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.