Expanding and Factorising Brackets

Algebra and Sequences

Multiplying through

Expanding removes a bracket by multiplying the outside term by every term inside it
Each product keeps the sign that belongs to its term
- 5a(2a − 7) gives 10a² − 35a
A negative outside the bracket changes the sign of every term inside
- −3y(2y − 4) gives −6y² + 12y
Writing the negative term in brackets while multiplying keeps the sign rules visible

Exam tip

"Expand and simplify" is the wording in 14 of the 43 papers, and the two verbs are two separate marks. Expanding earns the first; collecting the like terms afterwards earns the second, so an answer left as 6x + 8 − 2x + 5 has stopped one step short of the mark it was going for.

Worked example

Expanding a single bracket

Expand 4x(3x − 5).

Solution:

Expanding removes the bracket by multiplying the term outside it by every term inside
The bracket holds two terms, so there are two products to write
Take each inside term with the sign in front of it: +3x and −5
4x × 3x = 12x², because 4 × 3 = 12 and x × x = x²
4x × (−5) = −20x, negative because one factor is negative
The answer is 12x² − 20x
The two terms are unlike, one in x² and one in x, so nothing collects and this is the simplest form
Check with a value: x = 2 gives 8 × 1 = 8 in the question, and 48 − 40 = 8 in the answer
The minus sign belongs to the 5 and travels with it, which is where most sign errors in this topic start