Factorising (Edexcel GCSE Maths): Revision Notes
Factorising
What is factorising?
Factorising is the opposite process of expanding brackets. When you expand brackets, you multiply terms out to create a longer expression. Factorising does the reverse - it takes a longer expression and puts it back into brackets by finding common factors.
Understanding this reverse relationship is crucial - if you can expand brackets, you already have the foundation for factorising. The key is recognising what factors were originally taken out.
Basic factorising method
When factorising expressions, you need to follow a systematic approach to ensure you identify all common factors correctly. This methodical approach helps prevent errors and ensures you don't miss any common factors.
Step-by-step method
- Find the biggest number that divides into all the terms in the expression
- For each letter in turn, identify the highest power (such as x, x² etc.) that appears in EVERY term
- Write the brackets and fill in all the components needed to recreate each original term
- Check your answer by expanding the brackets to ensure it matches the original expression
Always start with the biggest number that divides into all terms. Missing this step is a common mistake that leads to incomplete factorisation.
Worked examples
Worked Example 1: Factorise
- The biggest number that divides into both 3 and 6 is 3
- The highest power of x that appears in both terms is x
- This gives us:
- Check: ✓
Worked Example 2: Factorise
- The biggest number that divides into both 8 and 2 is 2
- The highest powers that appear in both terms are x and y
- This gives us:
- Check: ✓
Difference of two squares (D.O.T.S.)
The difference of two squares is a special type of factorising that occurs when you have "one thing squared" minus "another thing squared". This pattern has a quick and reliable factorisation method that you should memorise.
D.O.T.S. stands for Difference Of Two Squares. This acronym helps you remember when to apply this special factorisation technique.
The D.O.T.S. formula
When you see an expression in the form , you can factorise it using:
This formula only works for subtraction between two perfect squares. It does not work for addition ( cannot be factorised using real numbers).
D.O.T.S. examples
Worked Example 1: Factorise
- Here a = x and b = 1
- Answer:
- Remember that 1 is a square number
Worked Example 2: Factorise
- Here a = 3p and b = 4q (since and )
- Answer:
- You need to recognise that 9 and 16 are perfect squares
Worked Example 3: Factorise
- First, take out the common factor of 3:
- Then apply D.O.T.S. to where a = x and b = 5y
- Answer:
Using D.O.T.S. in simplifying fractions
The difference of two squares pattern often appears in algebraic fractions, and recognising it can help you simplify complex expressions. This is a common exam technique that tests your ability to spot the D.O.T.S. pattern.
Worked Example: Simplify
- The numerator is a difference of two squares where a = x and b = 6
- So
- The denominator
- Therefore:
The common factor cancels out, leaving a much simpler expression.
Important reminders
When factorising, the components you take out and place at the front are called common factors. The terms that remain inside the brackets are what you need to multiply the common factors by to recreate the original expression.
Always check your factorisation by expanding the brackets - if you get back to the original expression, your factorisation is correct. This is particularly important with difference of two squares questions, as they can appear in various forms within exam questions.
Be especially careful with difference of two squares in fraction form, as this is a common exam technique for testing your ability to spot and apply the D.O.T.S. pattern.
Remember!
Key Points to Remember:
- Factorising is the reverse process of expanding brackets
- Always look for the biggest common factor first, then consider highest powers of variables
- D.O.T.S. follows the pattern
- Check your answer by expanding the brackets back out
- D.O.T.S. is often used in simplifying algebraic fractions