. Solution : f{x) = a:3 — 9a;2 + 24a;+16 /, {x) = 3a;« - 18a; + 24 = 0; or, /,(a:) = a;2 — 6a; + 8 = 0; whence, a; = 4 or 2, critical values. /i(a;-Aa:). ^^4 , = (4 - A a;)* - 6(4- A a:) + 8 = - ( A a; = o f /i(a;+ Aa;), ^^4 > = (4 + A a;)* - 6(4 + A a;) + 8 = + ■) A « = o ) .'. f{x) is a minimum when a; = 4 But f{x)x = 4 = 43 - 9 X 42 + 24 X 4 + 16 = 32. .*. Minimum value of f{x) = 32 /i(a;-Aa;), ^^g . = (2 - a a;)»- 6(2 - A a;) + 8 = + j A a; = o j fi(x+ Ax)^ ^^2 I = (3 + A a;)« - 6(2 + A a;) + 8 = - I A a; = o I .*. f(x)x = 2 is a maximum But f(x)x = 2 = 2» - 9 X 22 + 24 X 2 + 16 = 36 .*. Maximum value of f(x) = 36. The value of f(x)x = a is best obtained by synthetic division, as in Art. 106. EXERCISE 101. Find the maxima and minima values of : 1. 4:a^-15a^-}-12x-l 5. x^ -dx^ - 9x-\-6 2. 23^-21a^-{-36x-20 6. (a; - 1)* (a; + 3)^ 3. x^-{-6x + 6 7. (re - af {x + bf 4.ic2_6a; + 5 B. x^ -3x^ -\-3x-\-H 388 ADVANCED ALGEBRA. 9. a^-^Qx-6 11. a^-^3(?-{-U \0. a^ — Qx — h 12. a;* + a;3 + a;2 — 16 13. Show where a line a feet long must be divided so that the rectangle of the two parts may be the greatest possible. 14. Find the altitude of the maximum cylinder that can be inscribed in a sphere whose radius is r. Suggestion.— Let B C = Xy BD = r — x, and AB — y^ then, 2/^ = (^ + ^) (r — a;) = r* — ic*, and /(a;) = F=ir2/2 X 2 a; = 3 ir a; (r^ — jr*) /i(a;) = 2ira; X (-2 a;) + (r3_a;8)x2ir = whence, x = -^ ^J~^ o 2 and, 2/2 _ ^3 _ 3.3 _ ,.2 o 2 /- 15. Find the altitude of the maximum cylinder that can be inscribed in a cone whose altitude is a and whose radius is J. 16. Find the volume of the maximum cone that can be inscribed in a given sphere. 17. Find the area of the maximum rectangle that can be inscribed in a square whose side is a, 18. What is the maximum convex surface of a cylinder the sum of whose altitude and diameter is a constant a ? 19. Find the altitude of the maximum cylinder that can be inscribed in a right cone whose altitude is a and the radius of whose base is J. 20. Eequired the area of the maximum rectangle that can be inscribed in a given circle. 21. Required the greatest right triangle which can be constructed upon a given line as hypotenuse. CHAPTER XII. THEORY OF EQUATIOJ^S, Introduction. 708. Equations of the first and second degree have already been treated, and need no further attention here. 709. Jerome Cardan, an Italian mathematician (1501- 1576), published in 1545 a method of solving cubic equa- tions, now known as *' Cardan's Formula." But, as this formula is not finally reducible when the roots of an equa- tion are real and unequal, it is not of much practical value. 710. Eene Descartes, a French mathematician (1596- 1650), transformed the general bi-quadratic equation so as to make its solution depend upon that of the cubic equa- tion ; but, as he invented no new method of solving the latter, the same difficulties are encountered in the applica- tion of his rule as are met in Cardan's. 711. Nicholas Henry Abel, a Norwegian mathematician (1802-1829), demonstrated, in 1825, the impossibility of a general solution of an equation of a higher degree than the fourth. Previous to that date many such solutions were attempted. 712. The real roots of numerical equations of any de- gree are, however, attainable through laws and principles to be developed in this chapter. 390 ADVANCED ALGEBRA, Normal Forms. 713. Theorem I. — Every equation of one unhnown quantity with real and rational coefficients can he trans- formed into an equation of the form of Ax^-\-Bx^-'^-{-Cx^-^-i- .... +i; = 0, in which A and all the exponents of x are positive in- tegers, and each of the remaining coefficients, including L, is either an integer or zero. Note. L may be regarded the coefficient of a:P. Demonstration. — 1. If the equation contains fractional terms, it may be cleared of fractions. 2. If there are any terms in the second member, they may be transposed to the first member. 3. All terms containing like exponents of x may be collected into one term by addition. 4. If A is negative, both members may be divided by — 1. 5. If X contains negative exponents, both members may be multi- plied by X with a positive exponent numerically equal to the greatest negative exponent. 6. If X contains fractional exponents, x^ may be substituted for x, in which m is the L. C. M. of the denominators of the fractional ex- ponents. The roots of the transformed equation will he the mth root of the roots of the original equation. 1. The terms may now be arranged according to the descending powers of x. 714. The equation A af-i-Bx''--' + (7:r"-2-f . . . . + X = 0, is known as the first normal form of an equation of one unknown quantity, and will hereafter be represented by Example. — Transform 3x^+-r—S-[-7 x~^ = - + -r- x^ o xt into the first normal form, and compare the corresponding roots of the two equations. NORMAL FORMS, 391 Solution: Given 3a;§ + -r — 8 + 7a;— f = — + -j. (A) xi ^ xt Clear of fractions, 9x + 12 — 24:X^ + 21 x—^ = 4xi + 9xi (B) Transpose and collect terms, 9a; + 21a;-i-28a;i-9a;* + 12 = (C) Multiply by xt , 9a;t + 21-28a:f-9a;l + 12a;i = (D) Puta; = a;«, 9a;9 + 21 - 282;* - 9^:^ + i2a:3 _ q (E) Rearrange terms, 9a;9 + 0a;8 + 0a;' + 0a;«-28a;5-9a:4 + 12a:3 + 0a:2 + 0a: + 21 = (F) The roots of (A) = Vof the roots of (F). 715. An equation that contains all the powers of x, from the highest to the lowest, is called a Complete Equa- tion, An incomplete equation may be written in the form of a complete equation by supplying the wanting terms with coefficients of zero. Thus, a;^ — 4a;^ + 2a; — 5 = may he written 2:^ ± ar* — ^7? ±0x^-\-2x — b = 0. 716. Theorem II, — The equation F^ {x) = ?nay be transformed into an equation of the form of x^-\-p,x''--' +p,x--'-\-., . .+^„ = 0, in lohich the coefficient of a:" is unity ^ and each of the remaining coefficients is either an integer or zero. Demonstration.— Take Fn{x) = Ax"" + Bxf^-'^ ^Cx!^-'^ ■¥ + L = ^ ^ X Ax"" Bx»-^ Cx»-^ Put X = -r, — r- + .„ , + -j^r^T + +L = A A" A'*-^ A»-2 Multiply by .4„_i , a;« + Bx^-^ + A Cx'^-^ + + J.«-^ X = Put pi for B, Pi iov AC, p„ for J.«-» L, x"* + pi X^-^ + Pi X^-^ + + Pn= This is the second normal form of an equation of one unknown quantity, and will hereafter be represented by /« {x) — 0. ADVANCED ALGEBRA. 717. Cor. 1, — Each root of /„ {x) = is A times as great as the corresponding root of F^ {x) — 0. 718. Cor, 2, — The coefficient of the second term of /„ {x) is the same as the coefficient of the second term of F^ {x), and the succeeding coefficients of /„ {x) are obtained ly multiplying the succeeding coefficients of F^ix), in order, ly A, A^ , A^, ^"~^. Note. — If terms are wanting, supply them with coeflScients of 0. Example. — Transform the equation Ax^ — dx^-\-2a^ — 7 = into an equation of the form of f(x) = 0. Solution : Given Fix) = Aafi -Sx^ + Ox^ + 2x^ + Ox-7 = 0, thenwm fix) = x^-dx^ + 4:x0a^ + Px2x^ + ¥x0x-4^x'7=z0 [718] or, fix) = x^-dx^ + d2x^- 1792 = 0. The roots of fix) = are 4 times as great as those of Fix) — 0. EXERCISE 102. Transform the following equations into equations of the form of f{x) = 0. Compare the roots of the trans- formed equation with the roots of the original equation. 1. Sa^-\-2x^-da^i-l!x^6 = 2. 2x' + 4:a^-x^-{-x^-1l = 3. 4:X^-\-3a^-5x^+llx-l=:0 4. Sx^-2x^ + 3xi-2x^-{-4:X^-2 = 5. x-^ 4_ 2 ic-^ + 3 x-^ - x-^ -^x-^-2x-^ + 2 = 6-lx+^x-i+lxi-^lx-^ + 3 = 7. x^'-dx^^2x^-\-dx-^-\-2 = 8. |^t+J^t_ 1^ + 1 = DIVISIBILITY OF EQUATIONS. Divisibility of Equations. 719. Theorem III. — If a is a root of F^ (x) = 0, then X — a is a factor of F^ (x). For, let F. {x) ^ {x - a) = F^_, {x) + ^ then, {i^„_i {x)\{x-a)-\-r = F^ {x) = but, x — a = 0, since x = a. r = 0; whence, F„ (x) -^ (x — a) = i^„_i (x), 720. Cor. 1. — If a is an integral root of F^ {x) = 0, it is a divisor of the absolute term of F^ {x) [163]. 721. Cor. 2. — If X — a is a factor of F^ {x), then a is a root of F^ {x) = 0. For, F^{x) = {F,_, (x)} (a; - a) = ; whence, x — a = 0, and x = a. 722. Cor. 3* — If x is a factor of F^ {x), then zero is a root of Fr,{x)-0. Number of Roots. 723. Theorem IV, F^ {x) = has at least one root. The demonstration of this theorem may be found in special treatises on the Theory of Equations. It is too long and tedious to be introduced here. 724. Theorem, V. F^ {x) = has n roots and only n. For, F^ {x) — has at least one root. [T. IV.J Let a = one root of F^ {x) = ; then, F„ (x) = { F„_t (x)] {x — a} =0 [T. IILl .-. F,,_^{x) = 0, Let b = one root of i^„_i (x) = ; [T. IV.] then, F„_^ (x) = {F^^s (x)] {x-b}=0 [T. III.] .-. F^_,{x) = 0. 394: ADVANCED ALGEBRA. Now, as F^ {x) = is of the nth. degree, and each time a root is removed by division the degree is lowered by unity, it follows that n roots and only n can be removed before F^{x) reduces to an absolute factor. Therefore, Fn {x) = has n roots and only n. 725. Car, F^ (x) = may he written A{x — a){x — b){x — c) (x — l) = Q; or simply {x — a){x — b) (x — c) (x — l) = 0, in which there are n factors of the form of x — r, the second terms of which are the roots of F^ {x) = with their signs changed, and may he positive or negative, fractional or integral, rational, irrational, or imaginary, subject only to restrictive conditions explained hereafter. Relation of Roots to Coefficients. 726. OThearem Vl.—If F^ (x) = be put in the form of x^ + B^ a;»-i + Ci a;*-^ + . . . . -|- Xi = 0, 5^^ dividing both members of the equation by A, the coefficient of x"", then will 1. Bx = the sum of the roots vnth their signs changed. 2. Ci= the sum of the products of the roots taken two together. 3. Di = the sum of the products of the roots with their signs changed, taken three together. 4' El = the sum of the products of the roots taken four together. And so on to 5. Li = the product of all the roots with their signs changed. Demonstration : Let the n roots of the equation he a, b, c, I; then, Fn (x) = x'* + Bi x^-^ + Ci x^-^ + + Li = {X - a){x - b){x - c) . . . . (X - I) [725]. After which the theorem is a direct inference from the binomial for- mula [587], and the principle that " changing the signs of an even IMAGINARY MOOTS. 395 number of factors does not change the sign of their product " [page 26, Ex. 3]. Cor. — Changing the signs of the alternate terms of F^ {x) = changes the signs of its roots. Imaginary Roots. 727. Theorem VII, — Imaginary roots can enter F^ {x) = only in conjugate pairs. For in this way only will their sum and the sum of their products be real [657], as they must be [713]. 728. Cor, 1, — The product of the imaginary roots of Fn {x) = is positive. For the product of each pair is positive. Thus, (a -\-bi) {a '-bi) = a^-\- W. 729. Cor, 2, — When all the roots of F„ {x) = are imaginary the absolute term is positive. Suggestion. — For the equation is then of an even degree. 730. Cor, 3, F^ {x) = has at least one real root oppo- site in sign to the absolute term, when n is odd. 731. Cor, 4, F„ (x) = has at least two real roots, one positive and the other negative, if n is even and the absolute term is negative. 