IGCSE Maths Expanding & Factorising Notes
Expanding removes brackets by multiplying out; factorising is the reverse, writing an expression as a product of factors.
Key concepts
- Expanding
- Multiply every term inside a bracket by the term outside; use FOIL for double brackets.
- Common factor
- Factorise by taking out the highest common factor of every term.
- Difference of two squares
- a² − b² factorises to (a + b)(a − b).
Key formulas
Double bracket (FOIL)
(a + b)(c + d) = ac + ad + bc + bd
Difference of two squares
a² − b² = (a + b)(a − b)
Worked example
Expand and simplify (x + 4)(x − 2).
- 1Multiply First terms: x × x = x².
- 2Outer and Inner: x × −2 = −2x and 4 × x = 4x.
- 3Last terms: 4 × −2 = −8, then collect: x² + 2x − 8.
Answer: x² + 2x − 8
Common mistakes
✗ Forgetting to multiply the outside term by every term in the bracket.
✓ Multiply the outer term by each term inside, one at a time.
✗ Sign errors with negative terms.
✓ Track signs carefully — a negative times a negative is positive.
✗ Not factorising fully.
✓ Check whether a bracket or remaining terms factorise again.
Quick check
Try these, then reveal the answers.
Expand 3(2x − 5).Reveal answer
6x − 15
Factorise 6x + 9.Reveal answer
3(2x + 3)
Factorise x² − 16.Reveal answer
(x + 4)(x − 4)
Exam tips
- Always simplify by collecting like terms after expanding.
- 'Factorise fully' means take out the highest common factor and keep going.
- Check an expansion by re-factorising, and a factorisation by expanding back.
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Frequently asked questions
What is the difference between expanding and factorising?+
Expanding multiplies brackets out into separate terms; factorising rewrites terms as a product of brackets — they are reverse operations.
What does FOIL stand for?+
First, Outer, Inner, Last — the order for multiplying two brackets so no pair of terms is missed.
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