# Factor Factor[poly]

factors a polynomial over the integers.

Factor[poly,Modulusp]

factors a polynomial modulo a prime p.

Factor[poly,Extension{a1,a2,}]

factors a polynomial allowing coefficients that are rational combinations of the algebraic numbers ai.

# Details and Options • Factor applies only to the top algebraic level in an expression. You may have to use Map, or apply Factor again, to reach other levels.
• Factor[poly,GaussianIntegers->True] factors allowing Gaussian integer coefficients.
• If any coefficients in poly are complex numbers, factoring is done allowing Gaussian integer coefficients.
• The exponents of variables need not be positive integers. Factor can deal with exponents that are linear combinations of symbolic expressions.
• When given a rational expression, Factor effectively first calls Together, then factors numerator and denominator.
• With the default setting , Factor[poly] will treat algebraic number coefficients in poly like independent variables.
• Factor[poly,Extension->Automatic] will extend the domain of coefficients to include any algebraic numbers that appear in poly. »
• Factor automatically threads over lists, as well as equations, inequalities and logic functions.

# Examples

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## Basic Examples(3)

Factor univariate polynomials:

Factor multivariate polynomials:

Factor polynomials over the integers modulo 2:

## Scope(13)

### Basic Uses(6)

A univariate polynomial:

A multivariate polynomial:

A rational function:

A non-polynomial expression:

Factor threads over equations and inequalities:

Factor a polynomial over the Gaussian integers:

Factor a polynomial over an algebraic extension:

Factor a polynomial over the integers modulo 3:

Factor polynomials over a finite field:

Factor a polynomial over an extension of a finite field:

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

Some non-polynomial expressions can be factored:

Factor 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(3)

When modeling behavior with polynomials, it is important to determine when the polynomial evaluates to zero. For example, suppose the cost to produce a video game system is modeled by the following expression:

Also suppose the revenue can be modeled by the equation:

If we wish to know the number of units we must sell before making a profit, we calculate the difference:

Then we solve to find where the profit function is zero using Factor:

This reveals to us there is a zero for profit at :

Find a number which is equal to its square:

Subtract from both sides of the equation:

Use Factor to find when a polynomial is zero:

The only numbers that are equal to their square are thus and :

Compute the greatest common divisor of two polynomials:

We can see they share a common factor of . Confirm this result using PolynomialGCD:

## Properties & Relations(3)

Expand is effectively the inverse of Factor:

FactorList gives a list of factors:

FactorSquareFree only pulls out multiple factors:

## Neat Examples(2)

The first factoring of where a 2 appears as a coefficient: