Polynomial Equations and the Remainder Theorem

Before relating roots to coefficients we need one basic tool: the Remainder Theorem. It says that if a polynomial $f(x)$ is divided by the linear polynomial $x-a$, the remainder is always a constant, and that constant is exactly $f(a)$:

The proof is short: $f(x)-f(a)$ is a sum of terms of the form $c\,(x^k-a^k)$, and every such term is divisible by $x-a$ because $x^k-a^k=(x-a)(x^{k-1}+x^{k-2}a+\cdots+a^{k-1})$; so $f(x)-f(a)$ is divisible by $x-a$, which rearranges to the stated identity.

In particular, $x-a$ divides $f(x)$ exactly (remainder $0$) precisely when $f(a)=0$, i.e. precisely when $a$ is a root of $f(x)=0$. This is the Factor Theorem, the special case of the Remainder Theorem that lets us confirm a candidate root and then depress (reduce the degree of) the equation. …