Factoring in Algebra

Factoring rewrites an expression as a product of simpler factors. It is the reverse of expansion. Factoring exposes multiplicative structure that can make equations easier to solve and expressions easier to analyse.

Check for a greatest common factor first. Factoring it out can simplify the remaining expression.

On this page (8 sections)

Start with the greatest common factor

For $6x^3+9x^2$ , the greatest common numerical factor is $3$ , and the smallest common power of $x$ is $x^2$ . Factor their product from both terms.

6x^3+9x^2=3x^2(2x+3)

Each original term is recovered on expansion: $3x^2\cdot2x=6x^3$ and $3x^2\cdot3=9x^2$ . This factorization is an identity, valid even at $x=0$ ; it does not divide an equation by the possibly zero factor.

Factoring monic trinomials

For a monic trinomial $x^2+bx+c$ , seek two numbers whose sum is $b$ and product is $c$ . The reason comes from expansion:

(x+r)(x+s)=x^2+(r+s)x+rs

For $x^2+5x+6$ , the numbers $2$ and $3$ have sum $5$ and product $6$ .

x^2+5x+6=(x+2)(x+3)

The side factors determine four partial products

x

2

x

3

x^2

2x

3x

6

Total area:

x^2+2x+3x+6=(x+2)(x+3)

Diagram explained

The top side is split into x and 2; the left side into x and 3. Each region is the product of its corresponding side lengths. This is a schematic model for positive x, not a numerical scale drawing.

The area model uses positive side lengths. The algebraic identity itself holds for all real $x$ . When a constant is negative, the factors may have opposite signs: $x^2+x-6=(x-2)(x+3)$ .

When the leading coefficient is not 1

For $2x^2+7x+3$ , the product of the leading coefficient and constant is $2\cdot3=6$ . Split the middle term using numbers with product $6$ and sum $7$ : $6$ and $1$ .

Split the middle term and group

$2x^2+7x+3=2x^2+6x+x+3$
Rewrite 7x as 6x + x.
$2x(x+3)+1(x+3)$
Factor each pair of terms.
$(2x+1)(x+3)$
Factor the common binomial.

Expansion is the final check. Guessing factors can work, but showing how the middle term is produced makes the method easier to reproduce.

Difference of squares

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

The two cross terms cancel when the product is expanded.

9x^2-25=(3x)^2-5^2=(3x-5)(3x+5)

Do not apply this pattern to a sum of squares. Over the real numbers, $x^2+1$ has no real linear factors.

Factor completely

A factorization is complete only relative to the allowed coefficient system. Continue while a non-constant factor can still be factored in that system.

2x^3-8x=2x(x^2-4)=2x(x-2)(x+2)

The factor $x^2-2$ has no rational linear factors, but over the real numbers it factors as $(x-\sqrt2)(x+\sqrt2)$ . State or follow the number system in the question.

Factoring an expression is not the same as solving an equation. The product $2x(x-2)(x+2)$ is an equivalent expression; setting it equal to zero would create the equation whose roots are $-2$ , $0$ and $2$ .

Practice questions

Factor $x^{2} + 5x + 6$ .
Factor $9x^{2} – 25$ .
What should you check before trying a trinomial pattern?

Check your answers

$2+3=5$ and $2\cdot3=6$ , so $x^2+5x+6=(x+2)(x+3)$ .
$9x^2-25=(3x)^2-5^2=(3x-5)(3x+5)$ .
Check for a greatest common factor in all terms before trying another pattern.

Sources and further reading

OpenStax: Factoring trinomials. Definitions, conventions and related examples.
OpenStax Elementary Algebra 2e. Language of algebra, equations, inequalities and factoring.
OpenStax Intermediate Algebra 2e. Quadratic equations, factoring and algebraic techniques.
Wolfram Language: Algebraic Transformations. Symbolic simplification, expansion and factoring reference.