For students and parents · O-Level E-Math & A-Math
Factorising in A-Math, the Four Methods and When Each Wins
Choosing the right method to write an expression as a product.
Mrs Eileen Toh
Founder & Curriculum Architect · ex-MOE · 3 min read · Updated 29 Jun 2026
Factorising means writing an expression as a product of simpler factors, and it underlies solving equations, simplifying fractions and sketching curves. Students who struggle here usually struggle because they reach for one method and force it, rather than reading the expression and choosing. There are four moves worth knowing, and recognising which one a given expression wants is the skill.
Method one, take out the common factor
Always the first look. 6x2 + 9x = 3x(2x + 3). Pull out the largest factor shared by every term before doing anything else; many later methods only work once the common factor is gone.
Method two, difference of two squares
If you see one square subtracted from another, a2 - b2 = (a - b)(a + b). So x2 - 9 = (x - 3)(x + 3). The signs of the two brackets are always one plus and one minus. This pattern is easy to miss when the squares are disguised, for example 4x2 - 25 = (2x - 5)(2x + 5).
Method three, the quadratic
For x2 - 5x + 6, find two numbers that multiply to +6 and add to -5, which are -2 and -3, giving (x - 2)(x - 3). When the x2 carries a coefficient, the same idea with the cross-method: 2x2 + 7x + 3 = (x + 3)(2x + 1), and 6x2 - x - 12 = (2x - 3)(3x + 4). Multiply the brackets back to confirm, every time.
Method four, the factor theorem for cubics
A cubic will not factorise by inspection, so use the factor theorem: if substituting x = a makes the expression zero, then (x - a) is a factor. For x3 - 2x2 - 5x + 6, try x = 1: 1 - 2 - 5 + 6 = 0, so (x - 1) is a factor. Divide it out and factorise the resulting quadratic to get (x - 1)(x - 3)(x + 2). The trial values worth checking are the factors of the constant term, here the factors of 6.
The teaching point
Factorising is a choice, made by reading the expression: a common factor first, then the difference of two squares if you see two squares, the quadratic methods for a quadratic, and the factor theorem for a cubic. The mistake is method-blindness, trying the quadratic cross-method on a cubic, or missing a common factor that would have made everything simpler. Read first, then factorise.
If your child knows each method but cannot tell which one a question wants, that is precisely the structure-first skill GPA's Secondary Math programme builds, on real O-Level algebra. A short diagnostic consult will show where the choosing breaks down.
Every factorisation on this page was checked independently before publishing.