LIBRARY UNIVERSITY OF CALIFORNIA ST. LOUIS EXHIBIT Z\ 6^ J/,^MJL ^y^-^ *s^-*^c^-*t. y9i>4^»j::4^ : J^3 Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/advancedalgebraOOwelcrich ADVANCED ALGEBRA BY WM. T. WELCKER Graduate of West Point, PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CALIFORNIA, AUTHOR O^ " WELCKER'S MILITARY LESSONS," ." WELCKER'S PRIMARY ARITHMETIC" (unpublished), "WELCKER'S PRAC- TICAL ARITHMETIC" (unpublished), Etc. SAN FRANCISCO: W. M. HINTON & CO., Book and Job Printers 536 Clay Street. Entered according to Act of Congress in the year 1880, by William Thomas VVelcker, in the office of the Librarian of Congress at Washington, PREFACE. This small volume contains what remains of the course in Algebra, after matriculation, to the students in the Colleges of Civil Engineering, Mines, and Mechanic Arts in the University of California. It is intended as a continuation of the excellent work on algebra by Mr. John B. Clarke, of the Mathematical Department of the University; and it is thought it will, in connection with Clarke's Algebra, or with any work of similar scope, furnish a good and sufficient preparation for those who intend to pursue the higher mathematics. The constant aim and endeavor throughout has been so to present the various topics discussed as to render them easy of comprehension by the undergraduate student. WM. T. WELCKER. Berkeley, Caxifornia, July, 1880. J0514(> CONTENTS ARTICLE. PAGE. Summatiou of series of fractious of certain forms 2-5 1-5 Method of De Moivre 6-7 5-10 If a is a root, first member divisible by a and conversely. 9 11 Elimination 10-14 12-17 Every equation has at least one root 17 18 Every equation of the inth degree has m roots 18 19 Composition of Equations 20 20 Every equation of an odd degree has at least one real root 28 24 Every equation of even degree and absolute term negative has two real roots of different signs 29 25 Every equation has an even number of real positive roots when absolute term positive; an odd number if term is negative 30 25 Changing signs of alternate terms changes signs of roots 31 26 Des Cartes' Kule " 32 27 De Gua's Criterion 34 29 To transform to an equation whose roots shall be multi- ples of former roots 36 30 To clear of fractions and yet keep coefficient of highest power unity 37 30 To transform to an equation whose roots shall be recip- rocals of the former roots 39 33 Eecurring Equations 42-43 34 To transform to an equation whose roots shall be squares of the former roots 44 34 To transform to an equation where the roots shall be greater or less by a certain quantitj' 45 35 Another mode of finding transformed coefficients 46 37 Synthetical Division 47 38 To transform to an equation wanting a second or any other particular term 49 41 Derived Polynomials 50 43 VI ' CONTENTS. ARTICLE. PAGE. Relations of Derived Polynomials to the roots of an equation 51 45 To discover equal roots, if any 52-53 47-48 Kational Integral Function . 53 50 Any term of such function can be made to contain the sum of all which precede or succeed 54 50-51 Law of Continuity 56 51 Limits of Roots, definitions of 57 52-53 MacLaurin's Limit 53 Ordinary Superior Limit of Positive Roots 59 54-55 Inferior Limit of Positive Roots 60 55-56 Superior Limit of Negative Roots 61 56 Inferior Limit of Negative Roots 62 56 Newton's Limit 63 56 Budan's Test of Imaginary Roots 64 58 If the substitution of p and q give different signs there is one real root between them 65 61 When there may be an odd number of roots between them and when an even number 66 61 Sturm's Theorem. Its Object 67 62 Enunciation of Sturm's Theorem 68 63 Three Lemmas 69-71 64 Demonstration of Sturm's Theorem 72-74 65-71 Cardan's Solution of Cubic Equations 86 73 Solution of Equations of the 4th Degree 88 77 Waring's Method 89 77 Waring's Method applicable only to equations having two, and but two, imaginary roots 90 78 Occasional solution of higher equations 91 80 Cubic equations having special relations between the co- efficients 92 81 Detection of whole number roots 93 83 Solution of recurring equations 97 86 Recurring equations of higher even degrees solved by one of a degree half as high 98 87 Exponental Equations 99 90 Approximate solutions of higher numerical equations. . . 100 90 Horner's Method 101 90 Demonstration of Horner's Method 102 91 Manner of shortening the calculations 103 95 Newton's Method 105 100 Fourier's Conditions 106 101 Trigonometric solution of cubic equations 107 103 PpCIPLES OF ALGEBilA, CHAPTEE I. Summation of Series. Art. 1. In addition to the treatment of this subject in Chirk's Algebra, a little more will be here added. It was there shown that when a series w^as so constituted that each term might be derived from the expression J^ by giving a constant value to i^ and suitable values to q and k, the sum of n terms of the series could be found by forming two auxiliary series from the expressions^ and ~ — respectively, subtracting n terms of the second from k-\7p r J the corresponding n terms of the first, and then dividing the difference by I?. Art. 2. If d series of fractions have the form its sum is equal to the difference between a se- k{k-\ij){L-VAp) ries whose terms Jiare the form — , . and another whose terms J tare the form 77— — ,/, , ^ , divided by 2p. For q _ q _ q{k±p){k-i-2p)-qk(k-j-p) _ kik-yp) (k-\-p){k-]-2p) k{k-fp)\k-\~2p) ~ - k(k-^py)ik-^2p) = kik.^pm-2pY -^^ *^^^ ^'^'^''^ by 2p gives the form proposed. If any term of the pro- Z PRINCIPLES OF ALGEBRA. posed can be found in tliis way, the sum of n terms of that series may be had by taking the difference between the sums of the n corresponding terms of the two auxihary se- ries and dividing it by 2p. Ex. 1. What is the sum of the series 3 9 J^ 21 ^ 5.8.11 + 8.11.14 + 11 AAAl + 14.17.20 ^ ^^^ ' Comparing with the formula above, we see that (/^= 3, 9, 15, etc.; p = d; and k=5, 8, 11, 14, etc. Therefore the auxiliary series are: ^'^^ k{k-\-p)' 5T8 + 8.11^ii:i4 -^' • ' • ' From 3(2?t— 3) • . 3(2/1— 1) + (3n— 1) (3n+2) + (3?i+2)(3n-L5) 3 9 (yfc+p)(yt+2j9) • 8.11 ' 11.14 ' ' ' * ' 3(2n— 3) 3(2?i— 1) (3?i+2)(3H + 5) ^ (3?H-5)(3?7-f8) and the sum : 3 3(2»— 1) -I ^ 1 A_r — _ 3(2»— 1) 1 5 . 3L 5 . 8 "~ (3n4-5)(3n+8) J to infinity. 2.315.8 (3n-f5)(3n-f8)J ' 8.11^11.14 1 14.17 Now the sum of this last series to infinity r= -^. Hence the sum of n terms of the proposed series is: lr£_ 27 1—1 -] 1 2140 (3H-f5)(3?i+8)J +24' and when n = oc, r 2 11 1 1 n n' 2 40~ /„ , 5\/ , 8 (3 + .)(^+'). J. _ 1 1 13^ +"24"8U+24~ 240' SUMMATION OF SERIES. ii The sum of n terms of Example 1 = Irl 2n—l ^^ 1,1, - 77^ — ,^ , ^,,» — r-^v -\~ 4- -, 1 -.T 4- . . to n — 1 terms 2L40 (3n4-5)(3?x+8)J '^8.11^11.14^ Ex. 2. What is the sum of an infinite number of terms Of the series 172^ + ^^ + 37475 + etc ? Ans. IJ 1 4 Ex. 3. What is the sum of the series -|- ^-^-^ 4- ' 1.0.0 0.0.7 5T7T9 + 779711 + ^''•' *° '"''"'y • '^'''- 24- Art. 3. If a series of fractions have the terms of the form Yjj- — r/i—rrr-^jT-n—s > its sum will be equal to the difference between the sums of two series whose terms have respectively the forms g and ^ /ciki-p)(k^2p) {k+p)(k-i-2p){k-{-dp) divided by 3p. In these formulas p is constant, and q and k have suitable values assigned to them. The proof of the above is easily seen by performing the subtraction: k{k+p)(k^2p) {k-{-p){kJr2p)(k^3p) " ( k+p)(k-\r2p)(qk^ dpq) - (k^p)(kJr2p) qk k{kJrPnk+2py{k+Sp) 3pg k(k+p)(kJr2p)(k^dpy Ex. 4. What is the sum of n terms and of an infinite Qb 8, g2 rj.2 number of terms of the series - ^ ^ . -[- ^ ^ . ^ -|- 1.2.3.4 ' 2.3.4.5 ' 3.4.5.6 + etc? PRINCIPLES OF ALGEBRA. Here g --= 6^ T, 8^ 9^ etc.; p = l; k=l, 2, 8, 4, 5, etc. Hence the auxiliary series are: 2.3.4"^ 3.4.5"^ ' ■ ' ' n(n+l)(/i+2) '^ (n+l)(7i4-2)(n + 3) 3L1.2.3 0i+l)(n+2)(7i+3)J"^3\2.3.4^ and /S' =^ „ . ^ ^ „ (n+l)(n+2)(n+3) (0:5 + 4:576 ^^'^*"^^^| The sum of this last series to infinity is, by preceding^ methods ^^T^; 1 6 — 1/13 15 17 , , i \ + 3 (2X4 + 3.1:5 + 4.5.(5 "■ ^*<'-' **' " *^™^^-) Whenn = oo,S=l(6 + ;-^) = |. Ex. 5. AVhat is the sum of the series i:ir5 .T + 375^ +5.7 .1,11' "^ ^"'^'^''y ■ ^"'- 7^2 ■ Art. 4. A consideration of the laws of the preceding series and the mode of their summation will show that the sum of any series of fractions of the form k{k+p){/c+2pY . . . . ik-i-mp) is equal to — of the difference between two series of mp fractions of the forms, respectively, and k{k-^p) . . [^f(m— l)/jj ik-^p)(k-\-2p) . . . [k~\ mp StiMMATlON OF SERIES. 5 Art. 5. If a series of fractions has the terms as fellow: k'^'kik^py' k{k^--p){kyip)'^ ' ' ~^k{k^p) . . . (k~{-mp) (in which the last term is a general term), its sum will be equal to the difierence of two series of fractions, having re* spectiveh^ the forms: e(c4-/>) . . . (c^-mp) r(c'4-p) . . . [c- [ -(m+l) y>] k(kJrp) . .-. [k^(m-l)p] ^"^ k{k-\-p) . . . (k^mp) divided by k — c — j), as will be seen by performing the in- dicated subtraction. 2 2 4 Ex. 1. Find the sum of vi terms of the series ._ -[- ^^ -|- 2.4.6 ir5r7 + «'*'- Here p = 2, c = 2 and k==S, and the auxiliary series are : ,. , 2.4 ^ 2.4.6 2.4.6 .... [2-i-2m— 2] , ^^-^+-3X+ • • • 3Xr . . .l3+2,^-4]""^ ^4_^ 2.4.6 2.4.6 .... 2m-f2 3 '' 3.5 + • • • 3.5.7 .... (3-[-2m— 2)' ^==-A/o 2.4.6 . . . .(2m+2) v_ — 1V'^~3.5.7 . . . .(2m+l)/ 2.4.6 .... (2w-f-2)_ 3.5.6 . . ! . (2m + l) Ex. 2. Find sum of ??i terms of - -f -^ 4. 7i;^T^^ + etc. 2 ' 2.4 ' 2.4.6 ' 1.3.5.7 . . . (2 m+l) 2.4.6 . . . 2m Art. 6* Some special modes of summation: METHOD OF DE MOIVRE. Assume a series whose terms converge to and involve fJie powers of an indeterminate x; place the series equal to S and midtiply both members of this equation by a suitable binomial, trinomial, etc., which involves the powers of oc with constant coefficients; then assume x so that the binomial, trinomial, etc., may be equal to zero and transpose some of the first ter)n.^; PRINCIPLES OF ALGEBRA. their sum will be found equal to the tium of iJie remaining terms. Ex. 1. Let it be required to find tbe sum of ^—^y-; — ,■ i. . ^ It .it -[- —j-}- etc., to infiiiity. Place ,S'.^1 j^-r „ - 74 etc., and multiply both members by ,r — 1. We get , . / -. X x*' a? x^ .S (.r— l)--.r-^ ~-f-4-j-r etc. -1-2-3-4-5- ^*^- Whence by addition H {x~\)= — l-rY^2"^/3~^3~4"^r5 -}- etc. Now suppose that x =^1 and we oet — 4. _[_ — \^ — ^^ ° 1-2^2.3^3.4^4.5 -p etc. = 1. Here the first term, — 1, is transjDosed and is equal to the sum of the proposed series to infinity. Ex. 2. If we had multiplied by the binomial x"^ — 1 we would have found the sum of —-_}_--- ^——_j_- -[-etc. i.o JL.'± 0.0 4.0 and of r—- — 2T— i+TT^ — etc., to infinity. 1.3 2.4 ' 3.5 ^ Ex. 3. Let the multiplier be the binomial 2^ — 1. >S' = """"^ 2 ^ 4 "^ 4 ~^ ^*^' *^ i^^^i^y 2.r— 1 ,S (2^-lH 2a;-f ^V^l'-f^' -f etc ir J?^ r* ^' 1 — r — - — - — - -|- etc. ; adding we obtain 2 o 4 5 .S (2-^'—l)=— 1+^79-1-2:3 -r374-|-4;5-r etc. Now make 1 -^ 'Ix — 1 = 0, whence x = , and we ofet 2 ° ' 1.2.2 ' .) o o2 ~f o !< 03 + F-FTTrf etc. =^ 0, and the sum of the se- 2.0.2 o .4: . Ji 4.5.2 SUMMATION OF SERIES. 7 Art. 7. It will be observed that when the multiplier is a binomial the resulting series, before the value of .r is assigned, will consist of fractions having two factors in the denominators; if the multiplier is a trinomial there will l)e three factors, etc. 5 6 Ex.4. The sum of the series^ 2 3 2'^+ 07472'+ 7 1 To prove this let the multiplier be (2.r — 1) (.r — 1) = 2j?' — 3.r+l: then .S(ar'-3^+l) = 2^'+^|-+|:+_+ etc. „ Zx^ Z3? Zx' Za? IT Hi ^ "7 nr^ 4-l+o4""Q"i~V~7 k^Vt."]" 6tc., which by addition is i5 O 4 O D Bx ^'^^ 5x' 6x' Ix' S (^'-3x+l) = 1-0+1-273+270+0. 5-f-4.X6+ etc. Now if we make the factor 2x — 1 = 0, we get x= 1 5 6 7 _^ 5 2^ 1.2. 3. 2^+2. 3.4. 2»+3. 4: 5. 2^+ "~1.2.2' 4 -'4 ' Had we made the second factor equal or ,c = 1 we 5' 1 6 should have found the sum of the series ^--^-[" o ^ 4" L. n. o ^.0.4 l-etc. =f-l ^ 3.4.5 "22 1 X x^ * W- Ex. 5. Suppose --I — 4— — r^;r- + ,' »' + ^tc. = »S and that we use the multiplier ax — h, we will get (!!i+J!fL_L JIfLj. etc \ b bx b.v^ bar^ V m 77J-L-/- m-\-2^ m-{-3r PRINCIPLES OF ALGEBRA. And by addition we find S (ax—b) = - -^ i l''' + ''l^~''L^^ - m 7n(m-[-r) {m^2i-) a — (m-}-r)6 „ — {m^-r) (m+2r)" "^ + ^^^- "^"^^ ^^'^^^ ^''^ — 6 = 0, but retaining for the present x in the second member we liave alter transposing the series — ^—r- ,- — x 4- m w(m-fr) ' ( m+2r) ft — (??i+r) b ^ (m-f 3r) a — (7?i-[-2r)6 3 _ (7n-^r) (m-l-2r) ^ ^ (?7i+2ry(»i+3r) ^ ^" ^^^'^ — . Now if any particular numerical series be proposed and it can be shown to coincide with the above when suit- able values are substituted for the various letters we will be able to find its sum from the formula. Suppose it were required to sum the series o • q"1" Q~f: • o2"l~r"> • 03 ~r ^^^' Sy inspection we find that L . o o 0.00 ,0 . / o .r = - and since ax — b = 0, a = 36; m =^ 1, r = 2 and o from the first numerator {ni-[-r) a — mb = 2. *. 3a — b = 2; 18 3 1 or 3a — o ^ = 2.-.- « = 2, and a = and /> ^=-. These So 4 4 several values being substituted in the formula, term after term, build up the j^roposed series, the sum of which 6 1 1 theiefore = — = — r. m 1 4 The foregoing gives the sum of the series to infinity, but if it be required to find the sum of a finite number, ?i, of terms we may proceed as follows : -Take the series to n-\-l terms, and multiply by ax — b as l)efore, 1 X x^ x"-^ ^ x" m. ' m-\-7' ' 77i-f-2r ' " * in-^{n — !)?■ '^ in-^-nr * ax — b ax ax^ ax""^ ax" m m-{-r~^ ^ m-f{7i — l)r^ m-^rw b bx bx^ bx" , ,,„, {ax — b)S' m m^r m-\-2r ' ' ' m-\-nr SUMMATION OF SERIES. M whence by adding and transposing we get: (m+r)a--mb i,n+^r)a--(rn+r)b ^, +etc.,tonthterm = 7n{m^r) ^ (m-fr)(m+2r) b ax"'^'^ SUax — b) A ■ . Then, using the values of the ^ ' m m-^-nr letters belonging to the numerical series last considered, 13 1 to wit: m = l, r=2,x==^, ^^^I» ^^^i' ^^^ we have f 4. f2131 41' S = sum of n terms of ^.^ + — .^, + 5^.- _|_ . . . = 1 1 4 4.3«(l+2?i)' Ex. 6. Let it be required to find the sum of the infinite ^^"■'^^ 1.2.3-4 + 2.3.4*8 + 3X516 + 4.5.6*32 + ^^''• Here J, the square and higher powers of which are pres- ent, is represented by x in the series 1 -{---|--_j---j-etc. 2 ij 4 Moreover, there are thi^ee factors in the denominators of the coefficients. Let us, then, multiply by the trinomial ax^ — bx-{-c, and we get S{ax'—bx^c) , , l^tA.- l^U/ IjUj U.// , bx^ bx^ bx^ -bx--^--^-^-eic. ^ ajc^ ax\ , «'^ H-^^ + X+^^'^- Whence adding: c -}- -j-o~^ H ^ n 3 ^' ^~ 12a-86-6c 20^156+12c ^^^ ^^.^^ 2.3.4 ^ 3.4.5 ^ ' y ' S(ax^—bx-^G). Now making cLic^—bx^G = 0, and in substituting that value of x = iy previously quoted, we get ^ — 2 + c = 0; also, from the first and second numerators of the proposed 10 PRINCIPLES OF ALGEBRA. series, compared with those over the same denominators in the formulas, we get two more equations: 6a — 36+ 2c = 19 and 12a— 86+6c' = 28. From these we find a = 6, 6 =: 7, c = 2, and these placed in the formula, build up the proposed series. After transposing the first two terms, 19 1 28 1 29 1 2 and -3, we find: ^-^-3.^ + ^J^-^ + SXsTG + '^'^ ^^ infinity = 3— 2 = 1. We may determine the sum of n terms of this series in the manner of the last example. It is: 1 4+n (7iH-l)(n-f2)2-'- To determine the multiplier to be used for any particu- lar case, assume the series up to the term — — : . n-j-2 ]Sx. 7. Find the sum of x-^2x'-{-3x^-^4:x'-\- etc., to 00 . Multiply the proposed series by x^ — 2a?-}- 1. X S = 1— 2.r-[-^=' Ex. 8. Find the sum of .r+ 4^^=^-1-9^-1- 16^*+ etc., to 00 . Use 1 — Sx-{-Sx^ — x^ as a multiplier of the series pro- posed. Ans. x{l-\-)x CHAPTEK II. Elimination. Art. 8. In order that an equation may be solved, it must contain only one unknown quantity. If it is identical it may have an infinite number of solutions, but if it is a common equation it will have only a limited number of roots. If then we have a single equation containing more than one unknown quantity, and the equation is not iden- tical, but is indeterminate, we must attribute values to all the unknown quantities save one, before solution. If we have two or more simultaneous equations we must eliminate so as to obtain a single equation with one unknown quantity. ELIMINATION 11 The methods which are used when the equations are of the first degree will, upon application to those of the high- er degrees, be found to fail to give results practically use- ful. The method of the greatest common divisor is that usually employed with higher equations, and to its discus- sion we will proceed after establishing a principle in the nature of equations which is Art. 9. If (I is a root of an equation luhose second viem- ber is zero, the first member ivill be exactly divisible by the bi- nomial X — a. Let the equation be oc'" -|- Px""'-^ Qx'""'-^ Rx'""^^ 4- Tx -{- U= 0, and let a be a root of the equation; then X = a or, X — a =^ 0. Suppose the first member to be divided by ;r — a and that we continue getting terms of the quotient until the remainder is without an x, or is independent of x. Let this remainder z= II, and the quotient (which may consist of one or of several terms) = Q, then {x — a) ^ -f- i? = it"«-{- Px"'--'^Qx"'-'-{- -\-Tx-\-V =^. But since the second member = the first must = 0, and since a is by supposi- tion a root; a; — a = 0, and this leaves i^ = 0. The re- mainder being = 0, the division was exact, which proves the theorem. The converse is also true. If the first member of an equa- tion whose second is zero, is divisible exactly by the binomial x — a, then a is a root of the equation. Since the division is exact there will be no remainder, and (X — a) Q = x'"-{-Px'"-'-\-Qx'"-' j^ Tx ^ U. Now Avhatever value of x makes the second member of this equa- tion = is a root of the proposed equation; but x ^= a effects this, by reducing the first member to zero, and hence a is a root of the proposed equation. These properties belong of right to the general Theory of Equations, to the discussion of which we will soon pass, but being necessary to the understanding of the principles of elimination now to be examined have been introduced here. 12 PRINCIPLES OF ALGEBRA. Art. 10. When two or more equations are simultaneous they have values for the unknown quantities entering them which are the same in all the equations. These are of course common to them all; and there may be values in the different equations which are not common. The com- mon roots make the equations compatible and are known as compatible roots. If thus b were a root common to two equations in x and y, and if b were substituted for y in the first members of the tw^o equations, they would become polynomials in x only; moreover, since b is compatible with some value of a: in both the first members, if we call that value of ^ = a then those first members will have, from Art. 9, a com- mon divisor, x — a. Having substituted in the two first members the known value of y, let it be supposed that the process for obtaining the H. C. D. were applied to them; it would terminate of course in a remainder ^= 0. If, then, without substituting the value of y, and even without knowing it, we apply the process for finding the highest common divisor, we obtain, after a sufficient num- ber of operations, a remainder in y only. Now, if we Jmd known and substituted the proper value of y, this remainder, which we will call R, should be = 0. Hence R=y\i/) =: is a true equation. This equation is called the Jinal equa- tion in y. Now, among the roots of this final equation in y will be found all the compatible roots or values of y, and when they are substituted in the last preceding divisor placed equal to zero, they will give the corresponding values of x. That is, such will ordinarily be the result. But if the pre- ceding divisor, upon the substitution of the values of y, becomes zero at once, so that we cannot obtain the corres- ponding values of x, we proceed to the next preceding one, which wall be ordinarily of the second degree with regard to X, and give two values of x to each one substituted for y. If this fails, we proceed to the divisor last before this, and so on. But usually the substitution of the values of y, found from ELIMINATION. 