Factoring Calculator

About This Tool

This factor calculator helps you factor algebraic expressions and polynomials into simpler expressions whose product equals the original input. It can be used for expressions with a greatest common factor, trinomials, differences of squares, grouping patterns, and sums or differences of cubes.

The tool is designed for algebraic factorization. If you only need the positive and negative integer factors of a whole number, use a separate Factors Calculator instead.

How to Use the Factoring Calculator

1. Enter the algebraic expression.
2. Use ^ to enter exponents when necessary.
3. Check that all signs and parentheses are correct.
4. Click calculate.
5. Review the completely factored form.
6. Open the steps to see which factoring method was applied.
7. Expand the answer to verify that it matches the original expression.

For example, enter x^2 + 5x + 6. The factorization calculator identifies two numbers that add to 5 and multiply to 6, then returns (x + 2)(x + 3).

What Does Factoring Mean?

Factoring reverses multiplication. It changes an expanded expression into a product of simpler factors without changing its value.

For example:

Expanded form: x² + 5x + 6

Factored form: (x + 2)(x + 3)

Multiplying the two binomials returns the original polynomial. A factor calculator can make this process faster while also showing the algebraic rules behind the result.

How to Factor an Algebraic Expression

A useful factoring order is:

1. Look for the greatest common factor.
2. Count the number of terms.
3. Check for special patterns.
4. Factor trinomials when possible.
5. Try grouping for expressions with four or more terms.
6. Repeat until every factor is completely factored.
7. Expand the result to confirm the answer.

Starting with the greatest common factor often makes the remaining expression easier to recognize.

Factoring Out the Greatest Common Factor

The greatest common factor, or GCF, is the largest numerical or algebraic factor shared by every term.

Example:

6x + 12 = 6(x + 2)

Both terms share a factor of 6. Removing it first leaves a simpler binomial inside the parentheses.

Another example:

8x³ + 12x² = 4x²(2x + 3)

The factor calculator should identify both the numerical GCF and the lowest shared power of the variable.

Factoring Trinomials

A trinomial contains three terms. For a monic quadratic such as:

x² + 5x + 6

Find two numbers that:

• Add to 5
• Multiply to 6

The numbers are 2 and 3, so:

x² + 5x + 6 = (x + 2)(x + 3)

For trinomials where the leading coefficient is not 1, additional steps such as splitting the middle term may be needed.

Factoring the Difference of Squares

The difference-of-squares pattern is:

a² − b² = (a − b)(a + b)

Example:

x² − 9 = (x − 3)(x + 3)

This rule applies only when two perfect squares are separated by subtraction. A sum such as x² + 9 cannot be factored over the real numbers using this pattern.

Factoring by Grouping

Grouping is often useful when an expression contains four terms.

Example:

3x³ + 6x² + 2x + 4

Group the terms:

(3x³ + 6x²) + (2x + 4)

Factor each group:

3x²(x + 2) + 2(x + 2)

Then factor out the shared binomial:

(x + 2)(3x² + 2)

A factorization calculator can help identify the repeated binomial when the grouping pattern is not immediately obvious.

Factoring Sums and Differences of Cubes

Use these identities:

a³ + b³ = (a + b)(a² − ab + b²)

a³ − b³ = (a − b)(a² + ab + b²)

Example:

x³ − 8 = (x − 2)(x² + 2x + 4)

The quadratic factor in this example cannot be factored further over the real numbers.

Factoring Examples

These examples show why choosing the correct method is important. The factor calculator evaluates the expression structure before presenting the factored form.

Factoring vs Finding Factors of a Number

These are related but different tasks.

• Factoring an algebraic expression: x² + 5x + 6 = (x + 2)(x + 3)
• Finding factors of 12: 1, 2, 3, 4, 6, 12

This page focuses on algebraic expressions and polynomials. Use a Factors Calculator when you need the divisors or factor pairs of an integer.

Factoring vs Expanding

Factoring changes a sum into a product, while expanding changes a product into a sum.

x² + 5x + 6 ⇄ (x + 2)(x + 3)

Expanding the factored result is one of the easiest ways to check whether the answer is correct.

Factoring vs Solving

Factoring rewrites an expression. Solving finds values that make an equation true.

Example:

• Factor: x² − 9 = (x − 3)(x + 3)
• Solve: x² − 9 = 0 gives x = 3 or x = −3

A factor calculator may help with the first step of solving a polynomial equation, but the equation must still be set equal to zero and solved.

Common Factoring Mistakes

Avoid these errors:

• Forgetting to remove the GCF first
• Using the difference-of-squares rule on a sum
• Choosing numbers that multiply correctly but do not add correctly
• Making sign errors
• Stopping before the expression is completely factored
• Cancelling terms across addition
• Confusing integer factors with polynomial factors
• Failing to expand the answer for verification

Related Calculators

• Simplify Calculator
• Factors Calculatorcoming soon
• Prime Decomposition Calculatorcoming soon
• GCF Calculatorcoming soon
• Polynomial Calculatorcoming soon
• Quadratic Formula Calculatorcoming soon
• Expand Calculatorcoming soon
• Equation Solvercoming soon

Start Factoring

Enter an algebraic expression above and use the factor calculator to review its factored form, the method applied, and the steps used to reach the answer.