Multiplying Out Brackets Revision Notes for AQA GCSE Maths

Multiplying out brackets

Learning to expand brackets is a fundamental skill in algebra that helps simplify expressions and solve equations. When you multiply out brackets, you're essentially distributing terms across the expression to create an equivalent but expanded form.

Single brackets

When working with single brackets, there's one key principle to remember: the term outside the brackets must be multiplied by every single term inside the brackets. This is sometimes called the distributive property.

infoNote

The distributive property is the foundation of expanding brackets. It ensures that no term is missed when multiplying out expressions.

How it works

Take the expression outside the brackets and multiply it by each term inside, making sure to keep track of positive and negative signs. Each multiplication creates a separate term in your final answer.

lightbulbExample

Worked Example: Single Bracket Expansion

Example 1: $4a(3b - 2c)$

Multiply $4a$ by $3b$ : $4a \times 3b = 12ab$
Multiply $4a$ by $-2c$ : $4a \times (-2c) = -8ac$
Final answer: $12ab - 8ac$

Example 2: $-4(3p² - 7q²)$

Multiply $-4$ by $3p²$ : $-4 \times 3p² = -12p²$
Multiply $-4$ by $-7q²$ : $-4 \times (-7q²) = +28q²$
Final answer: $-12p² + 28q²$

Double brackets

Double brackets are more challenging because you need to multiply every term in the first bracket by every term in the second bracket. This creates four separate multiplications that you then combine.

The FOIL method

A helpful way to remember this process is using FOIL:

First: Multiply the first term in each bracket together
Outside: Multiply the outside terms (first term of first bracket × second term of second bracket)
Inside: Multiply the inside terms (second term of first bracket × first term of second bracket)
Last: Multiply the last term in each bracket together

infoNote

The FOIL method is specifically designed for binomials (expressions with two terms). It's a systematic way to ensure you multiply all terms correctly.

lightbulbExample

Worked Example: Double Bracket Expansion Using FOIL

Expanding $(2p - 4)(3p + 1)$ :

Following the FOIL method:

First: $2p \times 3p = 6p²$
Outside: $2p \times 1 = 2p$
Inside: $-4 \times 3p = -12p$
Last: $-4 \times 1 = -4$

This gives us: $6p² + 2p - 12p - 4$

The final step is combining like terms. The two $p$ terms $(2p$ and $-12p)$ combine to give $-10p$ , so our final answer is $6p² - 10p - 4$ .

Common mistakes with squared brackets

chatImportant

Critical Mistake to Avoid:

A frequent error occurs when dealing with squared brackets like $(3x + 5)²$ . Remember that this means $(3x + 5)(3x + 5)$ , NOT $3x² + 5²$ .

When you expand it properly: $(3x + 5)² = (3x + 5)(3x + 5) = 9x² + 15x + 15x + 25 = 9x² + 30x + 25$

Always expand squared brackets fully - there's no shortcut!

Triple brackets

When you encounter three brackets, don't panic! The strategy is to multiply two brackets together first, then multiply that result by the remaining bracket.

infoNote

With triple brackets, you can choose any two brackets to multiply first. The result will be the same regardless of which pair you start with.

Method for triple brackets

Choose any two brackets to multiply together first
Expand those two brackets using the FOIL method
Take your result and multiply it by the third bracket (treating it like a single bracket problem)

lightbulbExample

Worked Example: Triple Bracket Expansion

Expanding $(x + 2)(x + 3)(2x - 1)$ :

Step 1: First, multiply $(x + 2)(x + 3)$ :

$x \times x = x²$
$x \times 3 = 3x$
$2 \times x = 2x$
$2 \times 3 = 6$
Result: $x² + 3x + 2x + 6 = x² + 5x + 6$

Step 2: Then multiply $(x² + 5x + 6)$ by $(2x - 1)$ :

$x² \times 2x = 2x³$
$x² \times (-1) = -x²$
$5x \times 2x = 10x²$
$5x \times (-1) = -5x$
$6 \times 2x = 12x$
$6 \times (-1) = -6$

Step 3: Combine like terms: $2x³ - x² + 10x² - 5x + 12x - 6 = 2x³ + 9x² + 7x - 6$

bookmarkSummary

Key Points to Remember:

When multiplying single brackets, distribute the outside term to every term inside
For double brackets, use FOIL: First, Outside, Inside, Last
Always combine like terms at the end
$(a + b)²$ is NOT equal to $a² + b²$ - you must expand it fully
With triple brackets, multiply two brackets first, then multiply by the third
Keep track of positive and negative signs throughout your calculations

Multiplying Out Brackets (AQA GCSE Maths): Revision Notes