Lesson plan

Factor theorem

In algebra, the factor theorem connects polynomial factors with polynomial roots. Specifically, if f(x) is a (univariate) polynomial, then x - a is a factor of f(x) if and only if f (a) = 0 (that is, 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 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,\ldots,X_n) and g(X_2, \ldots,X_n) are multivariate polynomials and g is independent of X_1 , then X_1 - g(X_2, \ldots,X_n) is a factor of f(X_1,\ldots,X_n) if and only if f(g(X_2, \ldots,X_n),X_2, \ldots,X_n) is the zero polynomial.

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Article summary

In algebra, the factor theorem connects polynomial factors with polynomial roots. Specifically, if f(x) is a (univariate) polynomial, then x - a is a factor of f(x) if and only if f (a) = 0 (that is, 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 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,\ldots,X_n) and g(X_2, \ldots,X_n) are multivariate polynomials and g is independent of X_1 , then X_1 - g(X_2, \ldots,X_n) is a factor of f(X_1,\ldots,X_n) if and only if f(g(X_2, \ldots,X_n),X_2, \ldots,X_n) is the zero polynomial.

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