Little Bézout Theorem

Theorem
Let $\map {P_n} x$ be a polynomial of degree $n$ in $x$.

Let $a$ be a constant.

Then the remainder of $\map {P_n} x$ when divided by $x - a$ is equal to $\map {P_n} a$.

Proof
By the process of Polynomial Long Division, we can express $\map {P_n} x$ as:
 * $(1): \quad \map {P_n} x = \paren {x - a} \map {Q_{n - 1} } x + R$

where:


 * $\map {Q_{n - 1} } x$ is a polynomial in $x$ of degree $n - 1$


 * $R$ is a polynomial in $x$ of degree no greater than $0$; that is, a constant.

It follows that, by setting $x = a$ in $(1)$, we get $\map {P_n} a = R$.

Hence the result.

Also known as
This theorem is sometimes referred to as the Polynomial Remainder Theorem.

Some sources call it merely the Remainder Theorem, but there is more than one theorem so named.