13 the final equation in ?/, in the preceding divisor, will not at once reduce it to zero, but will give a polynomial in x which will be a common divisor of the first members of the original equation after the value of y has been substituted in them. Now, this polynomial in x should be -equal to zero, because, being a common di\isor of the first mem- bers of the original equations, it contains that factor of the form X — a, or the product of such factors, belonging to the compatible values of x, and probably other factors beside. This divisor, a f{x), might, if Ave knew the factors compos- ing it, be put under the form of {x — ci)f^{x), or (,r — a) (x — c){x — d)f'(x), according to its nature, and of course the substitution for x of the values, a, c, d, etc., would reduce it to zero. This divisor, then, /(.r)=^ 0, is a true equation, from which we ought to obtain the compatible values of x. Art. 11. Having obtained the final equation in y, if by factoring its first member, or in any other way, we can solve it, we do so; substitute the roots for y in the last preceding divisor, or in the one before that, as the case may require, obtain the values of x, and verify both by substituting them in the original equation. Art. 12. Foreign Roots. — We will thus obtain all the compatible roots, but we may get also others. For if in preparing the dividends at any time, we have found it necessary to multiply any by y, or any f{y), we may thus have introduced foreign values of y, which will appear among the roots of the final equation in y. This might have been done, for instance, to avoid having y appear in any denominator of the quotient. For if we denote the first member of the first equation by A and of the second equation by B, and by Q the quotient of A-i-B and by E the remainder, we shall have: ^1 =: 0, i? = 0, and A == BQ^R, and from the next division: B = RQ'^R', R = -K' Q "+ ^", 6tc. Now, since the equations are simul- taneous and their first members have a common divi- 14 PRINCIPLES OF ALGEBRA. sor, the remainders R''\ E'' , B' , etc., will on, substituting the values of y and x be found successively= 0, and finally R ^= 0. Now, if Q, or any quotient, should have y in its de- nominator the substitution of its value in such denomina- tor might reduce it to and make (^> = oo , and then al- though B = 0, BQ would not be ^^ 0. The supposition on which the process is founded is that A and B are al- ready (or have been made) whole with respect to y. In this way foreign roots may have been introduced into the final equation. They may be detected by trial in the proposed equations and rejected. Furthermore, it may happen that all the proper values may not be found by means of the final equation, since we may have suppressed some factors in the process for find- ing the H. C. D., which w^ould reduce to zero on the substitu- tion of the proper value of y . Such factors should be placed := and the values of y substituted, and the values of X found from the resulting equations. If the final remainder should be independent of y, it of course is not zero, and the equations have no compatible roots. Art. 13. Mention has already been made of a solution of the final equation in y when it was possible to detect its factors. Similarly when w^e can detect the factors of the first members of the proposed equations we may shorten the process. Suppose all the factors of each to have been discovered: they will be of tw^o kinds, commoyi and those not common. Among the common some may be altogether in X, some altogether in y and some functions of both x and y. Likewise the same three species may exist among the factors not common. Now any one of these factors being placed = will sat- isfy the equation to which it belongs. Suppose we first consider those which are common to the two first members. Those in x only will give a limited number of values for x and any values whatever for y provided they are finite; ELIMINATION. 15 those in y only will give a limited member of values for y and leave .r indeterminate, and those which are functions of X and y both will give an infinite number of sets of val- ues for .r and y. Second. Of those not common we cannot place two in X only, or in y only, equal to zero at the same time, for it would not be true unless one was equal to the other multi- plied by some constant factor. And this is contrary to the supposition that we have already considered all that were common. There remain those not common and which con- tain both X and y. To these we should apply the process for the H. CD., and we will obtain a limited number of values for x and also for //. EXAIMPLES. 1 . Let the equations be x^ — ^yx'^-}-{3y^ — Sy-\-l)x — 2/*+ y'^ — 2y=^0, and x^ — 2«/.r~j-]/'* — ^=0. First Division. of^—Syx'-i-(Sy'-y-i-l)x—f-{-y'—2y \ x'—2yx+f—y x^ — 2yx'^-{-{y^ — y)x \ x — y —yx'^{2y'^l)x—f-{-y'—2y —yx^-{-2y^x —y'+y'' This division was performed without preparation. So, likewise, with the Second Division . x^ — 2yx-]~y'^ — y \ x — 2y x"^ — 2yx • I ^ y'^ — y and y"^ — y=0 is the final equation in y . Its roots are .v=0 and y=l . Hence we have the systems and ?/=l x=--2 2. Let the equations be oif-{-y^=0, and x^-{-xy-\-y^ — 1=0 No preparation will here be necessary. 16 PRINCIPLES OF ALGEBRA. First Division. x^-\-x'^y-\-y'^x — X | x — y ~yx^—y\x^x-{-y^ ~~ " —yx^—y^'x—y^-i^y x-^2y'—y Second Division. x^J^yx-{-yl—l I x^2y'—y x^—yx^2y^x \ x-\-2y — 2y^ (2y—2y')x-\-y'—l '' ^ {2ij-2f)x-l f ^6,/-2f 4:i/^ — 6y+3/ — 1 and this placed = gives the final equation in y. It is evident upon inspection that y = l and y = — 1 are roots of this equation, and the other four are ±:J;/ldz|/ JZ3] y = 1 and y = — 1 give, upon substitution in the origi- nal equations, ,t = 1 and ^=—1, values to be expected, as the equations are symmetrical. 3. Let the equations be: x^-\-2yx^-\-2y{y — 2)x-{-y* — 4 = 0sLndx^^2xy-{-2y'—5y-^2 = 0. First Division. x^-j-2x^y^2y^x—4:yx^y^—4: \ x''-^2xy^2y'—^y^2 a^-\-2x^y-]-2y'^x — 5yx-{-2x \ x (2/— 2)^+17^— 4 and this remainder may be factored thus, {y — 2)[.r-j-2/-l-2], and the factor y — 2 laid aside. Second Division. x'-^2yx^2^y^—^y^2 \ x-^y^2 xf^r !/-^+2^ I x^{y—2 ) (y—2)x^2y^—by^2 {y—2)x-^ y^ —4 2/^— 52/+6 which, placed = 0, gives y = 2 and i/ = 3. y = 2 gives x = Q and x = — 4. y = ^ gives xt= — 1 and x = — 5 TrENERAL THEORY. 17 from the second equation, but upon trial with the first equation only the roots x = — 4, y = 2, or ^= — 5, y = d, are found to be compatible roots. ' The suppressed factor y — 2 gives ?/ = 2; a value also found from the final equation. The foregoing treatment of this subject is mainly taken from the excellent discussion of Elimination in Hackley's Algebra. Art. 14. Labatie and Sarrus have perfected a method of elimination b}^ which foreign roota are not introduced into the final equation. This mode is quoted by Hacldey and by Todhunter in his Theory of Equations; but it is doubt- ful whether any advantage is gained over the simplicity and ease of trying the roots in the original equations and rejecting such as do not verify them. CHAPTER V. Nature or General Theory of Equations. Art. 15. An equation, as we know, is an algebraic ex- pression of the equality of two quantities. (And before it can be solved must contain a single unknown quantity.) This statement is true even of an identical equation which is true for any value of the unknown quantity or quantities entering it; for the mind, in the act of attributing a value to any unknown quantity in the equation, may be sup- posed to regard that, for the time being, as the only one . Now the two equal quantities, i. e., the two members, of every equation, may be placed in the first member leaving the second member zero; and the polynomial, [after it has been arranged with reference to the descending powers of the unknown quantity, may be divided by the co-efficient of the highest power, and so at the same time may be the second member, placing the equation in the form 3 18 PHINCIPLES OF ALGEBRA. j.-_j_P;^— ^_I_g.r— _i^i^./;"'--|- . . . ,-^Tx^ U= 0. . . .(1) This general form, which is often called the reduced form, has the co-efficient of x'" unity and P, Q, B, T, etc., any quantities not transcendental; they maybe algebraic or nu- merical, whole or fractional, positive or negative, rational, irrational, real, imaginary or zero. When any co-efficient is zero the corresponding power of the unknown quantity is usually absent. The equation is ihen incomplete; but when all the powers are present from the highest to the zero power the equation is complete. The co-efficient of the zero power of the unknown quan- tity is called the absolute tei^m of the equation. Art. 16. The form/ (x) = x"'^Fx"'-'^ Qx'"-'^ + Tx-\'U'= above described is the most convenient for ex- amining the nature of equations, but many of the proper- ties of equations which will be demonstrated are true when the equation has not been reduced to this form. And, on the other hand, many proijerties will be demon- strated only of equations having real co-efficients and even of those having their co-efficients numbers. Art. 17. Every equation has at least one root. Much ingenuity and mathematical skill have been used in de- monstrating this proposition by algebraic analysis, but it seems unnecessary for it is almost if not quite axiomatic. Since an equatio;i is an algebraic expression of the equal- ity of two quantities, or of the fact that their difference is = 0, there mast be some qaantity, or value of the unknown, such that when its different powers have been multiplied by the appropriate co-efficients and the sum of all the pro- ducts taken, the result shall be zero. OtJiei-wise there would be no equation; the truth would not have been told by the algebraic expression. The requisite value of the unknown quantity may be a real quantity or an imaginary expression ; and it is called a root of the equation. GENERAL THEORY. 19 Art. 18. Every equation of the mth degree has m roots and no more. We have just seen that every equation has at least one root, and we already know that an equation of the first de- gree has one root; also, that an equation of the 2d degree has two roots, and it is now to be proved that an equation of the mth degree has m roots; that is, the number of roots is equal to the number of units in the exponent which shows the degree of the equation. It has been proved in Art. 9 that if a is a root of an equation the second mem- ber of which is zero the first member will be exactly divis- ible by .r — a. Now suppose that a is a root of the equation af'-J^Px^-'^Bx^'-'-^-Bx^-^'-] . -fTar+r=0, then we shall have x'"-{-Px'"-'-i- Qx""-'-]- Tx^ U^{x — a) [x'^-'-i- PV-^4- ...._!_ T'x^ U'].... (1) Now this can be satisfied by placing x — a = 0; and also by placing a^^'-^-fP'^p"'-"-] -^T'x-\-U' = 0, which is a new equation, and it also has at least one root. Suppose that this root is 6, then, as before, we have x*^~^ _^p.^^-i_l_ _ _ _ ^j^T'x f U'= {x—b) \x^'-'-^P''x^'-^^Q''x^-' -j- • . . -f T'^x-j- f/"] which can be satisfied by placing x — b = 0; and also, by placing ^— -fP^'j:— ^-f- . . . . -f- T''x^ W = 0; and this is a new equation having at least one root, which may be called c, and when the corresponding factor X — c is divided out we shall, as in the previous cases, have a new equation. The degree of this equation will be m — 3. Continuing this process until the original first member has undergone m — 1 successive divisions we shall have a quo- tient of the first degree, of the form x — Z, which, placed equal to zero, gives an equation of the first degree, with one and but one root. Thus the total number of roots is r», and the continued product of the corresponding factors formed by subtracting each root from x will be equal to the original first member, so that we shall have the equation 20 PRINCIPLES or ALGEBRA. X — c) {X — d) (;r — /) ... (2) And there can be no more roots than m; for if there could be another and it were h, different from a, 6, c, d. . ly there would be a factor x — k which, multiplied into the product of all the others, would give for the first mem- ber a different polynomial, and one of a degree higher by unity; hence k vannol he a root. This fact may also be seen thus: If k is a value of .r, let it be substituted in the continued product, {x — a){x — h) ix — 0) (X — I) = 0, and we derive ik — a){k — b) {k — c) ik — /), which cannot be zero, because none of the factors are zero; whereas, when a true root, as a, b, c, etc., is substituted, there will always be one factor which vanishes. Thus the theorem is seen to be true. Art. 19. Equal Boots. — It may happen that one or more of the factors x — a, x — h, etc., shall be repeated, in which case the corresponding roots will appear as often in the equation; these are called equal roots. Thus, (r — a)^ (dp — b)ix — c)'^ = is an equation of the 6th degree, which has three equal roots, a, and two equal roots, c. It will be shown further on that when an equation has equal roots they may be discovered and the first member divided by the product of the factors belonging to them, thus depressing or reducing the degree of the equation. This operation is spoken of as " dividing out" the roots. COMPOSITION OF EQUATIONS. Art. 20. When the roots of an equation are a, b, c, d, 6, .... Z, we have seen that af"-\-Pii(^''~'^Qx/''-''-^Tx-\-U = {x—a)^x—b){x—G) . . (x—l) ..'.... (1) Now, if the multiplications indicated in the second member be performed, the result will be as follows: GENERAL THEORY. 21 xr-\-x' \—h ab -\- etc. , -f X'' -{-ac -\-ad -f6c -\-hd bS ztabcd. . + • • + (ibcde. . ±:abce . . ±abcf . .o ri" ztbcde ft) ,__ bcdf..^ S- CO and since the equation of which this is the second member is identical, we have, from the principle of Indeterminates' Coefficients : P = —a—b—c—d . . —k—l; or,— P = a+Z>+c . . +A:-f-/. g=a6+ac-f . . -i-bc^ . . -\-kL E=: — abc — abd— . . — ikl; or, — R = abc-{-abd-\- . . -]-ikl. S==abnd-\~abce-\- . . . gikl. U= ±iabcde .... kl; or, =fi U= abode . . . . kl. The quantities a, b, c, d, e . . . . k, I, all appearing with the negative sign, the product of an even number of them is plus and of an odd number minus, which accounts for the double sign wherever it appears, because in those cases the number of factors is not known. From these results the following important relations of the roots of an equation to its coefficients are manifest, to wit; 22 PKINCIPLES OF ALGEBRA. First. The co-efficient of the second term (with its sign changed) is the algebraic sum of the roots. Second. The co-efficient of the third term is the sum of the combinations of the roots in groups of two . Third. The co-efficient of the fourth term (with its sign changed), is the sum of the combinations of the roots in groups of three; and so on. Fourth. The co-efficient of the absolute term (with its sign changed when it is even numbered, i.e., when the de- gree of the equation is odd), is the continued product of the roots . Art. 21. DEDUCTIONS . Since the absolute term is the product of the roots it will be exactly divisible by any root; and, also, when there is •no absolute term one of the roots is zero . Further, when there is no second term it is because the sum of the posi- tive roots is exactly equal to the sum of the negative roots. Art. 22. When the roots of an equation are all posi- tive, the terms will be alternately positive and negative; because the product of an even number of negative terms is plus and of an odd number is minus . Art. 23. Since the first member of an equation of which the second = is composed by multiplying together the factors {x — a), {.r — b), etc., it will have m factors or divis- ors of the 1st degree; and since any two of them may be mul- tiplied together, giving a factor of the 2d degree, any three giving a factor of the third degree, and so on, there will be m(m — 1)^. . . ,, ^T , m(m—l){ni — 2\ . -5^-^— Mivisors of the 3d degree,— ^^ — ' \ ^ divisors of the 4th degree, and so on. Art. 24. When a, b, c, etc., are the roots of an equation that equation is (x — a){x — b){x — c). . . .