The remainder, factor, and rational roots theorems

One of the consequences of the division algorithm is that the value of a polynomial $p(x)$ at a point $a$ can be found by taking the remainder when $p(x)$ is divided by $(x-a)$:

Since $p(x)=(x-a)q(x)+r$ using the division algorithm, we get $p(a)=(a-a)q(a)+r=0+r=r$.

Carrying out this division using synthetic division turns out to be exactly the calculation done in evaluating $p(x)=p_nx^n+p_{n-1}x^{n-1} +\ldots + p_0$ as

$$p(a)=((\ldots (p_n a +p_{n-1})a+p_{n-2})a+p_{n-3})a + \ldots)a +p_0$$

a way of evaluating polynomials which turns out to be very efficient in terms of operation count.

A corollary of the remainder theorem is the factor theorem:

Theorem 2   The number $a$ is a root of the polynomial $p(x)$ if and only if $(x-a)$ divides $p(x)$ evenly.

This follows because $p(a)=r=0$ precisely when $(x-a)$ divides $p(x)$.

Another result which follows from the remainder theorem is the rational roots theorem:

Theorem 3   If $\frac{a}{b}$ is a rational root of the polynomial $p(x)=p_nx^n+\ldots+p_0$ with integer coefficients, then $a\vert p_0$ and $b\vert p_n$.

This follows because $(bx-a)$ will divide $p(x)$ only if $b$ divides $p_n$ and $a$ divides $p_0$. Notice that this theorem tells possible rational roots, not known rational roots. It gives a list of numbers to check.