Form of any Equation. CHAPTER VIII. GENERAL THEORY OF EQUATIONS. SECTION I. Composition of Equations. 266. Any equation of the nth degree, with one unknown quantity, when reduced as in art. 118, may be represented by the form n-1 n-2 A x2 + B x1-1+C x*−2 + &c. + M = 0. If this equation is divided by A, and the coefficients BC Α' Α' &c., M A represented by a, b, &c., m, it is reduced to n-1 x2+ax2-1+ bxn−2+ &c. + m = 0. 267. Theorem. If any root of the equation x2+axn−1 + bxn−2+ &c. + m = 0 is denoted by x', the first member of this equation is a polynomial, divisible by x-x, without regard to the value of x. Proof. Denote x-x' by x1, that is, If this value of x is substituted in the given equation, if P z is used to denote all the terms multiplied by x[1], or Form of any Equation. by any power of [1], and Q used to denote the remaining terms, the equation becomes P+Q= 0. Now the given equation is, by hypothesis, satisfied by the value of x. which is divisible by x 1, or its equal x-x'. 268. Corollary. The preceding division is easily effected by subtracting from the polynomial хп x2+axn−1 + bxn−2+ &c. +m, the expression x'"+ax'n-1+bxn−2+ &c. + m = 0, which does not affect its value, but brings it to the form x2 — x' n+a (xn−1 — x' n−1)+b (xn− 2 — x' n − 2)+&c., of which each term is, by art. 49, divisible by x quotient is, by art. 50, xn−1 + x' Ꮖ n-2 α -x'. The n-3 269. Corollary. The first term of the preceding quotient is x-1; and if the coefficients of x-2, x13, &c., are denoted by a', b', &c., the quotient is Number of the Roots of an Equation. and the equation of art. 267, -1 -2 is (x − x') (xn−1 + a' xn−2 + b′ xn−3+ &c.) = 0; which is satisfied either by the value of x, x = x', or by the roots of the equation x2-1+ a' x2-2 + b' x2 - 3 + &c. = 0. If now x" is one of the roots of this last equation, we have in the same way xn-1+a'xn−2+ &c. = (x-x") (xn−2+a" xn−3+&c.) = 0, and the given equation becomes -3 (x — x') (x — x'') (xn−2+ a'' xn−3+ &c.) = 0; which is satisfied by the value of x", x= x"; so that x" is a root of the given equation. By proceeding in this way to find the roots "", xv, &c., the given equation may be reduced to the form (x — x') (x — x") ( x − x'") ( x − x1) &c = 0, — . - — in which the number of factors x x', x-x", &c. is the same with the degree n of the given equation; and, therefore, the number of roots of an equation is denoted by the degree of the equation; that is, an equation of the third degree has three roots, one of the fourth degree has four roots, &c. But all these roots are not necessarily real or rational; they may, on the contrary, be irrational or even imaginary. 270. Scholium. Some of the roots x', x'', x''', &c., are often equal to each other, and in this case the number of unequal roots is less than the degree of the equation. Imaginary Roots. Thus the number of unequal roots of the equation of the 9th degree, = (x-7)(x+4)3 (x — 1)5 — 0, is but three, namely, 7, —4, and 1, and yet it is to be regarded as having 9 roots, one equal to 7, three equal to -4, and five equal to 1. 271. Corollary. The equation would appear to have but one root, that is, but it must, by art. 269, have n roots, or rather, the nth root of a must have n different values, must be divisible by x-1, and we have x3 — 1 = (x — 1) (x2 + x + 1) = 0. Now the roots of the equation are x2+x+1 = 0 x={(−1+√−3), and = § (—1——3). Imaginary Roots. Hence the required roots are x=1,={(−1+√—3), and = } (—1—√—3). 3. Find the four roots of the equation must be divisible by x- 1, and we have x5 — 1 = (x — 1) (x4 +x3+x2+x+1)=0. Now the roots of the equation x2+x3+x2+x+1=0 can be found by the following peculiar process. Divide by x2, and we have |