Cambridge (CIE) IGCSE — International Maths Extended
Expanding & Factorising Brackets
The big idea: Expanding removes brackets by multiplying everything out;
factorising is the exact reverse — it puts an expression back into brackets.
Once you can do both fluently, you can flip between the two forms whenever a problem needs it.
📋 Summary — What This Chapter Covers
- Expanding single brackets — multiplying one term by everything inside a bracket, e.g.
3x(x+2)
- Expanding double brackets — using FOIL or a grid to multiply two brackets together
- Expanding triple brackets — expand two brackets first, then multiply the result by the third
- Factorising out terms — pulling the highest common factor (HCF) out to the front
- Factorising by grouping — splitting a 4-term expression into two pairs, each with its own common factor
- Factorising simple quadratics (where
a = 1) — finding two numbers that multiply to c and add to b
- Factorising harder quadratics (where
a ≠ 1) — using the "multiply ac, add to b" trick plus grouping
- Difference of two squares — spotting
a² − b² and instantly factorising to (a+b)(a−b)
- Deciding which method to use — a decision process for any factorising question you're handed
1. Expanding Single Brackets
Think of the term outside the bracket as a "delivery van" — it has to visit every single term
inside the bracket and multiply with it. Nothing inside gets left out.
3x(x + 2) means "3x multiplied by the whole bracket (x + 2)."
So 3x multiplies the x, AND 3x multiplies the 2.
Watch the signs!
This is where most marks are lost. Remember:
− × − = + (two negatives make a positive)− × + = − (a negative and a positive make a negative)
Tip that saves marks
It genuinely helps to put brackets around negative terms in your working, e.g. write
4x × (−3)
instead of
4x × −3. It stops sign errors creeping in when you're moving fast.
Worked Example
Expand −7x(4 − 5y)
Multiply −7x by each term inside:
(−7x) × 4 + (−7x) × (−5y)
= −28x + 35xy (negative × negative = positive)
Answer: −28x + 35xy
Expand & Simplify (multiple brackets added together)
Sometimes you'll be given two separate bracket expressions added or subtracted, like
4(x + 7) + 5x(3 − x). The process is simple: expand each bracket completely
separately first, then collect like terms at the end.
Worked Example
Expand and simplify 3x(x + 2) − 7(x − 6)
Expand each bracket separately (careful — the second bracket has −7 in front, not +7):
3x × x + 3x × 2 + (−7) × x + (−7) × (−6)
= 3x² + 6x − 7x + 42
Collect like terms (6x and −7x):
Answer: 3x² − x + 42
Practice Question 1
Expand and simplify: 2(x + 5) + 3x(x − 8)
Practice Question 2
Expand: −5x(3x − 4y)
2. Expanding Double Brackets
Now the "delivery van" idea doubles up: every term in the first bracket has to multiply
every term in the second bracket. For two brackets each with 2 terms, that's always
4 multiplications in total.
Method 1: Using a grid
3. Expanding Triple Brackets
twice
Strategy
Worked Example
Expand (2x − 3)(x + 4)(3x − 1)