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 \(p(a)=r\).