Factorisation
1. What is Factorisation?
Factorisation is the process of writing an algebraic expression as a PRODUCT of its FACTORS.
6x² + 9x = 3x(2x + 3). The factors are 3x and (2x + 3).
'Factorisation is the REVERSE of expansion. While expansion REMOVES brackets, factorisation INTRODUCES them.'
2. Method 1 — Taking Common Factor
Find the HCF of ALL terms and factor it out.
Worked Example: Factorise 12x³y — 18x²y² + 6xy³
HCF of 12, 18, 6 is 6. HCF of x³, x², x is x. HCF of y, y², y³ is y. Common factor = 6xy.
12x³y — 18x²y² + 6xy³ = 6xy(2x² — 3xy + y²)
Worked Example: Factorise 4a²b — 6ab² + 8a²b²
Common factor = 2ab. = 2ab(2a — 3b + 4ab)
3. Method 2 — Factorisation by Grouping
When there is NO common factor in ALL terms, GROUP terms that have a common factor.
Steps:
- Group terms with common factors.
- Factor out the common factor from EACH group.
- Look for a COMMON BINOMIAL factor and factor it out.
Worked Example: Factorise 2xy + 3x + 2y + 3
Group: (2xy + 3x) + (2y + 3) Factor each group: x(2y + 3) + 1(2y + 3) Common binomial: (2y + 3) = (2y + 3)(x + 1)
Worked Example: Factorise a² + ab + ac + bc
Group: (a² + ab) + (ac + bc) = a(a + b) + c(a + b) = (a + b)(a + c)
4. Method 3 — Using Identities
Recognise expressions that match standard identities.
| Identity | Factorised Form |
|---|---|
| a² + 2ab + b² | (a + b)² |
| a² — 2ab + b² | (a — b)² |
| a² — b² | (a + b)(a — b) |
| x² + (a+b)x + ab | (x + a)(x + b) |
Worked Example: Factorise 16x⁴ — 81y⁴
= (4x²)² — (9y²)² = (4x² + 9y²)(4x² — 9y²) = (4x² + 9y²)(2x + 3y)(2x — 3y)
5. Method 4 — Splitting the Middle Term
For quadratic expressions of the form ax² + bx + c:
- Find two numbers p and q such that p + q = b AND p × q = ac
- Split the middle term bx as px + qx
- Factor by grouping
Worked Example: Factorise x² + 7x + 12
a = 1, b = 7, c = 12 Find p, q: p + q = 7, p × q = 12 p = 3, q = 4 (or 4, 3)
x² + 3x + 4x + 12 = x(x + 3) + 4(x + 3) = (x + 3)(x + 4)
Worked Example: Factorise x² — 5x + 6
p + q = —5, p × q = 6 p = —2, q = —3
x² — 2x — 3x + 6 = x(x — 2) — 3(x — 2) = (x — 2)(x — 3)
Worked Example: Factorise 2x² + 7x + 3
a = 2, b = 7, c = 3 ac = 6. Find p, q: p + q = 7, p × q = 6 p = 6, q = 1
2x² + 6x + 1x + 3 = 2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1)
Worked Example: Factorise 6x² — x — 2
a = 6, b = —1, c = —2 ac = —12. Find p, q: p + q = —1, p × q = —12 p = —4, q = 3 (or p = 3, q = —4)
6x² — 4x + 3x — 2 = 2x(3x — 2) + 1(3x — 2) = (3x — 2)(2x + 1)
6. Choosing the Right Method
| Expression Type | Method |
|---|---|
| ALL terms share a factor | Common factor |
| Four or more terms with groups | Grouping |
| Perfect square trinomial | a² ± 2ab + b² identity |
| Difference of two squares | a² — b² identity |
| Quadratic (x² + bx + c) where a = 1 | Split middle term |
| Quadratic (ax² + bx + c) where a ≠ 1 | Split middle term (find ac) |
| Four terms, two pairs | Grouping |
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'Stopping too early — leaving a common factor' | Check each factor: is it FULLY factorised? |
| 'Sign errors in grouping' | When factoring —(x — y), write as —x + y. Double-check signs |
| 'Forgetting the 1 when factoring' | When all terms cancel, keep 1: a + ab = a(1 + b), not a(b) |
| 'Wrong splitting numbers' | Always VERIFY: p + q = b AND p × q = ac. BOTH conditions must hold |
ICSE Exam Focus (6–8 marks)
- 2-mark questions: Factor out common factor
- 3-mark questions: Factorise by grouping or using identities
- 4-mark questions: Split the middle term (a = 1 or a ≠ 1)
- 6-mark questions: Multi-method factorisation (three or more steps)
Self-Test
Q1. Factorise: 15ab — 10a²b² + 20ab² A1. Common factor = 5ab. = 5ab(3 — 2ab + 4b).
Q2. Factorise: 6xy — 4x + 9y — 6 A2. Group: (6xy — 4x) + (9y — 6) = 2x(3y — 2) + 3(3y — 2) = (3y — 2)(2x + 3).
Q3. Factorise: 9a² — 25b² A3. = (3a)² — (5b)² = (3a + 5b)(3a — 5b).
Q4. Factorise: x² + 9x + 20 A4. p + q = 9, p × q = 20. p = 4, q = 5. x² + 4x + 5x + 20 = x(x+4) + 5(x+4) = (x+4)(x+5).
Q5. Factorise: 3x² + 10x + 8 A5. ac = 24. p + q = 10, p × q = 24. p = 6, q = 4. 3x² + 6x + 4x + 8 = 3x(x+2) + 4(x+2) = (x+2)(3x+4).
Q6. Factorise: 4x² — 12x + 9 A6. (2x)² — 2(2x)(3) + 3² = (2x — 3)².