(x — 1)=0. Suppose the roots of an equation are 1, 2, 3, 4: the equa- tion is (x—l){x—2)(x—3)(x—^)=x'—10x''^S5x'— 50x^24: =0. GENERAL THEORY. 23 EXAMPLES. 1. Form the eciiuitiou whose roots are 3, 7 and — 6. 2. Form the equation in which the roots are 9, 5, — 1, and — 3 . 3. What is the equation of which the roots are — 3, 2+ 1 —1 and 2—1—1 ? Ans. a^—x'—l.r~\- 15=^0. Art. 25. Since in the reduced equation U=abcde . . ./, T=abcde. .k-^-ahcde. .h-^-ahcde. .g, -f etc. , where the terms of the value of 7" are composed of m — 1 letters each, if we divide the latter by the former we get T abcde ....h , abode.... g , ^ 1111 u=abrde:::.m^^d;^d^:z:gi^'^'-=^b^-, which is real but not necessarily positive. If either of these be substituted in an equation for j", it is apj^arent that the results w^ill be partly real and parti} imaginary, unless some of the coefficients could furnish the necessary factor to make the imaginary quantities disappear from the product. But the coefficients in this article are supposed to be real. Consequently there must be another imaginary root of the proper form, to cause the product to be real. If, now, a thiril imaginary root should enter into the com- position of the equation, a fourth, and of the necessary form, must enter to keep the product real. There cannot he, therefore, an odd number of imaginary roots. If the coefficients are further supposed to be all rational, it is evident by the same course of reasoning that all irra- tional roots must enter by pairs. Art. 28. Hence evert/ equation of an odd degree has at least one real root, with a sign different from that of the ab- solute term. The imaginary roots are of the form a -f |/ — t)^ and a — |/ — 6% or else^ '^+v^ — b and a — y ' — b, and their products result in the sums of positive quantities. And this positive sum is a factor of the absolute term and exercises no influence on the sign of that term . And so of the product of all of the imaginary pairs . This leaves the one real root to give sign to the absolute term which, of course, is the opposite of its own. (See Art. 20.) It is also true that every equation of an odd degree hav- ing rational co-efficients will have at least one rational root; the sign may or may not be the same as that of the abso- lute term. GENERAL THEORY. 25 Art. 29. Erery cquaiion of c/ven degree and having real ro-effieien(f<, with ifs abi^olufe term negative, will have at leant tivo real rootn, one poHitire and the other negative. The pro- ducts of the pairs of imaginary roots will exert no inflneu.^e on the si.L>n of the absolute term, and if all the 7'oots were imaginary the absolute term would he pot^itive, but as it is not positive there must be at least two real roots and such that their product will be negative, they, therefore, must have diiferent signs. Art. 30. Every equation will have an even number of real positive roots if the absolute term is positive; and an odd number of sueh roots if the absolute term is negative. First. When the degree is even and the absolute term positive. The degree being even the number of the abso- lute term is odd, and, therefore, it is the continued product of the roots just as it stands. In this case that product is positive. If there are any imaginary roots, the quadratic factors belonging to the pairs will exert no influence on the sign of the absolute term . The total number of roots being even, the number of real roots must be or even . Now the product of the real roots must be positive and if there is any real root negative there must be another one negative to neutralize the influence of the sign, otherwise the sign of the absolute term would be changed. Hence the number of the real positive roots is even, which proves the theorem for this case. It is apparent that the number of negative real roots would also be even. Second. When the degree is even and the absolute term negative. Here, also, the absolute term, as it stands, is the product of the roots; and if there are imaginary roots they exert no influence on the sign. The number of real roots is even, and since the product is negative there must be at least one which is negative. This may be con- sidered as set aside for the moment. There now remain for consideration an odd number of real roots, whose product must be positive, and if there is among these a negative 4 26 PRINCIPLES OF ALGEBRA. root, there must also be another of the negative sign to neutralize its effect; in other words, if there aie any nega- tive roots among those dov^^ being con.-idered, there must be an even number of them; consequently, the number of real positive roots is odd. Third. AVhen the degree is odd and the absolute term positive. In this case the sign of the absolute term must be changed to give the continued product of the roots; that is, the product is negative. We know (Art. 28), that for such an equation there is one real and negative root. Let that be set aside ns necessary to change the sign of the ab- solute term. Th^^ total number of roots remaining is even, and the number of real roots remaining is even . More- over the product of these remaining roots must be positive and consequently if there are among these any negative roots there must be an even number and therefore the number of real positive roots must be even . Fourth. When the degree is odd and the absolute term negative (By Art. 28) there is one real positive root. The total number remaining is even, and the number of real roots remaining is even; but their product must be posi- tive as the real root set aside is positive; consequently among these remaining real roots if there are any negative roots there must be an even number of theui, likewise an even number of positive real roots, which, with the one set aside, makes the number of real j^ositive roots odd. Thus the theorem is established. Art. 31. // the signs of the alternate terms of an equa- tion be changed, the roots of the new equation ivill be the same as those of the former equation but with opposite signs. Let the equation be .r"^Px"'-'-^Qx"'"^^ . . . .^Tx-\-U =^0... (1) If we change the alternate signs, beginning with the second, we have .r-— P.r'"-^-f ^.r— "'-^ . ^-Tr±U =0....(2)., and beginning with the first, we have — x»'^Px"'"'^ — Qx"*"'^^ dz Tx^ U~0 (3), which equa- tions, (2) and (3), are merely one and the same. Now sup- pose -\-a to be a root of (1) and to be substituted in it for x. GENERAL THEORY. 27 The result will be a'"^Pa"'-'-\-Qa'"-'-{- . :..^Ta^U=Q. (4). Now if ^ rt is a root of (2) and (3) it must, on 'substitution in the one or in tlie other (as may be suitable, for they are merely two forms of the same thing), give eq. (4). When m is even use (2) and when ??i is odd use (3) . Therefv^re —a is a root of (2) or else of (3). Hence the principle . Changing- the signs of all the terms would not affect the roots, since it would simply be multiplying both members of the equation by — 1. • DESCARTES' RULE. Art. 321. No equation van have more positive roots than there are variations in the signs of its terms, nor more nega- tive roots than there are permanences of those signs . To demonstrate this assume the equation x"'±lPx"'~'±: Q.r'"-'±E.r"'~^± . .,±Tx±: U=0; in which the signs come in any order that may be prescribed. Now suppose that we introduce one more positive root, which will be- done by multiplying by x — a, and note the effect on the signs. The product will be ±F i x^±Q — a :T^Pa ,w— / ±E ,r'«-'zb ±U\ X H=r 1 =pC7a=0, Now so long as the co-etficient in the upper line is greater than the one in the lower line it will determine the sign of the total co-efficient of that term; if we suppose then in the first case, that all the upper were greater than those be- low we would have the same number of variations and per- manences as in the original equation, but having to come down at last to =p Ua, there is one more variation than in the original equation. If the lower co-efficients are all greater than those above they will give sign to the terms; but the signs froin the second toward the right, being always the opposite of the signs of the original first member, the number of changes of sign and of permanence, or repeti- tion of sign, will be the same. But one more variation was introduced when we descended at the second term . When a co-efficient in the lower line is affected with a sign coi4rary to the corresponding one above and is also 28 PRINCIPLES OF ALGEBRA. greater than that above, there is a change from a perma- nence of sign to a variation, for the lower co-efficient gives sign to the term, and we know that it is different from that of the preceding term above which is here sup- posed to be the same as that of the co-efficient above in this term . Hence each time we descend to the low^er line in order to determine the sign there is a variation which is not found in the original equation, and if, after descending, we remain in the lower line throughout, the number of permanences and variations of sign henceforth will be the same as in the given equation because the signs are al- ways the opposite of those above . If we ascend again to the upper line, we might make either a permanence or a variation; but suppose the worst, and that always there would be a permanence, it would merely offset the variation gained in coming down, and it will be necessaiy to come dow^n at last, making a variation at that time. , Therefore, the effect has been to produce one more variation than the original equation had; and so it would be upon the intro- duction of every positive root. Similar reasoning would show that the multiplication by the factor x-\-a, belonging to a negative root, would neces- sarily introduce a permanence of sign . And since the in- troduction of every positive root brings a variation, and the introduction of every negative root brings a permanence, the Rule of Descartes is shown to be true . Art. 33. When the roots are all real the number of positive roots Avill be the number of variations, and the number of negative roots the number of permanences . Suppose that the degree of the equation was m; then, the complete number of terms being ?7i-]-l. and n represent- ing the number of variations and p the number of perma- nences, m=n-\-p. Again, suppose that k = the number of positive roots and r = the number of negative roots. We shall have : m = k-^r; hence n-^-j^ = k-\-r, and n — k = r—^. Now, GENERAL THEORY. 29 by Descartes' Kule, k cannot be >/i; nor can it be less, because that would make, in the second member, r^p. which the Rule forbids? Therefore n = k and p :=::! r. DE GUa's criterion. Art. 34. If a term of an equation is absent between two lerins Jiarlng like signs, there are two imaginary i^oots. The absent term having for a co-efficient, we have a right to supply it either as -^0 or — 0. Suppose the order of signs to be: -|- -P — 1 , and for writing -|- or — , we have : -| — | i j and _!__!_ 4_ . In the upper line are 5 variations, 2 permanences. In the lower, 3 variations, 4 permanencies. Now, if all the roots are supposed to be real, there will by the first arrangement be 5 positive roots and 2 negative; by the second, 3 j)ositive and 4 negative. There are, then, two roots which have changed about, being in one case positive and in the other negative. But both suppositions being legitimate, we have two real roots, which are both positive and negative, which absurdity shows them to be imaginary. Where the terms between which the zero term is found have contrary signs, we can predicate nothing about the nature of the roots, because in that case the number of variations and permanences will be the same, whether we suppose the absent term to be positive or neg- ative. EXAMPLES. How many imaginary roots in — 30 PRINCIPLES OF ALCtEBEA, CHAPTER IV. Transformation of Equations. Art. 35. The changing the form of an equation, au J yet preserving the equation is an operation not only allow- able but often of the greatest convenience. We have seen already (Art. 31), that the signs of the toots of an equation may all be changed by changing the signs of the alternate terms; that is, the changing of the signs of the terms in this manner gives another equation whose roots are numerically the same, but have opposite signs to those of the first equation. Art. 36. To transform an equation into another equatiov whose roofs shall be some multiple of the r^oots of the first . Let the equation be x"" + Px"""^ Qx'"-'^ f Tx^ ?7= 0, . . . .(1) and suppose its roots to be a, b, c, etc. It is required to produce an equation of which the roots shall be ka, kb'kc, etc, y Make?/=A'.r . ' -x^^-, and this, substituted for x, gives '•'^. + Pp^.+ Q^JZ+ +4+ ^='^' ■■■■ <^'- Multiplying by t", we get: y'-J^Pky'--'-{- Qkfif"-^^ . . . . -L Tt"-'yJ^ Uk-" -= (8). In this equation, since if=^kx, the roots are ka, kb, kc, etc. Art. 37. This transformation leads to one of the most important, which is To clear an equation of fractions and yet keep the co-effi- cient of the highest power unity . F H Let x'"^~x'""^Qx"'-'-^~x"'-^-^ etc., = be an equa- fc (J tion having fractional co-efficients, and j)lace y=gkx, that is, equal to x multiplied by the least common multiple of the denominators. Then TRANSFORMATION OF EQUATIONS. 31 jr , P.V-- Qy"- By'"-' , ^^^ _o Multiply by ^""Y/"', and we have ir-\-gF]r-'-{^kYQir-'-{'FcfRy'''-' -f- etc., =0. This is an equation of the reduced form, wherein (if the roots of the original equation are a, b, c, etc.) the roots are kga, kgb, hjc, etc. If the denominators are numbers we may obtain a trans- formed equation of greater convenience by assuming for Ay/ a number less than the L.C.M. of the denominators, but which shall be such a product of prime factors of the denominators as shall secure, after the substitution, an en- tire quotient in each co-eflficient. This will be a matter of inspection and discretion to be used in each example. For instance take the equation -*--|^+lV~i-9M) - «■ ^''^^ 9000istheL.C.M. of the denominators, but the 3d and 4th powers of 9000 are inconveniently larger . But = 2x3; 12 = 2=x3; 150=2x3x5% and 9000 =- 2^'X3'^X5^ Suppose that in the example we make y = 2x3x5'^ = 30.r; we shall obtain : y' _ ^f . W _ _Jy^ _ _ 13 _ 2%3\5^ 6. 2^3^ 5=^"^ 12.2^3^5^ 150.2.3.5 9000 ' Now', the denominator 9000 = 2^ 3'. 5=* is the most diffi- cult one to provide for, and yet it will disappear when we multiply by 2\ 3*. 5*. The result will be : ,;*_5. 5,y3^5. 3. 5y_7. 213^5//— 13.2.3^5 = 0; or, ■,/^_25?/^+375i/'^— 1260^— 1170 = 0. If the roots of this equation can be found, those of the first will result from the relation y= 30.r. EXAMPLE 2. 7 11 25 ,r^— ^.r"'-[-— .r— ^r,=0 . If we make ?/ = 2 X 3r, we get if— 6 Ob 72 14?/'^-fll?/— 75=0. 32 / . PRINCIPLES OF ALGEBRA, EXAMPLE 3. , 13 , 21 3 32 ., 43^ 1 ^ 12 ^40 225 600 800 12 = 2^3 40 = 2^5 225 = 3^5^ 600 = 2^3.5^ 800 -= 2^5^ The prime factors being 2, 3 aud 5 ifc might appear that 2.3.5 would be a proper multiplier for X, but on trial we would find at the third term 21.2^3^5^/ 21.3^5^^ • ;, n ^ .• x> . -^ o3 f^ o3 Q3 K3 ^^ o — ' ^^ irreducible traction. But ii we Jt.O.ji .0.0 2i use the product 2^3.5 we shall obtain ,/_ 65?/*+ 1,8902/^—30,7201/2— 928,800?/— 972,000 = in which X = ^^ V . 60 ^ If the equation has the co-efficient of the highest isomer dif- ferent froyn unity; divide through by that co-efficient and then proceed as before . Suppose 3?/^~5?/-|-r=0, or i/^-i- -^/^—^^/^-^ = . Put..r=3?/. • .^ -^+ l = ^'^'- ^ -35-^+ryr=:0. Art. 33. From equation 3, Art. 36, we see that if the second term of an equation is exactly divisible by k, the third term by F, the fourth by P, etc., its roots will have a common divisor k. And any equation may be transformed into another of which the roots are - of those of the former by dividing the second term by /:;, the third by P, the fourth by k^, etc. This would give at once the result of making the multiplier 7 instead of k in the transformation of Art. 36. k For an example take the equation .x^ — 8^^ — 5^-4-84=0. . .(1) and let the second co-efficient be divided by 2, and the succeeding co-efficients respectively TRANSFORMATION OF EQUATIONS, 3li 5 84 1)V 4 and 8. We i^et y'^lr'—-x -[- ^ '= . . . .(2) . The 4 o 7 3 roots of equation (1) are 7, — 3 and 4, and ■^, — - and 2 will verify eq. (2). Art. 39. To trniixform an equation into anotlier, tlie_ roots of which shall be the reciprocals of those of the first. Substitute for .r in the equation ' y x"'-^Px"'-'-{'Q.r"'-'^ + 2:r+C7=0, and the result is ^^. + ^_, + ^.+ +^+ t7=0; whence, by clearing and reversing the order of the terms and dividing by ^. y + ~y"- + §r-'+. ..+^y' + ^y+\j=o. in which, if the original roots were a, 6, c, these roots are \' \' l '^- Art. 40. If any term is wanting in the given equation, there will be one wanting in the transformed equation at the same distance from the last as the other was from the first term. If the original equation is wanting in the second term, the one next to the last will be wanting in the trans- formed one, because the latter coeJB&cients are equal to the former divided b}^ C7, and-=r=r 0. Art. 41. If an equation be transformed by making a: = --, and the transformed equation should have the coeffi- V cients identical with those of the given equation, but in re- versed order, the two equations are one and the same. This is evident upon sight, and therefore their root^ must be the same. If the roots of the original equation were a, b, c, d, etc., the roots of the transformed one must also be a, 5 84 PRINCIPLES OF ALGEBRA. h, c, etc. But we know that the roots are also , r, -, a b' (' 1 1 T etc.: hence the roots of both are: a, - b «' ' b> c. c etc. ai^—px^-\-qx^—px-\-l = 0, ./-^-f (/j^'+l = 0, .r*-f 1 =0 in which the coefficients are: 1 — j^ +? — P +!» 