Factorise: \(6x + 12\)
Look for the greatest common factor.
Both terms share a factor of 6. So \(6x + 12 = 6(x + 2)\).
Factorising Expressions
Factorising involves rewriting expressions by taking out common factors or reversing expansion. It is closely linked to expanding brackets and solving equations, and is a key skill in GCSE Maths.
Overview
Factorising means writing an expression as a product using brackets.
It is the reverse of expanding brackets.
\( 6x + 12 = 6(x + 2) \)
In most GCSE questions, the first step is to find the highest common factor (HCF) and take it outside the bracket.
What you should understand after this topic
- Understand what factorising means
- Find a common factor
- Take the highest common factor outside a bracket
- Check answers by expanding back
- Handle negative terms correctly
Key Definitions
Factorise
Write an expression as a product using brackets.
Factor
A value or term that multiplies with another.
Common Factor
A factor shared by every term in the expression.
Highest Common Factor
The largest factor that goes into every term.
Bracket Form
The rewritten expression after a factor is taken outside.
Expand
The reverse process of factorising.
Key Rules
Find the HCF first
\(8x + 12 = 4(2x + 3)\)
Every term must divide exactly
\(15a + 20 = 5(3a + 4)\)
Variables can be common factors too
\(6x^2 + 9x = 3x(2x + 3)\)
Check by expanding back
\(2(x + 5) = 2x + 10\)
Quick Pattern Check
Number factor only
\(10x + 15\)
Variable factor included
\(12x^2 + 4x\)
Negative term included
\(9x - 6\)
Take out the full HCF
\(14y + 21\)
How to Solve
What does factorising mean?
Factorising means rewriting an expression into bracket form. It is the reverse of expanding brackets.
\( 3x + 6 = 3(x + 2) \)
Factorising is the opposite of expanding brackets.
Step 1: Find the highest common factor (HCF)
Look for the largest number and any variables that all terms have in common.
\( 8x + 12 \)
Both 8 and 12 are divisible by 4.
So the highest common factor is \(4\).
Exam tip: Always take out the largest common factor.
Step 2: Factor out the common factor
\( 8x + 12 = 4(\;\; ) \)
\(8x \div 4 = 2x\)
\(12 \div 4 = 3\)
Answer: \(4(2x + 3)\)
Step 3: Check by expanding
\(4(2x + 3) = 8x + 12\)
So the factorised answer is correct.
Exam habit: Expanding is the fastest way to check your answer.
When variables are part of the factor
\( 15x^2 - 5x \)
The number HCF is \(5\).
Both terms also contain \(x\).
So factor out \(5x\).
Answer: \(5x(3x - 1)\)
Common structure to recognise
Most factorising questions follow the same pattern.
Find the HCF.
Divide each term by the HCF.
Write the bracket.
Check by expanding.
Key idea
Everything inside the bracket should multiply back to the original expression.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Factorise \( 6x + 12 \).
Edexcel
Factorise \( 9a - 3 \).
Edexcel
Factorise \( 4y^2 + 8y \).
Edexcel
Factorise \( 5p^2 - 10p \).
Edexcel
Factorise \( 3x^2 + 6x + 9 \).
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on algebraic fluency and recognising common patterns.
AQA
Factorise \( x^2 + 7x \).
AQA
Factorise \( x^2 + 9x + 20 \).
AQA
Factorise \( x^2 - 5x - 14 \).
AQA
Factorise \( 2x^2 + 7x + 3 \).
AQA
A student factorises \( x^2 + 6x + 9 \) as \( (x + 9)(x + 1) \). Explain the mistake and give the correct answer.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, special products, and algebraic structure.
OCR
Factorise \( x^2 - 16 \).
OCR
Factorise \( x^2 + 10x + 25 \).
OCR
Factorise \( 4x^2 - 9 \).
OCR
Factorise \( 3x^2 - 12x \).
OCR
Factorise completely \( 6x^2 - x - 2 \).
Exam Checklist
Step 1
Look for the highest common number factor.
Step 2
Check whether a variable can also come outside.
Step 3
Divide each term carefully to fill the bracket.
Step 4
Expand back to check the answer.
Most common exam mistakes
Not using the HCF
Taking out a smaller factor when a bigger one is possible.
Missing a variable
Taking out only the number and forgetting the common letter term.
Sign mistakes
Writing the wrong sign inside the bracket.
No check
Not expanding back to make sure the factorised form is correct.
Common Mistakes
These are common mistakes students make when factorising expressions in GCSE Maths.
Not taking out the highest common factor
Incorrect
A student factorises but leaves a larger common factor inside.