732. Car. 5, — The sign of F^ {x) for any real value of X depends on the real roots of F^ (x) = 0. For the product of x — {a-\-b i) and x — {a — bi) = {x — aY + b^, a positive quantity ; and this is true of every pair of factors containing conjugate imaginary terms. ' 733. Cor, 6, — Every entire function of x with real and rational coefficients may be divided into real factors of the first or second degree. 396 ^^ VANGED ALGEBRA. Fractional Roots. 734. Theorem VIII, — JVo root of /„ {x) = can he a rational fraction. Take fn(x) = x^ + PiX^-''-{-p2X--^+ ....+jo^ = [716]. If possible, let a; = t- , a rational fraction in its lowest terms. Then, by substitution, Jni- Jn-1 -1- Jn-2 "h • • • • "f ^« " ^. Multiplying by 5**"^ and transposing terms, we have an integer, which is impossible. Scholium. — From this theorem it follows that the ra- tional fractional roots of F^ {x) = may be obtained by transforming F^ {x) = into /« (x) = and dividing the roots of the latter equation by A, the coefficient of ic"~^ in the former. Relations of Roots to Signs of Equation. 735. Theorem. IX. — If F„ (x) = has no equal roots, then Fn (x) will change sign-if x passes through a real root. For, take F^{x) =\x — a){x — b){x — c) {x—l) = [7^5] ; conceive x to start with a value less than the least root and continually increase until it becomes greater than the greatest root. At first, every factor of F^ (x) is nega- tive, but, at the instant it becomes greater than the least root, the sign of the factor containing that root will be- come plus, while the others remain minus ; whence, F^ {x) will change sign. It will, moreover, retain its new sign until it passes over the next greater root, when it will again change sign, and so on. RELATIONS OF ROOTS TO SIGNS OF EQUATION. 397 736. Cor, 1, — If for any two assigned values of x, Fr, {x) has different signs, one, or, if more than one, an odd number of roots of F^ [x) = lie between these values, 737. Cor. 2, — If for any two assigned values of x, F^ {x) has the same sign, either no root or an even number of roots of Fr, {x) = lie between these values, 738. Some of the properties of F^ {x) = 0, already dis- cussed, are beautifully illustrated by the following graph. Y 1. It is seen that y = Fn{x)=:0 when a; = 1, 2, 3, and 5. There- fore, these values of x are roots of i^„ {x) = 0. 2. Immediately before a; = 1, 2/ is positive, and immediately after x = l, y is negative ; immediately before x = 2, y is negative, and immediately after x = 2, y is positive, etc. ; illustrating that when x passes over a real root, Fn (x) changes sign. 3. At a; = 3 two values of y become zero ; therefore, two roots become identical, or, in other words, 3 is twice a root. Were the abso- lute term of i^„ {x) so changed as to make y somewhat less, the a:-axis would cross the graph twice between a; = 2 and a; = 4, once before a: = 3, and once after, thus proving conclusively the duality of the root 3, when y = 0. 4. Immediately before a; = — 2 and a; = 6, the graph approaches 398 ADVANCED ALGEBRA. the ic-axis, but in each case makes a turn before reaching it, prevent- ing, thereby, equal roots or unequal real roots. These turns locate the position of imaginary roots. The truth of this statement becomes manifest when we suppose the absolute term of Fn {x) to so change as to cause y to gradually decrease, the a;-axis will gradually arise and finally touch the graph at ic = — 2, thereby making two equal roots, and, if y continues to decrease, the ic-axis will cross both branches above the turn at x= —2, making two unequal real roots. The student will be interested in observing the changes in the roots if the absolute term of the equation so changes as to cause the a;-axis to gradually move from the position Aix\ to the position A^x^- 5. It must not be assumed, however, that imaginary roots always denote a turning point in the graph of the equation. Such may or may not be the case. 739. If any two successive terms in a complete equa- tion have like signs, there is a permanence of sign ; if unlike signs, a variation of sign. Thus, in the equation cc« - 5 a;5 + 8 a;* + 7 a;3 - 3 rc2 + 2 re - 5 = there are five variations and one permanence. 740. Theorem X. — No complete equation has a greater number of positive roots than there are variations of sign, nor a greater numher of negative roots than there are per- manences of sign. Demonstration : Let the following be the successive signs of a com- plete equation : + -- + + — There are here two permanences and three variations. To intro- duce another positive root, the equation must be multiplied by x — a. The signs of the product will readily appear from the following work : + — - + + — -I- — + — — + + — — + + -- + + — The double sign denotes a doubt, growing out of an ignorance of the relative numerical magnitudes of the terms added. Now, a careful inspection will show that, whether we regard both doubtful signs negative, both positive, or one negative and the other RELATIONS OF ROOTS TO SIGNS OF EQUATION. positive, the number of permanences will not be increased, but the number of terms is increased by one ; therefore, the number of varia- tions must be increased by at least one. Since the introduction of a positive root introduces at least one variation, it follows that the num- ber of positive roots can not exceed the number of variations. In a similar manner, by introducing the factor x -\- a, it may be shown that the number of negative roots can not exceed the number of permanences of sign. This is Descartes' celebrated rule of signs. 741. Car, 1, — If all the roots of an equation are real, the number of variations equals the number of positive roots, and the number of permanences equals the number of negative roots. 742. Cm*. 2, — An equation whose terms are all positive can have no positive roots, 743. Cor. 3. — An equation tvhose terms are alternately positive and negative can have no negative roots. Limits of Roots. 744. A nnmber known to be equal to or larger than the largest root of an equation is called a superior limit to the roots of the equation. 745. A number known to be equal to or smaller than the smallest root of an equation is called an inferior limit to the roots of the equation. 746. Theorem XI. — If the first h coefficients of F^ (x) are positive, and P is the smallest of them, then, if Q is numerically the largest subsequent coefficient, \/ ^ + 1 is a superior limit to the roots of F^ {x) = 0. Demonstration : It is evident that the case in which x must have the greatest value to make Fn (x) = when the first h coefficients are positive, is the one in which these coefficients are all equal to the least one of them (P), and the remaining n + 1 — h coefficients are all 400 ADVANCED ALGEBRA. negative and each- equal to the greatest among them {Q). Therefore, the value of a; is a superior limit to the roots of Fn {x) — 0, if Pa^ + i — Paari + i-k _ Qx'^ + ^-^—Q or, z — z X— 1 X— 1 or, Pa?' + i-A(a;*-l) = ^(a;» + ^-*-l) or if, P(x^ - 1) = ^, since 1 > (l - ^^^^^ + 1. = l/| 747. Cor, — If the signs of the alter^iate terms of an equation he changed, then loill the superior limit to the roots of the transformed equation, with its sign changed, be the inferior limit to the roots of the original equation [726, Cor.]. Equal Roots. 748. Theorem XII, — If F^ {x) = has equal roots, it may he separated into two or more equations with unequal roots. This is a direct inference from Art. 701. Commensurable Roots. 749. The integral and rational fractional roots of F^ (x) = are called its commensurable roots. 750. Problem 1. To find the commensurable roots of Fn {X) = 0. Solution : Pursue the following line of investigation : 1. Determine the number of roots the equation has [724]. 2. Determine how many roots may be positive and how many negative [739]. 3. Determine the limit to the positive and the negative roots [746, 747]. COMMENSURABLE ROOTS. 401 4. Determine what integral numbers may be roots [720]. 5. Find and remove the integral roots by synthetic division [719, 105]. 6. Determine whether there are any equal roots [701], and if so, remove them by synthetic division. 7. Find the rational fractional roots from the equation resulting from the removal of the integral roots, and according to Theorem VIII, Scholium. niustrations. — 1. Find the commensurable roots of F^ (x) = 24 cc* + 122 a;3 + 5 a;2 - 26 a; - 5 = 0. Solation : 1. This equation has four roots, all real, or two real [724, 731]. 2. There are one variation and three permanences of sign ; there- fore, there can not be more than one positive nor more than three negative roots [739]. 3. The only integral roots possible are +1, — 1, +5, and — 5 [720]. 4. The largest positive root < 4/ — + 1, or < 2 [746]. 5. Neither + 1 nor — 1 is a root, since F^ (x) is not divisible by either a; — 1 or a; + 1, as witness : -1)24 + 122+ 5- 26- 5 - 24- 98 + 93- 67 + 98- 1)24 + 122 + 24 + 93+ 67- 5-26- 146 + 151 + 72 5 125 146 + 151 + 125 + 120* Note. — It is evident that when + 1 is a root of Fn ix) = 0, the sum of the positive coefficients must equal the sum of the negative coefficients ; and, if — 1 is a root, + 1 is a root if the signs of the alternate terms are changed. These facts determine a more expe- ditious method of testing whether either + 1 or — 1 is a root. 6. — 5 is a root, as witness : -5)24 + 122+ 5-26-5 -120-10 + 25 + 5 + 2-5-1 7.' The resulting equation, after removing the root — 5, is F^ (x) = Mx^ + 2 x^ — 5 X — 1 = 0, which has no integral roots. Transform Fa (x) = into an equation of the form of /a {x) = 0, f3(:x) = afi + 2x^- 120 a: - 576 = [733, Sch.]. 402 ADVANCED ALGEBRA. 8. /s {x) = has three roots [724], only one of which can be posi- tive, and the largest positive root possible is ^^77 = 24. 9. The divisors of 576 not exceeding 24 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24. From the relative values of the positive and negative co- efficients it will be seen at a glance that a; > 6. 10. + 8 and + 9 are not roots, but + 12 is a root, as witness : 4-8)1+ 2-120-576 + 8 + 80- -320 + 10- 40- -896* + 9)1 .+ 2- 120- -576 + 9 + 99- -189 + 11- 21- -765* hl2)l .+ 2- 120- -576 + 12 + 168 + 576 14+ 48 11. The resulting equation, after removing the root + 12 from /a ix) =z 0, is /a (a:) =: a;2 + 14 a; + 48 = ; whose roots are found to be - 8 and - 6 [3311. 12 8 12. The four roots of Fiix) = Q are, therefore, —5, +04.' "oi* and - ^ [733, Sch.], or -5, + -^, - g , and - ^. 2. Find the commensurable roots of /^ (a;) = a;^ -f 3 a;« - 1 2 a;5 - 3 6 a:* -f 48 a:^ ^ 144 a;2_g4^_ 192 = 0. Solution : It may readily be found by synthetic division that + 2, — 2, and — 3 are the only integral roots of this equation. The resulting equation, after removing these roots, is fi (a;) = a:* — 8 a:* + 16 = 0. If any of the roots of this equation are integral, they must be equal to one or more of the roots already found. They may, however, all be incommensurable or imaginary. Factoring ft (a;), we have {x^ — 4) (a;* — 4) = ; whence, a; = + 2, — 2, +2, — 2. Therefore, all the roots of /t (x) = are ±2, ±2, ± 2, and — 3. EXERCISE 103. Find the commensurable roots of : 1. a^-^x^-\-'ix-b = ^ 2. a:3_6a;2-f-10a;-8 = 3. a^ - 11 2:2 -f 41 2; - 55 = COMMENSURABLE ROOTS. 