1 -i-q -fl* 1 -]-lj ^^^ of the kind whose roots are of the form a and -. a Art. 42. If we have an equation of an odd degree, or one of an even degree without its middle term, and the signs of the corresponding terms, counting from first to last, and from last to first, are opposite, the roots will also be of the form a, -. Because, if we obtain the transformed a equation, and then change the signs throughout, we do not affect the roots at all (Art. 31), and yet it becomes identi- cal with the original equation, and must therefore have the same roots. For example, let j^-^px'^-{-px — 1 = 0. Substituting - for x, we have, after clearing, ^—Pl/+py^—/ =■ 0; or, y^—py'^py—1 = 0. Equations whose roots are of the form a, , b, , etc., are called recurring equations. Art. 43. A 7'ecurring equation of an odd degree must have 1 for a root when the absolute term is — 1, and — 1 for a root whe'u the absolute term is -f 1; because these numbers being substituted for x will satisfy it. j^et o(f — px^-\-qo(^^xi^'^px — 1 = 0, and substitute -fl for .r, we get 1 — p-^q — q^p — 1 =^ 0. The other roots (Art. 42) will be of the form a, -, 6, - , etc. Art. 44. To transform an equation into another of which the roots shall be the squares of the roots of the first. TRANS FORMATION OF EQUATIONS. 35 Let us assume, for convenience, that in the first member of the equation the even numbered terms are negative, and transpose all the negative terms to the second member. We shall have: x"'-[-q.r"'-^'~\'ii.r"'~^^ etc., =i px"'~' -\-r.r'"'^ -\- etc. Square both members, and we have: .r^'"+ 2c/,/-""-^ -f (^'H- 2«).r^'"-^-|- etc. ,= jo V^^-^-j- 'Ipraf'"- ^+ etc. And therefore ,](f"'^{1q—p'yr""-''^(q'-\-2s—'lpr)x""-*^ etc., = 0. Now this is a true equation, as we have a right to square both members. Let y = x^, and substitute in the last equation; the result is: y"'^{^q — p^)y'"~^-\-{(f-^'l>i —2pr)y'"~--\- etc., = 0, an equation whose roots are the squares of those of the first. EXAMPLE. Let x^-^-Sx'^ — 6x — ^8 ^ 0. In this by transposition we have x^ — Qx = 8 — 3,^"^, and by squaring, x^ — 12x*^36x^ rz= dx* — 48a7'^-|-64; whence x^ — 21j7*-|-84-r^— 64 = 0, and placing y = x\ 2/^— 21//'^-]- 84?/ — 64 ==0. By trial we find — 1 is a root of the given equation, and "dividing it out," we find the others to be —4 and 2. Squaring these, we get 1, 16 and 4, which are roots of the new equation and will verify it. Art. 45. To transfoi^m an equation into anotfier whom roots shall be greater or less than those of the first equation by any given quantity. First place, if necessary, the equation in the reduced form: ^'« _j_ Px"''-'-{- g.^'«-^-]- + Tx^ U=0, (1) Let x^ be any given quantity, and makey±x'=x. The new equation in y will have its roots greater or less than the roots of the original one by x\ Let us use the -f sign only; the results of substituting y — x^ would only differ in sign at the appropriate places. Substituting y-f-x' for x, we obtain: {y+x'r^P(y-\-xT-'-{-Q{y+^r'-'^-- + T{y^x')+U=0. KoW) develop by the Binomial Theorem, and arrange ac- m PRINCIPLES OF ALGEBRA. cording to the a. trending powers of the unknown qiianfifi/ ij, (which is done merely for subsequent convenience), there will result: + + -f -h + + - H^ s ^ •* § ■» i f ■^o + + + + -U ^ §' jo T + ^ i—i Ss s s 5: > > i i 1^ + fcO + 11 I + + + And this is the required transformed equation, but with the usual order of the terms reversed. If P\ C/ , etc., rep- TRANSFORMATION OF EQUATIONS. 37 resent the values of the co-efficients from y'""' down, and the usual descending order be resumed, the equation will be: y"'-\-P']r-'+Q'y"'-'^ . . • •+2"'2/+ 6^ =0, ... .(3) Art. 46. A method of arriving at the values of the transformed co-efficients, P' , Q', E\ etc., which is preferred by some as being shorter, is as follows: Divide the first member of the equation to be transformed by X minus the differ enee betiveen the old and new roots, the re- mainder ivilt be tJie new absolute term; divide, by the same, this quotient, and the remainder icill be the co-efficient of the first power of the new unknown; divide, by the same, the last quotient obtained and the remainder will be the next co-effi- cient in order, and so on to the last co-effia;=-f (/.r — r=0. Ans . //^— (3^+p)//'-f (3f?'''— 2/x^+f7)//— ((^ fjx^-^— ^^+r)=0 . 3. Find the equation whose roots are greater by 2 than those of ;r*--2.r^+5.r^+4a:— 8=0. Ans. if—lOy'-^^lif— 72//+36=0. 4. Find the equation whose roots are greater by 1 than those of x^— 5.r'~0.r— 2=0. Ans. //*— 4//'+//"'=0. As // is twice a factor of every term of this transformed equation, let us "divide out" //^, and we have .//^ — 4y-j-l = 0, whose two roots are 2-\-\/3 and 2 — yS, and as the four roots are 0, 0, 2-l-i/3, 2 — 1/3, if we subtract 1 from each we get the roots of the given equation, — 1, — 1, l-j- l 3, 1-, 3. Art. 49. 2h transform an equation into another ivantbig the second or any i^ctrticular term . From equation (2) of Art. 45 we see that the coefficient of y"'-' (which is the second term in the usual arrangement) is mx'-\-P. Now, since x' is entirely arbitrary, we can P give it such a value that mx'-\-P =^0, .'. x' = ; that m 4 42 PRINCIPLES OF APGEERA. is, minus the coefficient of the second term divided by the exponent which denotes the degree of the equation to be transformecl. All we have to do is to substitute for x the P quantity y . The equation resulting will have roots P greater by — than the original roots. m To cause the third term to be absent from the new equa- tion, we must place the coefficient of ;v'"~', which is ~^-^- ,2''^-j-(m — l)P.r'-l Q, = 0, and solve this quadratic to get the requisite value of .t\ To cause the fourth term to be absent it will be neces- sary to solve an equation of the third degree; the next coefficient would give an equation of the fourth degree, and so on upward. These equations would be difficult or impossible to solve . EXAMPLES . 1. Transform cr^ — 6x'^-f7 = into an equation where the second term is absent. Ans. y^. — 12y — 9 = 0. 2. Transform x* — 8.r'— 5j--[-12 = into an equation whose second term is wanting. Ans. y'—24:y''—Gdy—4:G = 0. It sometimes happens that the same value of x' will sat- isfy both the equations arising from jDutting the coefficients of the second and third terms = 0. In this case those two terms will vanish simultaneously. As an example : ,x^-|- 4./'^-J- (jx^-\-3x^ 4 = 0. p 4 Here ^= -r- = — 1 = x\ and substituting y-\-x' = m 4 y — 1 for ,x: . y^-^fJr 6/- 42/+1 4:y^—12y'-\-12y—4: Gf-12y-\-Q 3?/-3 +4 2/* — 2^-j-4 =^0 is the transformed equation . TRANSFORMATION OF EQUATIONS. 43 Or by successive divisions: n.if-\-'Sx 2;r'^-|-2.r x-^1 .r+1 1 3j;''-|-3ic X jfl ,0 ,-1 Giving the remainders 4, — 1, 0, 0, and consequently the equation ^' — ^-{-4 = 0. The same example by Synthetic Division is as follows: 1 +4 +6 +3 +4 I -1 —1 —3 —3 —0 1 -1-3 -j-3 -}-0,-j-4 —1 —2 - 1 1 -h2 -Tl,^! —1—1 1 -f 1,-f-O —1 i,H-0 Here the remainders are as before, and the e({uation is DERIVED POLYNOMIALS. Art. 50. By examining equation (2) Art. 45, we see that the coefficient of y" is simply the first member of the equation which was transformed with a dash placed on x. Omit the dash, and let this be denoted by f(x). The coefficient of i/ is formed ffom f{x) by multiplying each coefficient by the exponent of ./■ in the term and di- minishing that exponent by unit3^ Let this coefficient be denoted by/'(.x). 44 PRINCIPLES OF ALGEBRA. The numerator of the coefficient of if is formed from f\-r) by the same law as that by which f\-r) av.is derived fromy(;r). The denominator is 2 or 1.2. Let it be denoted by /_::«. •^ 1.2 And the law by which any of these coefficients is ilerived from its immediate predecessor is: .Multiply each term of the preceding coejficient by the expo- nent of X in that term, diminuih this exponent by unity and divide the alyebraic sum of the results by the number of pre- ceding coefficients. These coefficients, after the tirst, are derived from their immediate predecessors by the same law as that by which the coefficients of the Binomial Formula are built \\]}. The student of Calculus will recognize the numerators as differential coefficients of the iirst, second, third order, etc. They are called derived functions, or derived polyno- mials. Thus f^{x) is the Jirst derived polynomial off{x). /"(.r) is the second deriiied polynomial . f^^'{x) is the third derived poly )iomial . etc., etc., etc. Mark the distinction between the derived polynomials and the coefficients of tlie cleveJopment in equation (2) Kvi. 45. The tirst coefficient is the original first member with x^ in place of x; the second coefficient is the first derived polynomial; the third coefficient is \ of the second derived polynomial; the fourtli coefficient, ov .^ o of the third de- rived })olynoniial, etc. EXAMPLES. Let 'dx'-\-QiX^—Zx'-^'2x-\-l = 0. f\x) = 12^+18:c'^— 6a'+2. /"(jc) = 36£c^4-36.r— 6. /•'"(ic) = 72.T-f 36. /--(«.) = 72. ^- ^ ^ — JL — 0; whence clearing" of TRANSFORM VnON OF EQUATIONS. 45 The last terms having .if, the terms into wliich is mul- tiplied do not appear in the sacceediny; derived polynomi- als. In this way 1, 2, —6, 3() and 72 are successi 'ely dropped, which terminates the series. Let it be required to transform the equation 3./-^' I 15.r^-l 25./'— 3 = into one wanting- the second term. First placing it in rhe 25 3 fractions (Art. 37), v/M-15//-h75v/-27 = 0. — P J{-'') ^=^ //!4-i%''-i-'^^// — '^'* ^^"^^ -^^ = — — = — ^5 . • . /(.r) = —125 [-375—375—27 = —152 /'(.r) ^ 3v'4 3()v+75; /'(.r) .^ 75— 150 -j 75 = ^T) = %+15; q^^ -15+15^0 r '(')__. /'"W-i_i 2.3 " 2:3 ~ Hence the equation is k^ — 152 ■■= 0, an equation wanting both the second and third terms. 2. Transform .r' — 10y-[-7,r^-[-4.r — 1) =: into an ecpia- tion wanting the second term. Ans. = 0, each of its single roots will be twice a root of the equation proposed, and all of its repeated roots will appear once more frequently in the proposed equation. Having found all the equal roots, make a continued product of the binomial factors coirresponding to these, and divide f{x) by it; this will lower or depress the degree of the equation as many units as there are equal roots and render it far easier to be solved, and may even bring the depressed equation within the limits of those which we know how to solve directly and exactly. EXAMPLES. 1. Find the equal roots in the equation ^7_3^6_|_9_^3_i9^4_|_27.^_33^2_|_27.^_9 ^ q . The first derived polynomial is : And the H.C.D. between this and the first member is: Placing this equal to zero, and finding the H.C.D. be- tween this first member and its first derived polynomial j we get X — 1. Then (x — 1)^ is a factor of the first member of the secondary equation and {x — 1)^ is a factor similarly in the first member of the original equation. There are now known to be three roots = 1. Dividing {x — 1)^ out of a;*— 2ar»-|-4^'^— 6,r+3 = 0, we have 07^+3 = 0, and x = ~V — 3. {j(f-\-2>y will be a factor of the original first mem- ber and the product of the factors corresponding to the equal roots of the jjroposed equation is {x — lf{x^-\-^f --= original first member. 2. What are the equal roots in 2a;*~12a;»+19x=*— 6^+9 = 0? The first derived polynomial is 8x^—S6x^-\-SSx — 6, and the H.C.D. =07— 3. 50 PRINCIPLES OF ALGEBRA. There are two roots = 3, and the others, after dividing out and dei^ressing to an equation of the second degree, are found to be — ^r — and ^ — . 3. Find the equal and other roots of the equation x^-\- -f 2^— 12j*— 14a^-i-47j-^-f 12.r— 36 =- 0. Ans. two = 2, two ^= — 3 and also 1 and — 1. 4. What are the roots of .x^-f 4.r*— 14a?''— IT.i-— 6 = ? Ans. three = — 1, and besides 2 and — 3. CHAPTER VI. Limits and Places of Roots. Art. 53. A rational integral function of x is one in ivhich the exponents of x are lohole numbers and the coefficients are independent of x . Thus fix) = x"" -f Px'"-'-{- Qx'"-'-\- . . . .-\-Tx'-\- Vx -f U, in which m is a positive whole number (integer) and P, Q, T, etc., are independent of x, is a rational integral func- tion of X of the With degree. Art. 54. In any rational integral function ofx arranged according to the descending powers of x, any term which is present may be made to contain the sum of all ivhich follow it, as many times as we please, by taking x large enough. And any such term may be made to contain the sum of all ivhich precede it. by taking x small enough. In fix) = xr-\-Px'''-'-\-Qx"'-'-\- . . -\-Sx'"-"-^-'-\-Tx'"-*'-{- . . . . . .-\-U, Sx'"-"-^' will be the /«th term, and may contain the sum of all which follow it, if x be large enough. If it can be made to contain something larger than that sum, it will, of course, contain that sum. Now, suppose all the LIMITS AND PLACES OF ROOTS. 51 terms after Sx'"-**-^' to have the largest coefficient among them. Let it be L; then X(,r"'-"-{-.r"'-''-'-j-,r"'-"-'-f -f j^;-{-l) > ^^^ sum of the terms following the nth, and = jr,/__ \ Divide the /ith term by this: L{x"'-''-^' — 1) X — i _S ^ x'"-i'{x~l) S x-1 By increasing x we may increase the numerator indefi- nitely, and at the same time make the denominator as near unity as we please . Consequently the ?ith term will con- tain those that follow as mnjij times as desired. This proves the first j)art of the proposition. Suppose we make x = --, then increasing t/ diminishes .r. We have: ~['^-{-Py-{-Qif^- .... +^2/"-^+2'r+ • • • -^-WV The series within the brackets is such that any term, as Sy'*~', may be made to contain the sum of all which pre- cede it, 'i^-\-Py-\-Qf/-{~ etc., as often as we j)lease, by taking y large enough, which means .r small enough. This is evidently shown by the same reasoning as in the first case, and establishes the second branch of the proposition. Art. 55. The first term of the function may be made to contain the sum of all of its successors any number of times. Art. 56. A variable quantity is said to increase or de- crease under the law of continuity, when, in passing from one designated state to another, it passes through every intermediate state without interruption . A taper burning away, a cask of fluid being discharged by a cock, a plant growing, present instances. Let it be shown that if x increases or decreases under the law of continuity, that f{x) will increase or decrease under the same law. 52 PRINCIPLES OF ALGEBRA. When .T = a, let /(a) designate the corresponding state of the function; when x = h, f{b) the state of the function then corresponding, etc. Suppose that x' were a certain value of X, and give it a small increment, u. "We see from the development (2) of Art. 45, that we shall have: f{x^^u) =/(^')+"A.rO+ J^ ...... . J^ ........ 1.2-^ ^ ''^i.2.3'^ (•^;+--- 1.2.3.. f"'\x'). In (2) of Art. 45, ^--— f'"\x') \ . Z . o . . .Ill Now, if we transpose the term f{x') to the first member, we have: f(x'+u)-f{.x') = uf'{:^')+^"{.r')- IL The first term in the second member (which is present) may be made indefinitely greater than the sum of all which here follow it (they Avould precede it in the arrangement of Art. 55), by taking the increment, n, small enough. But when u is taken extremely minute, although the first term in the second member will contain the sum of the following terms an indefinite number of times, the first term itself becomes indefinitely small. Hence the differ- ence between the states of the function, f{x'-\-u)—f{x'), be- comes inappreciable. Hence, when the successive increments of X are indefinitely small, and x varies under the law of continuity, the function of x will vary under the same law. LIMITS OF ROOTS. Art. 57. The limits of the roots of an equation are values between which all the roots exist. A superior limit of the positive roots is any number or quantity of their kind greater than the greatest of them. LIMITS AND PLACES OF ROOTS. An inferior limit of the poi^itive rootfi is any number or quantity of their kind less than the least of them . A superior limit of the negative roots is a number or quan- tity of their kind tvhich is negative but numerically gr-eater than all of the negative roots. An inferior limit of the negative roots is a number or quan- tity of their kind which is negative but numerically less than any of the negative roots. Since a root of an equation whose second member is zero, when substituted for the unknown quantity, will re- duce the first member to zero, if we put for the unknown quantit3' any quantity greater than ^he greatest of the j^os- itive roots, the first member, when reduced, will be found greater than zero, that is, positive. This number and all greater than it, that is, all b^etween it and -J- go , would be superior limits of the positive roots of the equation. The smallest of such limits which is attainable is, of course, the one required for practical use, as a general thing. It may be j)roved that the greatest coefficient plus 1 is a superior limit of the positive roots; and even the greatest negative coefficient plus 1 is such a limit, and often a better one, because it may be smaller. This last is called MACLAURIN S LIMIT. In the equation f{x) =. x"'-{-Px"'-'-{-Qaf" "-f ■^-Tx-\- C/' = 0, . . . .(1), let the first term be positive and the others either positive or negative as may happen. Let N be the greatest negative coefficient, and suppose what would be the most unfavorable case which could happen, that all the other coefficients excej^t the first were equal to it and all negative. These negative terms would then form a geo- metrical progression with the ratio x, and it would be necessary only to put for x a value which would make i\V— 1) .r'">iV(~r'""'-[-.r' f ^+1) = x-l Now, if in the inequation «;"'<— ^-^ — r—^ we place a:* — 1 equal 54 PRINCIPLES OF ALaEBRA. to N, or X = ^Y-f 1, we shall satisf}' it, having .r'" >.>:"'— 1. Hence the greatest negative coefficient plus 1 i.s a superior limit of the positive roots of an equation. Art. 58. Since changing the signs of the alternate terms would make the positive roots all negative (and the negative all positive) it is evident that ttie greatest negative coefficient of the transformed equation plus unity would be a superior limit of the negative roofs of the equation; that is, intrinsically less than all of the roots. It would be numer- icallj'^ greater than any negative root. ORDINARY SUPERIOR LIMIT OF POSITIVE ROOTS . Art. 59. When the first term is followed immediately by one or more other positive terms, a closer limit may be obtained. Let us suppose .r"'~" to be the power of .r in the first neg- ative term, and take the most unfavorable case which could happen, that is, that all the succeeding terms are negative and all have the greatest coefficient among them. Let S be that coefficient. Then if we can make a;'">;SV"'""-|- Sx"'~*'~'-\- . . . .