Learning Objectives
You should be able to:
- Explain what factorisation means.
- Recognise factorisation as the reverse of expanding brackets.
- Find the highest common factor of algebraic terms.
- Factorise expressions containing numbers and variables.
- Check an answer by expanding the brackets again.
Introduction
This lesson develops Factorisation from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🎬 Take out the common factor
Look for the greatest factor shared by every term.
🤔 Think About This...
Imagine a school buys:
- 3 exercise books
- 3 pens
We could write this as:
Both groups contain 3.
Instead of writing the 3 twice, we can take it outside:
📖 What is Factorisation?
Factorisation means rewriting an expression as a product of factors.
In simpler language, it means taking out the number, letter or expression that is common to every term.
➡️ Expanding
becomes
⬅️ Factorising
becomes
🕵️ The StudyNest Thinking Method
👉 What is common to every term?
The common factor could be:
- a number,
- a variable,
- or both a number and a variable.
Where possible, take out the highest common factor.
🔍 Finding the Highest Common Factor
Before working with algebra, let us review common factors using numbers.
Consider:
The factors of 6 are:
The factors of 12 are:
The greatest factor they share is 6.
📝 Worked Examples
Example 1
Factorise:
🔍 StudyNest Question: What is common to both terms?
Both 3x and 6 are divisible by 3.
Take 3 outside the brackets.
Example 2
Factorise:
🔍 StudyNest Question: What is the highest common factor of 10 and 15?
The highest common factor is 5.
Example 3
Factorise:
🔍 StudyNest Question: What is common to both terms?
The highest common factor of 8 and 12 is 4.
Example 4
Factorise:
🔍 StudyNest Question: What number and variable are common to both terms?
Both terms contain 3 and x. The common factor is therefore 3x.
🌍 Real-Life Example
Imagine a school buys:
- 12 exercise books
- 12 rulers
Instead of writing:
We can factor out the common number:
Factorisation helps us write expressions more neatly by taking out what is common.
⚠ Common Mistakes
Taking out a factor that is not common to every term.
Always check every term before factorising.
Forgetting to divide every term after taking the common factor outside.
Not taking out the highest common factor (HCF).
Example:
Practice Questions
Factorise the following expressions.
- 4x + 8
- 15y + 20
- 9a − 12
- 5m + 25
- 6x + 18
- 8p + 12p²
- 14k − 21
- 16x² + 8x
✅ Show Answers
- 4(x + 2)
- 5(3y + 4)
- 3(3a − 4)
- 5(m + 5)
- 6(x + 3)
- 4p(2 + 3p)
- 7(2k − 3)
- 8x(2x + 1)
🧠 Did You Know?
⭐ What You Can Do Now
- ✅ I understand that factorisation is the reverse of expanding brackets.
- ✅ I can identify the highest common factor.
- ✅ I can factorise expressions with numbers.
- ✅ I can factorise expressions with numbers and variables.
- ✅ I can check my answer by expanding the brackets again.
🎉 Well Done!
You can now recognise common factors and rewrite expressions in a simpler form.
🔍 Find what is common.
➡ Take it outside the brackets.
✔ Divide every term by the common factor.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
📈 Your Progress
Lesson 8 of 22
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.