The reverse of expansion
Factorisation means writing an expression as a product of factors. It undoes expansion. For example, expansion turns 3(x + 4) into 3x + 12; factorisation turns 3x + 12 back into 3(x + 4).
Key idea / 要点
First look for the highest common factor (HCF) of all terms and take it outside the bracket: ab + ac = a(b + c).
Common factor
To factorise 6x + 9, the HCF of 6 and 9 is 3, so 6x + 9 = 3(2x + 3). Check by expanding: 3·2x + 3·3 = 6x + 9. For 4x2 + 6x, the HCF is 2x, so 4x2 + 6x = 2x(2x + 3).
Worked example / 例题
Factorise x2 + 5x + 6.
Find two numbers that multiply to 6 and add to 5: they are 2 and 3.
So x2 + 5x + 6 = (x + 2)(x + 3).
Difference of two squares
When you see a2 − b2, use a2 − b2 = (a + b)(a − b). For example x2 − 16 = (x + 4)(x − 4), because 16 = 42. This only works with a minus sign between two perfect squares.
Remember / 记住
- Always take out the common factor first.
- Check by expanding your answer.
- a2 − b2 = (a + b)(a − b), but a2 + b2 does not factorise like this.