# PolynomialRemainder

PolynomialRemainder[p,q,x]

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

# Details and Options • The degree of the result in x is guaranteed to be smaller than the degree of q.
• Unlike PolynomialMod, PolynomialRemainder performs divisions in generating its results.
• With the option Modulus->n, the remainder is computed modulo n.

# Examples

open allclose all

## Basic Examples(1)

Find the remainder after dividing one polynomial by another:

## Scope(2)

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

PolynomialRemainder also works for rational functions:

## Options(1)

### Modulus(1)

Use a prime modulus:

## Applications(1)

Euclid's algorithm for the greatest common divisor:

## Properties & Relations(3)

For a polynomial , , where is given by PolynomialQuotient:

Use Expand to verify identity:

To get both quotient and remainder use PolynomialQuotientRemainder:

PolynomialReduce generalizes PolynomialRemainder for multivariate polynomials:

## Possible Issues(1)

The variable assumed for the polynomials matters:

Introduced in 1988
(1.0)