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PolynomialMod[poly,m]

gives the polynomial poly reduced modulo m.

PolynomialMod[poly,{m1,m2,…}]

reduces modulo all of the mi.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
CoefficientDomain  
Modulus  
Applications  
Properties & Relations  
See Also
Tech Notes
Related Guides
Related Links
History
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PolynomialMod

PolynomialMod[poly,m]

gives the polynomial poly reduced modulo m.

PolynomialMod[poly,{m1,m2,…}]

reduces modulo all of the mi.

Details and Options

  • PolynomialMod[poly,m] for integer m gives a polynomial in which all coefficients are reduced modulo m.
  • When m is a polynomial, PolynomialMod[poly,m] reduces poly by subtracting polynomial multiples of m, to give a result with minimal degree and leading coefficient.
  • PolynomialMod gives results according to a definite convention; other conventions could yield results differing by multiples of m.
  • Unlike PolynomialRemainder, PolynomialMod never performs divisions in generating its results.
  • PolynomialMod takes the following options:
  • CoefficientDomain the type of objects assumed to be coefficients
    Modulus 0modulus to assume for integers
    TrigFalsewhether to do trigonometric as well as algebraic transformations

Examples

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Basic Examples  (2)

Reduce a polynomial modulo 2:

Wolfram Language code: PolynomialMod[3x ^ 2 + 2x + 1, 2]

Reduce a polynomial modulo another polynomial:

Wolfram Language code: PolynomialMod[3x ^ 2 + 2x + 1, x ^ 2 + 1]

Scope  (6)

Reduce a polynomial modulo an integer:

Wolfram Language code: PolynomialMod[3x ^ 3 + 21x ^ 2 - 7x + 55, 9]

Reduce a multivariate polynomial modulo an integer:

Wolfram Language code: PolynomialMod[35x ^ 3 + 21x ^ 2y ^ 2 - 17x y ^ 3 + 55z - 123, 19]

Reduce a polynomial modulo a polynomial:

Wolfram Language code: PolynomialMod[3x ^ 3 + 21x ^ 2 - 7x + 55, 2x ^ 2 - 7]

The difference of the original polynomial and the result is divisible by the modulus:

Wolfram Language code: Cancel[(3x ^ 3 + 21x ^ 2 - 7x + 55 - %) / (2x ^ 2 - 7)]

Reduce a polynomial modulo a polynomial with complex coefficients:

Wolfram Language code: PolynomialMod[3x ^ 3 + 21x ^ 2 - 7x + 55, x ^ 2 + I]

Reduce a polynomial modulo a polynomial and an integer:

Wolfram Language code: PolynomialMod[3x ^ 3 + 21x ^ 2 - 7x + 55, {2x ^ 2 - 7, 9}]

Reduce a polynomial modulo two polynomials and an integer:

Wolfram Language code: PolynomialMod[3x ^ 3 + 21x ^ 2y ^ 2 - 7x y ^ 3 + 55, {2x ^ 2 - 7, x y - 3, 9}]

Options  (3)

CoefficientDomain  (2)

With the default CoefficientDomain->Rationals, integer coefficients can be inverted:

Wolfram Language code: PolynomialMod[x ^ 2, 2x - 1]

With CoefficientDomain->Integers, PolynomialMod does not invert integer coefficients:

Wolfram Language code: PolynomialMod[x ^ 2, 2x - 1, CoefficientDomain -> Integers]

Modulus  (1)

Reduce a polynomial modulo a polynomial over the integers modulo 3:

Wolfram Language code: PolynomialMod[x ^ 2, 2x - 1, Modulus -> 3]

Applications  (1)

Reduce all coefficients of a polynomial modulo an integer:

Wolfram Language code: PolynomialMod[1234(x + y) ^ 7 - (x - y) ^ 5, 7]

Properties & Relations  (5)

For univariate rational polynomials, PolynomialRemainder is the same as PolynomialMod:

Wolfram Language code: PolynomialMod[2x ^ 3 + 3x ^ 2 + 4x + 5, 6x ^ 2 - 7]
Wolfram Language code: PolynomialRemainder[2x ^ 3 + 3x ^ 2 + 4x + 5, 6x ^ 2 - 7, x]

PolynomialRemainder considers all polynomials to be univariate in the specified variable:

Wolfram Language code: PolynomialRemainder[x ^ 2, x + a, x]

For multivariate polynomials, PolynomialMod picks its own variable order:

Wolfram Language code: PolynomialMod[x ^ 2, x + a]

The main variable here is a:

Wolfram Language code: PolynomialMod[a ^ 2, x + a]

PolynomialRemainder considers parameters to be invertible:

Wolfram Language code: PolynomialRemainder[x ^ 2 + a ^ 2, a x - 1, x]

PolynomialMod does not invert symbolic expressions:

Wolfram Language code: PolynomialMod[x ^ 2 + a ^ 2, a x - 1]
Wolfram Research (1991), PolynomialMod, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialMod.html (updated 2003).

Text

Wolfram Research (1991), PolynomialMod, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialMod.html (updated 2003).

CMS

Wolfram Language. 1991. "PolynomialMod." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/PolynomialMod.html.

APA

Wolfram Language. (1991). PolynomialMod. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolynomialMod.html

BibTeX

@misc{reference.wolfram_2026_polynomialmod, author="Wolfram Research", title="{PolynomialMod}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialMod.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_polynomialmod, organization={Wolfram Research}, title={PolynomialMod}, year={2003}, url={https://reference.wolfram.com/language/ref/PolynomialMod.html}, note=[Accessed: 13-June-2026]}