Factorising (AQA GCSE Maths): Revision Notes
Factorising
What is factorising?
Factorising is the opposite of expanding brackets. When you expand brackets, you multiply out the terms. When you factorise, you put the expression back into brackets by taking out common factors.
The relationship between expanding and factorising can be shown as:
- Expanding:
- Factorising:
This relationship is key to understanding why factorising works and helps you check your answers.
Step-by-step method for factorising
To factorise an expression successfully, follow these four clear steps:
1. Look for the largest factor Find the largest factor that you can take out of every term in the expression. This is called the highest common factor (HCF).
Example: In , both terms can be divided by 4, so the largest factor is 4.
2. Write the factor outside brackets
Place this largest factor outside the brackets, keeping the plus or minus sign from the original expression.
Example: 12x - 8 = 4(\text{___} - \text{___})
3. Work out what goes inside the brackets Divide each term in the original expression by the factor to find what needs to go inside the brackets.
Example: and , so
4. Check your answer Expand the brackets to make sure you get back to the original expression.
Example: and , so ✓
Partial vs complete factorisation
You need to understand the difference between partial and complete factorisation:
Partial factorisation means you haven't taken out the largest possible factor:
This expression has only been partly factorised because you could still take out 'a' from the terms inside the brackets.
Complete factorisation means you have taken out the largest possible factor:
The two factors here are and , and no further factorisation is possible.
Worked examples
Worked Example 1: Basic Factorisation
Factorise
Solution: Both terms contain the factor 'y', so:
Check: and , so ✓
Worked Example 2: Complete Factorisation
Factorise fully
Solution: Both terms contain the factors 2 and p:
- The largest common factor is
- and
- Therefore:
Check: and , so ✓
Understanding factors in expressions
When you write an expression with brackets, both parts of the factorised expression are factors of the original expression.
For example:
This means that , , , and are all factors of the original expression .
Exam tips
Critical Exam Advice:
- Always check your factorisation is complete by looking for any remaining common factors inside the brackets
- Problem-solving skills are essential throughout your exam, so practise identifying when factorisation is needed in different contexts
- When the question asks you to "factorise fully", make sure you take out the largest possible factor
- Always expand your answer to check it matches the original expression
Remember!
Key Points to Remember:
- Factorising is the opposite of expanding brackets - use this to check your answers
- Find the largest factor that can be taken out of every term in the expression
- Complete factorisation means taking out the biggest possible factor, not just any factor
- Always check your work by expanding the brackets to get back to the original expression
- Look for common factors in both numbers and letters when factorising algebraic expressions