Factorising

Factorising is the reverse of expanding brackets and is a key skill in GCSE Maths used to simplify expressions and solve equations. Factorising means writing an expression as a product of factors. When taking out a common factor, write it outside the brackets and divide each original term by it to find the terms inside.

In this lesson, we will learn how to factorise expressions by taking out common factors step by step.

What does factorising mean?

Factorising involves finding a common factor in each term and placing it outside a bracket.

Examples of factorised expressions:

3(x+4)3(x + 4)

5(a−2)5(a − 2)

x(x+3)x(x + 3)

When expanded, these give back the original expressions.

Example Question:

We will work through this step by step:

Factorise:

6x+96x + 9

Exam Tip

Always look for the greatest common factor (GCF) — the largest number and variable that divides into every term.

Factorising fully is important for gaining full marks.

Method — Factorising by Taking Out a Common Factor

To factorise:

  1. Identify the highest common factor (HCF) of all terms
  2. Divide each term by that factor
  3. Write the common factor outside the brackets and the divided terms inside

Worked Example

Factorise:

6x+96x + 9

Step 1: Find the common factor

Both terms are divisible by 33

Step 2: Divide each term by 33

6x÷3=2x6x ÷ 3 = 2x

9÷3=39 ÷ 3 = 3

Step 3: Write the factorised form

3(2x+3)3(2x + 3)

Worked Example 2 (With Variables)

Factorise:

4x+124x + 12

Step 1: Common factor is 44

Step 2: Divide each term

4x÷4=x4x ÷ 4 = x

12÷4=312 ÷ 4 = 3

Step 3: Write result

4(x+3)4(x + 3)

Worked Example 3 (Including Variable Factor)

Factorise:

5x2+10x5x² + 10x

Step 1: Common factors are 55 and xx

Step 2: Divide each term by 5x5x

5x2÷5x=x5x² ÷ 5x = x

10x÷5x=210x ÷ 5x = 2

Step 3: Write result

5x(x+2)5x(x + 2)

Worked Example 4 (With a Negative Term)

Factorise:

8x−128x − 12

Step 1: Common factor is 44

Step 2: Divide each term

8x÷4=2x8x ÷ 4 = 2x

−12÷4=−3−12 ÷ 4 = −3

Step 3: Write result

4(2x−3)4(2x − 3)

Final answer:

The factorised expression is written as a product of factors and brackets.

Check your solution (important habit)

Expand your answer to ensure it matches the original expression.

Example:

4(2x−3)→8x−124(2x − 3) → 8x − 12 ✔️

Think Like a Mathematician

Factorising is the opposite of expanding.

Being able to move between these forms is essential for solving algebra problems efficiently.

Common mistakes to avoid

  • Not taking out the greatest common factor
  • Forgetting to divide every term
  • Sign errors inside brackets
  • Leaving the expression partially factorised

Always check by expanding.

Try one yourself

Factorise:

9x+69x + 6

Think carefully before checking.

The highest common factor (HCF) of 9 and 6 is 3.

Divide each term by 3: 9x ÷ 3 = 3x and 6 ÷ 3 = 2.

Write 3 outside the bracket and the quotients inside.

Answer: 3(3x + 2).

Check by expanding: 3 × 3x + 3 × 2 = 9x + 6.

Exam Style Question

Factorise:

12x2+18x12x² + 18x

The coefficients 12 and 18 have HCF 6, and both terms contain x. The HCF of the terms is 6x.

Divide the terms by the common factor: 12x² ÷ 6x = 2x and 18x ÷ 6x = 3.

Write the common factor outside the bracket.

Answer: 6x(2x + 3).

Check by expanding: 6x × 2x + 6x × 3 = 12x² + 18x.

Frequently Asked Questions

Why is factorising important?

Factorising makes expressions easier to solve, simplify, and analyse. It is especially important when solving quadratic equations and simplifying algebraic fractions.

What does factorising mean in algebra?

Factorising means writing an expression as a product of factors by taking out common terms.

Next Lesson

Continue → Solving Linear Equations

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