Correct
Always take out the highest common factor (HCF), including both numbers and variables, to fully factorise the expression.
Not dividing every term
Incorrect
A student takes a factor outside but does not divide all terms by it.
Correct
Every term must be divided by the factor outside the bracket. If one term does not divide correctly, the factor is wrong.
Sign errors inside brackets
Incorrect
A student gets the signs wrong after factorising.
Correct
Check signs carefully when dividing terms. For example, \(-6x\) divided by \(3\) gives \(-2x\), not \(2x\).
Missing variable factors
Incorrect
A student takes out only numbers and leaves variables inside.
Correct
Look for variables that appear in every term. For example, \(4x^2 + 2x\) can be factorised as \(2x(2x + 1)\).
Not checking by expanding
Incorrect
A student finishes factorising without verifying the result.
Correct
Always expand the brackets to check your answer matches the original expression. This confirms the factorisation is correct.
Try It Yourself
Practise factorising algebraic expressions into simpler forms.
Foundation
Foundation Practice
Factorise by taking out common factors.
Factorise: \(5x + 10\)
What number goes into both terms?
Both terms are divisible by 5. So \(5x + 10 = 5(x + 2)\).
Factorise: \(8y - 4\)
Take out the largest factor possible.
The common factor is 4. So \(8y - 4 = 4(2y - 1)\).
Factorise: \(9a + 3\)
Find the highest number that divides both terms.
3 is the highest common factor. So \(9a + 3 = 3(3a + 1)\).
Factorise: \(7x^2 + 14x\)
Both terms contain x.
The common factors are 7 and x. So \(7x^2 + 14x = 7x(x + 2)\).
Factorise: \(4x^2 + 8x\)
Factor out both the number and the variable.
Common factors are 4 and x. So \(4x^2 + 8x = 4x(x + 2)\).
A student writes \(6x + 12 = 6x(x + 2)\). What is the mistake?
Check what should be inside the bracket.
Only common factors are taken out. The correct answer is \(6(x + 2)\), not \(6x(x + 2)\).
Factorise: \(3y^2 + 6y\)
Factor out the common number and variable.
Common factors are 3 and y. So \(3y^2 + 6y = 3y(y + 2)\).
Which expression factorises to \(5(x + 3)\)?
Expand to check.
\(5(x + 3) = 5x + 15\).
Factorise: \(10x + 5\)
Look for the greatest common factor.
The highest common factor is 5. So \(10x + 5 = 5(2x + 1)\).
Higher
Higher Practice
Factorise quadratics and recognise patterns.
Factorise: \(x^2 + 5x + 6\)
Find two numbers that multiply to 6 and add to 5.
2 × 3 = 6 and 2 + 3 = 5, so \(x^2 + 5x + 6 = (x + 2)(x + 3)\).
Factorise: \(x^2 + 7x + 10\)
Find numbers that multiply to 10 and add to 7.
5 × 2 = 10 and 5 + 2 = 7, so \((x + 5)(x + 2)\).
Factorise: \(x^2 - 9\)
This is a difference of squares.
\(x^2 - 9 = (x + 3)(x - 3)\).
Factorise: \(x^2 - 5x + 6\)
Look for numbers that multiply to 6 and add to -5.
-2 × -3 = 6 and -2 + -3 = -5, so \((x - 2)(x - 3)\).
Factorise: \(2x^2 + 7x + 3\)
Split the middle term.
2x² + 7x + 3 = (2x + 1)(x + 3).
Factorise: \(3x^2 + 10x + 8\)
Split the middle term carefully.
3x² + 10x + 8 = (3x + 4)(x + 2).
A student factorises \(x^2 + 6x + 9\) as \((x + 3)(x + 6)\). What is the mistake?
Check multiplication and addition conditions.
3 × 3 = 9 and 3 + 3 = 6, so the correct factorisation is \((x + 3)^2\).
Factorise: \(x^2 + 4x + 4\)
Look for a perfect square.
This is a perfect square: \((x + 2)^2\).
Which expression factorises to \((x + 4)(x + 5)\)?
Expand to check.
\((x + 4)(x + 5) = x^2 + 9x + 20\).
Factorise: \(x^2 - x - 6\)
Find numbers that multiply to -6 and add to -1.
-3 × 2 = -6 and -3 + 2 = -1, so \((x - 3)(x + 2)\).
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is factorising?
Rewriting an expression as a product of factors.
How do I factorise?
Find the highest common factor and take it outside brackets.
Why is factorising useful?
It helps solve equations.