Expanding and Factorising Brackets

Algebra and Sequences

Four terms with no factor common to all of them may still factorise in pairs
Factorise the first two terms and the last two separately, aiming to produce the same bracket in both
That shared bracket is then taken out as a common factor
- ab + 4a + 3b + 12 becomes a(b + 4) + 3(b + 4), which factorises to (b + 4)(a + 3)
If the two brackets do not match, try pairing the terms differently

Worked example

Factorising by grouping

Factorise xy + 3x + 2y + 6.

Solution:

Four terms with nothing common to all of them, which is the signal to group them in pairs
Take the first two together and the last two together, keeping the order
First pair: xy + 3x, whose common factor is x, giving x(y + 3)
Second pair: 2y + 6, whose common factor is 2, giving 2(y + 3)
The two brackets are now identical, and that is the check that the pairing worked — different brackets mean the terms should be paired the other way round
The expression reads x(y + 3) + 2(y + 3), which is (y + 3) multiplied by x and by 2
Take the common bracket out: (y + 3)(x + 2)
Check by expanding: xy + 3x + 2y + 6, which is the expression we started from