-\-Sx-\-S, it will be more than sufficient to make the first member jiositive, because, in fact, x"' would be increased by the addition of the other positive terms. Divide both members of the inequation by x^'\ and we get: ♦ 1<^+ A_ , A , -|--'^'_ . -^ There are n terms before the first negative, term and let us suppose X = \/'8-\-\\ representing the value of \fS by >S", whence ^ == S"", and .r = !+*§', the second member of the inequality will become: 1-11 ~r /o/ \-i\m—i I /.C' 1 1 Xw *^ (5' _|_1)« -r (6^'-j-l)«-'-^ '••••• ' (5f' +1)"'-^ ' (*S^'+1)' LIMITS AND PLACES OF ROOTS. 55 We have here a geometrical progression, in which the j&rst term is — ^ ^^ with a ratio . ^, . ... . Its sum, there- (/^ -j- 1) \^ ^r J-) fore, is: S'" S'" S"* S'" 1 -i"'^ -'- which is the difference between two proper fractions, and therefore less than 1, as was required. The quantity l,^ S-\-l will consequently make the first member of the given equation positive, and be a superior limit of the pos- itive roots. This result may be stated in common language thus : Extract that root of the greatest negative coefficient of which the index is the number of terms before the first negative term; increase this by 1, and the res\dt will be a good superior limit of the positive roots -of the equation. If any term is absent, it must be counted to determine the index of the root. If n = 1, the second term is negative, and ^V'S-^-l = S-\-l, the same as in Art. 57. EXAMPLES . Find a superior limit of the positive roots of x*-\-llx^ — 2Bx—ei= 0. Here n = 3, and the limit is ^^67-|-l. The* cube root of 67 is between 4 and 5, and hence 5-f 1 will be the limit. 2. ^*+llx=^— 25r— 61 = 0. Limit = 1^61+1, or 5. INFERIOR LIMIT OF POSITIVE ROOTS. Art. 60. If in any equation we make x = ~, the roota y 56 PRIJsCIPLES or ALGEBRA. of the transformed equation being reciprocals of tliose in the first, the greatest positive root of the transf onned will be the reciprocal of the least positive root of the given equation . Hence to obtain the inferior limit of the positive roots: Substitute - for x; find the superior limit of the positive roots of the transformed equation ; its reciprocal will be the limit required. SUPERIOR LIMIT OF NEGATIVE ROOTS. Art. 61. This, as already indicated (Art. 58), will be the superior limit of the positive roots of an equation whose roots have signs opposite to those of the given equa- tion. This transformed equation can be had by making X = — y, or by changing the signs of the alternate terras. This limit is numerically superior, but not algebraically or in fact. INFERIOR LIMIT OF NEGATIVE ROOTS. Art. 62. Take the reciprocal of the last transformed equation, that is, put x = — -, and find a superior limit of the positive roots of this equation, it will be the required limit, because, since x = , we have y = , and y X the greatest positive value of y will correspond to the least {numericaUy considered) negative value of x . Newton's limit. Art. 63 . Any number, which on being substituted for the unknown quantity in the first member of an equation and in its derived polynomials, makes them all positive, is a superior limit of the positive roots. If the roots a, b, c, I of f{x) = be diminished by x', that is, if we make x = x'-{-y, we shall have, eq. (2), Art. 45: LIMITS AND PLACES OF ROOTS. 57 = 0. If such a value be i^laced in this equation for x' as to make all the terms positive, we know that all its roots, that is, values of ?/, must be negative, and from the rela- tion X = x'^ii, we have y = x—.r\ so that y being nega- tive, x'^x, and consequently, whatever value will make the first derived polynomial, /'(.r'), positive will make posi- tive the original first member, where the coefficients are the same, but x takes the place of a quantity greater by.'z;'. EXAMPLE. Find a sui)erior limit to the positive roots of x^ — 5.r'^-[- lx—1 = 0. We need not retain the dashes upon^the x, but write : f{oc) = dx^—Wx-\-l. 1.2.3 Beginning at the last derived function in which x ap- pears, and substituting the smallest whole number which will make it positive, Ave see that 3 makes it positive. Like- wise the next before it, and so on to the last. 3, then, is the limit. EXAMPLE 2. What is the superior limit of the roots of x^ — 5.^* — 13a;' + 17a;''— 69 = 0? We have derived polynomicda as follow (after dividing out their appropriate denominators): 5a;^— 20./— 39a;'^4- 34r . IOj-^— 30./^— 39j'fl7. 10a;'— 20a;— 13. 5a'— 4. 1. 58 PRINCIPLES OF ALGEBRA. 1 placed for x gives 5 — 4=1, positive; but fails in the next above. 2 fails, but 3 gives a positive result. 3, when tried in/"(.r), fails, and so does 4, by a single unit, 5, being tried, gives -[-, and being tried in f\-r), fails, and so does 6. And 7 is found to be the required limit. It will be perceived that this has given us the smallest limit in whole numbers, and it will always give us a closer limit than any of the j)revious methods. The amount of comjDutation confines its use to cases where closeness of limit is im^^ortant. It was invented by the immortal New- ton, who has shed brilliant and enduring light upon all of the man}^ branches of learning to which he addressed himself. Sudan's test of imaginary roots. Art. 64. If the roots of an equation be reduced by a quantity r, and the transformed equation shows a loss of m variations of signs, and if the reciprocal equation be re- duced by -, and this transformed equation shows n varia- tions, which were not lost but w^hich remain; then there are m — n imaginary roots between r and 0. Because , in reducing the roots of the equation by /', all positive roots less than r will have become negative*, and there will be as many positive roots between and r as there have been j)ositive roots changed into negative, which is to say, as many as there have been variations lost, whereas, in reducing the roots of the reciprocal equation by -, no positive root greater than - will be changed. But should a different result appear, it would indicate the ex- istence of imaginary roots, the number of which within these limits will be the number of variations lost by the * The factors of the first degree belonging to negative roots are of the form x-^c, x+d, etc., and in the multiplication which builds up the first member of the reduced equation, they exercise no influence on the signs or number of variations. LIMITS AND PLACES OF ROOTS. 59 first transformation minus the number not lost in the one last described. Now suppose /• was a superior limit of the positive roots; when we reduced by r, the number of lost variations would be equal to the number of positive roots, provided they were all real. And in the recii)rocal equation would be an inferior limit of the positive roots, and when it was transformed by reducing the roots by the quantity , the transformed equation would show no loss of variations, provided the j^ositive roots were all real. A different result would show that there was an absurdity or contradiction about some of the roots, which we would therefore per- ceive to be imaginary. And these would appear to be pos- itive. And the number of imaginary roots thus discovered would be the number of variations lost in the transforma- tion of the original equation minus tke number not lost or which remain in the transf or oration from the reciprocal equation. Again take the original equation and change the alter- nate signs; the positive roots will be turned into negative, and the negative into positive roots. Proceed with this as with the original equation, and we shall discover the num- ber of imaginary roots, apparentl}^ negative. EXAMPLE . Find the number of imaginary roots in Since this equation is of an odd degree, with the abso- lute term negative, there is at least one real root i)ositive, and since there is but one variation, there is but one such root. We need not look among the positive roots for im- aginary roots, but according to Sudan's Test, we change 60 PRINCIPLES OF ALGEBRA. the alternate signs and have which 1 is a superior limit. COEFFICIENTS OF DIRECT EQUATION. 1 —3 4-2 4-3—2 -h2 I 4-1 .3j.^_|_2.r^ 1 3,,^_2.r 4-2, of COEFFICIENTS OF RECIPROCAL EQUATION . 2 --2 -}-3 -1-2—3 -hi I -f-1 +1 —2 ±0 +3 +1 + 2 ±0 +3 -1-5 +2 —2 ±0 +3 +l,+3 + 1 -1 -1 +2 ±0 -1-3 -rS +2,-1-3 + 2-^2+5 +10 -1 -1 +2,+3 _|_1 ±0 —1 -1-2 +5-i-10, + 12 +2+4-^9 ±0-l,+l +1+1 + 4+9,+19 + 2+G +i,±o +1 +6,+15 +2 +2 +8 1 +2 ±0 -1-1 -}-3 -f3 2 -f 8-hl5+19-l-12-f 3 Since in the coefficients of the transformation from the direct, the third, ±0, is between two terms of like signs, we know from De Giia's Test, that there are two imagin- ary^ roots in the trAnsformed equation; we may therefore use the plus sign, which shows 4 variations lost. In the transformation of the recij)rocal equation there are none left; hence 4—0 = 4, the number of imaginary roots. 2. In QC" — 10jr*-[-6j:-[-l=:0, how many imaginary roots? Ans. All real. 3. How many imaginary roots in .r* — 4..>;^-|-8.r^ — 16,r-[- 20 = 0? Ans. None. 4. In .r* -f-.r'-f j""^4-3.r — 100 = 0, how many imaginary roots ? Ans. 2 imaginary roots and 2 real with opposite signs. PLACES OF REAL ROOTS. Art. 65. It has been shown that if x varies under the law of continuity that /(.:tT)=a;'"-[-P.r"'-'-f . .-J-T^-j- f/will do LIMITS AND PLACES OF ROOTS. 61 SO likewise. Let us sii])pose that we had substituted for x in /(.*•) a number, p, and the result was greater than 0, or -{-. Then, if x decreases under continuity, it will after a time, come upon the value of one of the roots, when the result will be 0. Continuing to decrease, its value (say 7) will give a result less than or — and consequently we say that if two number's, p and q, when miihd'diited for the unknown quanUty in the first member of an equation of which the second member is 0, give results with opposite signs, there is at least one real root tjetween p and q. A quantity may change its sign by passing through in- finity as well as through 0. Let :r = -; here as y decreas- es, .r increases; when // is very small, r becomes very great; when // =^ 0, .v =z :c ; when ?/<0, or negative, x be- comes negative; but in the rational integral function, which is the first member of the equation, no finite value of X, as between p and 7, could make f{x) = cc . There might be more than one root between p and q. Moreover, if there are roots between p and q, the substitu- tion of p and q will not necessarily produce results with contrary signs, for, Art. 66. When an odd number of routs lie between p and q, their substitution loilt give results having opposite signs; when an even number of roots lie between them the results will have tJie same signs. Suppose that there were several roots, a, b, c, etc., be- tween 2^ aiicl q, and some others besides. Let the product of the factors of the first degree with respect to these latter roots be Y; then we shall have : /(,r) = {x-a)(x-b){x-c) X r= 0. Substitute for x first p and then q; let Y' be what Y be- comes on the substitution of p, F" the result of substitut- ing q in Y. Now Y' and F" will have the same sign; otherwise, by Art. 65, there would be another root, or 62 PRINCIPLES OF ALGEBRA. roots, lying between p and 7, which is contrary to the sup- position. If we now make the sul)stitiitions, and for convenience write the tirst result over the other, we shall have: {p~a){p—mp—^^) X>" {ri—a)l>7, then a, b, c, etc., willjbe and >7, so that all the quotients will be negative except ,7,,. Now, if the number of roots a, b, c, etc., between p and q is even, the product of these fractions will be positive, and the first result divided by the second will have a posi- tive quotient, that is, the results of the substitutions of p and q will have the same sign, and the contrary will be true when ct, b, c, etc., are odd in number. THE THEOREM OF STURM. Art. 67. But the best of all the modes 3'et discovered of determining the character and places of the roots of an equation is the celebrated theorem of Sturm, contributed in 1829 to the scientific world by that eminent French mathematician. The object of *S/wrm\s ilieorein is to dis- cover the number of real and imaginary roots in any eqwx- tion, and the places of the real roots. Sturm's Theorem does all that is accomplished b}^ the methods which have thus far been examined, and more be- side. Still those methods should be preserved, because they are sometimes sufficient for the purpose in hand and of easier application than the theorem of Sturm . This theorem deals with the signs of certain functions of the unknown quantity, which are : the first member of the LIMITS AND PLACEfi OF ROOTS. 63 equation, its first derived polynomial and certain others which are formed in the following manner: First free the equation of equal roots, if it has any; then apply to /(.r), the first member, and to/'(^), its first derived j^olynomial, the process for finding the H.C.D., but with this differ- ence — after each remainder has been found, change its sigri, and during the intermediate operations neither introduce nor suppress any factor but a pcMtive one. Let /(.r) = Nx"^ -L.rx'"-'^Q.r"'-'-\- -[■Tx'\-. U =^ be the equation, and designate f{x) by V, and/'(.r) by Vi, and by — V^, — T3, — r^, — Tv, the remainders of the various divisions wherein the quotients were: (>,, Q.,, Q^, .... Q^-j- We shall have the following equations: F_= F_a._-T; (1) Tv cannot be 0, otherwise there would be a CD. between f{x) and /"(./•), which is contrary to the supposition . It must, then, be a number, because the oj^eration is to be carried on until the last remainder is independent of r. Art. 68. Letyl and B be two numbers, and A<^B. Let A be substituted for x in the expressions, V, F,, K^, T^, etc., and the signs of the results recorded; then substitute J9 and record the signs. The sign of K,. will always* re- main the same, being independent of x. Then the theorem declares that The number 0/ varlal Ions in the first series of signs, dimin- ished by the number of variations in the second, will be equal to the number of real roots between A and B. 64 PEIIS'CIPLES OF ALGEBEA. Art. 69. To show this, it will be convenient first to establish three lemmas, as follow : FIRST. No two coxHi'vutire functions, V, Fj, F, etc., ran hccoinr for the same value attributed to x. Let us take any equation out of the group (1), as and suppose that V„_i and F„ should both vanish for a value of x, then from the equation, F;,.y.i, would also be zero. And the next equation of the series, having F,, and Kt-i-u both 0, would give F„.|._, = 0, and so on. Thus they would all vanish and the last equation would give F,. =0, which cannot be . SECOND. Art. 70. When amj one of these functions becomes 0, the one before it ivill have a different sign from the one fol- lowing it for the same value of x. as is shown by taking any one of the equations, as F.^-^ Fs Q^ — K* and letting F^ =0 . F = V THIRD. Art. 71. If a number almost equal to one of the real roots of the equation be substituted for the unknown quantity in the first member, and likewise in its first derived polynom- ial, the results will have contrary signs; but if the substituted quantity be greater than this root by an extremely small amount the signs of the results will be the same. Let us suppose that a were a root and the added small quantity ii. Let a-\-u and a — u be substituted for .'■: then b}^ (2) Art. 45 we shall have: /■(«+") ==/(«)+/'(")«+/"(«) j^-f THEOREM OF STURM. 65 f(a-u) = /•(«) -r{a)u.+f"(a)^'^-. . . . ytn—i In these f{a) -= 0, and as a is a very minute quantity, the first terms of those portions of the series which remain will far exceed in value the sum of the remaining terms (Art. 54) and will give sign to the series. These series will then become: «[/'('')+l"2 /"'(«)+.■ ■ • • + «"-'] ■ • • -(3) and -u\f'(ay-^^f"(a)+ . . . , + «'-'] ... .(4) The upper one will be positive and the lower will be negative, and /'(«) positive all the time, as they are here shown. In all cases, it is apparent that the sign of the lower series will be different from f\a), and that of the upper the same. But (4) and (2), which are the same, are the result of substituting a — u in F, while (3) and (1) are the result of substituting a-\-u. f\a) is the result of sub- stituting a for X in Fi. The truth of the lemma is, then, demonstrated. Art. 72. Then to demonstrate the theorem of Sturm, let us suppose a varying quantity, A^ which at the outset is less than the least of the real roots of F = 0, Fi = 0, F^ =- 0, . . . . F„ = 0, . . . . and F,_i = 0, that is, of all the equations formed by putting Sturm's functions equal to 0. Let it be substituted for x in all of them, and record the signs of the results. Afterwards suppose A to grow: after a time it will be equal to the least of the roots mentioned above, and some one of the functions Fi, Tg, etc., will van- ish. But, as its sign agrees with the one before it and dis- agrees with the one after it (Art. 70), or else agrees with the one behind it and disagrees with the one before it, the number of variations will not be affected. And this will be true even if two, or several, of the func- 66 PRINCIPLES OF ALGEBRA. tions vanisli at the same time; because the same conditions as those just described would hold when the vanishing func- tions were separated from each other by intervals in the series of them. And this must be the case because no two consecutive ones can vanish simultaneously (Art. 69). This will continue until the varying value of A arrives in the close neighborhood of a root of V =. 