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  (4)

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

Polynomial quotient and remainder over the integers modulo :

Polynomial quotient and remainder over a finite field:

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:

PolynomialQuotient and PolynomialRemainder:

PolynomialReduce generalizes PolynomialQuotientRemainder for multivariate polynomials:

Wolfram Research (2007), PolynomialQuotientRemainder, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialQuotientRemainder.html (updated 2023).

Text

Wolfram Research (2007), PolynomialQuotientRemainder, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialQuotientRemainder.html (updated 2023).

CMS

Wolfram Language. 2007. "PolynomialQuotientRemainder." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/PolynomialQuotientRemainder.html.

APA

Wolfram Language. (2007). PolynomialQuotientRemainder. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolynomialQuotientRemainder.html

BibTeX

@misc{reference.wolfram_2025_polynomialquotientremainder, author="Wolfram Research", title="{PolynomialQuotientRemainder}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialQuotientRemainder.html}", note=[Accessed: 22-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_polynomialquotientremainder, organization={Wolfram Research}, title={PolynomialQuotientRemainder}, year={2023}, url={https://reference.wolfram.com/language/ref/PolynomialQuotientRemainder.html}, note=[Accessed: 22-February-2025 ]}