403 4. a:3 + 6a:2_j_i4^_{.12 = 5. :r*-3a;3_2a;2 + 12a:-8 = 6. 12:^3 _|_3a;2_3^._2 = o 7. a:* + 2a:3-7a;2-8a; + 12 = 8. 2r* - 8 a;3 + 10 a;2 + 24 a; + 5 = 9. a;5 + 3a;4-3a:3-9a:2-4a;-12 = 10. a;*-5ic3 + 3a;2 + 2iC + 8 = 11. 8a:3_i6^8_3^_|_2i = o 12. 16a;^-48a:3_^ 32^:2 _^12a;-9 = 13. 3a;5 + 2a:*-21a:3-14a:2 + 36a; + 24 = 14. 9 a;5 + 81 :r* + 203 a:3 + 99 a,-2 - 92 a; - 60 = 15. 18a;5 + 9a;* + 22a;3 + lla;2-96a;-48 = 16. a:* + 4ic3_i3 2;3_28a; + 60 = 17. ar* + 2a:3_ii^2_12a; + 36 = 18. ic5 + 4a;* + a:3_iQ^_4^_|.3^0 19. 3 a;6 + 22 a;5 + 8 a;* - 42 :z;3 _ Y9 ^2 _ 52 ^ _ 12 = Incommensurable Roots. 751. The incommensuraUe roots of an equation are best sought for after all the commensurable roots have been removed by division and the resulting equation trans- formed into an equation of the form of /« {x) = 0. 752. The first step necessary in the search for the values of the incommensurable roots of an equation is to find the number and situation of such roots. Jacques Charles Frangois Sturm, a Swiss mathemati- cian (1803-1855), discovered a method of doing this in 1829, known as Sturm's method. 404 ADVANCED ALGEBRA. 753. Sturm's Series of Functions. — Assuming that f^ (x) = has no equal roots, this eminent mathematician formed a series of functions, as follows : The first two terms of the series are /„ (x), and its first derivative, which we will now represent by /«_i (x). The other functions, and which are called Sturmian functions, are derived as follows : Divide /„ (x) by /„_i (x), and represent the remainder with its sign changed by /«_2 {x). Divide /„_i {x) by /„_2 {x), and represent the re- mainder with its sign changed by /„_3 (x) ; continue this process until the last remainder with its sign changed is an absolute term. Eepresent this remainder /o {x). There will then be n-\-l of these functions, as follows : /n (^), /«-l {X), fn-2 {x) ....fo [x). Caution. — Care must be taken in the operation of successive divis- ion not to reject any negative factors except in the remainders. 764. Relation of the terms of Sturm's series of func- tions. — If we put qi, qz, q^ as the successive quo- tients obtained in finding the Sturmian functions, it is evident that /»(^)=/«-l(^)^l-/n-2W (1) /„_! {X) = /._2 (X) q2 - fn-3 (^) (^) fn-2 (X) = fn-3 (x) §'3 " /«-4 (^) (3) /„_3 (x) = ./;_4 (x) q, - /„_5 (x) (4) /„_4 (x) = /«_5 (x) q, - /„_6 (x) (5) etc., etc., etc. 755. Fundamental Principles. 1. No two consecutive functions can vanish, i. e., be- come 0, for the same value of x. For, if possible, \Qtx = a make /„_2 = and /„_3 = ; then will /„_4 = [754, 3], and hence, too, /n_6 = INCOMMENSURABLE ROOTS. 405 [754, 4], and so on until lastly /« (:?;) = ; but /o {x) is the absolute term and can not be zero. Therefore, etc. 2, If any one of the functions intervening hetween /„ {x) and /o {x) vanishes for any value of x, the two ad- jacent functions have opposite signs for this value. Thus, \i x=:a causes /„_3 {x) to vanish, /„_2 (x) — -f.-,{x) [754, 3]. 3. If any value of x, as x = a, causes any intervening function to vanish, then will the number of variations and the nuniber of permanences in the signs of the functions he the same for the immediately preceding and the imme- diately succeeding values of x, i. e., for x = a — and x = a-\- <^ . For the two adjacent functions will have opposite signs when x-=a [755, 2], and will not change their signs for any value of x from x = a — c> and a; = « -|- o , since no root of either can lie between these values [755, 1]. But the function in question does change its sign, since x passes over a root of the function in going from a; = a — o to ^ = « + o . If the signs of the three functions for x = a — <^ are +? +> — > for x = a-\-o they will be + , — , — , which in either case form one permanence and one variation. Similarly, +, — , — will change to + ,+,—; — , +, + will change to —,—,+; and — , — , + will change to — , +, +. 4' If any value of x causes /„ {x) to vanish, then will one variation in the signs of the functions be lost in pass- ing from the immediately preceding value of x to the im- mediately succeeding value. {X - «„) + (1st term) (*-««) + (2d term) {x-a,) + (3d term) 406 ADVANCED ALGEBRA. Let f,{x) = {x — «i) {x - ttg) {x — a^) (x — a,) ; then /,_! (a;) = {x — ttz) {x — as) {x — a^) . (x — tti) {x — Us) (x — a^ . {x — «i) {x — a^ {x — a^ . {x — «i) {x — ttz) {x — a^) (x — «„) + (4th term) (x — tti) (x — ttz) {x — a^) x—a,^_i^ (nth term) Now, if x equals, say as, then will fn(^) and all the terms of /„_i (x), except the third, vanish. Now, the third term of /„_i (x) contains all the factors of fn (x) except x — a^. Therefore, if x is infinitesimally less than %, x — % will be negative and /„ (x) and /„_i {x) will have opposite signs or will form a variation ; but, if x is infinitesimally greater than as, x — as will be positive and /„ (x) and /„_i (x) will have like signs or will form a permanence. Therefore, a variation is lost in passing from tts — O to «3 -j- O . 766. These principles are true if /„ (x) contains imagi- nary roots as well as when all the roots are real, since the signs of the functions depend wholly upon the real factors they contain [732]. Sturm's Theorem. 757. The number of variations of sign lost in the terms of the Sturmian series, as the value of x continuously changes from a to b, a being less than h, equals the num- ber of real roots of f„ (x) = lying between a and b. Demonstration. — For each time the value of x, in ascending from a to b, passes over a root of /» {x) = 0, there is lost one variation of sign [755, 4] and only one [755, 3]. INCOMMENSURABLE ROOTS. 407 758. Cor, 1. — The theorem is equally true for F„ {x) = 0, there being nothing in the demonstration of it to restrict its application to /« {x) = 0. 769. Cor, 2, — The difference between the number of variations when + oo and — cc are substituted for x in the series is the number of real roots in the equation. 760. Cor. 3, — The difference between the number of variations when and + oo are substituted for x is the number of positive roots, and, when and — oo are sub- stituted for X, the number of negative roots. 761. Eemark 1.— It is evident that the sign of the absolute term of a function is the sign of the value of the function, when a; = 0. 762. Remark 2. — The sign of the first term of a function is the sign of the value of the function, when a; = ± oo . For, Ax^ = Acc^-^ .x> Bx:^-'^ + Ca;"-* + Dai^-^ + + Lx""-^ > Bx^-^ + Cx:^-^ + Dx?"-^ + + L, when a; = ± oo . 763. Remark 3. — The sign of the value of a function for any integral or decimal value of x is best determined by the method ex- plained in Art. 106. Illustration. — Find the sign of i^4 (a:) = 3 a:* — 2 a;^ + 7a;2 — 3a;-8 when x=l'2. Solution : The value of i^4 {x) when a: = 1-2 is + -3808, as witness: 1.3)3 _ 2 +7-3-8 3-6 + 1-92 + 9-984 + 8-: 1-6 + 8-92 + 6-984 + -3808* .'. The sign of F^{x) is +.' Note. — In practice it is usually not necessary to make the last multiplication and addition to determine the sign of the value. 764. Remark 4. — Though it is not usually best to apply Sturm's method of solution to equations before the commensurable roots have been removed by division, on account of the great labor involved in deriving and evaluating the different functions when the equation is of a high degree, yet such a course may be pursued. If there are equal roots, the fact will appear in deriving the functions, and if there are integral or fractional roots they will be discovered in evaluating the functions to determine their signs. 408 ADVANCJSn ALGEBRA. Example. — Determine the number and situation of the real roots in /s (a:) = o^ — 12 a;^ + 57 a; — 94 = 0. Solution : /a {x) = a;^ — 13 a;2 + 57 a; - 94 /2(a;) = 3a;2-24a; + 57 fi{x) = —x + ^ foix)=- Substituting in these functions as follows, we shall have : For x=+Go, + + — — one variation. For X = 0, _ 4- 4. _ two variations. For x= —cOy _ + + _ two variations. There is, therefore, one real root between and 4- oo . There is no negative root. Therefore, there are two imaginary roots. To find the situation of the real root, we proceed as follows : For re = 1, we have — + + _ two variations. For a; = 2, we have — + + _ two variations. For ic = 3, we have — + ± _ two variations. For aj = 4, we have + + — — one variation. Therefore, there is one real root between 3 and 4, or the first figure of the real root is 3. To find the next figure, we proceed as follows : For X — 3-1, we have — + _ — two variations. For a; = 3*2, we have — + _ _ two variations. For X = 3-3, we have — + _ _ two variations. For X = 3-4, we have + + — — one variation. Therefore, the root lies between 3*3 and 3'4, or the first two fig- ures of the root are 3-3. By a continuation of this process the root might be extended to any number of figures. A more expeditious method, however, is known, and will be explained hereafter, for extending a root after a sufficient number of figures have been found to distinguish the root from any other root lying near it. Thus, if an equation had the two roots 3*1256. . and 3*1234. . ., the first four figures of each root only would be found by Sturm's theorem. When it is known, as in the above example, that only one real root lies between two numbers, it becomes necessary only to study the signs of /„ (x), since passing over the roots of the intermediate functions does not cause a change in the number of variations. The same conclusion will be reached by the simple application of Art. 735, since /a (x) changes sign between a; = 3 and a; = 4. INCOMMENSURABLE ROOTS. 409 EXERCISE 104. Find the number and situation of the real roots in the following equations : 1. a:3_4a;2-6a; + 8 = 4. x^-10a^-[-Qx-\-l = 2. x^ + 6x^'-Sx + 9 = 5. 2:r*-lla;2^82,_16_0 3. a^-{-3s^-'6x-[-2 = 6. xf" - Ux^ + Ux -3 = Horner's Method of Root Extension. 765. In 1819, W. G. Horner, an English mathema- tician, published an elegant method of extending a root of an equation to any desired number of places, after a sufiBcient number of initial figures have been found by- other methods to distinguish the root from other roots of the equation. This method is based upon the following principle : 766. Principle, — If F^ {x) he continuously divided hy X — a, the successive remainders will he the coefficients in inverse order of an equation whose roots are a less than the roots F^ {x) = 0. Demonstration : Take En (x) = A a?— ^ + 5a:«-2 + + Jx^ + Kx + L = (A) Put Xi + a = X, or Xi=x — a; En {Xi + a) = A(xi + «)«-! + i? (a;, + a)«-2 + + J{Xi + af 4- K{xi + a) + i = (B) Expand terms, bracket coefBcients of like powers of Xx , and rep- resent the coefficients of the transformed equation by -4j , ^i , . . . . «/i , Kx, Li', then, En {Xx) = Ax a;i»-i + Bx Xx^-^ + . . . . Jx Xx^ -{■ KxXx-\- Lx = (C) Now, the roots of (C) are evidently a less than those of (A). Substitute Xi = x — a m. (C), En{x -a) = Ax{x- ay + Bx(x- a)"-! + + Ji{x- af + Kx{x-a) + Lx = (D) Of 1 HE -NIVERSITY \ 410 ADVANCED ALGEBRA. Now, Fn (x — a) is evidently equivalent to Fn (x), and will leave the same remainder when divided by a; — a as will Fn (x) -i-{x — a). But, if Fn {x — a) is continuously divided by x — a, the successive remainders will he Li, Ki, Ji, £i, and Ai, or the coefficients of Fn (xi) in inverse order. Therefore the theorem. Applications. 