Factoring Calculator - Factor Polynomials Step by Step

Factoring Calculator

Factor polynomials (quadratic, cubic, etc.) using GCF, difference of squares, and more. Learn factoring methods with step-by-step explanations for algebra students.

100% FreeStep-by-step SolutionsMultiple Methods

Polynomial Factoring Calculator

Enter a polynomial expression to factor (e.g., x^2 + 5x + 6, 2x^2 - 8, x^3 - 3x^2 + 2x)

Polynomial Expression *

Quick Input:

Use format: x^2 + 5x + 6, 2x^3 - 8x^2 + 6x, etc. Variable can be x, y, z, etc.

Factoring Examples

Click on any example to automatically fill the calculator

AC Method

Simple quadratic factoring

Expression: x^2 + 5x + 6

Difference of Squares

Difference of squares

Expression: x^2 - 9

Perfect Square

Perfect square trinomial

Expression: x^2 + 6x + 9

GCF + Factoring

Quadratic with GCF

Expression: 2x^2 + 8x + 6

GCF Method

Cubic with GCF

Expression: 3x^3 - 12x^2 + 12x

GCF + Difference of Squares

Difference of squares with GCF

Expression: 4x^2 - 16

What is Factoring?

Factoring is the process of breaking down a polynomial into a product of simpler polynomials (factors). It's the reverse of polynomial multiplication and is fundamental in algebra.

Why Factor Polynomials?

Solve Equations: Factor to find zeros using Zero Product Property
Simplify Expressions: Cancel common factors in fractions
Graph Functions: Find x-intercepts and behavior
Advanced Topics: Partial fractions, calculus applications

Example: $x^2 - 5x + 6 = (x - 2)(x - 3)$
This means $x^2 - 5x + 6 = 0$ has solutions $x = 2$ and $x = 3$

Common Factoring Methods

1. Greatest Common Factor (GCF)

Factor out the largest common factor first.

6x² + 9x = 3x(2x + 3)

2. Difference of Squares

$a^2 - b^2 = (a + b)(a - b)$

x² - 9 = (x + 3)(x - 3)

3. Perfect Square Trinomial

$a^2 \pm 2ab + b^2 = (a \pm b)^2$

x² + 6x + 9 = (x + 3)²

4. AC Method (Grouping)

For ax² + bx + c, find factors of ac that add to b.

2x² + 7x + 3 = (2x + 1)(x + 3)

Applications of Factoring in Algebra

Solving Equations

Quadratic equations: ax² + bx + c = 0
Zero Product Property: if ab = 0, then a = 0 or b = 0
Finding roots and x-intercepts
Higher-degree polynomial equations

Simplifying Expressions

Reducing rational expressions
Canceling common factors
Simplifying complex fractions
Partial fraction decomposition

Function Analysis

Finding zeros and y-intercepts
Graphing polynomial functions
Determining domain restrictions
Optimization problems

Learn More About Factoring

OpenStax Intermediate Algebra

Free textbook chapter on GCF, grouping, and polynomial factoring techniques

Khan Academy

Video lessons on factoring quadratics and polynomials

Purplemath

Comprehensive guide to factoring techniques with worked examples

Frequently Asked Questions

How do you factor a polynomial?

1) Factor out the Greatest Common Factor (GCF) first, 2) Count the terms — binomials may be a difference of squares, trinomials may factor as (x+p)(x+q), 3) For four or more terms, try grouping, 4) Check whether any factor can be factored further.

What is the difference of squares formula?

Difference of squares: a² − b² = (a + b)(a − b). Examples: x² − 9 = (x + 3)(x − 3); 4x² − 25 = (2x + 5)(2x − 5). Both terms must be perfect squares separated by subtraction.

How do you factor a quadratic ax² + bx + c?

Find two numbers that multiply to ac and add to b. Split the middle term and factor by grouping. Example: 2x² + 7x + 3 → numbers 1 and 6 → 2x² + 6x + x + 3 → (2x + 1)(x + 3).

What is factoring by grouping?

For four or more terms, group terms in pairs, factor the GCF from each pair, then factor out the common binomial. Example: x³ + 2x² + 3x + 6 → x²(x + 2) + 3(x + 2) → (x² + 3)(x + 2).

What is sum and difference of cubes?

Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²). Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²). Mnemonic SOAP: Same sign, Opposite sign, Always Positive for the trinomial.

Quadratic Formula Calculator

Solve quadratic equations ax²+bx+c=0 using the quadratic formula with step-by-step solutions and discriminant analysis.

Greatest Common Factor Calculator

Find GCF/GCD of two or more numbers using Euclidean algorithm and prime factorization methods with step-by-step explanations.

Simplify Calculator

Simplify fractions, radicals, and mathematical expressions with step-by-step solutions.

Radical Calculator

Simplify square roots, cube roots, and nth roots with step-by-step solutions using prime factorization.