# 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[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)

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:

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:

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: