CHAPTER XLVIII.
Theory of Equations.
559. General Form of an Equation of the nth Degree. Let p_0 x^{n} + p_1 x^{n-1} + p_2 x^{n-2} + \ldots + p_{n-1}x + p_n, be a rational integral function of # of n dimensions, and let us denote it by f(x); then f(x)= 0 is the general type of a rational integral equation of the nth degree. Dividing throughout by p_0, we see that without any loss of generality we may take
x^{n} + p_1 x^{n-1} + p_2 x^{n-2} + \ldots + p_{n-1}x + p_n = 0
as the general form of a rational integral equation of any degree.
Unless otherwise stated the coefficients p_1, p_2, \ldots p_n, will always be supposed rational.
If any of the coefficients p_1, p_2, p_3, \ldots p_n, are zero, the equation is said to be incomplete, otherwise it is called complete.
560. Any value of x which makes f(x) vanish is called a root of the equation f(x) = 0.
561. We shall assume that every equation of the form f(x)=0 has a root, real or imaginary. The proof of this proposition will be found in treatises on the Theory of Equations; it is beyond the range of the present work.
562. Divisibility of Equations. If a is a root of the equation f(x)=0, then is f(x) exactly divisible by x-a.
Divide the first member by x-a until the remainder no longer contains a Denote the quotient by Q, and the remainder, if there be one, by R. Then we have
f(x) = Q(x-a) + R=0.