0, that is, the origi- nal equation, when the signs of V and Fj will be different, giving a variation, and after it passes the value of the root, and — u becomes -{-u, they will agree in sign, giving a permanence, or losing one variation (Art. 71). If it be supposed that A continues to grow until it ar- rives in close proximity to another real root of T= 0, the same thing will take place, and when V becomes zero and emerges with a change of sign, it will have passed another rooty ayid another variation will have been lost . And so every time a real root is passed, a variation of the signs of the functions F, Fj, Fg, etc., will have been lost, until A has grown greater than the greatest real root of the original equation. The number of variations lost will be equal to the number of real roots between^ and B, and as they were taken as the numerically superior limit of the negative roots and the superior limit of the positive roots, the total number of real roots will be known. This number subtracted from the exponent which shows the de- gree of the equation will give the number of imaginary roots. This last must of course be or an even number, and thus the theorem is found to be true . Art. 73. If in finding Fj, F^, etc., any one of the functions placed = should give an equation all of the roots of which were imaginary (and this fact would be known by the function remaining of one sign for all real values of x,'^ the work need proceed no farther. The polynomial function remaining always remaining of * Note.— When the first member is constantly of the same sign for all" real values of X, we infer that all the roots are imaginary, because if one value resulted in plug and another in minus, there would be a real root between the numbers substituted. THEOBEM OF STURM. 67 the same sign and the last one, F^, being constant, none of those between them can change sign. And therefore, if any loss of variations takes place, it must- do so among those which precede. This conclusion will be reached by considering the chain of equations which connect the poly- nomial spoken of with V^. Both ends of the chain remaining of a constant sign, the nature of the connection is such as to prevent all interme- diate functions from changing. Or otherwise thus: Take the equations 'r—4: == 'r—3 Vr-5 ' r—2 • • • • (1) 'r-3 ^^ ''^r-2 Vr-2 ' /--/ .... (2) Vr..-^ = Yr.-. Qr... - V, (3) Suppose that it was found that F^..2 would not change sign. Since the quotients ^i, Q.^ .... Q^..j would have no influence on the sign, as the values of x are not substituted in them, if we transpose V^ to the first member then thai member being fixed in sign, F^..i , would be so likewise. If F^..3 were found constant in sign we might eliminate V^i out of (2) by substituting its value (3) and a single equation would result with F^_2 and quantities fixed in sign; hence F^_2 could not change sign and in the same way it could be shown that F^_i could not change sign. If V^_i were found to be constant in sign, F^_3 and V^_^ could be eliminated by substituting the values from (2) and (3) whence it would be seen that F^_i must not change sign; then in succession the other two. And so for any function. Art. 74. After the varying value of x has passed above and below the greatest and least roots, the further increase or decrease can make no difference in the signs; and this is true up to any extent even to -f oo and — oo . The sub- stitution of -j- X and — x v^ill cause the functions to take their signs from their first terms, and will be found conve- nient because we need substitute in the first term only . This will give the number of the real roots . But in ad- 68 PRINCIPLES OF ALGEBRA. dition to this we may wish to know their places, that is, be- tween what whole numbers they may lie . To do this we substitute for J and B, and 1, 1 and 2, 2 and 3, 3 and 4, etc. Thus having found one or more roots between 2 and 3 we know that the roots are 2 -\- sl fraction less than unity. For the negative roots we substitute — 1 and 0, — 2 and — 1, — 3 and — 2, and so on. Having found a root be- tween — 3 and — 2 it will be ^ — ^3 plus a fraction, etc. Its initial figure will be — 2, as — 2.57. EXAMPLE. 1. 8^-3 _6j;— 1=0. f'{x)=8a^—6x—l f^(x):=24:x'^ — 6. Suppress the positive factor 6 in/''(^) and we have Fi=4^"^ — 1. 8a^—6x—l I 4:x'—i 8a?~-2x I 2x — 4,x — 1. Here changing sign we have 4:X-\-l=V2. Multiplying V^ by the positive factor 4 and we get 16x^ — 4 to be divided by 4.:r-f 1. 16^^—4 I 4a:+l 16ar^+4ar | 4a:— 1 4:X — 4 —40^—1 — 3 and -\- 3 and -f 3 = F"^. Hence the poly- nomials are minus oo . plus go . V==Sx'—6x—l — F,= 4a^— 1 -f V,= 4.x -f 1 — Vr= ■ +3 + Hence 3 variations have been lost and all the roots are real. To determine their places: substitute and 1, and — 1 . 3 variations. variation and 1 — + ~ -f- + -f + + One variation lost and hence there lies between 1 and one real root, which is zero plus a fraction. There are no more positive real roots becjiuse -f- 1 gives the THEOREM OF STURM. 69 same signs as -f- tween 1 and -{- oo and — 1 00 , and, therefore, no real root can lie be- + Two variations lost, and there are two negative roots, between — and 0. We have, then, 3 roots, 1 positive and 2 neg- _j_ -|- ative. The signs being the same for — 1 and — CO we know that plus 1 and min- us 1 are the smallest limits in whole numbers. Example 2. x' — 4.x^ — 6^ -f 8 = 0. Here V = x^ — Aaf — Gx -f 8 F,= dx' — 8^ — 6 Multiplying F by the positive factor 3, and proceeding as above indicated. dar^—Ux'— 18x^24: I 3j?»— Sot— 6 3^^- Sx'—Qx. x,—l 4:X^ — 12er-}-24; suppress -|-4 and multiply by 3: 3^^ 8^+ 6 —17^+12 .-. V,= llx—12 S,jc'—Sx—6 1 17.7r— 12 17 3^,-100 Slar'^— 136^—102 Blx'— S6x — lOO^r— 102 17 —1700^—1734 —170007—1200 — 534 Then we have: V = af—lx'—Gxi-S Fi = 3a;'^— 8^— 6 V, = llx—12 V, = +534. For X = -{-oo ,+ + + +, no variations. For X = — 00 , 1 1- , 3 variations. 3 — = 3 variations lost, and 3 real roots. 70 PRINCIPLES OF ALGEBRA. V V, K Fa Var. For x = 0, -\ \- 2 x = l, + -f 1 x=2, + + 1. x = 3, + + 1 ^ = 4, - + + + 1 ^=5, + + + H- ^ = 0, + + 2 x=—l,-\- + h 2 x=—2, \ h 3 Between — 2, which gave the same signs as — x , and — 1, there was 1 variation lost; hence the root is — 2 -[- a fraction, or a root whose initial figure is — 1. Between and -f 1 a variation was lost, and there is a root whose initial figure is 0. At 4 there was 1 variation and at 5 none; hence a root 4 -f- a fraction. Ex, 3. 2x' — 11./;= -f 8x — 16 = 0. Here V= 2^* — ll^^-f 8^ — 16 V,= 4:x:' — 11^ +4 V,^^nx'—12x +32. If this were placed equal to zero the two roots would be found to be imagin- ary. If the trinomial were a true square we should have 4(11^^ _|- 32) = ( — 12xy which it is not. V, will not change sign for any real value of x and we will proceed no further in getting Sturm's functions . -f- ^ gives -j- -\- -]-; no variation — 00 gives -j- [- ; 2 variations . • . 2 — = 2 and there are 2 real roots, and of course 2 imaginary roots. X ^= gives ^ -j- X = gives \- -{- X == 1 gives \- X = — 1 gives — + + X = 2 gives — + -h ^' = — ^ gives [- ^ = 3 gives + + + oc =^ — 3 gives -|- 1- and the initial figures of the real roots are 2 and — 2. Ex. 4. x^ -j- 11^^ — 102^ + 181 = 0; in which Uvo of the roots are nearly equal. THEOREM OF STURM. 71 The functions are V = x^ + llaf — 102x -f 181 V, = Saf -\-122x — 102 V, = 12207 — 393 F^= -\- number. — 00 gives 3 variations and -f- oo none; so there are 3 real roots. X = gives -j -f- and so do a? = 1, 2 and 3, but X = 4: gives no variation, therefore there are two posi- tive roots lying between 3 and 4. Their initial figures are 3. Let the equation be transformed into one whose roots are less by 3 (Art. 45). The functions of this equation will be: V, = 3if^4:0y—9 V, = 122?/— 27 Fa = -|- number. Now in these substitute y = 0, y = 0.1, y = 0.2, y = 0.3, etc., and we find: y= gives + — — -f-, 2 variations; y = .1 gives -f — — -f , 2 variations; • y = .2 gives -|- -f, 2 variations; 2/ = .3 gives -}- -j- -j- -|-, no variation; two positive roots between .2 and .3, and of the proposed equation between 3.2 and 3.3. Transform the equation into another whose roots shall be less than the roots of the last by 0.2, and we have: F = 8^-|-(20.6).s'^— (•88)8+.008 F, = 'ds'-\-(4:1.2)s—.88 F, = 122s— 2.6, or 61s— 1.3 Fz = -\- number. Substitute s = 0, the signs will be : s = .01, the signs will be: s = .02, the signs will be : s =. .03, the signs will be One positive root between .01 and .02, also one bet\wen .02 and .03, and for x we have a; = 3.21 and x = 3.22. Their sum = 6 . 43. . • . _ H _ 6.43 = —17 . 43, = the third root, which is negative . + — +, 2 var. + — +, 2 var. h> 1 var. + + + +, no var. 72 PRINCIPLES OF ALGEBRA.. CHAPTER YI. EXACT AND DIRECT SOLUTION OF EQUATIONS! Art. 84. We know that equations of tlie first and sec- and degrees admit of direct and exact solutions; the first presenting a single root and the latter two roots . It will be now shown that such equations of the third de- gree as have two imaginary roots can be solved direc Jy and exactly; and that equations of the dth degree of ivhich two, and only two, of the roots are imaginary can be directly and exactly solved. Above these equations there are no means of exact and direct solution, at least none as yet have been discovered, and it is believed that none can be discovered . Propositions for the exact solution of equations of degrees higher than the 4th have occasionally been presented, with plausibility, but the practical results have been such as not to invalidate the accuracy of the statement above. Art. 85. Let x^ -{- Px'' ^ Qx -\- M = 0; this is a gen- eral representative of equations of the 3d degree, and any equation of the 3d degree will be a particular case of this general form. But as the difficulties of solving equations of a degree higher than the second are sufficiently great at best it will be well to diminish them by removing as many terms as possible. Now as we can exchange any complete equation for an- other wanting the second term, let us write • x^ -\-px -}- q = (1) and form for it the functions of Sturm, we shall have x^-^-jJX-^q I 3x^-{-p 3 ~ U V=3(^-\~px-\-q Fi^3^^+^ V^= —2px—Sq V,=-4:p'—21q' Sx'-\-dpx^Sq 3x^-\-px 2px^dq . • . F,^—2px—Sq cardan's soldtion. ' 73 Asrain : 3./^+p 1 -2 2p 1- 6px'-\-dqx oxSq -3-r, 9^ —dqxi-2p' 2p —18pqx-\-4:p^ —18pqx—27q' 4:2f^21q -27(/ The roots of this equation, it being of the 3d degree, must all be real, or else one must be real and two imaginary. Since 2^ and q are not numbers, but the general representa- tives of any real coefficients of the first and zero powers, let us see what relations they must have in order that all the rooU: shall be real. There must result from the substitution of — 00 3 variations, and from the substitution of -|- 00 no variations, in order that all the roots may be real. When -f 00 is substituucd in V the result is -f ; in V^ it is -f , but in V2 it will not be -[- unless p is negative. Then p must be negative; but ^ will not be -\- unless p is negative and moreover has such a value that 4j/ is greater than 27r/. With this condition F3 will be plus and remain so. The first term of K will also have a positive coefficient; so that the substitution of — will give the signs 1 \-; 3 variations, and the substitution of -|- ^ gives no variations. We see, then, when ^- > | or v-j > ft) that all the roots will be real. cardan's solution of equations of the third degree. Art. 86. If the equation is complete, let it be trans- formed to another wanting the second term . It will take the form a^^px-i^q = 0, (1) 10 74 PRINCIPLES OF ALGEBRA. Let the unknown quantity be placed equal to the sum of two other unknown quantities, ij and z; then x = y-\-z; x' = ]f^2yz{y^z)^z' .-. ;f^-dy{,j^z)2-{y'^z') ^ {), iin&* replacing y-\-z by^^ in the second term, we have: ^_3^,^-(y;3j„,3)_0, (2) This has the same form as eq. (1), and by comparison: p = -Syz, ... (3), and q = -(y'+^), or y'-^^ ^-q,.. (4) — P — P^ Since 2 = ., , 2^ =^ ^rr\, and this in the value of — q Sy 21i/' ^ gives y— ^3 = —q; • • • y'-{^qy' = ^■ Since this is a trinomial equation, we have: '^-'V-lMf+D and, since the equation y-^z = ^ is symmetrical with re- spect to y and z, z will have the same values. But not to repeat the same value for both, we will take the first for y and the second for z, and adding them together, we get for the value of x: which is the celebrated Formula of Cardan. (f ^ If 4 "^27, Under each grand radical sign there is the same indi- cated square root, which will be real when p is positive, p3 and also when p is negative, provided that ^ is less than J-. 2" is of course always positive. But the inequation ~ <^ J- > ^y -^I't- '^^j shows that all the roots cannot be real, therefore one loill be r^eal and two imaginary . Cardan's formula will then solve the cubic equation in such a case. If the last inequation should prove to be an equation in cardan's solution. 75 any case, then the indicated square root would vanish and X = 2UJ — I ; which is real, and Cardan's formula would apply in this case also. But if we seek the greatest com- mon divisor between the first member of the equation and its first derived polynomial, the remainder is 4:p-{-21(f, and if this = 0, there is a CD. of the first degree with respect to .r, and therefore two of the roois are equal. If all three of tlie roots were equal, the equation would re- duce to a binomial equation of the form (.r — of = 0, and X = a, and all the roots would be = a. But when p is negative and jy^^j^, Cardan's Formula fails, and this is the condition, that all the roots are real, but not equal. All are unequal. Art. 87. Since every quantity has 3 cube roots, and since x = sum of two cube roots, it might at first sight ap- pear that the cubic equation had nine roots, because each one of the first set might be taken in conjunction with each of the 3 in the second set, making 9 in all. Now we know that every equation of the third degree has three roots and no more, and this appearance of niae roots must be de- ceptive. To explain this, let us remember that the three cube roots of any quantity, as a^ are of the form a, — ^ and ~ '- ', and so of the values of y and z. But it will be remembered (eq. 3, Art. 79), that ^2 = — ^, a real o quantity, since p and q, the coefficients of the proposed equation, are supposed to be real. If, then, we take one of the cube roots which are equal to y and one of the cube roots which are equal to z to make a value of x, that is, to make up a root of the equation, we must so take them that their product will be real. We could take the two rational 76 PKINCIPLES or ALGEBRA. and real cube roots, but when we take an}' one of the imag- inary expressions we must take another, but differing in sign before the radical part of it, so that the product would be the difference of two squares and real. This restriction would allow us to form two values only of x out of the im- aginary parts, which with the one made up of the real parts would give the number of values of x, three, and only three. In the solution of numerical examples, it will in general suffice to substitute in Cardan's Formula for q and i^ their appropriate values; but sometimes greater simplicity may be obtained by treating the example in the same manner that Qi?^'px^q =r was treated in the deduction of Car- dan's Formula. EXAMPLES. 1. Solve the equation a^- 9j^-f 28./— 30 -:: 0. Ans. 3, 3 f 1— 1, 3— V ^. The transformed equation is i/^-f ^ =" O? ^i^ which p --= 1 and 5 =^ . • . Cardan's Formula becomes: — •^•\/l/(a7) •'• ^^ = 0.. one root, and as x = w-f 3, x = 3. This " divided out" of the equation leaves an equa- tion of the second degree with roots as above. 2. Solve the equation ,r^— 7.r'-f 14r— 20 =- 0. Ans. 5, 1+1 — 3, 1 — I — 3. , 7 344 ^ The transformed equation is //' — ~u — -^^ == 0. And this cleared of fractions gives: ?/ — 21?/ — 344 = 0. One root of this is 8 and w= v 8 ^ , ,7 8-|-7 = 5. waking's method. 77 Ex. 3. Solve x^ — 6.^^^ + lO.r — 8 = 0. Here ic^ — ^u — 4 == 0, and the two radicals in Cardan's formula give 1.5774- . . . and 0.422-f ; their sum is 2 =w. Ex. 4. ^ + 1.r +12 = 0. Ex. 5. x^ —AHx^-'. 128. Ex. 6. a'' — 3./^ — 2.V' — 8 = 0. Let a.-^ = ^. Remark — When ^_<;^^ and the latter is negative, the values of x are apjyareyitly imaginary though known in fact to be all real. This is called the irreducible case because Cardan's formula fails, and no means have yet been discovered to surmount the difficulty by Algebra alone. SOLUTION OF EQUATIONS OF THE FOURTH DEGREE. Art. 88. The equations of this degree admit of direct and exact solution by the methods now known only when they have hvo, and only two, imaginary roots. If their roots are all real or are all imaginary we do not know how to solve them exactly, but in the case of numerical equations can resort to some method of approximation. Descartes and Waring have each demonstrated an excel- lent method of solving equations of the fourth degree hav- ing only tw^o imaginary roots. We will here reproduce DR. WARING's METHOD OF EXACT SOLUTION OF EQUATIONS OF THE 4th DEGREE. Art. 89. Let the proposed equation be ^* -f 2pr^ = qx^ -f rx -f- s (1) Now {x' + pr -t- ny = x* -{- 2p.7^ + (p' -f 2?i) x" + 2pnx -f n^ (2) If therefore we should add to both members of eq (1) the quantity (p' -{- 2n) x' -j- 2pnx -j- ''^^ the first member would be a perfect square. The second member becomes (p^ -}- 2n -{- q) of -j- {2xn -j- r) a? -[- {n'' + «); which, being in reality a trinomial arranged according to the descending powers of x, will be a perfect square if 4 times the product of the extreme terms = the square of the middle term; 78 PRINCIPLES OF ALGEBRA. that is if 4 {p^ + %i + q) {ii^ -f s) = {2pn + rf ; leaving off for the moment the powers of x. Performing the opera- tions indicated, transferring all the terms to the first mem- ber and arranging according to the undetermined quantit}^ n we get 8/i^ 4- Iqri' + (8.S' — Apr) n + Aqs + ipK^ — r^ = 0. . . .