GCSE MATHS • HIGHER TIER
Factorising Basics Made Simple
Learn how to find the highest common factor, place it outside a bracket and check each factorised expression by expanding it again.
Clear method
Exam practice
Worked proofs
Proof in Four Moves
1
Find
Identify the highest common factor shared by every term.
2
Divide
Divide each term by that common factor.
3
Write
Place the common factor outside a bracket and the quotients inside.
4
Check
Expand the bracket to confirm that the original expression returns.
What You’ll Learn
Find the HCF
Identify the highest numerical factor shared by all terms.
Include Common Variables
Recognise when a variable or power also belongs in the common factor.
Factorise Negative Expressions
Choose a negative outside factor when it makes the terms inside the bracket clearer.
Check by Expanding
Reverse the factorisation to verify that it reproduces the original expression.
Key Representations
Reverse the Distributive Rule
ab + ac = a(b + c)
Use the Highest Common Factor
6x + 12 = 6(x + 2)
Include Shared Variables
6x² + 9x = 3x(2x + 3)
Negative Common Factor
−3x − 9 = −3(x + 3)
Worked Proof: Consecutive Numbers
1. Identify the HCF
The highest common numerical factor of 15 and 10 is 5. Both terms also contain x, so the HCF is 5x.
2. Divide Every Term
15x² ÷ 5x = 3x and 10x ÷ 5x = 2
3. Write the Factorised Form
5x(3x + 2)
4. Check by Expanding
5x × 3x + 5x × 2 = 15x² + 10x
More Worked Examples
1
Numerical Common Factor
6x + 12
2. Divide both terms by 6: x and 2.
2
Different Coefficients
10a + 15
2. Divide each term by 5: 2a and 3.
3
Negative Common Factor
−3x − 9
2. Check: −3 × x = −3x and −3 × 3 = −9.
4
Three Terms
4x + 8y + 12
2. Divide every term by 4: x, 2y and 3.
- Common GCSE Mistakes
- Not using the highest common factor
- Forgetting a shared variable
- Dividing only one term
- Sign errors with negatives
- Leaving a common factor inside
- Exam Tips
- Find the highest common numerical factor before writing the bracket.
- Check whether every term also shares a variable or power of a variable.
- Divide every term by the chosen factor and keep each sign attached.
- If the first term is negative, consider taking out a negative common factor.
- Expand your final bracket to check that the original expression returns.
Skills Used in Algebraic Proof
Ready to Practise?
Printable Worksheet
Practise numerical, negative and variable common factors using printable questions with answers
Interactive Quiz
Test how accurately you identify the highest common factor and complete each bracket.
Exam-Style Questions
Apply basic factorising skills to longer GCSE algebra questions.
Watch the Factorising Basics Tutorial
Follow a clear step-by-step explanation of finding the highest common factor, dividing every term, writing brackets and checking by expanding.
Frequently Asked Questions
Factorising means rewriting an expression as a multiplication, usually by placing a common factor outside a bracket. For example, 3x + 12 factorises to 3(x + 4).
They are reverse processes. Expanding removes brackets by multiplication, while factorising rewrites an expanded expression as a product containing brackets.
Find the largest number that divides every coefficient, then include any variables shared by every term. For 18x² + 12x, the highest common factor is 6x.
Expand the bracket again. If the result matches the original expression exactly, the factorisation is correct. For example, expanding 5(2a + 3) gives 10a + 15.
