Expanding Brackets | CIE IGCSE Maths: Core Revision Notes 2025

Expanding One Bracket

How do I expand a bracket?

The expression 3x (x + 2) means 3x multiplied by the bracket (x + 2)
- 3x is the term outside the bracket
  - this is sometimes called a factor
- and x + 2 are the terms inside the bracket
Expanding the brackets means multiplying the outside term by each term on the inside
- This will remove (get rid of) the brackets
- 3x (x + 2) expands to $3 x \times x + 3 x \times 2$ which simplifies to $3 x^{2} + 6 x$
Beware of minus signs
- Remember the rules
  − × − = +
  − × + = −
- It helps to put brackets around negative terms

Worked example

(a)

Expand

4 x (2 x - 3)

Multiply the $4 x$ term outside the brackets by both terms inside the brackets

$4 x \times 2 x + 4 x \times (- 3)$

Simplify

$8 x^{2} - 12 x$

(b)

Expand

- 7 x (4 - 5 y)

Multiply the $- 7 x$ outside the brackets by both terms inside the brackets

$(- 7 x) \times 4 + (- 7 x) \times (- 5 y)$

Simplify and remember that multiplying two negatives gives a positive

$- 28 x + 35 x y$

Expand & Simplify

How do I simplify brackets that are added together?

First expand both brackets separately
- $4 (x + 7) + 5 x (3 - x)$
  - The first set of brackets expands to $4 \times x + 4 \times 7$ which simplifies to $4 x + 28$
  - The second set of brackets expands to $5 x \times 3 + 5 x \times (- x)$ which simplifies to $15 x - 5 x^{2}$
  - So $4 (x + 7) + 5 x (3 - x) = 4 x + 28 + 15 x - 5 x^{2}$
Then collect like terms
- - The other two terms are not like terms
- So $4 (x + 7) + 5 x (3 - x) = 19 x + 28 - 5 x^{2}$

Worked example

(a)

Expand and simplify

2 (x + 5) + 3 x (x - 8)

Expand each set of brackets separately

You can keep negative terms inside brackets

$2 \times x + 2 \times 5 + 3 x \times x + 3 x \times (- 8)$

Simplify each term

$2 x + 10 + 3 x^{2} - 24 x$

Collect like terms (the 2x and the -24x)

$- 22 x + 10 + 3 x^{2}$

(b)

Expand and simplify

3 x (x + 2) - 7 (x - 6)

Expand each set of brackets separately
Be careful: the second set of brackets has a -7 in front, not +7

$3 x \times x + 3 x \times 2 + (- 7) \times x + (- 7) \times (- 6)$

Simplify each term
Remember that multiplying two negatives gives a positive

$3 x^{2} + 6 x - 7 x + 42$

Collect like terms

$3 x^{2} - x + 42$

Expanding Double Brackets

How do I expand two brackets using FOIL?

Every term in the first bracket must be multiplied by every term in the second bracket
- Expanding (x + 1)(x + 3) requires 4 multiplications in total
A good way to remember all the multiplications is FOIL
- F = First: multiply together the first terms in each bracket
- O = Outside: multiply the first term in the first bracket by the last term in the last bracket
  - Visually, these are the outer terms
- I = Inside: multiply the last term in the first bracket by the first term in the last bracket
  - Visually, these are the inner terms
- L = Last: multiply together the last terms in each bracket
It helps to put negative terms in brackets when multiplying
Simplify the final answer by collecting like terms (if there are any)

How do I expand two brackets using a grid?

You may prefer a more visual method using a grid
To expand (x + 1)(x + 3), write out the brackets as row and column headings of a grid
- They can be in either direction
- Remember to write the appropriate sign in front of each term
x +1

x

+3
For each cell in the grid, multiply the term in the row heading by the term in the column heading
x +1

x x² x

+3 3x 3
Add together all the terms inside the grid to get the answer
- x² + x + 3x + 3
Collect like terms
- x² + 4x + 3

How do I expand a bracket squared?

Remember that a square number is a number multiplied by itself
Write (x + 3)² as (x + 3)(x + 3) and use one of the methods above
- With FOIL: (x + 3)(x + 3) = x² + 3x + 3x + 9
- Then collect like terms: x² + 6x + 9
Do not make the common mistake of saying (x + 3)² is x² + 3²
- This cannot be true, try substituting in x = 1
  - you would get (1 + 3)² = 4² = 16 on the left
  - but 1² + 3² = 1 + 9 = 10 on the right

Worked example

(a)

Expand

(2 x - 3) (x + 4)

Using FOIL, multiply together the first, outer, inner and last terms

$F O I L 2 x \times x + 2 x \times 4 + (- 3) \times x + (- 3) \times 4$

Simplify each term

$2 x^{2} + 8 x - 3 x - 12$

Collect like terms (the 8x and -3x)

$2 x^{2} + 5 x - 12$

(b)

Expand

(x - 3) (3 x - 5)

Using FOIL, multiply together the first, outer, inner and last terms

$F O I L x \times 3 x + x \times (- 5) + (- 3) \times 3 x + (- 3) \times (- 5)$

Simplify each term

$3 x^{2} - 5 x - 9 x + 15$

Collect like terms (the -5x and -9x)

$3 x^{2} - 14 x + 15$

Worked example

Expand ${(2 x + 3)}^{2}$ .

Remember that the answer is not (2x)² + 3²
Rewrite the expression as two separate brackets multiplied together

$(2 x + 3) (2 x + 3)$

Using FOIL, multiply together the first, outer, inner and last terms

$F O I L 2 x \times 2 x + 2 x \times 3 + 3 \times 2 x + 3 \times 3$

Simplify each term

$4 x^{2} + 6 x + 6 x + 9$

Collect like terms (the 6x and 6x)

$4 x^{2} + 12 x + 9$

Expanding Brackets (CIE IGCSE Maths: Core)

Revision Note

Expanding One Bracket

How do I expand a bracket?

Worked example

Expand & Simplify

How do I simplify brackets that are added together?

Worked example

Expanding Double Brackets

How do I expand two brackets using FOIL?

How do I expand two brackets using a grid?

How do I expand a bracket squared?

Worked example

Worked example

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Author: Mark

	x	+1
x	x²	x
+3	3x	3