1. Transform aH^ -\- x^ -}- x^ -^ 3 x — 100 = into an equation whose roots are 2 less than those of the given equation. Foniii ( + 3 Famii + 1 + 1 + 3 -100 + 2 + 6 + 14 + 34 + 3 + 7 + 17 - 66* + 2 + 10 + 34 + 5 + 17 + 51* + 2 + 14 + 7 + 31* + 2 The transformed equation is x^ + 9 x^ + 31 x^ + 51x — 66 = 0. Explanation. — Dividing by x — 2, by synthetic division, the co- efficients of the first quotient are 1 + 3 + 7 + 17, and the first re- mainder is — 66, the absolute term of the transformed equation. Dividing 1 + 3 + 7 + 17 again by + 2, the second quotient is 1 + 5 + 17, and the second remainder, or the coefficient of x in the transformed equation, is + 51. Dividing 1 + 5 + 17 again by + 2, the third quotient is 1 + 7, and the third remainder, or the coefficient of x^ in the transformed equation, is + 31. Dividing 1 + 7 again by + 2, the fourth quotient is 1, and the fourth remainder, or the coefficient of a;' in the transformed equation, is +9. Therefore, the transformed equation is x* + 9x^ + dlx^ + 51x — 66=0. Query. — Could you tell by inspection that the roots of the trans- formed equation are less than those of the original equation ? Query.— Since 1 + 9 + 31 + 51 > 66, can a;* ^. 9 ^^a + 31 ^s + 51 a; — 66 = have a positive root equal to or greater than unity ? Why, or why not f INCOMMENSURABLE ROOTS. 411 2. Transform rz;* + 9 a;^ + 31 a;^ + 51 a; - 66 = into an equation whose roots are '8 less than those of the given equation. 1+9 +31 +51 -66 (-8 -8 7-84 31-072 656576 -0-3424* 9-8 -8 38-84 8-48 82-072 37-856 10-6 -8 47-32 9-12 + 119-928* 11-4 -8 + 56-44^ 12-2* The coefficients of the first quotient are 1 + 9-8 + 38-84 + 82-072, and the first remainder is — 0-3424, which is the absolute term of the transformed equation. The second remainder, or the coefficient of rr, is 119-928. The third remainder, or the coefficient of x^, is 56-44. The fourth remainder, or the coefficient of ofi, is 12-2. The transformed equation is Q^ + 12-2 a;3 + 56-44 a;^ + 119-928 a: - -3424 = 0. 3. Transform ir* + 12-2 0^3 _|_ 56-44 a;2 + 119*928 a; - -3424 = into an equation whose roots are '002 less than those of the given equation. 1 12-2 56-44 119-928 --3434 (-003 •002 -024404 -112928808 + -240081857616 12-202 56-464404 120-040928808 -'102318142384* •002 -024408 -112977624 12-204 56-488812 120-153906432* •002 -024412 12-206 56-513224* •003 12-208* The coefficients of the first quotient are 1 + 12-202 + 56-464404 + 120-040928808, and the first remainder, or the absolute term of the trans- formed equation, is — -102318142384. The second, third, and fourth remainders, or the coefficients of x, x^, and x^, are 120-153906432, 56-513224, and 12-208. The transformed equation is ic* + 12-208x3 + 56-513224 a;2 + 120-153906432 a; - -102318142384 = 0. 412 ADVANCED ALGEBRA. 4. The integral part of one of the roots of a^ -f x^ + ^^ + 3 a; — 100 = is 2. Extend the root. Form. 1 +1 + 2 + 3 + 1 + 6 + 7 + 10 + 3 + 14 H + 17 + 34 + 51* - 100 . 1 2-802 - 34 - 66* + 2 65-6576 + 5 + 17 + 14 + 31* -0-3424* + 2 31-072 82-072 37-856 119-928* + 7 + 2 + 9* 7-84 . 38-84 8-48 47-32 9-12 56-44 * •240081857616 - -102318142384* •8 •112928808 120-040928808 •112977624 120-153906432 * + 9-8 •8 10-6 •8 •024404 56-464404 •024408 56-488812 •024412 56-513224* 11-4 •8 12-2* •002 12-202 •002 12-204 •002 12-206 •002 12-208* Explanation. — 1. Transform the equation into one whose roots are less by 2. The new equation is ocf^ + Q x^ -{- ^Ix"^ -^^ bl x — QQ = 0. The roots corresponding to the one we are considering will now be a decimal. 2. Since (-1)2 = -01 ; (-1)3 = -001 ; and (-l)* = •OOOl, the first three terms are small in comparison to 51a:; therefore, 51 ic = 66 nearly, whence 51 may be taken as a trial divisor to find the next figure of the root, considerable allowance being made for the omitted terms. At first we would be tempted to try "9 for the value of x. But, upon transforming the equation into one whose roots are less by -9, we shall find that the absolute term will become positive, which shows that "9 INCOMMENSURABLE ROOTS, 418 is a superior limit. We therefore use -8 for the next term of the root, and transform the equation into one whose roots are '8 less. The transformed equation is x^ + 12'2a;3 + 56-44 a;^ + 119-9282; - -3424 = 0. The root of this equation is now less than -1. 3. Omitting the first three terms of the equation on account of their smallness, and using the coefficient of x as a trial divisor, we see that the root is less than '01 and is about -002. The next figure of the root is therefore 0, and the following one 2. Transform the equa- tion into one whose roots are less by '002 ; the resulting equation is Q^ + 12-208 a;3 + 56-513224 a;^ + 120-153906432 a; - •102318142384 = 0. The work may be extended as far as we please. 76 7o Bemark 1. — When the number of decimal places in the absolute term becomes equal to the number of such places desired in the root, we may begin to drop one figure in the preceding term (trial divisor), two in the next preceding term, and so on toward the left. When all the figures of the first term are exhausted, the remaining fig- ures of the root may be found by simply dividing by the trial divisor. 768. Bemark 2.— The absolute term after each transformation must be negative, else would the last figure of the root used be too large (a superior limit). 769. Bemark 3. — The method may be applied with equal facility to extending an integral root after a sufficient number of initial fig- ures have been obtained by trial or by Sturm's Theorem to distinguish the root from others of the equation. It may be used with exactness whenever there is an exact root ; hence, the incorrectness of the title "Horner's Method of Approximation" given the method by most authors. 770. Bemark 4. — The negative roots are the numerical equiva- lents of the positive roots of the equation resulting from changing the signs of the alternate terms, and may be found accordingly. 5. Solve a? - 1728 = 0, or find the Vi728, Solution. (12 + 10 10 + 100 100 200 300* -1728 1000 -728* 10 20 + 728 10 30* 64 "364 32 414 ADVANCED ALGEBRA. 6. Extract the 5tli root of 4312345 to thousandths, e., solve approximately x^ — 4312345 = 0. Solution. 1 -4312345 1 21-229 20 400 8000 160000 3200000 20 400 800 8000 24000 82000 48000 80000 * 160000 -1112845* 20 640000 40 1200 1200 800000 * 20 84101 884101 884101 - 228244 * 60 2400 1600 4000* 20 4101 88304 972405 * 80 84101 4203 88304 4806 198220-84882 20 101 100* - 30028-15168 * 4101 102 18699-2416 991104-2416 1 101 4208 103 4306 92610* 18877-3264 1 886-208 1009981-5680 * 102 98496-208 1 103 104 4410* 890-424 94886-682 894-648 1 21-04 104 1 4481-04 21-08 95281-280* 105* 4452-12 21-12 •2 105-2 •2 4473-24 21-16 105-4 4494-40 * -2 105-6 •2 105-8 •2 106-0* The number of decimal places in the second remainder is greater than the number required in the root; therefore, the remaining fig- ures may be found by dividing the remainder by 1009981-568 [767, Rem. 1]. INCOMMENSURABLE BOOTS. 4-15 EXERCISE 108. Solve : 1. x^ - 704 a; - 58425 = 2. a? - 15348907 = 3. x^-\-Za?'-^x-'i = 4. iC*-4a;3-6a:2 + 32a;-26 = 5. ir*-19a;3 + 24a;2 + 712a;-40 = 6. 2^5 + 12 a;* + 59a;3 + 150a:2 + 201a; + 94 = 7. 3a;* + 24a:3 + 68a:3_^g2^_964 = 8. Find the cube root of 2 9. Find the fifth root of 5 10. a:3 4-iir - = y\ (1) then, a;2 + 2 + -2 = 2/^ ; (2) or, a;2 + |, = 2/2-2; JO 3 1 and, a;3 + 3a;+ - + ^ = 2/«; ^'^'+^+K'^-'^)=^' a;3+ J-=y3_3y. (3) Substitute (1), (2), and (3) in (C), (2/3 - 32/) + 4(2/2 - 2) - 3y + 2 = 0; whence, j/^ + 4 2/* — 6 2/ — 6 = 0. 422 ADVANCED ALGEBRA. EXERCISE 107. Solve : Z. QC^ — dx^-\-^x — l = 4. ic* + 3a;=^ — 3a;— 1 = 5. 2a;*-5a;3 + 4a;2~5a; + 2 = 6. a;s + 5a;* + 10a;3 + 10:z;2_j_5^_^l_0 7. 6a;5-ir*-432;-'' + 43a;2 + ir-6 = 8. bx'>-\-ll7^-^%x?-^^x^-^llx-{-b = Reduction of Binomial Equations. 785. A Binomial Equation is an equation of two terms, one of which is absolute ; as, a;" ± « = 0. 786. Every Mnomial equation can te reduced to the form 2/** ± 1 = 0. Demonstration : Take the general binomial equation a;* ± a = 0. Put ^"^ for X, y a r + a = ; whence, 2/" ± 1 = 0. 787. r ± 1 = is a recurring equation, and may be so solved. Illustrative Solutions.— 1. Solve a:* + 1 = 0. (A) Divide by x^, ^^ + ^ = (1) Put ^^\ = y (3) Square x^ + 2+l, = y^ Transpose, ^' + ^, = y'-^ Substitute in (1), y^-2 = Factor, (y + ^2 ), (y - a/s") = ; whence, y — ± \/2. Substitute in (2), x + -= ± \/2 (3) X whence, x = ^ {\/2 ± ^/'^), or - ^ {^ T ^-2)- chapter xiii. determij^a:n'ts akb probabilities. Introduction. 788. In the polynomial «! 1)2 Ci — «i ^3 C2 + «8 ^3 ^1 — «2 ^1 ^3 + ^3 ^1 ^2 — ^3 ^2 ^1 ^ (-^) it will be seen : 1. That the letters a, h, and c of each term are ar- ranged in natural order. 2. That the subscript figures, 1, 2, and 3, are dis- tributed among the letters in the six different terms in as many ways as possible, using all in each term and making no repetitions. 3. That the first term contains no inversions of sub- script figures, they advancing in natural order from left to right ; the second term contains one inversion, 3 standing before 2 ; the third term contains two inversions, 2 and 3 both standing before 1 ; the fourth term contains one in- version, 2 standing before 1 ; the fifth term contains two inversions, 3 standing before 1 and 2 ; and the sixth term contains three inversions, 3 and 2 both standing before 1, and 3 standing before 2. 4. That in the positive terms there is an even number of inversions (zero being regarded an even number), and in the negative terms there is an odd number of inver- sions. 424 ADVANCED ALGEBRA. (B) 789. If we now arrange the nine different quantities found in (A) in a square, as follows : «i a2 a^ h h h ; Ci Cg C3 form all the possible products of them taken three to- gether, using in each product one and only one from each row, and one and only one from each column ; arrange the factors of the products in the natural literal order ; consider those products positive which have an even num- ber of inversions of subscript figures, and those negative which have an odd number ; and take the algebraic sum of these products, we will have : «i ^2 C3 — a^ J3 (?2 + «3 ^3 Ci — «3 li C3 + a^ bi < 7 • T'^^'^'o^^' 831. JPrin. 6, — The chance of two or more independent events happening simultaneously is the product of their several chances of happening, 832. Car. 1, — The chance of two or more independent events failing simultaneously is the product of their several chances of failing, 833. Cor, 2. — The chance of one of two independent events failing and the other happening is the product of the chance that one fails and the chance that the other happens. 