Factor theorem

In algebra, the factor theorem connects polynomial factors with polynomial roots. Specifically, if ${\displaystyle f(x)}$ is a polynomial, then ${\displaystyle x-a}$ is a factor of ${\displaystyle f(x)}$ if and only if ${\displaystyle f(a)=0}$ (that is, ${\displaystyle a}$ is a root of the polynomial). The theorem is a special case of the polynomial remainder theorem.

The theorem results from basic properties of addition and multiplication. It follows that the theorem holds also when the coefficients and the element ${\displaystyle a}$ belong to any commutative ring, and not just a field.

In particular, since multivariate polynomials can be viewed as univariate in one of their variables, the following generalization holds : If ${\displaystyle f(X_{1},\ldots ,X_{n})}$ and ${\displaystyle g(X_{2},\ldots ,X_{n})}$ are multivariate polynomials and ${\displaystyle g}$ is independent of ${\displaystyle X_{1}}$, then ${\displaystyle X_{1}-g(X_{2},\ldots ,X_{n})}$ is a factor of ${\displaystyle f(X_{1},\ldots ,X_{n})}$ if and only if ${\displaystyle f(g(X_{2},\ldots ,X_{n}),X_{2},\ldots ,X_{n})}$ is the zero polynomial.