Factorising Expressions: KS3 Maths Guide

Factorising an expression means writing it as a product — something multiplied together — rather than a sum. It is the reverse of expanding brackets. At KS3 you factorise by taking the highest common factor (HCF) outside a single bracket.

What does factorising mean?

When you expand brackets you multiply out: 3(x + 4) = 3x + 12.

Factorising reverses this: starting from 3x + 12, you find what factor both terms share and write it outside a bracket.

3x + 12 = 3(x + 4)

The expression 3(x + 4) is the factorised form. The number (or letter) placed outside the bracket is called the factor or common factor.

Finding the highest common factor (HCF)

The HCF is the largest factor that divides into every term of the expression without leaving a remainder.

Worked example 1: numerical HCF

Find the HCF of 12 and 18.

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6

HCF = 6

Worked example 2: algebraic HCF with one variable

Find the HCF of 4x and 6x².

Numerical part: HCF of 4 and 6 = 2
Letter part: HCF of x and x² = x (the lowest power present)

HCF = 2x

How to factorise into a single bracket: step-by-step

Step 1 — Find the HCF of all the terms in the expression.
Step 2 — Write the HCF outside the bracket.
Step 3 — Divide each term of the expression by the HCF to find what goes inside the bracket.
Step 4 — Check by expanding — you should get back to the original expression.

Worked example 3: factorise 6x + 9

Step 1 — HCF of 6 and 9: factors of 6 are 1, 2, 3, 6; factors of 9 are 1, 3, 9. HCF = 3.
Step 2 — Write 3 outside the bracket: 3( ).
Step 3 — Divide each term by 3: 6x ÷ 3 = 2x; 9 ÷ 3 = 3. Inside the bracket: (2x + 3).
Step 4 — Check: 3(2x + 3) = 6x + 9 ✓

Answer: 6x + 9 = 3(2x + 3)

Worked example 4: factorise 10a − 15

Step 1 — HCF of 10 and 15 = 5.
Step 2 — 5( ).
Step 3 — 10a ÷ 5 = 2a; 15 ÷ 5 = 3. Note the minus sign carries through: (2a − 3).
Step 4 — Check: 5(2a − 3) = 10a − 15 ✓

Answer: 10a − 15 = 5(2a − 3)

Worked example 5: factorise 8x² + 12x

Step 1 — Numerical HCF of 8 and 12 = 4. Letter HCF of x² and x = x. Overall HCF = 4x.
Step 2 — 4x( ).
Step 3 — 8x² ÷ 4x = 2x; 12x ÷ 4x = 3. Inside: (2x + 3).
Step 4 — Check: 4x(2x + 3) = 8x² + 12x ✓

Answer: 8x² + 12x = 4x(2x + 3)

Worked example 6: three terms

Factorise 6p² − 9p + 15.

Step 1 — HCF of 6, 9, 15 = 3. No letter common to all three terms (the last term has no p), so HCF = 3.
Step 2 — 3( ).
Step 3 — 6p² ÷ 3 = 2p²; 9p ÷ 3 = 3p; 15 ÷ 3 = 5. Signs carry: (2p² − 3p + 5).
Step 4 — Check: 3(2p² − 3p + 5) = 6p² − 9p + 15 ✓

Answer: 6p² − 9p + 15 = 3(2p² − 3p + 5)

Worked example 7: letter HCF only

Factorise m³ + m².

Step 1 — Both terms share m² (the lower power). HCF = m².
Step 2 — m²( ).
Step 3 — m³ ÷ m² = m; m² ÷ m² = 1. Inside: (m + 1).
Step 4 — Check: m²(m + 1) = m³ + m² ✓

Answer: m³ + m² = m²(m + 1)

Checking your factorisation

Always expand the bracket back out:

Multiply the factor outside by every term inside.

Every sign (+ or −) inside must be multiplied by the factor.

The result should be identical to the original expression.

If anything differs — even one sign — go back and recheck the HCF or the division step.

Common errors at KS3

Error 1 — Not using the HCF.
Taking out a common factor that is not the highest one still gives a valid factorisation, but it is not fully factorised. For example, factorising 12x + 8 as 2(6x + 4) is partially correct, but the fully factorised form is 4(3x + 2).

Error 2 — Forgetting a term inside the bracket.
Leaving a term out of the bracket is a common slip. Count: the number of terms inside the bracket must equal the number of terms in the original expression.

Error 3 — Losing a negative sign.
When the original expression contains a minus, the inside term becomes negative: 3x − 9 = 3(x − 3), not 3(x + 3).

Error 4 — Writing 1 outside the bracket.
If the HCF is 1, the expression cannot be factorised further (with a numerical factor). Putting 1 outside a bracket does not simplify the expression.

How factorising fits the KS3 national curriculum

The Department for Education's KS3 Mathematics Programme of Study requires pupils to "use algebraic methods to solve problems" and to "manipulate algebraic expressions by expanding brackets and factorising." Factorising into a single bracket with a numerical or letter common factor is the expected KS3 skill. Factorising and expanding brackets are treated as paired skills within the statutory national curriculum for Key Stage 3 and 4, with both assessed in Year 8 and Year 9 algebra tasks.

Frequently asked questions

What is the difference between expanding and factorising?

Expanding means multiplying out brackets to write an expression as a sum of terms: 4(2x − 1) = 8x − 4. Factorising means going in reverse — writing a sum of terms as a product using brackets: 8x − 4 = 4(2x − 1). They are inverse operations.

How do I know if an expression can be factorised?

An expression can be factorised if its terms share a common factor (a number, a letter, or both). If the HCF of all the terms is 1 and no letters appear in every term, the expression has no common factor and cannot be factorised by this method.

Does factorising only work with two terms?

No. You can factorise expressions with two, three, or more terms, as long as all terms share a common factor. The HCF method at KS3 works for any number of terms.

What comes after factorising into a single bracket?

At GCSE you extend to factorising quadratics into two brackets, for example x² + 5x + 6 = (x + 2)(x + 3). The single-bracket method you practise at KS3 is the essential foundation for that more advanced skill.

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