11. A can solve 3 problems out of 4, B 5 out of 6, and C 7 out of 8. What is the chance that a certain problem will be solved, if all try ? Solution : Unless all fail, the problem will be solved. The chance that A will fail is i , that B wUl fail -^ , that C wiU fail g , that aU will fail -J X -^ X -^ = Yq^ • Therefore, the chance of success is -^ , 834. Dependent Events. — In a series of events, any assumed event is said to be dependent upon a preceding 4AS ADVANCED ALGEBRA. event, if the happening of the preceding event changes the probability of the happening of the assumed event. 12. Find the chance of drawing 3 white balls in suc- cession from a bag containing 5 white and 3 red balls. Solution : The chance of drawing a white ball on the first trial K is -5- . Having drawn a white ball, there remain in the bag 7 balls, o 4 of which are white. The chance of drawing a white ball on the 4 second trial is therefore -=- . Similarly, the chance of drawing a white Q ball on the third trial is -k . Therefore, the chance of drawing three 5 4 3 5 white balls in succession is q- x -=- x -^ [831] = -^r^ . Therefore, 835. Prin, 7. — The chance that a series of events should happen is the continued product of the chance that the first should happen, the chance that the second should then happen, the chance that the third should follow, and so on, 13. In one of two bags are 3 red and 4 white balls, and in the other 5 red and 3 white balls, and a ball is to be drawn from one or other of the bags. Find the chance that the ball drawn will be white. Solution : The chance that the first bag will be chosen is -^ . Then, a 4 the chance of drawing a white ball from the first bag is -y ; hence, the 14 2 real chance of drawing a white ball from the first bag is -^ of -=r = -=^ . Similarly, the chance of drawing a white ball from the second bag is 13 3 — of -Q = T5 • These events are mutually exclusive ; therefore, the \ ^ ^ . 3 3 53 chance required ^^ -y + jg = htr • 836. Inverse Probability. — When an event is known to have happened from one of two or more known causes, the determination of the chance that it has happened from PROBABILITIES. 449 any particular one of these causes is a problem of inverse prohalility, 14. It is known that a black ball has been drawn from one of two bags. The first of these bags contained in balls, a of which were black, and the second n balls, b of which were black. What is the chance that the ball was drawn from the first bag ? Solution : Suppose that 2 N drawings were made. The chance is that N were made from each bag. In the N drawings from the first bag the chance is that ~ x N were black balls. In the drawings from the second bag the chance is that — x iV were black balls. Therefore, in 2 JV drawings, the chance is that ( — H jN were black balls. Therefore, the chance that a black ball was drawn from the first bag is (— xiV") -^(— + —) N = — "'\ . 837. Theorem, — If an event is believed to have been produced by some one of the causes Pi, Pg? A^ P»> which are mutually exclusive, and Pi, Pzy Pzy Pn rep- resent the respective probabilities of these causes when no other causes exist, then the probability that P, produced the event is -^ t—^ "-p^ i-^ — . Demonstration. — Let N be the number of trials made in produc- ing the event. The first cause operated N x Pi times ; therefore, on the supposition that no other causes operated than those named, the probability that the event was produced by the first cause is iV X Pi X pi. Under similar restrictions, the probability that the event was produced by the second cause is iV x Pg x jsa ; by the third cause, N X Pz x ps; by the rth cause, N x Pr x pr ; by any one of the causes, N{Pipi + P^p^ + Psps + + P«i?«). Therefore, the real chance of its having been caused by the rth cause, or P, , is N X Pr X Pr _ Pr X Pr N{Pipi + Ps^s +.... + PnPn) ~ P\ P\ + P^Pi + + PnPn' 15. Four bags were known to contain 3 red and 4 white, 4 red and 3 white, 5 red and 1 white, and 4 red and 4 450 ADVANCED ALGEBRA. white balls respectively. A white ball was drawn at ran- dom from one of the bags. Find the chance that it was drawn from the second bag. Solution: Pi =P2 = Ps = P4 = i, i?i = y, i>2 = |, Pi = \, and i?4 = o" . Therefore, the required probability is 1 ^ i 4^7 T 9 2_/4 i 1 1\ :5 4V7 "^ 7 "*■ 6 ■*■ 2/ 3 35 838. Probability of Testimony.— HhQ following exam- ples illustrate how to deal with questions relating to the credibility of testimony : 16. A speaks the truth a times in W2, B J times in n, and c times in r. What is the chance that a statement is true which all affirm ? Which A and B affirm and C denies ? Solution : 1. The statement is either true or false. If true, all have spoken the truth ; the probability of which is — x — x — = . ^ ' ^ •' m n r mnr If false, all have lied ; the probability of which is \ m)\ n)\ r) mnr Hence, the probability of the truth of the statement is, ahc ^ {abc (m—a){n—b)(r—c))_ abc mnr ' {mnr mrir ) ~ abc + {m—a){n—b)(r—c)' 2. If the statement is true, A and B have told the truth and C has lied ; the probability of which is — x — xfl ) = — ^^ , ^ ■' m n \ r J mnr If the statement is false, A and B have lied and C has told the truth ; the probability of which is fl- ») (l- * ) (^) = '"'-"»"-^>' . ^ J \ mj\ nj\rj mnr Hence, the probability of the truth of the statement is, ab{r—c) ^ \ab{r—c) {m—a){n—b)c\_ a b (r—c) mnr ' { mnr mnr ) ~ ab{r—c) + {m—a){n—b)c 17. A, B, and tell the truth to the best of their knowledge and belief. A observes correctly 4 times out of 5, B 3 times out of 5, and C 5 times out of 7. What is the probability that a phenomenon occurred (which was PROBABILITIES. 451 just as likely to fail as to occur), provided all had equal opportunity of observing, and all report its occurrence ? What if A and B report its occurrence and its failure ? Solution : 1. The phenomenon either occurred or failed. If it oc- curred, A, B, and C observed correctly ; the probability of which is -^ X -^ X -=-. The inherent probability that it would occur is -^ . o o 7 -© Hence, the probability that the assumption that it occurred is correct • 1 1 1 A -A is 2 X g X g X ^ _gg. If it did not occur, all observed falsely ; the probability of which 12 2 is -^ X -=■ X ■=■ ; and the probability of the correctness of the assump- 11222 tion that the phenomenon failed is7rX-=-x-=-x-=- = r-— , Hence, 2 5 7 175 the chance that the phenomenon occurred is 5^ -^ ( ^^ + 7^ ) = -- . do \35 175/ 16 2. The probability of the correctness of the assumption that the , . 1 4 3 2 12 phenomenon occurred iS7rX-=-x-=-x-7=- = 77==. ^ 2 5 5 7 175 The probability of the correctness of the assumption that the , ..,,.1125 1 phenomenon failed \s -z -k -^ y. -^ y. -p^ = ^. Hence, the chance of the event is ^^ -«- f r^ "^ s^) ~ 17* Note. — For a fuller treatment of Choice and Chance than space will permit to give in this book, see Whitworth's " Choice and Chance." EXERCISE 114. 1. If A's chance of winning a race is — and B's chance 1 17 ■Z-. show that the chance that both will fail is rr. o 24 2. If the odds be m to w in favor of an event, show that the chance of the event is — ; — , and the chance m-{-n against the event is — ; — . 7n-\-n 3. If the letters e, t, s, n be arranged in a row at ran- dom, show that the chance of having an English word is — . M O 452 ADVANCED ALGEBRA. 4. Show that the chance that the year 1900 + 2?, in which X < 100, is a leap-year, is ^ . 5. A draws 3 balls from a bag containing 3 white and 6 black balls ; B draws 1 ball from another bag containing 1 white and 2 black balls. Show that A's chance of draw- ing a white ball is to B's chance as 16 to 7. 6. Show that when two dice are thrown the chance that the throw will amount to more than 8 is -5 . lo 7. Show that the chance of throwing exactly 11 in one throw with two dice is :7^ . lo 8. One purse contains 5 sovereigns and 4 shillings ; another contains 5 sovereigns and 3 shillings. Show that the chance of drawing a sovereign is —rj , if a purse is selected at random and a coin drawn from it at random. Show that the expectation of the privilege is 125. 2 Yig d, 9. There are three independent events whose several chances are ^, -r-, and ■^. Show that the chance that one of them will happen and only one is j^ . 10. If two letters are taken at random out of esteemed, show that the odds against both being e are the same as the odds in favor of one at least being e, y. 11. A letter is taken at random out of each of the words choice and chance. Show that the chance that they are the same letter is — . b 12. A bag contains 6 black and 1 red ball. Show that the expectation of a person who is to receive a shilling for every ball he draws out before drawing the red one is 3 shillings. PROBABILITIES. 453 13. Two numbers are chosen at random. Show that the chance is ^ that their sum is even. 14. An archer hits his target on an average 3 times out of 4. Show that the chance that he will hit it exactly 27 3 times m 4 successive trials is 77-7 . 15. A box contains 10 pairs of gloves. A draws out a single glove ; then B draws one ; then A draws a second ; then B draws a second. Show that A's chance of drawing a pair is the same as B's ; and that the chance of neither ^ . . . 290 drawmg a pair is ^^ . 16. Show that with two dice the chance of throwing more than 7 is equal to the chance of throwing less than 7. 17. Two persons throw a die alternately, with the under- standing that the first who throws 6 is to receive 11 cents. Show that the expectation of the first is to that of the second as 6 to 5. 18. A's chance of winning a single game against B is ^ . Show that his chance of winning at least 2 games out of 3 . 81 ^^-25- 19. A party of n persons take their seats at random at a round table. Show that it is w — 3 to 2 against two specified persons sitting together. 20. Show that the chance that a person with 2 dice will throw double aces exactly 3 times in 5 trials is \m) ^ \36J X 10. 21. There are 10 tickets, five of which are numbered 1, 2, 3, 4, 5, and the rest are blank. Show that the prob- ability of drawing a total of ten in three trials, one ticket 33 being drawn each time and replaced, is r^r^ . SUPPLEMEl^T. COJ^TIJfUED FBACTIOJfS, I. Definitions. 839. An expression in the form of a h-{-c d-\-e /+ etc., is a Continued Fraction, 840. The discussion in this section will be limited to continued fractions in the form of 1 «4-i J+i c+ etc., and -, -7-, -, etc., will be called Partial Fractions, a' h c 841. A continued fraction may be written in a more convenient form, as follows : 111 ill 842. When the number of partial fractions in a con- tinued fraction is finite, it is a terminating continued frac- tion ; when infinite, an interminate continued fraction. 843. If at some stage in an interminate continued frac- tion one or more partial fractions begin to repeat in the same order, it is called a periodic continued fraction. CONTINUED FRACTIONS. 455 844. A periodic continued fraction is pure when it contains no other than repeating partial fractions, and mixed when it contains one or more partial fractions be- fore the repeating ones. rrt. 1 1 1 1 1 1 is a pure periodic fraction ; 11111 1 i i is a mixed periodic fraction. 