Factor theorem

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In algebra, the factor theorem connects polynomial factors with polynomial roots. Specifically, if f ( x ) {\displaystyle f(x)} is a (univariate) polynomial, then x − a {\displaystyle x-a} is a factor of f ( x ) {\displaystyle f(x)} if and only if f ( a ) = 0 {\displaystyle f(a)=0} (that is, a {\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 a {\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 f ( X 1 , … , X n ) {\displaystyle f(X_{1},\ldots ,X_{n})} and g ( X 2 , … , X n ) {\displaystyle g(X_{2},\ldots ,X_{n})} are multivariate polynomials and g {\displaystyle g} is independent of X 1 {\displaystyle X_{1}} , then X 1 − g ( X 2 , … , X n ) {\displaystyle X_{1}-g(X_{2},\ldots ,X_{n})} is a factor of f ( X 1 , … , X n ) {\displaystyle f(X_{1},\ldots ,X_{n})} if and only if f ( g ( X 2 , … , X n ) , X 2 , … , X n ) {\displaystyle f(g(X_{2},\ldots ,X_{n}),X_{2},\ldots ,X_{n})} is the zero polynomial.