Polynomial remainder theorem

For a polynomial p(x), division by x − a leaves the constant remainder p(a). It connects polynomial evaluation directly to division.

Polynomial division: An algorithm that writes a polynomial as a divisor times a quotient plus a remainder. Dividing p(x) by x − a produces the quotient-and-remainder identity behind the theorem.
Polynomial root: An input at which a polynomial evaluates to zero. A candidate root a is verified by checking that the remainder p(a) is zero.
Polynomial ring: A ring whose elements are polynomials with coefficients in a specified ring. The theorem holds over coefficient rings where division by a monic linear polynomial is available.
Remainder theorem for polynomials: A broader theorem stating that division by any nonzero polynomial leaves a remainder of smaller degree than the divisor. Unlike the linear case, its remainder need not be a single value.
Polynomial evaluation: The process of finding a polynomial's value at a specified input. Substituting x = a makes the divisor term vanish and leaves the remainder.
Polynomial factorization: The expression of a polynomial as a product of lower-degree polynomials. A zero remainder identifies a linear factor that can be removed.
Division algorithm: A procedure expressing a dividend as divisor times quotient plus a smaller remainder. For a nonzero divisor x − a, the remainder must have degree less than one.
Horner's method: A nested multiplication scheme for evaluating a polynomial efficiently. It computes p(a) directly, while the theorem interprets that value as a division remainder.
Factor theorem: A polynomial has a as a root exactly when x − a is one of its factors. It follows immediately by setting the theorem's remainder p(a) equal to zero.
Rational root theorem: A theorem that restricts the possible rational roots of a polynomial with integer coefficients. Its candidates can be tested efficiently by evaluating p(a), the corresponding remainder.

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Sturm's theoremRelated: Evaluating sequence polynomials at endpoints determines the signs used in the count.