845. The fraction resulting from stopping at any stage is called a convergent, 2. The Formative Law of Successive Convergents. 846. In the continued fraction 111 1 1 1 i — = the first convergent. - , -T = r , -. = the second convergent. - , -T , - = . , , ..x — i — = the third convergent. a -{■ b -{- c {ab-\-l)c-{-a ^ It will be seen that t The numerator of the third convergent is the numer- ator of the second convergent multiplied by the denomina- tor of the third partial fraction, plus the numerator of the first convergent; and 2. The denominator of the third convergent is the de- nominator of the second convergent multiplied by the de- nominator of the third partial fraction, plus the denomi- nator of the first convergent. 456 ADVANCED ALGEBRA. Will these laws hold true in the formation of any con- vergent from the two preceding convergents ? P O Tf ^ Let p-, -~f -^, and -^ be respectively the (/i — 2)th, ^1 Vi -^1 ^1 {n — l)th, nth, and {n + l)th convergents ; and p, q, r, and s the denominators of the (n — 2)th, {n — l)th, nth, and (?i + l)th partial fractions. Suppose the laws to hold true in the formation of the convergent ^^ , then will ^ = 7Q^, ' (^) Now, from the nature of the continued fraction, -^ may 1 R ^1 be formed by putting r-\ — f or r in -^ . Therefore, S 2il ^_ (r+l)8+i- («r + l)e +^ ?P (sr + l)e. + ^ sA sB +Q _ ^(re+p)+e Therefore, if the laws are applicable in the formation of the nth. convergent, they are also applicable in the formation of the {n + l)th convergent. But we have seen that they do apply in the formation of the third convergent, and, hence, apply in the formation of the fourth convergent, and so on. Therefore, in general, 1. The numerator of the nth convergent equals the numerator of the (n — l)th convergent multiplied by the denominator of the nth partial fraction, plus the numera- tor of the {n — 2)th convergent ; and 2. The denominator of the nth convergent equals the denominator of the (n — l)th convergent multiplied by the denominator of the nth partial fraction, plus the denomi- nator of the (n — %)th convergent. PROPERTIES OF CONVEROENTS, 457 Example. — Find the first 8 convergents of the con- tmued fraction ^^3_^i^4^5^^ + 4 + 3- Solution : R _ rQ -\-P _\ i i 1? ^ ?1? ^ 3118 Ri~ rQx-^Px~ 2' 7' 9' 43' 324' 491' 2188' 7055" Properties of Convergents. 847. Take the continued fraction _1 1 1 1 «zx.ll .1>..11 whence, i^i<2/ I, .1 1^.111 whence, «+ - _^ - < « + -_^-^-_^.. . . ; and - , -T , - > Vi etc. Therefore, Prin. 1, — The successive convergents are alternately greater and less than the continued fraction {the odd orders being too great and the even orders too small). 848. The difference between the first two conyergents = 5— 7-T = , , . ^x = unity divided by the prod- a aJ + 1 a (a 5 + 1) ^ ^ ^ uct of their denominators. Is this a general law ? P Q J? Let -^f -TT, and ^- be the (w— l)th, wth, and {n-\-l)th. convergents, and p, q, and r the denominators of the 458 ADVANCED ALGEBRA. {n — l)th, nth, and {n -f l)th partial fractions, and let '^ denote difference hetioeen. Assume -— r^ -^ = — „ ^ — - = „ .. ; then will Ri'^ Qi QiRi QiRi _ PQ.^QPr _ _j_ " Gi^i "ft A* ^^ Therefore, if the law holds good for the difference be- tween the {n — l)th and wth convergents, it will also for the difference between the {n + l)th and the wth conver- gents. But we have seen that it does hold good for the difference between the first and second convergents, and, hence, it will for the difference between the next higher pair, and so on. Therefore, JPrin. 2, — The difference between any two consecutive convergents equals unity divided by the product of their denominators. 849. Since PQ^'^QP^ = 1 [848, A], P and P^ can not have a common factor, neither can Q and Q^. Therefore, Prin, 3, — Bvery convergent is in its lowest terms. 850. If we let ^=- represent the true value of the con- tinned fraction ; then will P U ^P Cmn 1 ^ Q ^ P Q ''' p[-u. 1, and Pi < ft , 1 ^ a; < p /^ /. . D X ; or. •• ft(^ft + i^i) ^A(^ft + A) -^ is nearer -^ than is ^-. Therefore, vi ^1 -^1 2*Hn. 4. — The MgJier the order of a convergent the nearer does it approach to the true value of the continued fraction. 852. Cor. — A continued fraction is the limit of its con- vergent s ; or, if y be a continued fraction and x its vari- able convergent, y = lim. x. 853. The denominators of successive convergents in- crease more rapidly than their numerators [846 ; 1, 2] ; therefore, of any two convergents, that is the greater which has the greater denominator. But may there not 460 ADVANCED ALGEBRA. be some other fraction, not a convergent, with smaller de- denominator, that is a nearer approximation to a continued fraction than a given convergent ? Suppose -Yjr not a convergent, and nearer to -^ than ■—, ^^ U M Q U ^' ^' and ifi < §1 ; then — ^ -^ < ^ ^ -^ ; M P ^ Q P MP^^M^P /^ < -^ -- D- ; or, TT-B < But MtPi < QiPi, since M^ < ft. ,'. MPi '^ MiP < 1; which is impossible, since M, Ml, P, and Pi are integral. Therefore, Brin* 5, — Any convergent is nearer the true value of a continued fraction than any fraction with smaller de- nominator. Problems. 851. 1. To reduce a common fraction to a terminating continued fraction. Since an improper fraction is equivalent to an integer and a proper fraction, it will be necessary only to investi- gate a method for extending a proper fraction. Let - = a proper fraction in its lowest terms. Divide both terms by h, and put for the improper frac- a c tion -T , the mixed number p-\- -r] then. h a 1 c 1 Divide both terms of -r by c, and put h _ c = ^ + d n, J a 1 1 q^d c PROPERTIES OF CONVERGENTS. 461 d c p Again, divide both terms of - by d, and put -7 = r + 77 ; then, _5 ^ 1 1 1 a ~ p -\- q -\- r -\- e_ d It will now be seen that the denominators of the suc- cessive partial fractions have been obtained as follows : b) a {p bp 11. d) c (r rd e etc. Since a and I are integral, they have a highest com- mon divisor, and the division will eventually terminate. Therefore, the continued fraction will be a terminat- ing one, Rule. — To reduce a proper fraction to a terminating continued fraction, find the highest common divisor of its terms by successive division, and use the quotients in regu- lar order for the denominators of the partial fractions, 855. 2. To reduce a quadratic surd to a continued fraction. Ulnstrations. — 1. Eeduce ^26 to a continued fraction. Solution : V36 = 5 + — = 10 + :4— . = 10 + "^ 10 + 1 ^ 10 + 1 10 + Jl_ X ■• '»^ = «+fo + K + ^+--=5+^ 462 ADVANCED ALGEBRA. 2. Eeduce a/iQ^ to a continued fraction. Solution : ^19 = 4 + — X Vl9-4 3 ccj 3 Vl9 + 2 , 1 iCi = —y= = - — = = 1 + — 5 ^19"+ 3 „ 1 ars = -7= = - — s = 3 + - VlQ - 3 2 iCs Scholiwm, — A quadratic surd may always le reduced to a periodic continued fraction if the expansion is carried sufficiently far, 856. 3. To reduce a periodic continued fraction to a simple fraction. The periodic continued fraction ili_lll __ p + q -h r ~ p-\- q -i- r -\-x~ qr + qx-^1 _^ pqr-\-pqx-\-p-{-r-\-x whence, {pq-{-l)x^-\-ipqr — q -\-p + r)x = qr-^-l The value of x found from this equation is the value of the continued fraction. 857. 4. To approximate the ratio of two numbers. Example. — When the diameter of a circle is 1, the cir- cumference is 3 •1415926 + . Approximate the ratio of the diameter to the circumference. Solution : -I oi^iKno« 10000000 1111 rp^ , ., 1 : 3.1415926 = 3^jj^= 3 ^ y ^ j^ ^ J ^.... [Prob. 1]. The successive convergents, which are also the successive approxi- ^ ^, ^. 17 106 113 , mationsof the ratio, are : -g, ^, ^, g^g, etc. PROPERTIES OF CONVERGENTS, 46_3 EXERCISE 1 18. Reduce to continued fractions : 125 ^' 317 140 100 106 ^' 213 ^" 999 ^' 729 6. ViO 6. Vl2 7. a/30 8. V57 9. -3183 10. 3-1416 11. 67°, 20', 30" Find the successive convergents of : "•UJ + ^ ^ 13 ^ ^ ^ ^ 4+5 3+1+9+1 i i "■2 + 3 15 ^ ^ ^ 16 ^ ^ Find the true value of : 17. -r 18. 4 i i _Q 1 i „ i 1 i 2 + 4 *^" 2 + 4 ^"-1 + 2 + 1 21. Find a series of common fractions converging to 1: y/J. 22. Express approximately the ratio of a liquid quart (57*75 cu. in.) to a dry quart (67*2 cu. in.). 23. The square root of 600 is 24*494897, and the cube root of 600 is 8*434327. Find a series of four common fractions approximating nearer and nearer to the ratio of the latter to the former. 24. The imperial bushel of Great Britain contains 2218*192 cu. in., and the Winchester bushel 2150*42 cu. in. Find the nearest approximation, that can be expressed by a common fraction whose denominator is less than 100, of the ratio of the latter to the former. 25. Two scales of equal length having their zero points coinciding also have the 27th gradation of the one to coin- cide with the 85th gradation of the other. Show that the 7th and 22d more nearly coincide than any other two gra- dations. 464 ADVANCED ALGEBRA. THEORY OF JfUMBEBS. Systems of Notation. I. Definitions. 858. Notation is the art of expressing numbers by means of characters. 869. A system of notation is a method of expressing numbers in a series of powers of some fixed number. 860. The order of progression on which any system of notation is founded is called the scale of the system, and the fixed number on which the scale is based is called the radix, 861. Any integral number, except unity, may be taken as the radix. When the radix is two, the scale and system are called binary ; when three, ternary ; when four, qua- ternary ; when five, quinary ; when six, senary ; when seven, septenary ; when eight, octary ; when nine, nonary ; when ten, denary or decimal ; when eleven, undenary ; when twelve, duodenary ; etc. 862. In the decimal or denary system, 56342 = 5x10,000 + 6x1000 + 3x100 + 4x10 + 2 = 5Xl0* + 6xl03 + 3Xl02 + 4xl0 + 2; , or, in inverse order, 2 + 4x10 + 3x102 + 6x10^ + 5x10*. In the octary system, 34725 = 3 X8* + 4X83 + 7x88 + 2x8 + 5; or, in inverse order, 5 + 2x8 + 7x82 + 4x83 + 3x8*. SYSTEMS OF NOTATION. 465 .*. In general, if r be taken as the radix, and ttQ, a-i, «3, «3 . . . . ««_! as the n digits of a number, reckoning in order from right to left, the number is represented by «•-! y-""^ + a^«-2 ^""^ + ««-3 ^""^ + .... + «2 r^ -}- «i r + «o 863. Theorem, — Any integral number may he expressed in the form of ar^'-^-h r"~^ + ^ ^*~^ + -\-p r^-\-qr + s, in which the coefficients are each less than r. Demonstration : Let N equal the number of units in any number, and r" the highest power of the radix less than N. Divide N by r", and let the quotient be a and the remainder N', Then N=ar'' + N'. Now, a is less than r, else r" would not be the highest power of r less than N; and JV' is less than r". Divide N' by r«-^ and let the quotient be b and the remainder iV". Then N' = Ir^-"^ + iV", in which 6 < r and N" < r«-i. In like manner, divide N" by r*-^, and let the quotient be c, and the remainder N'". Then iV" = cr'-^ + N"\ in which c + &r»-2 + + pr^ -^ qr ■{• s = any number; then, a + 6+....4-jp + g' + 5 = the sum of the digits. Now, a (r«-» —1) + b (r»-9 -1) + + p (r^— 1) + q (r— 1) = the difference between the number and the sum of its digits, and every term is divisible by r — 1. DIVISIBILITY OF NUMBERS, 469 869. Cor. — In the decimal system of notation, The difference letween a number and the sum of its digits is divisible by 9 or S. 870. Theorem III, — A number, N, divided by r—1, leaves the same remainder as the sum of its digits divided by r — 1, r being the radix. Demonstration. — Put s for the sum of the digits; q and q' for the quotients; and c and c' for the remainders. 