Algebra · Lesson 8 of 16

Factorising with a Common Factor

Factorising puts brackets back. Find the highest common factor of every term, take it outside and write what is left inside.

Secondary
Updated 14 November 2026

What you will learn

Find the highest common factor of the terms.
Write the factorised form.
Check by expanding.

Before you start: you should be able to expand a bracket and find the highest common factor. See Expanding Single Brackets.

The idea

Factorising is the opposite of expanding. You start with an expression and write it as a product with brackets.

Each pair shows an expression and its factorised form. Expanding the right-hand side gives the left.

Method

Find the highest common factor (HCF) of all the terms, including any letters.
Write the HCF outside the bracket.
Divide each term by the HCF and write the answers inside.
Check by expanding.

A rectangle of area 12 with sides 3 and 4: 12 = 3 x 4. Factorising 3x + 12 = 3(x + 4) is the same idea with a letter.

Taking out letters too

In $12x^2 + 18x$ both terms have a factor $6$ and a factor $x$ . The HCF is $6x$ , so the expression is $6x(2x + 3)$ .

Worked examples

Example 1: Numbers only

Factorise $6x + 12$ .

HCF of $6$ and $12$ is $6$ .
$6x \div 6 = x$ and $12 \div 6 = 2$ .

Answer: $6(x + 2)$

Example 2: With a letter

Factorise $x^2 + 7x$ .

Both terms contain $x$ .
$x^2 \div x = x$ and $7x \div x = 7$ .

Answer: $x(x + 7)$

Example 3: Number and letter

Factorise $12x^2 + 18x$ .

HCF is $6x$ .
$12x^2 \div 6x = 2x$ and $18x \div 6x = 3$ .

Answer: $6x(2x + 3)$

Tip: always expand your answer to check it. If you do not get the original expression, something is wrong.

Why it matters

Factorising helps you simplify fractions, solve equations and later solve quadratics. Next, solving equations.

Practice

Your turn. Type or choose an answer, then press Check. Getting one wrong is fine, that is what practice is for. You can try as many times as you like.