Sheet № 13 · Foundation + Higher · AQA · Edexcel · OCR
Factorising Expressions –
Factorising is one of the most important algebra skills you will need for your GCSE Maths exam. It appears on both Foundation and Higher tier papers across AQA, Edexcel, and OCR, and it underpins many other topics such as solving quadratic equations, simplifying algebraic fractions, and sketching graphs. On this page you will learn what f
§Key definitions
Question:
Factorise fully 12x²y + 18xy².
Check:
6xy × 2x = 12x²y ✓ and 6xy × 3y = 18xy² ✓
Answer:
6xy(2x + 3y)
Q1 (Foundation):
Factorise 15x + 25.
Q2 (Foundation/Higher):
Factorise x² + 7x + 12.
§Formulas to memorise
ab + ac = a(b + c)
x² + (a+b)x + ab = (x + a)(x + b)
a² − b² = (a + b)(a − b)
Single bracket (common factor) factorising — take out the highest common factor (HCF) of all terms.
Double bracket factorising — rewrite a quadratic expression as the product of two linear brackets.
Difference of two squares — a special pattern where a² − b² = (a + b)(a − b).
Identify every term — in the expression. For example, in 6x² + 9x there are two terms: 6x² and 9x.
Find the HCF — of the coefficients (numbers). The HCF of 6 and 9 is 3.
Find the HCF of the variables. — Both terms contain at least one x, so x is a common factor.
Write the HCF outside the bracket. — That gives 3x( ).
Worked example
Factorise fully 12x²y + 18xy².
Working:
⚠ Common mistakes
- ✗Not fully factorising. Students write 3(4x + 6) instead of taking out the full HCF to get 6(2x + 3). Always check whether the terms inside the bracket still share a common factor.
- ✗Forgetting to factorise the variable part. In 8x³ + 12x², the HCF is 4x², not just 4. Look at the lowest power of each variable present in every term.
- ✗Sign errors in double brackets. When c is negative, one of your two numbers must be negative and one positive. Write out all factor pairs carefully.
- ✗Mixing up factorising and expanding. The question says "factorise" but some students expand instead. Read the instruction twice.
- ✗Ignoring difference of two squares. If you see something like 25 − 4x², recognise it as (5 + 2x)(5 − 2x) rather than trying to use double brackets with an x² term.
✦ Exam tips
- →"Factorise fully" means take out the complete HCF. If the examiner writes "fully," they expect every common factor removed. You will lose a mark if anything common remains inside the bracket.
- →Always expand to check. It takes 20 seconds and guarantees your answer is correct. In a high-stakes exam, that is time well spent.
- →Look for difference of two squares first. It is quicker than double brackets and students often miss it. Any expression of the form a² − b² fits this pattern.
- →On AQA and Edexcel, factorising is often the first step in a multi-part question — for instance, factorise then solve. Getting this step right unlocks the remaining marks.