1. N= q{r-l) + c 2. s = q' (r — l) + c' .*. N—s = (q — q') (r — 1) + (c — c'). Now, iV — s is divisible by r — 1 [T. II], and (q — q') (r—1) is evidently divisible by r — 1; therefore, c — c' is divisible by r — 1. But c and c' are each less than r — 1 ; hence, c — c' = 0, or c = c', 871. Car, — In the decimal system, A number is divisible by 9, if the sum of its digits is divisible by 9. 872. Theorem IV, — If from a number, N, we sub- tract the digits of the even powers of r, and add those of the odd powers, the result will be divisible by r + 1. Demonstration. — Let iV= ar* + br^ + cr^ + dr + e. Add —a + b — c + d — e^ then, a(r* — 1) + b (r^ + 1) + c{r^ — 1) + d{r + 1), the result, is divis- ible by r + 1, since every term is divisible by r + 1. 873. Theorem, V, — If a number, N, be divided by r + 1, the remainder will be the same as when the differ- ence between the sums of the digits of the even and odd powers of r is divided by r-\-l. Demonstration.— Put d for the difference between the sums of the digits of the even and odd powers of r ; g and q' for the quotients ; and c and c' for the remainders ; then will iV= g(r + 1) + c, and d = q* {r + \) + c'. .-. N-d = {q-q'){r + l) + c-c'. Now, N — d is, divisible by r + 1 [T. IV], and {q — q') (r + 1) is evidently divisible by r + 1. Therefore, c — c' is divisible by r + 1. But c and c' are each less than r + 1 ; hence, c — c' = 0, or c = c'. 470 ADVANCED ALGEBRA. 874. Cor, — In the decimal system of notation, A number is divisible by 11, if the difference between the sums of the digits in the even and odd places is divis- ible by 11, Even and Odd Numbers. 875. An even number is a number that is exactly di- yisible by 2. 876. An odd number is a number that is not exactly divisible by 2. 877. If we let x represent any integral number in- cluding zero, and regard zero as an even number, it be- comes evident that the general formula for an even num ber is 2 x, and for an odd number 2 ic + 1. 878. Theorem, I, — The sum, of any number of even numbers is even. Demonstration. — Let 2 a:i , 2 iCa , 2 X8 , 2xn represent n even numbers ; then will their sum be 2a;i + 2a:a + 2a:8 + + 2xn = ^{xi + x^ -^ Xt + + a^n) an even number. 879. Jlieorem II. — The sum of an even number of odd numbers is even. Demonstration.— Let 2a;i + 1, 2a;a + 1, 2a;8 + 1 + + 2a;an + 1 represent 2» odd numbers ; then will their sum be (2a;x + 1) + {2xu + 1) + (2xs + 1) +.... + (2x2n + t) = 2xi +2Xi + 20^8 + + 2a;an+ 2n = 2 (a^i + a:a + iCa + . . . . + x^n + n), an even number. 880. Theorem III, — The sum of an odd number of odd numbers is odd. Demonstration. — Let 2xi + 1, 2ira + 1, 2a;8 + 1, 2a;9«+i + 1 represent 271 + 1 odd numbers ; then will their sum be (2a;i + 1) + (2a;a + 1) + (2a:8 + 1) + . . . . + 2a;3«+i + 1) = 2xi + 2a;a + 2a;8 + + Sa^sn+i + 27i + 1 = 2{xi-\-Xi + Xf\- + x%n+\ + n) + 1, an odd number. EVEN AND ODD NUMBERS. 471 881. Theorem IV, — The sum of an equal even number of even and odd numbers is even. Demonstration.— Let (2xi + 1) + (2a;a + 1) + . . . . + (3a:an + 1) = the sum ot %n odd numbers ; and 2a;'i + 2a;'a + +2ic'a« = the sum of 2 n even numbers ; then will their sum be {2(a;i + a:'x) + 1} + 1 2 (a^a + ic'a) + If + + \'i{Xin + x'^n) + 1}, which is even [T. II]. 882. Thefyrem V. — The sum of an equal odd number of even and odd numbers is odd. Demonstration. — Let (2a;i + 1) + (2a;a + 1) + + (2a'9» + i +1) = the sum of 2 ?i + 1 odd numbers ; and 2a;'i + 2a;'a + + 2a;'2» + i = the sum of 2n + 1 even numbers ; then will their sum be {2{Xi + x\) + 1\ + {2(2:2 + a;'a) + 1} + + {2{Xin+i + x'^n + i) +1}, which is odd [T. IIIj. 883. Theorem VI. — The difference bettoeen two num- bers, if both are odd or both even, is even. Demonstration. — 1. Let 2 x and 2 x' be two even numbers. Their difference is 2 a; — 2 a;' = 2 (a; — a;'), which is even. 2. Let 2 a; + 1 and 2 a;' + 1 be two odd numbers. Their difference is (2 a; + 1) — (2 a;' + 1) = 2 a; — 2 a;' = 2 (a; — x% which is even. 884. Theorem; VII, — The difference between an odd and an even number is odd. Demonstration. — Let 2 a; + 1 be any odd number, and 2 x' any even number. Their difference is (2 a; + 1) — 2aJ' = 2 (a; — a;') + 1, which is odd. 885. Theorem VIII,— The product of any number of even numbers is even. Demonstration. — Let 2 a:, , 2 a^a , 2 a^s , . . . 2 a:„ be n even numbers. Their product is 2(2*-ia;i , Xi, Xz, a;„), which is even. Cot, — Any power of an even number is even, 886. Theorem IX, — The product of any number of odd numbers is odd. 472 ADVANCED ALGEBRA. Demonstration. — Let SiCi + 1, 2a;a + 1, 2a;n + 1 be n odd num- bers. It is evident, from the nature of multiplication, that the prod- uct of these numbers will contain the factor 2 in every term, except the last, which will be 1. That is, the product will have the form of 2 a;' + 1, which is odd. 887. Cm; — Any power of an odd number is odd. 888. Theorem X, — The product of any number of odd and even numbers is even. Demonstration. — The product of the odd numbers is odd [T. IX], and may be represented by 2 a; + 1. The product of the even numbers is even [T. VIII], and may be represented by 2 x'. .-. The entire product is 2x' {2x+l) = 2(xx'+x'), which is even. Example. — It is required to divide one dollar among 15 boys, giving to each boy an odd number of cents. Is this question possible ? Prime, Composite, Square, and Cubic Numbers. I. Definitions. 889. A Prime Number is a number that can not be produced by multiplying together factors other than itself and unity. A prime number is divisible only by itself and unity. 890. A Composite Number is a number that may be produced by multiplying together other factors than itself and unity. A composite number is divisible by other factors than itself and unity. 891. A Square Number is one that may be resolved into two equal factors. 892. A Cubic Number is one that may be resolved into three equal factors. PRIMES. 473 893. Two or more numbers are prime to each other when they have no common factor, except unity. 2. Primes. 894. Theorem I, — The number of primes is unlimited. For, let n be the number of primes, and, if n is not unlimited, let p be the greatest prime number. Then will 2x3x5x7xllX .... XphQ divisible by all primes not greater than jo ; and (2 X 3 X 5 X 7 X 11 X X i?) -f 1 not be divisible by any prime not greater than j9. There- fore, (2X3X5X7X11X Xi?) + 1 is itself a prime greater than p, or is divisible by a prime greater than p. In either case, p is not the greatest prime. Therefore, n is unlimited. 895. Theorem II, — Every prime number, except 2 and 3, belongs to the form 6 ;^ ± 1. For, every number evidently belongs to one of the forms Qn, 6 w + 1, 6 w + 2, 6 ?^ + 3, 6 tj + 4, or 6 w + 5, in which n may be any integer including 0. Now, 6w, 6 ?i + 2, and 6 w + 4, are each divisible by 2, and 6 tj + 3 by 3 ; hence, these forms are composite, except when ^ = in 6 ?^ + 2 and 6^ + 3, in which case we have the primes 2 and 3. The only forms remaining to contain primes are Qn-\-l and 6^ + 5. But 6 w + 5 = (67^ + 6) - 1 = 6 (?i + 1)-1 = 6 7^' ■— 1. Therefore, the general form 6 7^ ± 1 contains all primes, except 2 and 3. Scholium, — It must not be inferred from this propo- sition that all numbers expressed by 6 7i ± 1 are prime. Thus, when w = 4, 6^ + 1 = 25; and when n = 11, 6 71 — 1 = 65. 474 ADVANCED ALGEBRA. Cor, — Every prime above S, increased or diminished hy unity, is divisible by 6, 896. Theorem III, — iVo rational formula can repre- sent primes only. For, if possible, let a-{-bx-\-cc(^-}-dcc^-\- be prime for all values of x. When X = m, let a-\-bx-^cx^-\-da?-\- =i?; then, p = a-\-bm-\-cm^-{-dm^-\- When X = m-{-np, leta-\-bx-\-cx^-\-da^-\- =g ; then, q = a-{-b{m-{- np) -{- c{m-\- npf + d(m-\- npY + . . . . = a-\-bm-\-cm^-\-dm^-}- -\-rp =z p -{- r p = p (1 -\- r)f a composite number. 897. Scholium, — The form n^-\-n-\-4:l is prime for all values of n from to 39 inclusive, and the form 2 ^^ + 29 for all values of n from to 28 inclusive. These forms have been discovered by trial, and are not demonstrable. 898. Theorem, IV, — If a number is not divisible by a factor equal to or less than its square root, it is a prime. For, let N=xxy be any number not prime. Then, if x = y, W= /, and y = VW, But, if ic > VW, then y < ViV, since x X y = J^- But JV is divisible by y. Therefore, if iV is not prime, it is divisible by a factor equal to or less than VJSf, Hence, too, if a number is not divisible by a factor equal to or less than its square root, it is prime. 3. Composites. 899. Theorem I. — If a number is a factor of the prod- uct of two numbers and is not a factor of one of them, it is a factor of the other. COMPOSITES. 475 Thus, let a; be a factor of a h, and not a factor of a ; then will it be a factor of 5. For, — may be reduced to a terminating continued fraction [854]. Let — be the conyergent next in yalue to -. Then, ^ ^ — = — [848, P.] ; whence, X q X qx ^ -^^ px'^aq = l; and hpx ^ abq =: h. Now, hp X and abq ?iie each divisible by x ; therefore, their difference, h, is divisible by x, 900. Car, — If a number is prime to each of two or more other numbers, it is prime to their product. 901. Theorem II. — Every composite number may be resolved into one set of prime factors and into only one set. 1. Any composite number {N) is the product of two or more factors each less than N, which are all composite, all prime, or some composite and some prime. As many of these as are composite are again resolvable into other factors less than themselves, and so on, until no factor is further resolvable into factors less than itself and greater than unity, at which stage all the factors are prime. 2. Let one set of prime factors oi N he a, b, c, , and, if possible, let another set be ^, g', r, ; then will axbxcx =pXqXrX Now, suppose a different from q, r, , then it is not contained in q X r X [900] ; it must, therefore, be contained in p, but this can only be when a=p, since p is a prime. But, it a=p, bXcX =qXrX ; from which it follows as before that b is identical with one of the factors in qXr X ; etc. 476 ADVANCED ALGEBRA. 902. Theorem III, — The product of any r consecutive numbers is divisible hy [r. ^ n(n — l)(n — 2) (n — r-\-l) . , , , , For, — ^ — 1 ^^ ' — - IS the product of r consecutive numbers divided by [r, and it is also the number of combinations of n things taken r together, which is evidently a whole number. 903. Cor, J.— The coefficient of the {r + l)th term of ^, , . .... . n{n-l)(n-2)....