Factor

Factor[poly]

factors a polynomial over the integers.

Factor[poly,Modulusp]

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 a_i.

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 Extension->None, 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 lists:

Factor threads over equations and inequalities:

Advanced Uses (7)

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:

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 (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:

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Factor

Details and Options

Examples

Basic Examples (3)