Factoring in Math: Definition, Methods & Examples

Definitionnoun

What is factoring in math?

Factoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x² + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.

In Spanish: factorización
Also called: factorization, factorising, factoring an expression
Grade level: Algebra 1

Grade 6Grade 7Grade 8Algebra 1Algebra 2CCSS 6.NS.B.4CCSS 7.EE.A.1CCSS HSA-SSE.B.3.aGEDSATACT

Factoring in a picture

Multiplying the side lengths gives the area; factoring starts from the area and finds the side lengths. The two blue strips, 3x and 2x, make the middle term 5x.

Formal definition and notation

To factor a number n is to write it as a product of whole numbers, n = a × b; prime factorization continues until every factor is prime. To factor a polynomial is to write it as a product of polynomials of lower degree, such as x² − 9 = (x − 3)(x + 3). A polynomial is factored completely when no factor (other than a constant) can be factored further over the integers. The factored form is equivalent to the original: both give the same value for every x.

Symbol	Read it as	Meaning
3(2x + 3)	three times the quantity two x plus three	6x + 9 with its greatest common factor, 3, factored out
(x + 2)(x + 3)	x plus two, times x plus three	a product of two binomials; two sets of parentheses side by side mean multiply
2² × 3	two squared times three	the prime factorization of 12

Key facts

Factoring is multiplication run backward: (x + 2)(x + 3) expands to x² + 5x + 6, so x² + 5x + 6 factors to (x + 2)(x + 3).
Start with the greatest common factor: 4x² + 10x = 2x(2x + 5).
Factoring solves equations: if (x + 2)(x + 3) = 0, one factor must be 0, so x = −2 or x = −3.
A sum of squares such as x² + 9 does not factor over the integers (or the real numbers).
Factoring numbers and factoring expressions are the same idea: both write a quantity as a product.

Factoring patterns

ab + ac = a(b + c)

ab + ac = a(b + c): common factor: pull out the GCF of every term
x² + bx + c = (x + m)(x + n): trinomial: find m and n with m × n = c and m + n = b
a² − b² = (a − b)(a + b): difference of squares
a² + 2ab + b² = (a + b)²: perfect square trinomial

Always look for a common factor first. Then check the number of terms: two terms may be a difference of squares; three terms may be a trinomial. Finish by multiplying back to check.

Try it: factor a number

Interactive

Factoring works for numbers too. Type a whole number or tap a sample to see all its factors and a factor tree that rewrites it as a product of primes.

This interactive needs JavaScript. The worked examples below show the same ideas step by step.

Factoring examples, step by step

Example 1: Factor out the greatest common factor

Warm-up

Factor 6x + 15.

Find the GCF of the coefficients 6 and 15: it is 3.
Divide each term by 3: 6x ÷ 3 = 2x and 15 ÷ 3 = 5.
Write the GCF outside the parentheses: 3(2x + 5).
Check by distributing: 3 × 2x + 3 × 5 = 6x + 15.

Answer: 6x + 15 = 3(2x + 5)

Example 2: Factor a trinomial

Core

Factor x² + 7x + 12.

Look for two numbers that multiply to 12 and add to 7.
Factor pairs of 12: 1 and 12 (sum 13), 2 and 6 (sum 8), 3 and 4 (sum 7).
Use 3 and 4: (x + 3)(x + 4).
Check: x² + 4x + 3x + 12 = x² + 7x + 12.

Answer: x² + 7x + 12 = (x + 3)(x + 4)

Example 3: Factor completely

Test-style

Factor completely: 2x² − 18. Then find the values of x that make the expression equal 0.

Take out the common factor 2: 2(x² − 9).
x² − 9 is a difference of squares, x² − 3², so it factors as (x − 3)(x + 3).
Completely factored: 2(x − 3)(x + 3).
The product is 0 when a factor is 0: x − 3 = 0 or x + 3 = 0, so x = 3 or x = −3.

Answer: 2x² − 18 = 2(x − 3)(x + 3); it equals 0 when x = 3 or x = −3.

Common mistakes

✗ Mistake6x + 9 = 3(2x + 9)

✓ Instead6x + 9 = 3(2x + 3), because 9 ÷ 3 = 3.

Factoring changes the form, not the value. Multiply back to check: 3(2x + 9) = 6x + 27, which is not the original.

