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Mathematica > Mathematics and Algorithms > Polynomial Algebra > Polynomial Division >

Built-in Mathematica Symbol

PolynomialRemainder

PolynomialRemainder[p, q, x]
gives the remainder from dividing p by q, treated as polynomials in x.

Unlike PolynomialMod, PolynomialRemainder performs divisions in generating its results.

Find the remainder after dividing one polynomial by another:

The resulting polynomial will have coefficients that are rational expressions of input coefficients:

PolynomialRemainder also works for rational functions:

Use a prime modulus:

Euclid's algorithm for the greatest common divisor:

Divide by the leading coefficient:

For a polynomial f, f=qg+r, where q is given by PolynomialQuotient:

Use Expand to verify identity:

To get both quotient and remainder use PolynomialQuotientRemainder:

PolynomialReduce generalizes PolynomialRemainder for multivariate polynomials:

The variable assumed for the polynomials matters: