EQUATIONS, General Theory of. The theory of equations finds its origin in efforts to solve the equations which arise in the appli cations of algebra to problems in pure geometry or in applied mathematics. In the exposition of this theory a rational integral algebraic func tion of x arises which may be defined as follows: . . . au-ix + an.
It is assumed here that the exponent n is a posi tive integer and that the coefficients a., a,, a,, . . an are algebraic numbers independent of x. If this polynomial, is put equal to zero, we have an equation of the nth degree. Any value of the variable x which makes the value of the polynomial zero is said to the equation') f (x) and is called a of the equation. Thus, —1 is a root of the equa tion a" + 2=0, because (— 1)' + (-1) 2 ==. 0.
Fundamental Theorems about That at least one root of the equation f =0 always exists is a fundamental theorem which it is somewhat difficult to establish rigor ously. The proofs usually given in elementary texts lack rigor. Among the most satisfactory demonstrations are the four given by C. F. Gauss and the one based on the theory of func tions, given by A. L. Cauchy. Granted that every equation of the nth degree has at least one root, it is easy to show that it has n roots and no more. An equation of the second degree (a (quadratic equation))) has two roots, one of the third degree (a °cubic equation() has three roots, one of the fourth degree (a ((manic° or equation() has four roots, and so on. The proof of this theorem may be outlined as follows: If ri is a root of f (x) =0, then f(x) is divisible by x— r, without a remainder, so that f(x) (x—r,)fs(x), where fi(x), the quotient, is of the (n— 1)th degree. If r, is a root of fi(x) =0, then in the same way fs(x) (x — 'Of (x), and f (x) (x —ri) —r.) f„ (x). Proceeding in this manner, the degrees of the successive quotients diminish by unity at every step, until finally a binomial quotient of the first degree of the form a.(x— re) is ob tained. We then have f(x)=a6(x — (x—r5) (x — re)= 0. There are here PI binomial factors and no more, each of which, when equated to zero, yields a root. In special cases some of these roots may be equal to each other. Such roots are called or roots.
There are important relations existing be tween the roots and the coefficients of an equa tion. From the equalities (x —ri)(x —rs) e — (rt + rI)x + rtra= ; ri)(x—r.) + r,)x' + (ry, + nn +re.)x Mel= 0; (x— (X-11). . . — rn) . . . re)x( +(firs +ran • • • -Fre-tre)xe_s — (-1)nrir, rn =O.
we see that in the equation f 0, when the coefficient a, of the second term is equal to minus the sum of the roots; the co efficient as of the third term is equal to the sum of the products of the roots, taken two by two; the coefficient of the fourth term is equal to, minus the sum of the products of the roots, taken three by three; and so on, until finally we arrive at the last coefficient, an, which is equal to i — 1)" times the product of all the roots.
The coefficients of the equation are said to be symmetric functions of the roots, that is, func tions in which any two roots may be inter changed without altering the value of the func tion. As an illustration take 2x' + + 6x —5=0. To make a.-- 1, divide through by 2. Then the sum of the three roots is —2, the sum of their products, taken two by two, is 3, the product of all three roots The roots of an equation may be complex (i.e., imaginary) quantities. (See ALGEBRA) . Thus the equation x' + x + 1=-0 has the two complex roots i(-1 and i(-1—iV 3), where i If the coefficients of the equa tion f(x)) are all real, then it can be shown that, if complex roots occur at all, they occur in conjugate pairs; that is, if a + ib is a root, then a —ib is likewise a root. From this it follows at once that no cubic or other equation of odd degree and with real coefficients can have all its roots complex. Considerable infor mation on the character of the roots can usually be secured from °Descartes' Rule of Signs,° which may be stated as follows: An equation with real coefficients has as many positive roots as it has variations in sign, or fewer by an even number. A variation is said to exist whenever two successive terms have opposite signs. Thus there are two variations in + +. The theorem may be proved from the consideration that every time that a new positive root is in troduced into an equation, by multiplying f(x) by (x— r), the number of variations is in creased by an odd nutnber. Applying Descartes' Theorem to the equation xi— + +2x2— 5=0, observe that the sequence of signs is + — + + There are three variat.ons; hence, the equation has either three positive roots or one. To apply the theorem to negative roots, we first transform the given equation into a new one whose roots are the same as those of the given equation, excepting in sign. This can be done by writing —x in place of x. The above sextic then becomes xi + + x' + 2x' —5=0. This transformed equation has one variation; hence, by Descartes' Rule (q.v.),. it has one positive root, and the given equation has one negative root. As the total number of roots is six and the number of real roots is four or two, it follows that either two or four of the roots are complex. By the same reason ing we can show that xi-1 has one posi tive and four complex roots and that .r4+ xi + 1=0 has all its roots complex. In some cases, as in x' + x'—x' + 5=0, Descartes' Rule gives but little information.