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Expanding BracketsEdexcel IGCSE Maths: Revision notes

Section 1

How do you expand a single bracket?

To expand a single bracket, multiply every term inside the bracket by the term outside it.

For example: 3(x+4)=3x+123(x + 4) = 3x + 12

With a negative multiplier, be careful with signs: −2(x−5)=−2x+10-2(x - 5) = -2x + 10

Always check: the number of terms inside the bracket should match the number of terms in your expanded answer.

Key termsexpandtermcoefficient
Example

Expand 5(2x−3)5(2x - 3): multiply both terms by 5, giving 10x−1510x - 15.

Common mistake

Forgetting to multiply the second term inside the bracket — 3(x+4)3(x+4) is NOT 3x+43x + 4.

Section 2

How do you expand double brackets?

Double brackets, like (x+2)(x+5)(x + 2)(x + 5), are expanded using FOIL: First, Outer, Inner, Last.

(x+2)(x+5)=x2+5x+2x+10=x2+7x+10(x+2)(x+5) = x^2 + 5x + 2x + 10 = x^2 + 7x + 10

Always simplify by collecting the like terms (the two middle xx terms) at the end.

Watch signs carefully when a bracket contains subtraction: (x−3)(x+4)=x2+4x−3x−12=x2+x−12(x-3)(x+4) = x^2 + 4x - 3x - 12 = x^2 + x - 12

Key termsFOILlike termsquadratic expression
Exam tip

Write out all four products before simplifying — don't try to combine terms in your head.

Common mistake

Sign errors are the most common mistake — a negative outside a bracket flips the sign of every term multiplied by it.

Section 3

What about squaring a bracket, e.g. (x+3)2(x+3)^2?

A common exam trap: (x+3)2(x+3)^2 is not x2+9x^2 + 9. It means (x+3)(x+3)(x+3)(x+3), so you must expand it fully using FOIL:

(x+3)2=(x+3)(x+3)=x2+3x+3x+9=x2+6x+9(x+3)^2 = (x+3)(x+3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9

The general pattern is: (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2

Key termssquareexpanding pattern
Common mistake

Never write (x+3)2=x2+9(x+3)^2 = x^2 + 9 — the middle term 2ab2ab is always missing if you skip this step.

Think of it like this

Think of (a+b)2(a+b)^2 like squaring a two-part length — you get four rectangular areas, not just two squares.

Section 4

How do you expand triple brackets?

For triple brackets, such as (x+1)(x+2)(x+3)(x+1)(x+2)(x+3), expand two brackets first, then multiply the result by the third bracket.

Step 1 — expand the first two: (x+1)(x+2)=x2+3x+2(x+1)(x+2) = x^2 + 3x + 2

Step 2 — multiply this by the remaining bracket: (x2+3x+2)(x+3)=x3+3x2+3x2+9x+2x+6(x^2+3x+2)(x+3) = x^3 + 3x^2 + 3x^2 + 9x + 2x + 6

Step 3 — collect like terms: =x3+6x2+11x+6= x^3 + 6x^2 + 11x + 6

This gives a cubic expression (highest power is 3).

Key termscubic expression
Exam tip

Never try to expand three brackets in one step — always reduce to two brackets first, then multiply again.

Example

Expand (x−1)(x+1)(x+2)(x-1)(x+1)(x+2): first (x−1)(x+1)=x2−1(x-1)(x+1)=x^2-1, then (x2−1)(x+2)=x3+2x2−x−2(x^2-1)(x+2)=x^3+2x^2-x-2.

Section 5

How do brackets connect to factorising?

Expanding and factorising are inverse operations. Expanding removes brackets; factorising puts an expression back into bracket form.

Checking your expansion by factorising the answer (or vice versa) is a reliable way to catch mistakes in an exam, especially with quadratics like x2+7x+10=(x+2)(x+5)x^2+7x+10 = (x+2)(x+5).

Key termsfactoriseinverse operation
Exam tip

If time allows, factorise your final expanded quadratic to check it matches the original brackets.

Must Know

  • Multiply every term inside a bracket by the term outside — never skip a term
  • FOIL (First, Outer, Inner, Last) expands double brackets
  • (a+b)2=a2+2ab+b2(a+b)^2 = a^2+2ab+b^2, NOT a2+b2a^2+b^2
  • Expand triple brackets two at a time, then multiply by the third
  • Always collect like terms at the end for the simplest form
  • Watch signs carefully — a negative outside a bracket changes every sign inside

That's the notes covered.

Carry on to the next subtopic.