(3); an equation of the 3d degree with respect to n. If we can solve this conditional cubic equation and get a value for ??, this value of n will be what is required to make the trino- mial (7/ -f- 2/1 -f q) a;^ -f (2p?2^ -j- r) .r -{- (n^ -[- s) a perfect square. The quantity involving n which w^as added to both members made the first member a perfect square, and with the value of n (which we still call n) suppose! to have been found from the cubic (3) both members will be exact squares : (^' -f 2^x + ny= {p" J- %i -f- q) .r' -[- (2/)/i -f r)x^{n'^s); taking their square roots we have x^ -{- px -[- n= ± \_\/x>^ -f- 2?i -j- 7. ^ -f -\/n^ -f ^J when the middle term of the trinomial which makes the second mem- ber is positive; and when that middle term is negative we have x\-\- px -{^ 7i = ± \j\/if -{- %i -\- q.x — \/ ii" -f s\ . In either case we have two equations of the 2d degree and will get 4 values of x, wdiich are the 4 roots of the original equation. Art. 90. This method can be applied only to those equations of the 4th degree wliich have two imaginary and two real roots. For let us suppose the roots to be repre- sented by a, h, c and d: the product of two of them, say ah, Yvill equal the absolute term of one of the quadratics, giving ah = 11 — y n' -[- s; . • . n — ah = \/n' + .s squaring which we get n' — 2'ihn -f- a'b' = n' -^ r . • . — 2ahn -f a'h' = s, but as s, the absolute term of the proposed equation, is equal waring's method. 79 to — abed, we have — 2ab)i -{- a^b^ = — abed .-. ab — 2/i = — cd .'. n = o • Similarly the other two values of n ac 4- bd ^ ad 4- be are n = ^ and n = j^— If now all the roots of tlie bi-quadratic, or equation of the 4tth degree, a, b, e and d should be real, the values of n, or the roots of the equaJon of the 3d degree would all three be real and we could not solve it. Again if a, b, e, d should all be imaginary, their pro- ducts, two and two, being real, the 3 values of n would all be real and we could noi: solve. But if two of the roots a^ b, c and d are real and two imaginary there will be one root of the cubic real and two imaginary and then Cardan's Formula would apply. Thus suppose that a and c were n , . • ,, , ^<^' + ^^ , -. , real, b and d imaginary; the value n = — ~o~~~ wotild be real because we should have the product of two real -|- the 23roduct two imaginary quantities, and the sum would be real. The other roots would evidently be imaginary. EXAMPLE. X* — 6x^ + 5x^ -f- 2a; — 10 == 0; by comparing this with the formula equation we find 2p = — 6orp= — 3 q = -5 r = — 2 s = 10 and equation (3) then becomes 8m' — 20n' + 56n + 156 ^- 0, which divided by 4 gives 2ii^ — 5/i^ -\- lin -f- 39 = 0. On solving this we find one root n — — _ Hence (.^^ -Sx-^^^' + lx +^| .-. .X' - 3^ -| = ± (^+^j .• . x^ — Ax 5 and also, the other quadratic, x'^ — 2x = — 2: From these we get x = — 1, 5, 1 -[- y/ — 1 and 1 — 80 PEINCIPLES OF ALGEBRA. The solution of the intermediate cubic results thus: n^ — - 71^ -f 7?i + y = 0; make n = | .♦. ^/^ —5y'' -f 28y + 156 = 0; make 2/ = ^-l-| .-.^3 + |^ ^ + ^-^? = 0. Make t==~.-. s' -\- Ills + 5222 = 0. In this last equation o for the purpose of applying Cardan's formula we note p = 177 and q = 5222, hence ^ = 205379 and ^. --= 6817321 and the algebraic sum of the two cube roots in the formula = -14^.. ... ^ = 3=-^-and^=/ + 3^--- n 0/ + n 5 3' - — 3 .'. n — y 2 3 ~ 2* EXAMPLE 2. • Find the i •oots of x' —^r^— 17^'^— 3a;— 60 = ^^^ ~ 17 2* Ans. —4, 5, ,/ EXAMPLE 3. -3, - -V— -3. Fin d the roots of x' +7.r^— 33^-^+107iP- -154- = 0. = - 15 2* Ans. 2, —11, l+i/- -6, 1- -/= -6. CHAPTER VII. Occasional Solution of Higher Equations. Art. 91. Beyond equations of the fourth degree there are no direct methods for exact solution; and as has been seen, the existing methods do not apply to all equations of the third and fourth degrees . But when equations of any degree OCCASIONAL SOLUTION OF HIGHER EQUATIONS. 81 HAVE EQUAL ROOTS those equal roots maybe discovered by Art. 53, and divided out, thus reducing the degree of the equation. If the re- duced degree is the first or second, the remaining root or roots may be directly found; if the reduced degree is the third and the equation has two imaginary roots, Cardan's Formula will apply and tlie equation may be directly solved; also when the resulting equation is of the fourth degree and has two, and only two, imaginary roots, the equation may be exactly solved. Art. 92. Also when an equation OF THE THIRD DEGREE HAS COEFFICIENTS WITH CERTAIN SPECIAL RELATIONS, the roots may be found exactly. Suj)pose the equation to be of the form a? J^ dx"" -{- kx = q ....(1) Jf we had a cubic equation of the following form x^ + ^px^ + ^p^x ==q .,..{2) it is evident upon inspection that if we add j/ to the first member it would become a perfect cube. Adding p^ then to both members we have or" + dpx^ + dp'x + / = (/ + p^ . . .(3) or {x -\- pY = q -\- p^ whence x = — p -j- ^^q-\-p^ • • • • (4) Comparing equations (1) and (2) we see that d = 3p and d k k = 3//, whence p ^ ^ and p^ = -. Squaring both meni- o o d d^ d^ bers of p = -we havep'^ = — .•. by addition, 2p^ = — + o a y 3 = -3- .-.p^+^Z-yg- (5) From this last formula the value of p may be found, which is necessary to transform equation (1) into equation 11 82 PKINCIPLES OF ALGEBBA. (2) and which in (4) will give a root of the proposed equa- tion . But this is on condition that p have the same value in d = ^p and ^' = 3^^ that is, that d? = ^k . When the coeffi- cients of x^ and x are such that the square of the coefficient of x"^ is equal to three times that of x, this method will ap- EXAMPLE 1. a^^l^x'^^lbx = —125. Here d = 15 and d' = 225, and 3^- = 3X75 = 225; and /225-f225 p =^ -^ =5, and x = —5+i3/— 125+125 = — 5, and this root divided out gives the quadratic ^--f 10a;-j-25= 0, of which the roots are — 5 and — 5. All three of the roots are real and equal in this equation . EXAMPLE 2. x'-{-lBx'-i-15x = 218. d Here p =^- = 5, and x = — ■p-fif^'^H-p^ =■ — 5-[- o #'218 4-125 = —5+1^/243 = — 5-[-7 =-- 2. The other roots are : —17+ V —147 ^ —17—1 —147 X = —^ — and X = EXAMPLE 3. 7 ^+-H3 3 Aus. X = ly X ^= — '^~^\l — Q' *^ ^^ — ^ — \/' It will be observed that when this relation holds betvv'een the coefficients, if the second term be made to disappear the third wall disappear also . Further, that when applica- ble it is so without respect to the nature of the roots, as imaginary or not. If, then, the intermediate cubic in the OCCASIONAL SOLUTION OF HIGHER EQUATIONS. 83 solution of an equation of the fourth degree should be of the class just described, it would enable us to solve that equation without respect to the nature of its roots, widen- ing by so much the field of apj)lication of Waring's and Descartes' methods. WHOLE-NUMBER ROOTS. Art. 93. If an equation has been placed in the reduced form, X'" + Fx'"-' + Qx'"-' _|- . . . . _j_ 7-^ _|_ C^= 0, (1) ft cannot have any roots which are fractions. By fractions is here meant irreducible fractions, or frac- tions whereof the two terms are prime with respect to each other. Such fractions containing "one or more of the equal parts of unity" are commensurable with unity. Whole nnmbers are, of course, commensurable with unity. See Art. 2G. Art. 94. Consequently, when we find no whole num- ber among the roots of an equation of the reduced form, since we already know that there are no fractions among them, we know that none of the rools are commensurable with unity. 3-|-|/2, f/1 ^ are specimens of quantities not commensurable with unity. Imaginary quantities are never commensurable with unity. Art. 95. The absolute term of the equation will con- tain as divisors all the roots, whether wh^le numbers or not; but it will usually contain many other divisors besides the roots. We could scarcely hope to find what the in- commensurable divisors of any absolute term are, bnt we can more easily discover those which are whole numbers; and of these, taking those lying between a superior limit of the positive roots, L, and a numerically superior limit of the negative roots, — U\ we can discover by trial which among them are roots. But thp labor of these substitutions may be much short- ened by the results of the following investigation. 84 PRINCIPLES OF ALGEBRA. Art. 96. Let a, a whole number, be a root. Then and transposing to the second member all the terms except U, and dividing by a, we have: - = —a^-^—Pa^-^— .... —Ea'—Sa—T (1) Since the second member contains none but whole num- bers, it is entire, and therefore — is entire, which is merelv a confirmatory of what we already knew. Now transposing T to the first member, and dividing by a, we obtain : — h 2'= — a— ^— Pa— •?— . . . ,—Ea—S, a and as the second member is entire, we see that the quotient of the absolute term divided by the whole number root, plus the coefficient ofx is also exactly divisible by that root. For — j- T substitute T\ transpose S and divide by a as a before, and the result is: — ^^ = —a-^-3—Pa^-4— .... —Qa—R, a and as this second member is entire, we see that the former quotient plus the coefficient of x^ is exactly divisible by the root. Now making — ^^^^— = S\ transposing — B and dividing by a as before, ijiere results: 0/_[_ 73 ^--L^ = —a^n-4_p^.n-s_ _Q a a whole number. Therefore the last preceding coefficient plus the coefficient of a^ is exactly divisible by the root. Proceeding in the same manner, when we shall have transposed all the terms save two, we shall have an equa- O' tion like — = — a — P = a whole number. Transposing ~4-P P' —P and dividing as before: a = — = — 1. This shows a OCCASIONAL SOLrTION OF HIGHER EQUATIONS. 85 that the last of the quotients (which is forn^d when the coefficient of the second term is transposed) is — 1 . Every divisor of the absolute term which will stand all of these successive tests is a root, and as it is supposed that we will try only those divisors which are whole numbers, we will discover all the whole-number roots . Having found them we divide them out, as in the case of equal roots, and solve, if possible, the resulting equation. We may form a table at the heads of the vertical columns of which are placed all the entire divisors which lie between the upper and lower limits, and then make a simultaneous trial of them all; rejecting all which in any of the succes- sive divisicHis give quotients not entire; that is, any which fail to stand all the tests. Having formed the table by writing the whole-number divisors between the limits in a horizontal row proceed by the RULE Divide the absolute term by each divisor setting the quotient immediately beneath the divisor. Form new dividends by ctdd^ ing the coefficient of x to the quotients. Divide these by the numbers on trial, setting the quotients immediately beneath the dividends. Form new dividends by increasing the last quoti- ents by adding to them the coefficient of x^. So proceed, al- ways forming new dividends by the addition to the last quotients of the next succeeding coefficient towards the first, and re- jecting any divisor which at any stage gives a fractional quoti- ent. All those which finally give a quotient which is minus unify are roots . EXAMPLE 1. Find the entire roots of the equation : 9.r'-|-30^^-f22^*+10.x'+17;r'— 20^+4=0. , , lO-r^ 22 . 10 _ 17 , 20 4' , . y ^'+^-+9-^ + :^^+ 9-^ — 9 '^+9=0; make ^=3.-. 2/6_|_io?/5_|.22?/*+302/'r|-153?/'— 540i/+324=0. Here L = 4: and — U' = — 31 And the divisors (which are whole numbers) of the absolute 86 PRINCIPLES OF ALGEBRA. term 324, are 4, 3, 2, 1, —18 and —27. -2, —3, —4, —6, —9, -12, 3 .108 2 162 1 324 — 1 -324 -Ti —3 —108 —648 +216 —4 —81 -6 -54 —9 —36 -12 -27 -18 —18 —27 —12 —432 —144 —378 -189 -216 —216 —864 +864 1017 —1017 —708 +354 —621 + 155 J —594 +99 —576 +64 -567 —558 +31 184 —102 9 -552 +20'-^ 9 +3 —36 —18 —63 —63 507 -253J 369 —123 252 —42 217 _24l 27 33 +11 12 +6 —33 —33 —11 —11 —987 +987 -93 +31 53 -171 — 1-Z + 2 24 —4 33 +11 28 + 14 1009 —1009 21 +7 24 + 12 -1 — 1 —999 +999 +6 — 1 The onl^r two divisors giving a final result of — 1 are y 1 -}- 1 and — 6 and placing these in ^ = ;^ we have x = - o o and X = — 2 which will satisfy the equation j^roposed. 2x' — 15^=^ + 8x' + GSx + 48 = 0. 2. X 3 Ans. — 2, — 2, — 1, 3 and 4. .^4_5^3_j_ 25^ — 21 = 0. • ^ , 1 + 1/29 ,1 Ans. 3,1, — —^ and— l/29 SOLUTION OF RECURRING EQUATIONS. Art. 97. These equations have their coefficients re- cur when counted from the first and last. Their roots are of the form «, -, b, -. etc. a b The student should now carefully review Articles 41 to 43, inclusive. Binomial equations of the form oi?-\-l = 0, x'^-\-l = 0, X* — 1 = 0, etc., are recurring equations. 1. Take x^ — 1 ^0; we know that -|-1 is a root of this equation; dividing it out, we get X'- x^-\-x-]-l, and the roots of x^-^x-{-l == are ^ and — — ^ . OCCASIONAL SOLUTION OF HIGHER EQUATIONS. 87 2. Take ,i^~{rl =^0; we know that — 1 is a root of this equation, and dividing it out we get ~|~ ■ = on^ — '^' fl; the roots ot r^ — i>t* -1 = are ^— and 7= 3. Take a:^ — 1 = 0; we know that this is composed of two factors of the second degree, to wit, x'^ — 1 and x^-\-l; hence placing these equal to zero and solving, we get the four roots, -f 1, — 1, -{-V—l and — l/^. 4. Take x'—l = 0; this is (.r^ fl)(.r'— 1) = 0, giving cases 1 and 2. 5. j:,-5_|_;i^ ^ is(x-{~l){x'—x^-\-.jc^—x^~l) = 0, giving^ -|-1 -^^ and X'* — a^-\-x- — ^+1 = 0. This last is a recur- ring equation of the fourth degree. But before examining ^*-fl =^i .x'*'-l-l = 0, etc., we will demonstrate the following principle: Art. 98. Every recurring equation of the fourth and higher even degrees may he solved by using one of a degree half as high . Suppose we had a recurring equation of the fourth de- gree: x'-{-Px'^Qx'+Fx+l =: 0, and that the roots were a, -, b and j ; then the factors of the first degrfee would be x — a, x , x — b and x — - , and a the quadratic factors would be x"^ — K^+^/i- + 1 and x^ — \b-\--jx-[-l . Put a-\ — = k and b-{-~ =zl; then we have x^^kx-{-l and x"^ — lx-\-l. These multiplied together give the equation of the fourth degree in w^hich x enters as a factor four times, while k enters only twice; therefore what- ever equation we may obtain for the value of k, from or by 88 PRINCIPLES OF ALGEBRA. means of the original equation, will be only of the second degree . If we multiply the quadratic factors involving k and /, and place the product equal to the original equation, we will form an identical equation, and equating the cor- responding coefficients, we could determine the values of k and /; and it would be found that none of the subordinate equations. would be above the second degree. Again, suppose the recurring equation to be of the sixth decree and put k = «+-, I = &+, and h ^= c-\-~ . The ° ^ a b c quadratic factors x^ — ^j?+1j ^~ — ?J^+1 and x^ — }iX'\-l mul- tiplied together give x^—{k^l^h)r'-^{kl^kh^lh-{-Z)x'— {klh^2k^ 2/+ 2h)x^^ {kl-\- kh-\- lh^^y—{k^ Z-f- h)x^ 1 . Put this equal to the first member of original equation, supposed to be a^^Px^-{-Qx^-^Rx^^Qx^-\-Fx-\-\^ and equat- ing the coefficients, we get: Q = kl-^kh-f-lh-^-S; B = —{klh^2k^2l-i~2h); in which, since P, Q, and E are known numbers, we have to determine the three unknown quantities, k, I and h from the three equations, one being of the first, one of the sec- ond and one of the third degree. The resulting equation would therefore be of the third degree, one-half the degree of the original equation. A similar investigation would evidently show a similar result for any equation of a higher and even degree . Returning to Example 5 — we there saw that one of the roots was — 1, and when this was divided out there resulted the recurring equation x^ — ^-f ^'^ — x-{-l =^ =x* — (k-\-l)x'^ -i-{kl^2)x'—{k^l)x-^l. From this 1 = k-{-l and also 1 = kl4-2 . • . k'—k = 1 and ^ = ^J^. But as a+- = k. H l±|/5±i/— lQ-f2|/5 4 OCCASIONAL SOLUTION OF HIGHER EQUATIONS. 89 and either of these four roots when substituted in x^-\;-l = 0, will satisfy it. For instance take the first: 1 + v/54-l/-10+2i/5 ^^^ raise it to the fifth power. 4'^ = 1024. In the numerator place 1 + /5 ^ a and i/— 10+2v/5 = d. Then (c-^df = c^+5cV«+10c^^*^+10c=(f+5ccZ*-l-cf and 4- o or rn o o or o ^co o^ S % a: a] %. i II if II li rf^ lo h- 1 o <^ -q o 1 r + n^ tf^ o 00 00 o o p .^ \ -x en o 2 \ 1 o + bO OX 1 3 \ en \ 1 M O + err CI + bO § I o + or + 1 O + OX OX — 1024, and this divided by the fifth power of the denom- inator gives — 1. which satisfies the equation. In a similar manner other recurring equations, like a^-\-l = and x"'-\-l = 0, may be solved whenever we caij solve an equa- 12 90 PRINCIPLES OF ALGEBRA. tion of half the degree, and this is true of all equations which are recurring, whether binomial or not. EXPONENTIAL EQUATIONS. Art. 99. Exponential equations, or such as have the unknown quantity as an exponent, sometimes, but rarely, admit of exact solution. It is assumed that the student is familiar with the solution of exponential equations by con- tinued fractions and logarithms. These equations do not fall within the class of algebraic equations, but of transcendental equations. CHAPTEK VIII. Approximate Solutions of Higher Numerical Equations. Art. 100. "When it is not practicable to solve an equa- tion by any of the modes which have been discussed, we must rest content with an approximation to the roots. For- tunately this can be had closely enough, for practical pur- poses, by the methods which we now propose to examine. We have seen that when an equation has some equal roots, or some that are whole numbers, we may discover them and divide them out, and reduce the degree of the equation; if it is a recurring equation, we shall have to solve one only half as high in degree; and in short, if by trial, chance or in any other way, we can discover one or more roots, we would immediately depress the degree. If after all it is of the fifth or higher degree, we can only approximate, and so likewise with those of the third and fourth degrees when they do not happen to have two imagi- nary roots and no more; unless there is a peculiar relation among the coefficients such as was examined in Art. 92. Horner's method. Art. 101. This method was first published in 1819. It is the invention of W. G. Horner, and is regarded by most APPROXIMATE SOLUTIONS. 91 mathematicians as the most satisfactory mode of approxi- mating to the real and incommensurable roots of an equa- tion having numerical coefficients. The method is as follows : 1st. Having found by Sturm's Theorem, or otherwise, the whole-number part of a root, and still better, having found in addition one or more of the figures in the decimal part, to transform the original equation into another whose roots shall be less by the part already found . 