Polynomial remainder theorem

In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states that, for every number ${\displaystyle r,}$ any polynomial ${\displaystyle f(x)}$ is the sum of ${\displaystyle f(r)}$ and the product by ${\displaystyle x-r}$ of a polynomial in ${\displaystyle x}$ of degree less than the degree of ${\displaystyle f.}$ In particular, ${\displaystyle f(r)}$ is the remainder of the Euclidean division of ${\displaystyle f(x)}$ by ${\displaystyle x-r,}$ and ${\displaystyle x-r}$ is a divisor of ${\displaystyle f(x)}$ if and only if ${\displaystyle f(r)=0,}$ a property known as the factor theorem.