Factor theorem

A polynomial has a factor of the form x − a exactly when its value at a is zero. It connects polynomial roots with linear factors.

Polynomial remainder theorem: Dividing a polynomial by a nonzero polynomial leaves a remainder of smaller degree; division by x − a leaves the constant f(a). Its remainder formula gives the factor theorem immediately: zero remainder means divisibility by x − a.
Synthetic division: A shortcut for dividing a polynomial by a linear expression of the form x − a. After testing a candidate with the theorem, synthetic division finds the remaining factors efficiently.
Polynomial: An expression formed from coefficients and nonnegative integer powers of a variable, combined by addition and multiplication. The theorem applies to polynomial expressions and their values.
Converse statement: A statement formed by reversing the hypothesis and conclusion of a conditional statement. The factor theorem includes both directions: a root gives a factor, and a factor gives a root.
Polynomial long division: An algorithm that divides one polynomial by another, producing a quotient and a lower-degree remainder. Dividing by x − a exposes the remainder whose value determines whether the factor exists.
Rational root theorem: A theorem restricting rational roots of integer-coefficient polynomials to quotients of factors of the leading and constant coefficients. It supplies candidates that the factor theorem can verify.
Linear polynomial: A polynomial of degree one, expressible as ax + b with a nonzero. The factor in the theorem, x − a, is a linear polynomial.
Fundamental theorem of algebra: Every nonconstant polynomial with complex coefficients has at least one complex root. It guarantees roots exist over the complex numbers, while the factor theorem tests a specified value.
Polynomial evaluation: The process of computing a polynomial's value at a specified input. The theorem reduces a factor test to evaluating the polynomial at a.
Polynomial factorization: The expression of a polynomial as a product of polynomials of lower degree. Repeated factor-theorem tests help reveal the product decomposition.

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Cubic functionRelated: It turns a known cubic root into a linear factor.