What is factorising?
Factorising is the reverse of expanding — you rewrite an expression as a product of factors. It is essential for solving quadratic equations, simplifying algebraic fractions, and higher-tier proof questions.
The four types you need at GCSE:
- Common factor — take out the HCF of all terms
- Quadratics with a = 1 — find two numbers that multiply to c and add to b
- Quadratics with a ≠ 1 — multiply ac, find two numbers, split and group
- Difference of two squares — a² − b² = (a + b)(a − b)
Always look for a common factor first before attempting any other method.
Common factor factorising
Worked Example 1
Factorise fully: 12x²y − 8xy²
1
Find the HCF of 12x²y and 8xy²: HCF = 4xy
2
Divide each term: 12x²y ÷ 4xy = 3x 8xy² ÷ 4xy = 2y
3
Write in factorised form: 4xy(3x − 2y)
Answer4xy(3x − 2y)
Factorising quadratics (a = 1)
For x² + bx + c, find two numbers that multiply to c and add to b.
Worked Example 2
Factorise: x² − 2x − 15
1
Find two numbers that multiply to −15 and add to −2: −5 and +3
2
Write in brackets: (x − 5)(x + 3)
3
Check by expanding: x² + 3x − 5x − 15 = x² − 2x − 15 ✓
Answer(x − 5)(x + 3)
Factorising quadratics (a ≠ 1)
For ax² + bx + c: multiply a × c, find two numbers that multiply to ac and add to b, split the middle term, then factorise by grouping.
Worked Example 3
Factorise: 6x² + x − 2
1
Multiply ac: 6 × −2 = −12
2
Two numbers that multiply to −12 and add to +1: +4 and −3
3
Split: 6x² + 4x − 3x − 2
4
Factorise in pairs: 2x(3x + 2) − 1(3x + 2) = (2x − 1)(3x + 2)
Answer(2x − 1)(3x + 2)
Difference of two squares
Worked Example 4
Factorise: 4x² − 25
1
Recognise: 4x² = (2x)² and 25 = 5²
2
Apply a² − b² = (a + b)(a − b): (2x + 5)(2x − 5)
Answer(2x + 5)(2x − 5)
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Factorise: 10x + 15Foundation
Q2Factorise: 6x² + 9xFoundation
Q3Factorise: x² + 7x + 12Foundation
Q4Factorise: x² − 5x + 6Foundation
Q5Factorise: x² − 9Foundation
Higher (Grade 5–7)
Q6Factorise: 2x² + 7x + 3Higher
Q7Factorise: 3x² − 11x + 6Higher
Q8Factorise: 4x² − 25Higher
Q9Factorise: 6x² + x − 2Higher
Q10Factorise: x² − 2x − 15Higher
Higher — Hard (Grade 8–9)
Q11Factorise fully: 12x²y − 8xy²Grade 8–9
Q12Factorise fully: 5x² − 20Grade 8–9
Q13Factorise: 4x² − 12x + 9Grade 8–9
Q14Factorise: 6x² − 13x − 5Grade 8–9
Q15Factorise fully: x³ − 4xGrade 8–9
Answers
Foundation (Q1–Q5)
Q15(2x + 3)(HCF = 5)
Q23x(2x + 3)(HCF = 3x)
Q3(x + 3)(x + 4)(two numbers: +3 and +4, multiply to 12, add to 7)
Q4(x − 2)(x − 3)(two numbers: −2 and −3, multiply to 6, add to −5)
Q5(x + 3)(x − 3)(difference of two squares)
Higher (Q6–Q10)
Q6(2x + 1)(x + 3)(ac=6; numbers +6,+1; split: 2x²+6x+x+3)
Q7(3x − 2)(x − 3)(ac=18; numbers −9,−2; split: 3x²−9x−2x+6)
Q8(2x + 5)(2x − 5)(difference of two squares)
Q9(2x − 1)(3x + 2)(ac=−12; numbers +4,−3; split: 6x²+4x−3x−2)
Q10(x − 5)(x + 3)(two numbers: −5 and +3, multiply to −15, add to −2)
Higher — Hard (Q11–Q15)
Q114xy(3x − 2y)(HCF = 4xy)
Q125(x + 2)(x − 2)(take out 5 first: 5(x²−4); then DOTS)
Q13(2x − 3)²(perfect square; check: 4x²−12x+9 ✓)
Q14(2x − 5)(3x + 1)(ac=−30; numbers −15,+2; split: 6x²+2x−15x−5)
Q15x(x + 2)(x − 2)(HCF x first: x(x²−4); then DOTS)
Common mistakes
Common Mistake 1
Not factorising fully (incomplete HCF)
For 12x² + 8x, writing 2x(6x + 4) is not fully factorised — 2 is still a common factor inside the bracket. The HCF is 4x, so the correct answer is 4x(3x + 2). Always check whether the bracket can be factorised further.
Common Mistake 2
Wrong signs in the bracket
For x² − 5x + 6, students often write (x + 2)(x + 3) or (x − 2)(x + 3). Check: the numbers must add to −5 (both must be negative): (x − 2)(x − 3). Always verify by expanding back.
Common Mistake 3
Factorising x² − 9 as (x − 3)²
x² − 9 is a difference of two squares, not a perfect square. The correct factorisation is (x + 3)(x − 3). (x − 3)² expands to x² − 6x + 9, which is completely different.
Common Mistake 4
Using the wrong pair of numbers in the ac method
Students find numbers that multiply to ac but forget to also check they add to b. For 2x² + 7x + 3 (ac = 6): pairs that multiply to 6 are (1,6), (2,3). Only +6 and +1 add to 7. Check both conditions before splitting.
Common Mistake 5
Forgetting to take out a common factor first
For 5x² − 20, students jump straight to DOTS and write (5x + ?) — but 5 is a common factor. Always remove it first: 5(x² − 4) = 5(x + 2)(x − 2). Missing the initial common factor usually leads to errors.
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