2d. To obtain the next figure of the root by dividing the absolute term by the next preceding coefficient and taking the first figure of the quotient for the required figure . 3d. Then to transform this equation into another whose roots shall be less by the decimal figure last obtained; to divide the absolute term of this equation by the coefficient which immediately precedes it, and take the first figure of the quotient for the next figure of the root. 4th. Again transform the equation into another of which the roots shall be less by the decimal figure last ob- tained, divide the last coefficient by the one immediately preceding for the next figure of the root, and so continue till the desired number of places in the approximate root shall be found. In this way we find the real positive roots, and if there are any which are negative, obtain them approximately by changing the alternate signs of the proposed equation, which will make the roots now being sought all positive, and proceed as before. If preferred, when one or more roots have been found, they may be divided out and the degree of the equation reduced. Demonstration . Art. 102. When an equation has been transformed into another of which the roots are less by the whole- number part of the original root, and still more if they are less by the whole-number part and one or more figures of 92 PRINCIPLES OF ALGEBRA. the decimal part, the remainder of the root, that is, the value of the unkno\Yn quantit}' in the transformed equa- tion, is a very small quantity indeed. Therefore its second and higher jDowers may be neglected in comparison with itself, and the first member of the^ transformed equation, which will be of the form may be, without appreciable error, taken to be T-\- U 0; and the first figure of the quotient (and j)erhaps more) will be the initial figure or figures of the true value of u, and will therefore be the next required figure in the value of the original unknown quantity. Let it be required to find the approximate roots of the roots of the equation : ^4_8^-}-14a;2_]_4^_g ^ 0. Sturm's functions of this, when reduced to their simplest form, are: 1 8 + + + 4 + 8 + + + + + + + + 3 1 + + + 4 + + + + 2 + to + + + 2 + ± + + 1 + + + + 1 + OS + + + + + V,= hx'—Vlx^Q V, = 76^—103 Vi=^-\- number Variations Since the number of variations lost between — 1 and -}-6 is the same as between — oo and -}-oo , all the roots are real and comprised between — 1 and 6. Between — 1 and there is 4 — 3 = 1 variation lost; there is one negative root, then, between and — 1, and since it is numerically less than — 1, we substitute in V, Fi, F2, etc., in succession, — .1, — .2, — .3, etc., until there APPROXIMATE SOLUTIONS. 93 is a gain of variation, and the last preceding number will be the first figure of the root . This root in the exam j)le is found between — .7 ond — .8; therefore .7 is the first fig- ure of th6 negative root. gives 3 variations, and -\-l gives 2; hence there is a positive root which is a decimal fraction. By the succes- sive substitution of .1, .2, .3, etc., its first figure is found to be .7. 2 gives 2 variations and 3 gives 1; hence there is a root whose first figure is 2; and as 5 gives 1 variation and 6 gives none, there is a root whose first figure is 5. This last fact may be known at once, because 5 in F gives a negative result and 6 in F gives a positive result. There is there- fore one real root between them, or else some other odd number of roots . Let us now proceed after Horner's manner to find this root, whose whole-number part is 5. The coefficients of the original equation are: —8 +5 +14 —15 +4 5 —8 1 5 —5 3 +5 —1 +10 —1 +45 ,-13 +2 +5 +7 +5 +9 +35 ,+44 ,+44 1 ,+12; and 13^-44 :^= .2 ; .2 is the next figure in the root, and the coefficients of the transformed equation, that is, one whose roots are less than those of the original equation by 5, are: 1, +12, -[-44, -f 44, —13. It will be observed that they run in a diagonal line from left to right ux)wards in the calculation of the transformation. Now let us get an equation whose roots are less by . 2 than those of the equation whose coefficients are: 94 PRINCIPLES OP ALGEBRA. 1 + 12 +44 +44 0.2 + 2.44 + 9.288 -13 + 10.6576 + 12.2 +46.44 +53.283, .2 + 2.48 + 9.781 ,- 2.3424 + 12.4 +48.92, + 63.072 .2 + 2.52 + 12.6, + 51. 44 .2 1 + 12.8 2.3424-^63.272 gives .03; hence 3 is is the next figure of the root, and transforming: 1 + 12.8 +51.44 +63.072 -2.3424 | .03 .03 + .3849 + 1.554747 +1.93880241 12.83 +51.8249 +64.626747 ,-.40359759 .03 + .3858 + 1.566321 12.86 +52.2107, + 66.193068 .03 + .3867 12.89, +52.5974 .03 1 + 12.92 .40359759-f-66.193068 gives .006, and 6 is the next figure of the root. Again transforming: 1+12.92 +52.5974 +66.193068 - .40359759 | .006 .006 + .077556 + .316049736 +.399054706416 12.926 +52.674956 +66.509117736,— .004542883584 .006 + .077592 + .316515288 12.926 +52.752548 +66.825633024 .006 + .077628 12.932,+52-830176 .006 1 + 12.938 and .004542883584--66.825633 gives .00006; hence 06 are the next two figures of the root. Again transforming: 1 + 12.944 +52.830176 +66.825633 -^.004542883656 L'^^l- .00006+ .0007766436 + .003169857158616 +.00390972817142951696 12.94406+52.8309526436 +66.828802857158616 ,—.00063315548457048304 .00006+ .0007766472 + .0031699037584992 12. 94412+52.83172930832,+66. 83197 2760917152 APPROXIMATE SOLUTIONS. 95 As we will not carry the approximation nearer than to G figures, it will not be necessary to go further than has been done, since we have now the means of getting the sixth figure, which is done by dividing .000533115M845704304 by 66. 8319727G09171152, and we get a number between 7 and 8, but being nearer to the latter, we put 8 as the sixth figure of the root, and we have 5.236068. In the same manner we can find the other roots. Art. 103. But as the number of decimal places be- comes inconveniently large, especially when a considerable number of decimal places are desired in the root itself, we must attempt some measure of relief. This may be had by simply using no more places of decimals than are neces- sary in each stage of the operations. Having decided on the number of decimal places that shall be in the root, we will remember that that number, or one or two more, will be sufficient to have in the divi- dends. Also that the number in the dividend minus the number of the place of the required figure of the root at any stage, will give the number of places that ought to be used in the divisor. Thus, if there are to be 6 places of decimals in the approximate root and we are multiplying by the third figure, if the otheiv factor, which is the divi- sor, has 3 places, the product will contain 6 places and give the dividend to the necessary extent. One or two places more may well be preserved, and all the others to the right dropped; but in multiplications of such reduced numbers we must at the first product on the right hand in every case, add on the figure which would have been '* car- ried " there had no figures been dropped. If the number of places of decimals can be thus curtailed in the divisor, and since that divisor is itself a product in which the last figure of the root is a factor, the coefficient preceding may be cut down to a still smaller number of places. Each coefficient, as we proceed from right to left, may have one figure more dropped than was done in the case of its immediate predecessor. In this way the coeffi- 96 PRINCIPLES OF ALGEBRA. cients in the left hand columns will soon and successively become constant, because all decimals would have to be rejected, until finally there may be left only the absolute term and also the penultimate coefficient, wbich latter sim- ply loses one figure from the right every time a new figure in the root is found. Art. 104. This matter may be illustrated by finding again the root 5.236068: 1 —8 + 5 4-14 —15 +4 —5 —8 i 5.236068 —5 —3 4-5 — 1 4-10 -1 4-45 —13* 10.6576 + 2 + 9 4-35 ,4-44* 9.288 —2.3424* 1.9388024 +7 +5 ,-f44* 2.44 53.288 9.784 — .4035975* .3990549 1*4-12* 0.2 46.44 2.48 63.072* 1.554747 —.0045426* .0040095 12.2 0.2 48.92 2.52 64.626747 1.566321 .0005331 12.4 .2 51.44* .3849 66.19306* • .31608 12.6 .2 51.8249 .3858 66.50915 .31656 1*4-12.8* .03 52.2107 .3867 66.826 12.83 .03 52.5974* .08 12.86 .03 12.89 .03 1 4-12.92* .006 52.68 .08 52.76 - 12.926 APPROXIMATE SOLUTIONS. 97 The places in which decimal figures have been dropped off, and partial amends made by increasing the last figure, will be perceived upon inspection . Next let the root of which the first figure is the whole numder 2 be found. 1—8 -{-U -1-4 — 8 I 2.7320508 4-2 —12 4 16 V —6 2 2 —8 8 —12 + 8 — 7.4599 —4 2 —6 . —4 — 4 — 6.657 .5401 —.50511759 —2 . 2 —10 .49 —10.657 — 5.971 .03498241 —.03411504 0.7 -9.51 .98 —16.628 — .209253 .00086737 .00085356 0.7 .7 —8.53 1.47 —16.837253 — .206679 .00001381 .0001366 1.4 _7.06 —17.04393 .0000015 .7 — .0849 — .01359 2.1 —6.9751 —17.0575 .7 .0858 — .0135 2.8 —6.8893 —17.0711 .03 .0867 2.83 —6.802 .03 .008 2.86 —6.794 .03 2.89 .03 2.92 98 PRINCIPLES OF ALGEBRA. The quotient of 8 divided by — 4 gives — 2, which is much too small, as may be found by trial; it must be in- creased, and it is found that 2.7 and 2.8, when substituted for X in the first member, give different signs, hence there is a root between them, and we take .7 for the second fig- ure of the root. We afterwards proceed as usual. .7 is a quantity too great to have its second and higher powers dropped as inappreciable. The root of which .7 is the first figure may be found thus: .7 -1-14 -{-4 —8 I .763932 —5.11 6.223 7.1561 —7.3 .7 8.89 —4.62 10.223 2.989 — .«439* .79211376 —6.6 .7 4.27 —4.13 13.212* — .010104 —.05178624* .03951341 —5.9 .7 .14* —.3084 13.201896 — .028392 —.01227283* .1184220 —5.2 .06 —.1684 —.3048 13.173504* — .002368 —.0043063* .000039472 —5.14 —.4732 13.171136 .06 —.3012 — 2412 .0042668* -5.08 •06 .7744* 13.15872 .0149 — 72 —5.02 —.789 13.158 .06 — 15 — 07 1_4.96* —.804 . 13.15,7,3 To obtain the fourth root, which is negative, we must transform the equation into another whose negative roots APPROXIMATE SOLUTIONS. 99 correspond to the positive roots of tliis, and conversely, by changing the alternate signs, and we have as follows: 1_|_8 -^14 —4 —8 I .7320508 0.7 6.09 14.063 7.0441 8.7 .7 20.09 6.58 10.063 18.669 — .9559, -- .89261841 9.4 .7 26.67 7.07 28.732, 1.021947 — .06328159, .06171029 10.1 .7 33.74, .3249 29.753947 1.031721 — .00157130 154632 10.8. .03 34.0649 .3258 30.785668, .069478 — .00002498 2473 10.83 .03 34.3907 .3267 30.85514,6, .06952 25 10.86 .03 34 7174, .0218 30.92467, 173 10.89 .03 34.739,2 .022 30.926,40 2 10.92 34.75,1 30.9,2,8 The algebraic sum of these roots is equal to 8, the coefficient of the second term with its sign changed, as should be the case: 5.236068 2.732050 8 .763932 8.732050 8 —.732050 8 EXAMPLES. 1. Find one root of ar' — 2x— 5 = . 2. Of x'+lOx'— 24^— 24:0 = 0. 3. Of ar'-{-2x*-\-d.j(^-]-3x'-\-5x— 6^321 8. Ans. 2.0945515. Ans. 4.898979. 0. Ans. 8.414455. 100 principles of algebra. Newton's method. Art. 105. In this mode of ai^proximating to the roots of numerical equations, it will be assumed that all the roots which are equal or which are whole numbers, have been found and divided out. Then having in some way, by Sturm's Theorem, by chance or otherwise, found a a number, or numbers, which differ but slightly from the roots sought, let this approximation, plus or minus a new unknown quantity be substituted for the original unknown quantity in the first member of the equation, and let all the indicated operations be performed. The new unknown quantity represents the difference between the approximate root which is being tried and the true root, and of course should be so small a fraction that all terms involving its powers higher than the first may be dro2:)ped as inapi3recia- ble, or at all events, producing no serious error. Then from this equation of the first degree, find the value of this difference, and add it to the trial root when that is too small, or subtract it when the trial root requires to be diminished . We have now an approximate root once cor- rected. But if this be not close enough to the truth, let it he used as a trial root, precisely as before, and the value of a second correction obtained. And this corrected root may be used for a third correc- tion, and so on to any desired extent. Let us take as an example o(^-\-Qx'^x — 10 = 0; in which it has been found by trial that 1.1 is an apj^roxi- mate root, being somewhat too small . Let u = the differ- ence between 1.1 and the true root; we will then have x=^ tt-f-1.1. For a moment let 1.1 be represented by r . • . x = r^u) x'=r'-ir2rit-^u'; x'^7''-i-dr'u-\-^rit'^u'; and we have : a^ = u^-^Su^r-\-3ur^-\- r^ -f 6it?' =- 6it' +12ur-[-6?-' -\-x = u-\-r —10 = —10 0; APPEOXIMATE SOLUTIONS. 101 and dropping the terms involving u^ and u^, we have: {Sr' + 12r -f- 1) ii -[- 7^ -]- Gr' -[- r — 10 r^ 0, whence u = 10 _ r — Gi'' — r^ — Q 2 _i ~fo "ITi — ^^^ restoring 1.1 in place of r we have 10 — 1.1 — 7.26 — 1.331 ^^rr^^r.^ u =- 17^3 ^ .0173303. .ajid.x = r-{- u = 1.1173303. Now if we substitute this value of the root for r in the formula above we would get a second val- ue of 1^ or a second correction. But taking 1.1173 as being one root and dividing it out we get x' + 7.1173j; + 8.95215929 = 0; and solving this x = — 3.55865 ±: j/ir"8;952r592~9 -f (— 3.558657T\ the other two roots are — 5 . 48526 — 1 . 63204 to which adding the first + 1 . 11730 — 6 . 00000 which is the coefficient of the 2d term with its sign changed as it ought to be. The product of these roots which ought to be 10 is 10.002252 -]-.... showing a good approximation. EXAMPLE 2. Find a root of x^ -\- x'' -{- .x — 100 = 0. An approximate root is 4.2, The first correction by Newton's Method gives 4 .265 . . which is a little too large and a second correction gives 4 . 264430 . The same root found by Horner's Method is 4.2644299.. Fourier's conditions. Art. 106. It may happen that after several corrections have been made by "Newton's Method it will be found that we had approached the true value of the root for a while and then receded from it; this arises from over correction. The value of u, the correction, was obtained approximately, as will be remembered, by dropping its higher powers, and 102 PRINCIPLES OF ALGEBRA. it may have thus happened that the value of u in some case was too large and when added to the approximate root has carried the value beyond the truth; and if this be again corrected in the same sense the new result will be still fur- ther off; and would be an approach to some other root. It is necessary then to know, not only that there is a root between certain limits, but also that there is no other root within those same limits. In fact it has been demonstrated by Fourier that 1 . The limits between which the required root exists must be so narrow as to contain no other root of the given equation ; nor yet of the other two equations obtained by putting the first and second derived polynomials equal to zero. 2. That the approximation must start /rom that value which makes the fir st member and its second derived polynomial have the same sign. But it is not deemed necessary to give the demonstration of these principles because their application takes away from that simplicity and expeditiousness which are characteristic of the method of Newton. By that method, as it is given above and as it was left by its immortal author, good, prac- tical results can always be obtained. If at any time it is suspected that the approximate root is departing from the true value instead of approaching it, the matter may be determined at once by substituting in the proposed equa- tion. If the student is desirous of taking more trouble than is involved in the simple application of Newton's Method it would probably be better, at once to apply Horner's Method. The demonstration of Fourier's Conditions may be found in Hackley's Algebra, Todhunter's Theory of Equations and elsewhere. TRIOONOMETKICAL SOLUTION. 103 CHAPTER IX. Trigonometrical Solution of Equations of the Third Degree. Art. 107. The Irreducible Case of these equations may be solved by calling in the aid of Trigonometry. From that branch of mathematics we know that, calling w any arc, cos2i^ = 2cos'^ti — 1, (1) Now, cos3ft = cos(i6-l-2 J-, and p is negative. Let X = rcosii, wherein r is a constant yet to be de- termined. Then cosu = , and substituting in (3) we ^ dx 1 have -^ — -r- -7Cos3w, = 0, whence a^ ~x — - cosdu = 0, (5) 104 PRINCIPLES OF ALGEBRA. By comparing (4) and (5), we see that — = ~p and r = 2^ -7.- . Also that - ^/^^• ^ - iiuu /■ = ^^1 -—^ . Also inat — J — =r — q- -r X rcos3i^ =: —q, or —^Xrco?,'^iL = —q: . ■ . cos3i6= — = 4: o pr / p / j>3. Now, since q and ;? are known and p is negative, we know the numerical value of the cosine of du and having found Sit from the Tables of Natural Sines and Cosines, we know it and can find its cosine . This be- ing multiplied by r gives the value of x, which is a root of the proposed equation. The numerical value of ?• comes of course from r =^ 2^ / P. \ 3" Thus we have one of the real roots; but the value 2 / — p^ = not only cos3it, but also cos(360° — - 3w) and ■■^ 27 cos(360^ -f 3u), consequently the cosines of the thirds of these latter two arcs will be the remaining roots, after hav- ing been multiplied by r.' Art. 108. Should it happen that 3?^ = 180' or any mul- tiple thereof, cos(360°— 3i0 and cos(360°-f 3^^) would be equal, and the roots corresponding would be equal; but they might have been discovered and divided out in the first instance, when no resort to Cardan's Formula would have been necessary. UNIVERSITY THIS BOOK IS DUU ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY i WILL INCREASE TO SO CENTS ON THE FOURTH ' DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. 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