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FactorList[poly]

gives a list of the factors of a polynomial, together with their exponents.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Advanced Uses  
Options  
Extension  
GaussianIntegers  
Modulus  
Trig  
Applications  
Properties & Relations  
See Also
Tech Notes
Related Guides
History
Cite this Page

FactorList

FactorList[poly]

gives a list of the factors of a polynomial, together with their exponents.

Details and Options

  • The first element of the list is always the overall numerical factor. It is {1,1} if there is no overall numerical factor.
  • FactorList takes the following options:
  • Extension Nonegenerators for the algebraic number field to be used
    GaussianIntegers Falsewhether to allow Gaussian integer coefficients
    Modulus 0modulus to assume for integers
    Trig Falsewhether to do trigonometric as well as algebraic transformations
  • FactorList[poly,Modulus->p] requires that p be prime.
  • FactorList[poly,Extension->{a1,a2,…}] allows coefficients that are arbitrary rational combinations of the ai.

Examples

open all close all

Basic Examples  (2)

List the irreducible factors of polynomials:

List the irreducible factors of multivariate polynomials:

Scope  (11)

Basic Uses  (4)

A univariate polynomial:

A multivariate polynomial:

A polynomial with multiple factors:

A rational function:

Advanced Uses  (7)

The irreducible factors over the Gaussian integers:

The irreducible factors over an algebraic extension:

The irreducible factors over the integers modulo 2:

The irreducible factors over a finite field:

The irreducible factors over an extension of a finite field:

A polynomial irreducible over factors after embedding in a larger field :

List factors of non-polynomial expressions:

List factors of a polynomial of degree :

Options  (7)

Extension  (4)

Factor over algebraic number fields:

Extension->Automatic automatically extends to a field that covers the coefficients:

Factor a polynomial with integer coefficients over a finite field:

Factor a polynomial with coefficients in a finite field:

Embedding in a larger field allows further factorization:

GaussianIntegers  (1)

Factor over Gaussian integers:

Modulus  (1)

Factor over finite fields:

Trig  (1)

Factor a trigonometric expression:

Applications  (2)

FactorList can be useful in determining the behavior of functions:

has roots at and , at it does not cross the axis, and at it crosses the axis:

Show that a polynomial is a perfect square:

Compute the factors and note that the constant factor is positive:

Extract the exponents of nonconstant factors:

Check that every factor has even exponent and thus is a square:

Properties & Relations  (3)

FactorList gives a list of irreducible factors:

This multiplies the factors together:

Factor gives a product of factors:

Expand combines all the factors back together:

FactorSquareFreeList gives a list of square-free factors:

FactorInteger gives a list of prime factors of an integer:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_factorlist, author="Wolfram Research", title="{FactorList}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/FactorList.html}", note=[Accessed: 11-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_factorlist, organization={Wolfram Research}, title={FactorList}, year={2023}, url={https://reference.wolfram.com/language/ref/FactorList.html}, note=[Accessed: 11-August-2025]}