Factorization — Definition, Formula & Examples

Factorization is the process of breaking a number or algebraic expression into a product of simpler pieces called factors. For example, 12 can be factored into

2 \times 2 \times 3

, and

x^2 - 9

can be factored into

(x-3)(x+3)

Given a number or polynomial expression $A$ , factorization is the process of expressing $A$ as a product $A = f_1 \cdot f_2 \cdots f_n$ where each $f_i$ is a non-trivial factor — meaning no factor is simply 1 or $A$ itself, unless $A$ is irreducible.

How It Works

Start by looking for the greatest common factor (GCF) shared by all terms and pull it out front. Next, examine what remains: if it is a trinomial like

x^2 + bx + c

, find two numbers that multiply to

c

and add to

b

. If it is a difference of two squares like

a^2 - b^2

, rewrite it as

(a-b)(a+b)

. Always multiply your factors back together to check your work.

Worked Example

Problem: Factor the expression

2x^2 + 8x + 6

completely.

Find the GCF: Every term is divisible by 2, so factor it out.

2x^2 + 8x + 6 = 2(x^2 + 4x + 3)

Factor the trinomial: Find two numbers that multiply to 3 and add to 4. Those numbers are 1 and 3.

x^2 + 4x + 3 = (x + 1)(x + 3)

Write the full factorization: Combine the GCF with the factored trinomial.

2x^2 + 8x + 6 = 2(x + 1)(x + 3)

Answer:

2(x + 1)(x + 3)

Why It Matters

Factorization is essential for solving quadratic equations, simplifying algebraic fractions, and finding roots of polynomials. In courses from Algebra 1 through Precalculus, nearly every chapter builds on your ability to factor quickly and accurately.

Common Mistakes

Mistake: Forgetting to factor out the GCF first, then struggling with large coefficients inside the remaining expression.

Correction: Always check for a GCF before trying other factoring methods. Pulling it out first makes the remaining expression much simpler to work with.