# PolynomialQuotientRemainder

PolynomialQuotientRemainder[p,q,x]

gives a list of the quotient and remainder of p and q, treated as polynomials in x.

# Details and Options • The remainder will always have a degree not greater than q.

# Examples

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## Basic Examples(2)

Find the quotient and remainder after dividing one polynomial by another:

The dividend is equal to the product of the quotient and the divisor plus the remainder:

Find the quotient and remainder for polynomials with symbolic coefficients:

## Scope(3)

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

Polynomial quotient and remainder over the integers modulo :

PolynomialQuotientRemainder also works for rational functions:

The quotient and remainder of division of by are and , where : and are uniquely determined by the condition that the degree of is less than the degree of :

## Options(1)

### Modulus(1)

Use a prime modulus:

## Applications(1)

Express the rational function as a polynomial and simple fraction:

The transformed rational function:

## Properties & Relations(2)

For a polynomial , :

Use Expand to verify identity:

PolynomialReduce generalizes PolynomialQuotientRemainder for multivariate polynomials: