8. The capital theorem as regards numerical equations thus is, every numerical equation has a numerical root; or for shortness (the meaning being as before), every equation has a root. Of course the theorem is the reverse of self-evident, and it requires proof; but provisionally assuming it as true, we derive from it the general theory of numerical equations. As the term root was introduced in the course of an explanation, it will be convenient to give here the formal definition. A number a such that substituted for x it makes the function x1^n - p1x^(n - 1) ... [+-]p_n to be = 0, or say such that it satisfies the equation [f](x) = 0, is said to be a root of the equation; that is, a being a root, we have a^n - p1a^(n - 1) ... [+-]p_n = 0, or say [f](a) = 0; and it is then easily shown that x - a is a factor of the function [f](x), viz. that we have [f](x) = (x - a)[f]1(x), where [f]1(x) is a function x^(n - 1) - q1x^(n - 2) ... [+-]q_(n - 1) of the order n - 1, with numerical coefficients q1, q2 ... q_(n - 1). In general a is not a root of the equation [f]1(x) = 0, but it may be so--i.e. [f]1(x) may contain the factor x - a; when this is so, [f](x) will contain the factor (x - a)^2; writing then [f](x) = (x - a)^2[f]2(x), and assuming that a is not a root of the equation [f]2(x) = 0, x = a is then said to be a double root of the equation [f](x) = 0; and similarly [f](x) may contain the factor (x - a)^3 and no higher power, and x = a is then a triple root; and so on. Supposing in general that [f](x) = (x - a)^[alpha] F(x) ([alpha] being a positive integer which may be = 1, (x - a)^[alpha] the highest power of x - a which divides [f](x), and F(x) being of course of the order n - [alpha]), then the equation F(x) = 0 will have a root b which will be different from a; x - b will be a factor, in general a simple one, but it may be a multiple one, of F(x), and [f](x) will in this case be = (x - a)^[alpha] (x - b)^[beta] [Phi](x) ([beta] a positive integer which may be = 1, (x-b)^[beta] the highest power of x - b in F(x) or [f](x), and [Phi](x) being of course of the order n - [alpha] - [beta]). The original equation [f](x) = 0 is in this case said to have [alpha] roots each = a, [beta] roots each = b; and so on for any other factors (x - c)^[gamma], &c. We have thus the _theorem_--A numerical equation of the order n has in every case n roots, viz. there exist n numbers, a, b, ... (in general all distinct, but which may arrange themselves in any sets of equal values), such that [f](x) = (x - a)(x - b)(x - c) ... identically. If the equation has equal roots, these can in general be determined, and the case is at any rate a special one which may be in the first instance excluded from consideration. It is, therefore, in general assumed that the equation [f](x) = 0 has all its roots unequal.