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.


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Basic Examples  (3)

Find the remainder after dividing one polynomial by another:

The difference of the dividend and the remainder is a polynomial multiple of the divisor:

If the dividend is a multiple of the divisor, then the remainder is zero:

Find the remainder of division for polynomials with symbolic coefficients:

Coefficients of the quotient are rational functions of the input coefficients:

Scope  (4)

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

Polynomial remainder over the integers modulo :

Polynomial remainder over a finite field:

PolynomialRemainder 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)

Euclid's algorithm for the greatest common divisor:

Divide by the leading coefficient:

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:

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


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


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


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


@misc{reference.wolfram_2024_polynomialremainder, author="Wolfram Research", title="{PolynomialRemainder}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialRemainder.html}", note=[Accessed: 24-May-2024 ]}


@online{reference.wolfram_2024_polynomialremainder, organization={Wolfram Research}, title={PolynomialRemainder}, year={2023}, url={https://reference.wolfram.com/language/ref/PolynomialRemainder.html}, note=[Accessed: 24-May-2024 ]}