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FactorList
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FactorList

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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[poly,Modulus->p] factors modulo a prime p.
  • FactorList[poly,GaussianIntegers->True] allows Gaussian integer coefficients.
  • FactorList[poly,Extension->{a1,a2,…}] allows coefficients that are arbitrary rational combinations of the ai.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

List the irreducible factors of polynomials:

In[1]:=1
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https://wolfram.com/xid/0j40876y-u0nh8d
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0j40876y-l73trb
Out[2]=2

List the irreducible factors of multivariate polynomials:

In[1]:=1
✖
https://wolfram.com/xid/0j40876y-fzmrax
Out[1]=1

Scope  (11)Survey of the scope of standard use cases

Basic Uses  (4)

A univariate polynomial:

In[1]:=1
✖
https://wolfram.com/xid/0j40876y-gafh68
Out[1]=1

A multivariate polynomial:

In[1]:=1
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https://wolfram.com/xid/0j40876y-zpigpr
Out[1]=1

A polynomial with multiple factors:

In[1]:=1
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https://wolfram.com/xid/0j40876y-nsumxj
Out[1]=1

A rational function:

In[1]:=1
✖
https://wolfram.com/xid/0j40876y-ldddyq
Out[1]=1

Advanced Uses  (7)

The irreducible factors over the Gaussian integers:

In[1]:=1
✖
https://wolfram.com/xid/0j40876y-qdqsij
Out[1]=1

The irreducible factors over an algebraic extension:

In[1]:=1
✖
https://wolfram.com/xid/0j40876y-g9fovb
Out[1]=1

The irreducible factors over the integers modulo 2:

In[1]:=1
✖
https://wolfram.com/xid/0j40876y-gpjf3d
Out[1]=1

The irreducible factors over a finite field:

In[1]:=1
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https://wolfram.com/xid/0j40876y-cjmue
In[2]:=2
✖
https://wolfram.com/xid/0j40876y-iw6si
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0j40876y-v1bve
Out[3]=3

The irreducible factors over an extension of a finite field:

In[1]:=1
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https://wolfram.com/xid/0j40876y-lfuqta

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

In[2]:=2
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https://wolfram.com/xid/0j40876y-d6ohpw
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0j40876y-7w11l
Out[3]=3

List factors of non-polynomial expressions:

In[1]:=1
✖
https://wolfram.com/xid/0j40876y-0nwpsf
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0j40876y-7demo
Out[2]=2

List factors of a polynomial of degree :

In[1]:=1
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https://wolfram.com/xid/0j40876y-cupbp5
In[2]:=2
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https://wolfram.com/xid/0j40876y-j0guy7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0j40876y-qygg3
Out[3]=3

Options  (7)Common values & functionality for each option

Extension  (4)

Factor over algebraic number fields:

In[1]:=1
✖
https://wolfram.com/xid/0j40876y-gfu
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0j40876y-o0w
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0j40876y-psb
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0j40876y-ffj
Out[2]=2

Factor a polynomial with integer coefficients over a finite field:

In[1]:=1
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https://wolfram.com/xid/0j40876y-dc1q2
In[2]:=2
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https://wolfram.com/xid/0j40876y-b5cq27
Out[2]=2

Factor a polynomial with coefficients in a finite field:

In[1]:=1
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https://wolfram.com/xid/0j40876y-cyuam1
In[2]:=2
✖
https://wolfram.com/xid/0j40876y-f1tsb1
Out[2]=2

Embedding in a larger field allows further factorization:

In[3]:=3
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https://wolfram.com/xid/0j40876y-gyr9d0
In[4]:=4
✖
https://wolfram.com/xid/0j40876y-ksmme
Out[4]=4

GaussianIntegers  (1)

Factor over Gaussian integers:

In[1]:=1
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https://wolfram.com/xid/0j40876y-tvg4r
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0j40876y-jadxu5
Out[2]=2

Modulus  (1)

Factor over finite fields:

In[1]:=1
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https://wolfram.com/xid/0j40876y-csebit
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0j40876y-3yy2v
Out[2]=2

Trig  (1)

Factor a trigonometric expression:

In[1]:=1
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https://wolfram.com/xid/0j40876y-4wmzy
Out[1]=1

Applications  (2)Sample problems that can be solved with this function

FactorList can be useful in determining the behavior of functions:

In[1]:=1
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https://wolfram.com/xid/0j40876y-ickor0
In[2]:=2
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https://wolfram.com/xid/0j40876y-1gdvpe
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0j40876y-b5wafg
Out[3]=3

Show that a polynomial is a perfect square:

In[1]:=1
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https://wolfram.com/xid/0j40876y-h02hzx

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

In[2]:=2
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https://wolfram.com/xid/0j40876y-jw9gwo
Out[2]=2

Extract the exponents of nonconstant factors:

In[3]:=3
✖
https://wolfram.com/xid/0j40876y-7ir07a
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0j40876y-n8g7v2
Out[4]=4

Properties & Relations  (3)Properties of the function, and connections to other functions

FactorList gives a list of irreducible factors:

In[1]:=1
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https://wolfram.com/xid/0j40876y-er0cqh
Out[1]=1

This multiplies the factors together:

In[2]:=2
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https://wolfram.com/xid/0j40876y-dk1lxk
Out[2]=2

Factor gives a product of factors:

In[3]:=3
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https://wolfram.com/xid/0j40876y-fs4vnu
Out[3]=3

Expand combines all the factors back together:

In[4]:=4
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https://wolfram.com/xid/0j40876y-cwxmk
Out[4]=4

FactorSquareFreeList gives a list of square-free factors:

In[1]:=1
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https://wolfram.com/xid/0j40876y-havj23
Out[1]=1

FactorInteger gives a list of prime factors of an integer:

In[1]:=1
✖
https://wolfram.com/xid/0j40876y-eeemm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0j40876y-05712
Out[2]=2
Wolfram Research (1988), FactorList, Wolfram Language function, https://reference.wolfram.com/language/ref/FactorList.html (updated 2023).
✖
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).

✖
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.

✖
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

✖
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: 27-March-2025 ]}

✖
@misc{reference.wolfram_2025_factorlist, author="Wolfram Research", title="{FactorList}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/FactorList.html}", note=[Accessed: 27-March-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: 27-March-2025 ]}

✖
@online{reference.wolfram_2025_factorlist, organization={Wolfram Research}, title={FactorList}, year={2023}, url={https://reference.wolfram.com/language/ref/FactorList.html}, note=[Accessed: 27-March-2025 ]}