✗ Mistakex² − x − 12 = (x − 4)(x − 3)

✓ Insteadx² − x − 12 = (x − 4)(x + 3)

The constant is −12, so the two numbers must have opposite signs: −4 × 3 = −12 and −4 + 3 = −1. Two negatives multiply to +12.

✗ Mistake2x² − 18 = 2(x² − 9), done.

✓ InsteadKeep going: 2(x² − 9) = 2(x − 3)(x + 3).

Factor completely means no factor can be broken down further. After taking out the GCF, check what is left for a pattern such as a difference of squares.

✗ Mistakex² + 9 = (x + 3)(x + 3)

✓ Instead(x + 3)(x + 3) = x² + 6x + 9. The sum x² + 9 does not factor over the integers.

Only a difference of squares factors into (a − b)(a + b). Multiplying back reveals the missing middle term.

Not to be confused with

Factoringvs.Factor

A factor is one piece of a product, like 3 or (x + 2). Factoring is the process that finds those pieces: 3x + 6 = 3(x + 2).

Factoringvs.Distributive property (expanding)

Expanding goes from product to sum: 4(x + 5) = 4x + 20. Factoring goes from sum to product: 4x + 20 = 4(x + 5).

Factoringvs.Prime factorization

Prime factorization is factoring a whole number until every factor is prime: 60 = 2² × 3 × 5. Factoring in general can stop earlier, as in 60 = 6 × 10.

Quick check: 3 questions

Question 1 of 3

Factor completely: 12x + 8

Show answer and explanation

Answer: C. The GCF of 12 and 8 is 4, and 4(3x + 2) = 12x + 8. 2(6x + 4) is equal but not complete, since 6x + 4 still has a factor of 2. The other two divide only one term.

Question 2 of 3

Which is the factored form of x² − 5x + 6?

Show answer and explanation

Answer: D. You need two numbers with product 6 and sum −5: −2 and −3. Choice A gives +5x, choice B gives −7x, and choice C gives −5x but a constant of −6.

Question 3 of 3

Which expression equals x² − 49?

Show answer and explanation

Answer: B. x² − 49 is a difference of squares, so it factors as (x − 7)(x + 7). Squaring a binomial adds a middle term: (x − 7)² = x² − 14x + 49. Choice D expands to x² − 48x − 49.

Factoring: frequently asked questions

What is factoring used for?

Factoring turns a sum into a product, and products are easier to work with. It solves quadratic equations: from (x − 4)(x + 1) = 0 you read off x = 4 or x = −1. It also simplifies fractions, such as 3x + 63 = x + 2, and with numbers it helps you find the GCF and LCM.

How can I check my factoring?

Multiply the factors back together. If you get the original expression, the factoring is right. For x² − 2x − 15 = (x − 5)(x + 3), expanding gives x² + 3x − 5x − 15 = x² − 2x − 15. A quick spot check also works: put x = 1 into both forms and compare the values.

Can every expression be factored?

No. Some expressions are prime, meaning they have no factors other than 1 and themselves over the integers. x² + 9 and x² + x + 1 are examples. For a trinomial x² + bx + c, if no factor pair of c adds up to b, it does not factor with integers.

Practice and tools

Math practicePractice on Effortless Math Factoring CalculatorFactor polynomials step by step FOIL CalculatorMultiply binomials to check your factoring Algebra TilesArrange tiles into a rectangle to factor

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 6.NS.B.4, CCSS 7.EE.A.1, CCSS HSA-SSE.B.3.a

Cite this entry

Effortless Math Team. “Factoring.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/factoring/

Flashcard

TermFactoring6x + 9 → 3(2x + 3)

DefinitionRewriting a number or expression as a product of factors, the reverse of distributing. x² + 5x + 6 = (x + 2)(x + 3).

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Factoring

What is factoring in math?

Factoring in a picture

Formal definition and notation

Key facts

Factoring patterns

Try it: factor a number

Factoring examples, step by step

Example 1: Factor out the greatest common factor

Example 2: Factor a trinomial

Example 3: Factor completely

Common mistakes

Not to be confused with

Quick check: 3 questions

Factoring: frequently asked questions

What is factoring used for?

How can I check my factoring?

Can every expression be factored?

Related terms

Practice and tools

Sources

Flashcard

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