# PolynomialQuotient

PolynomialQuotient[p,q,x]

gives the quotient of p and q, treated as polynomials in x, with any remainder dropped.

# Details and Options

• With the option Modulus->n, the quotient is computed modulo n.

# Examples

open allclose all

## Basic Examples(3)

The quotient of two polynomials:

The degree of the remainder is less than the degree of the divisor:

The quotient of by , with the remainder dropped:

If the degree of the dividend is less than the degree of the divisor, then the quotient is zero:

## Scope(4)

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

Polynomial quotient over the integers modulo :

Polynomial quotient over a finite field:

PolynomialQuotient 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(2)

When the divisor divides the dividend , then the quotient of by satisfies :

Use PolynomialGCD to check that divides :

Verify that :

In general, the quotient of by satisfies :

The degree of the remainder is less than the degree of :

Factor a polynomial by finding one root at a time:

Take a quotient by the first factor:

Find another root and compute the quotient:

Verify the obtained factorization:

## Properties & Relations(4)

For a polynomial f, f==gq+r, where r is given by PolynomialRemainder:

Use Expand to verify identity:

To get both quotient and remainder use PolynomialQuotientRemainder:

PolynomialReduce generalizes PolynomialQuotient for multivariate polynomials:

Use PolynomialGCD to find a common divisor:

Use PolynomialQuotient to see the resulting factorization:

For rational functions common divisors are not automatically canceled:

Cancel effectively uses PolynomialQuotient to cancel common divisors:

The Cyclotomic polynomials are defined as quotients:

## Possible Issues(2)

The result depends on what is assumed to be a variable:

The result from PolynomialQuotient depends on recognizing zeros:

This is a hidden zero:

The result is as if the hidden zero was not zero:

Wolfram Research (1988), PolynomialQuotient, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialQuotient.html (updated 2023).

#### Text

Wolfram Research (1988), PolynomialQuotient, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialQuotient.html (updated 2023).

#### CMS

Wolfram Language. 1988. "PolynomialQuotient." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/PolynomialQuotient.html.

#### APA

Wolfram Language. (1988). PolynomialQuotient. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolynomialQuotient.html

#### BibTeX

@misc{reference.wolfram_2024_polynomialquotient, author="Wolfram Research", title="{PolynomialQuotient}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialQuotient.html}", note=[Accessed: 13-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_polynomialquotient, organization={Wolfram Research}, title={PolynomialQuotient}, year={2023}, url={https://reference.wolfram.com/language/ref/PolynomialQuotient.html}, note=[Accessed: 13-July-2024 ]}