{n-r-\-l) the bmomial theorem is -^ — , [595] ; therefore, The coefficient of every term of the binomial theorem is integral when n is a positive integer, 904. Cor, 2, — If we represent n(n-X)(n-2) (n-r±V) ^^ ^^ .^ ^^jj^^^ ^^^^^ All factors of the numerator that are prime and are greater than r are divisors of q. 905. Theorem IV, — Fermafs Theorem. If p be any prime number, and a be a number prime to p, then a^~^ — 1 will be divisible by p. Demonstration : a^ = [1 + (a — 1)]p =^ 1 + p{a-l) + ^^^~^\ a-lf + . . . .+ {a- V)p (A) .-. aP — {a - \)P -1 = p{a -1) + --^g + ^*^^- = a multiple of p [901. 140, P.]. "~ (B) Let a = 3, then ap ~.{a — l)P — \ = 2/' — 2 = a multiple of p. Let a = 3, then aP - (a-\)P -1 = ^p -2p -I = {^P - 3) - {2P - 2) = a multiple of p. ... Si* — 3 is a multiple of p [157, P.]. PERFECT SQUARES, 477 By continuing this process, it may be shown by induction that aP — a is a multiple of p. But aP — a = a(aP—^ — 1) and a is prime to p; therefore, aP-^ — 1 is divisible by p. Perfect Squares. 906. Theorem I. — Uvery square number is of the form 3 m or 3 m + 1. For, every number is of the form of 3x or 3 a: ± 1. Now, {3xY = dx^ = d{3x^)=z3m; and {3x±lY=(9af±6x + l) = 3{3x±2)-\-l = 3m-{-l. 907. Theorem II. — Every square number is of the form 4^m or 4 m + !• For, every number is of the form of ^x, 4cX-\-l, 4a: + 2, or 4a- + 3. Now, {4.xf = 16a^ = 4.(4:3^) = 4:m; {4:X + iy =16a^-^ 82; + l = 4(4a;2 + 2:r) + l = 4m + l; {4:X + 2Y = 16a^-\-lGx + 4. = 4: (4: a^ + 4: x -\- 1) = 4:m; and {4:X-{-3y = 16 a^ -\- 24: x -{- 9 = 4: {4: x^ -{- 6 x -\- 2) + 1 =4/7i + l. 908. Theorem, III, — Uvery square number is of the form 5 m or 5 m ± 1. For, every number is of the form 6x, 5 ic ± 1, or 5 a: ± 2. Now, (5 xY =z 25x^ = 5 {6a^) = 5m; (5a; ±1)2 = 25a;2±10a; + l = 5(5a;2±2^) + l = 5m-\-l ; and (5.'c±2)2 = 25a;2±20a; + 4 = 5 (5 3^ ± 4:X + 1) -1 = 5 m — 1. 478 ADVANCED ALGEBRA, 909. Theorem IV, — If a^-\-l^ = (^ when a, h, and c are integers, then luill ale he a multiple of 60, For, 1. a^ and h^ can not both be of the form 3 m + 1, else would c^ be of the form 3m + 2, which is not a square. Therefore, either a or J is a multiple of 3 [906]. 2. a^ and h^ can not both be of the form of 4^ + 1, else would c^ be of the form 4m + 2, which is not a square. Therefore, either a ov h must be a multiple of 4, or each of them a multiple of 2 [907]. In either case, ahc is a multiple of 4. 3. a^ and ¥ can not both be of the form 5 m + 1 or 5 m — 1, else would (^ be of the form 5 m ± 2, which is not a square. Therefore, either a^ or h^ must be of the form 5 m, or one of the form bm-\-l and the other of 6 m — 1 [908]. In the former case, either a or 5 is a multiple of 5, and in the latter, c is a multiple of 5, and in either case, ahc is a multiple of 5. 4. Since ale is a multiple of 3, 4, and 5, and these numbers are prime to each other, ale is a multiple of 60. Scholium, — By means of this theorem and the formula a = V{c -{-l){c — I), rational values of a, I, and c may he determined ly inspection that will satisfy the equation a^^¥=:(^. 910. Problem. To determine the rational value of x that will render x^ +px + q a perfect square. Solution : Let x^ +px + q = (x + mf then, x^ + px + q = x^ + ^mx + m^ whence, x = ^ , in which m may have any rational value from -- oo to + oo . Illustration. — What value of x will render a;^ — 7 a; + 2 a perfect square ? PERFECT SQUARES. 479 Solution : Here p = — 7, g = 2, and let m = 5, 25-2 23 ,6 then, X = _^_-^Q = 1117 = - 1 17 . r, o 529 161 „ 3844 /62\2 911. Cor, 1, — For m > Vq and 2m p, m being positive, x will be positive, 912. Cor. 2. — Put m^ — q = n{p — 2m); then q = m^ — n {p — 2 m) ; X = n, an integer ; and oi?-\-px-\-q — x^ -\- p X -\- m^ — n (p — 2 m) = {n-\- my, an integer. 7f? — 913. Cor. 3.— Put X = -^ — — m; or p — 2m m=^ ± a/^ - q, then, x^-\-px-\-q = 0, and a; = — | :f a/ ^ which conforms to Art SSI. Perfect Cubes. 914. Theorem I. — Every cube is of the form 4om or 4m ±1. For, every number is of the form 4:X, 4:X-{-l, 4 a; + 2, or 4 a; + 3. Now, {4:xY = 64:X^ = 4.(16x^) = 4:m; {4:X-\-iy = 64:a^-\-A83^-\-12x-{-l = 4 (16 a:^ + 12 rc2 + 3 a;) + 1 = 4 m + 1 (4a;4-2)^ = 64a:3_^96^2_j_43^_|_8 = 4(16x3 + 24a;2 + 12a; + 2)=4m (4a; + 3)^ = 64ar^ + 144a;2 + 108a; + 27 = 4:{Ux^-\-36x^-i-27x+'7)-l = 4:m-l 480 ADVANCED ALGEBRA. 915. Problem. To determine rational values of x that will render a? + px^ ■\- qx + r a perfect cube. Solution : Put x^ -{■ px"^ + qx ^^ r = {x + mf \ then, (^ — 3m)a;2 + (5'-3m2)a; + (r — m3) = 0. (A) 1. Put bm—p, or m = -^p\ then _ m^ — r _ jp«-27r ^- q-'dm^- 27q-9p' ' ^^"^ (pS 27 r « \' ~ V 27g-9i?2 ) ' ■ Cor.—If9pq-27r-'2p^ = 0, or r = ^^^Il^^\ 3 Q '^* x^-\-p^ + qx-\-r = 0, and x = ^^—^^. 2. Put m* = r, and suppose r = ri% then m = ri ; and (A) will become (i? — 3 ri) x^ + {q — B Ti^) x = 0; whence, x= ^o ; and x^+px^ + qx + r = x^+px^ + qx + ri^; and (re + m)^ = ( ^^3^^ ) . Therefore, ^ — 3ri _o ) Cor, — If q =p n, x= —ri, and 7? -\-p :i^ -\-qx-\-rx ^^' , Scholium, — Other values under particular suppositions may he obtained hy putting 3m^ = q = 3qi^. EXERCISE 117. 1. Find which of the following numbers are prime : 19r, 251, 313, 281, 461, 829, 957. 2. Find the least multiplier that will render 3174 a perfect square. 3. Find the least multiplier that will render 13168 a perfect cube. PERFECT CUBES. 481 4. Find which of the following numbers are divisible by 9, which by 11, and which by both 9 and 11 : 11205, 24530, 342738, 25916, 558657. 5. Show that, if ^ + 3' is an even number, then is j9 — g also an even number, provided p and q are integral. 6. Show that every cube number is of the form 7 w or 7/^±l. 7. Find such a value of x as will render the wax rational. Suggestion. — Put ^/a x = p. 8. Find such values of a; as will render vax-\-h rational. 9. Prove that 2*" — 1 is a multiple of 15. ' 10. Show that no square number is of the form 3 » — 1. 11. Show that n{n-\-l){^n-\-l) is divisible by 6. 12. Show that {n^ -f 3) (w*+ 7) is divisible by 32, when n is odd. 13. Show that n^ — n is a multiple of 30. 14. Show that the fourth power of any number is of the form 5 m or 5 m + 1. 15. Every even power of every odd number is of the form 8 w + 1' 16. Show that every square can be expressed as the difference between two squares. 17. Show that a' -\-a and a' ^a are even numbers. 18. Show that every number and its cube leave the same remainder when divided by 6. 19. If ?^ > 2, show that n^ — 5n^-{-4:n is divisible by 120. 20. It n is a, prime number greater than 3, show that n^ — 1 is divisible by 24. 482 ADVANCED ALGEBRA. 21. Find such a value of x as will render Vox rational. Suggestion. — Let ^/a x = p, any rational quantity, then will re = — . 22. Find such a value of a; as will rationalize vax-{-h. Suggestion.— Put ^/ax + b = p, and prove x = — . 23. Find such a value of x as will rationalize Suggestion. — Put \/ax^ + bx = px, and prove x = g_ . 24. Find such a value of x as will rationalize Suggestion. — Put '\/ax^ + bx + c^ = px + c, and prove x = ^_ ^ . 25. Find such a value of a; as will rationalize Va^a^-\-bX'^c. Suggestion. — Put ^/a^ x^ + bx + c = ax + p, and prove a; = , _ ^ — . 26. Find such a value of a; as will render Vax^-{-bx-\-c rational when J^ — 4 « c is a perfect square. Suggestion.— Put \/ax^ + bx + c = 0, and b^ — 4ac=z q^, and -b ±q prove a; = -2^. 27. Find such a value of x as will rationalize , , p^-b Suggestion.— Put ^a^ ->cbx^ = px, and prove x = ^ . 28. Find such a value of a: as will rationalize Vaa^-^ba^-\-cx-\-dK Suggestion.- ^,_^^^ Put Va^ + 6a;« + ca; + d« = 2^ a:+d, and prove x = -^ d« ANSWEES. 1. l+x+x'+x' Exercise 87. 2. 1 3 + 2 4 '^'U 4. 1+x- -x^- -7? 6. -d + 5x- -2x^- -Saf 81 3. 1+x-x^-x* 5. l-x-x^ + 2a^ a a' a* 9. l+a: + a;2— a;8 Exercise 88. 1- 2- ^^+ 4^^+ 3^6^' 2. 1+ i^- |^«- Ix^ ^ , 1 109 , 109 3 ^ . 1 1 5 5 , 3' ^+ 6 ^- 216^ + 3888^ ^' ^+ 3 ^- 9 ^ "^ 81^ 27 3.27« "3.27* ^ a 1 23 25 „ I . X x^ ^ x^ ^'^^r2^^27T2^-2A2^ '-"^^^-^^I^l _ia; a;' 7 ^ a:*ic«5a;' 3a§ 9a^ 81al ^a^ ga*^ 81a8 Exercise 89. 1. Divergent. 2. Convergent; divergent; divergent. 3. Convergent. 4. Convergent. 5. Divergent. 6. Convergent; divergent; divergent. 7. Divergent; convergent; divergent. 8. Convergent. Exercise 90. 1. x^ + Qx^ + llx + Q 2. a;3_2a;2-9a; + 18 3. a;4+2a;3-7a;«-8a; + 12 4. a:4-37a;2-24a;+180 484 ADVANCED ALGEBRA. 5, a;4 + 8a:3 + 24a:2 + 32a: + 16 6. a;4_20a;3 + 150a:2-500a;+625 7. 16 a;4-16 2:3-64.1:2 + 4 a; +15 8. (a: + 2) (a; +3) (a: + 4) 9. (a:-3)(a:-4)(a:+5) 10. (a: + 2)(a:+3)(a: + l)(a;-l) 11. (a; + 2)(a:-3)(a: + 4)(a:-5) 12. (a; + 2)(a;-2)(a;+3)(a;-3)(a;+4) Exercise 91. 1. a4_i2a3a;2 + 54a2a:4_i08aa;« + 81a:4 2. 32 + 400 a: + 2000 x^ + 5000 x^ + 6250 a:* + 3125 a:* 3. a:6-18aa:^ + 135a2a:4_540a3a;3+i2i5a4a:2_i458a6a:+729a« 4. 128 a:!^ + 2240 x^^ + 16800 a;io + 70000 a? + 175000 a:« + 262500 a:^ + 218750 a:2 + 78125 5. a;4-40 xl + 700 3:^-7000 a;t + 43750 a:2-175000a:l +437500 x - 625000 a:i + 390625 6. 2187 a;¥ + 5103 at a:^ ^. 5103 ^3 ^^.^ + 2835 ai a:l + 945 a« a:^ + 189 a¥a:f + 21 a9a:§+a¥ 7. l__a:-ga:2-_a:3 8. at — -5- a~i ^ — 77 ^~ ^ ^^ — 01 ^~~ ^ ^ o 9 81 9.xt-|x-i-|x-5-j|g^-| 10. a;— 4 — — aa;~4 + —a^x—i — r-^a^a:~4 -o o Id 11. a~%x~% — — a~V Ja:~ (T + — a~V h^x~^ — 12. a:— ^ + -^ aix—^ + -^aia;— V + — a^a;"^ 14. 8-06225; 8*94427; 7*006796; 5-000980 15. 2449440a.-* 16. - -^^a^ 17. - ^a""^^' ,3, ^r(r+l)(r+2)(r + 3)^_,,^,,^_. Li 19. 2^ 20. iJ^ 21. 343 Exercise 92. 1. n(n + l); ■|n(n + l)(n+2) 2. (2n-l)«; ■|n(4w«-l) AKSWURS. 485 3. ^n(n + l)', ^n(n + l)(n + 2) 4. w(n + 3); ^n(n + l)(2n + 7) 5. w2; ^nin+l){2n+l) 6. 2n{2n-l); -n{n + l){4n-l) o o 1. n{n + l)in+2); -^n(n + l){n + 2)(n+S) 8. n(n + 4)(n+S); -jn(n + l){n + 8)in + d) 9. n(n + 2)(n + lf; ^n(n + l)(n + 2)(n + d)i2n + d) 10. {2n + l){2n + 3){2n + 5); -i(2w + l)(2n + 3)(2w + 5)(2w + 7) -IS^ o o n 1 J./ 1 1^ 1 1 1_\ 11 3?i + l' 3 3 V "^ 2 "^ 3 w + l w + 2 n + dj' 18 13 iri____i___V 1 8 V4 2(?i + l)(n + 2)y' 32 ^ 6 n + ll ^ 21 180 12(2n + l)(2n + 3)(2w + 5)' 180 15. i _ -_1^+^; 4 16. ^(9n^ + 10n^-Sn-4) 4 2(ri + l)(w + 2)' 4 12^ ^ ^^ n(n + l)(n + 2)(n+3) ^g _w_ [4 ■ n + 1 19. n + 1 K; 20. 2-978809 Exercise 93. 1. x = y-y^ + y^-y*+ 2. x = y+ -^y^+ -^y^+ 24 2/*+ 3. x = y—2y^ + Sy^—4:y^+ 4. x = y—y^+y^—y'' + b.x = (y-1) - 1 (i/-l)2 + 1 (^-1)3 _ ^ (y_l)4 + . . . . 6.x = (y-1) + 2iy-lf + liy-lf + 30(2/-l)*+ .... 7. a; = i-|2 + |^-^ = '17590144 8. a; = -00999999 Exercise 94. 1.^ = 3, ^=-1; ., i~^ , 2. » = 2, q = 2; , ^^"*"^^ ^ ^ ^ 1— 3a; + a;2 ^ ' ^ 1— 2aj— 2a;2 o *, ;t Q 2 + 8a; ^ . ^ 3-lOa; 3.i? = 5, 5= -3; , r:^.Q^2 4.i? = 4, ^=-5; l_5a; + 3«2 -./---,«- "' i_4a;+5a;2 486 ADVANCED ALGEBRA. TOO o 1—x—^x^ ^ o o A l-5x + Sx^ „ ^ ^ l-x-1809x^-Udlx-' 'J'P = ^^9 = ^; 1_3^_3^. 8. p = 2, ^ = — 3 ; 9.p=-2, q = 2; l-2ic + 3a;2 2 + 3a:-2208a;« + 1616ic9 10. i)=l, q=-l, r=l; 1— a; + a;2— ic* Exercise 95. , 2 3 „ 3 2 1. — TR + 7i 2. a:+2 a;-2 2a; + l 2a;-l ,12 3 ^3 5 3. - + -— r + — ^ 4. X x + 1 x + 2 x + 1 (x + l)^ 15 7 - 3 4 °' ^J_.q /'^_La\2 "'" /^j..q\3 ^' a;)"»''-icZa; 6. (?2/ = (a:+ -^ — a;-2)f?a; ^ , ad^a; 1 „ , 5dx 1 6,dy — — J- . r 1, dy=-^- . ^ 2ci (aa; + 6)t 3 (5aj + 6)f 8.