What does factorising mean?
Multiplying 3 × 4 gives 12; factorising 12 goes the other way, writing it as 3 × 4 or 2 × 2 × 3. In algebra, factorising means writing an expression as a product of simpler expressions.
Expanding 2x(x + 3) gives 2x² + 6x. Factorising 2x² + 6x gives back 2x(x + 3).
Factorising is essential for simplifying expressions, solving equations and dividing polynomials.
Method 1: common factors
Find the highest common factor of all the terms (numbers and variables) and take it outside a bracket.
Example. 12a²b + 18ab²
The HCF of 12 and 18 is 6; the common variables are a and b. So 12a²b + 18ab² = 6ab(2a + 3b).
Check by expanding: 6ab × 2a = 12a²b and 6ab × 3b = 18ab². ✓
The common factor can be a whole bracket: 5x(y − 2) + 3(y − 2) = (y − 2)(5x + 3).
Method 2: regrouping
When there is no factor common to all terms, group the terms in pairs that do share a factor.
Example. 2xy + 3x + 2y + 3
Group: (2xy + 3x) + (2y + 3) = x(2y + 3) + 1(2y + 3) = (2y + 3)(x + 1).
Example. 6ab − 4b + 6 − 9a
Rearrange first: 6ab − 9a − 4b + 6 = 3a(2b − 3) − 2(2b − 3) = (2b − 3)(3a − 2).
Method 3: identities
Recognise the shape of the expression:
Identity | Example |
|---|---|
a² + 2ab + b² = (a + b)² | x² + 10x + 25 = (x + 5)² |
a² − 2ab + b² = (a − b)² | 4y² − 12y + 9 = (2y − 3)² |
a² − b² = (a + b)(a − b) | 9m² − 16 = (3m + 4)(3m − 4) |
Example. x⁴ − 81 = (x²)² − 9² = (x² + 9)(x² − 9) = (x² + 9)(x + 3)(x − 3). Keep going until no factor can be split further.
Example. 2x² − 50 = 2(x² − 25) = 2(x + 5)(x − 5). Always take out a common factor first.
Method 4: x² + px + q
Look for two numbers a and b with a + b = p and a × b = q. Then x² + px + q = (x + a)(x + b).
Example. x² + 7x + 12: we need two numbers that add to 7 and multiply to 12. They are 3 and 4, so x² + 7x + 12 = (x + 3)(x + 4).
Example. y² − 5y − 14: sum −5, product −14. The numbers are −7 and 2, so y² − 5y − 14 = (y − 7)(y + 2).
Sign guide for the two numbers:
q positive, p positive: both positive.
q positive, p negative: both negative.
q negative: one positive and one negative; the larger one has the sign of p.
Dividing polynomials
To divide, factorise and cancel common factors.
Example. (x² + 7x + 12) ÷ (x + 4) = (x + 3)(x + 4) ÷ (x + 4) = x + 3.
Example. (10y³ − 40y) ÷ 5y(y − 2) = 10y(y + 2)(y − 2) ÷ 5y(y − 2) = 2(y + 2).
A monomial dividing a polynomial divides each term: (12x³ − 8x²) ÷ 4x = 3x² − 2x.
Common mistakes
(a + b)² is not a² + b²; the middle term 2ab is easy to forget.
Cancelling terms instead of factors: in (x + 5)/5 the 5s cannot be cancelled.
Stopping too early: x² − 9 inside a factorisation should be split again into (x + 3)(x − 3).
Summary
Take out the HCF first, then try regrouping, identities, or the sum-and-product method.
Check every factorisation by expanding.
To divide polynomials, factorise and cancel common factors.