# Remainder theorem

The remainder theorem states that if a polynomial $p(x)$ is divided by $(x-a)$, then the remainder is a constant given by $p(a)$.

The factor theorem follows from this immediately: $(x-a)$ is a divisor of $p(x)$ if and only if $P(a)=0$.

We prove the remainder theorem by writing $p(x)$ as $p(x)=q(x)(x-a)+r,$ where $q(x)$ is some polynomial quotient and $r$ is a (constant) remainder. Setting $x=a$ in this formula, we find $p(a)=r$.