What should I learn from this Further Maths lesson?

By the end of this lesson, you should be able to Expand single brackets by multiplying the term outside by each term inside; Expand and simplify expressions involving two or more brackets; Identify and collect like terms to simplify algebraic expressions; Apply expanding and collecting like terms to form and simplify expressions in context.

How should I revise Expanding Brackets and Collecting Like Terms?

Read each section, then close the notes and explain the idea from memory. Check your explanation against the page and finish with an exam question from Algebra.

Expanding Brackets and Collecting Like Terms

Expanding Brackets and Collecting Like Terms is part of Expanding brackets and collecting like terms in GCSE AQA Higher Further Maths.

What this lesson covers

Expanding Brackets and Collecting Like Terms: Learn to expand single brackets and simplify expressions by collecting like terms. This lesson covers the…

Expand single brackets by multiplying the term outside by each term inside
Expand and simplify expressions involving two or more brackets
Identify and collect like terms to simplify algebraic expressions
Apply expanding and collecting like terms to form and simplify expressions in context

Key ideas to keep in view

Use this lesson to stay inside Expanding brackets and collecting like terms while connecting the detail back to the wider Algebra topic.

Algebra
Expanding brackets and collecting like terms
GCSE AQA Higher Further Maths

Revision notes

Read the core explanations from Expanding Brackets and Collecting Like Terms before testing yourself from memory.

1. Why Expanding Brackets Matters

Algebra is the language of problem-solving, and brackets are everywhere in it. The moment you can confidently remove them and tidy up what remains, a huge range of GCSE questions open up — from forming equations to proving identities.

Think of a bracket like a set of instructions in a bag. The number or term outside the bag tells you to give that instruction to everything inside. Expanding means unpacking the bag completely — nothing gets left behind.

2. Expanding a Single Bracket

The rule is straightforward: multiply the outside term by the first term inside, then multiply the outside term by the second term inside. The bracket disappears and you are left with two separate terms.

Watch the signs carefully. If the outside term is negative, every term inside changes sign when you multiply. This is where most marks are dropped.

Worked example: Expanding a Single Bracket

Expand $3(2x + 5)$

Outside term: $3$ — This is the multiplier for everything inside the bracket.
$3 \times 2x = 6x$ — Multiply the coefficient: $3 \times 2 = 6$, keep the variable $x$. [M1]
$3 \times 5 = 15$ — Multiply the constant: $3 \times 5 = 15$.
$3(2x + 5) = 6x + 15$ — Both terms written without brackets. [A1]

Answer: $6x + 15$

Worked example: Expanding with a Negative Outside Term

Expand $-4(3x - 2)$

Outside term: $-4$ — The negative sign belongs to the outside term and affects every multiplication.
$-4 \times 3x = -12x$ — Negative times positive gives negative. [M1]
$-4 \times -2 = +8$ — Negative times negative gives positive — a very common trap.
$-4(3x - 2) = -12x + 8$ — Both terms written without brackets. [A1]

Answer: $-12x + 8$

3. Collecting Like Terms

Once brackets are removed, the expression often contains several terms. Collecting like terms means grouping and combining terms that share the same variable and power, so the expression is written as concisely as possible.

Worked example: Collecting Like Terms

Simplify $5x + 3 - 2x + 7$

$5x$ and $-2x$ — Both contain $x$ to the power 1, so they are like terms.
$5x - 2x = 3x$ — Subtract the coefficients: $5 - 2 = 3$. [M1]
$3$ and $7$ — Both are numbers with no variable, so they are like terms.
$3 + 7 = 10$ — Add the constants: $3 + 7 = 10$.
$3x + 10$ — The expression is now in its simplest form. [A1]

Answer: $3x + 10$

4. Expanding and Then Collecting Like Terms

Most exam questions combine both skills: expand the brackets first, then collect like terms to reach the simplest form. The two-stage approach — expand, then simplify — keeps working tidy and reduces errors.

When an expression has two separate brackets added or subtracted, expand each bracket independently before combining anything. Trying to shortcut this step is the most reliable way to drop marks.

Worked example: Expand and Simplify: Two Brackets

Expand and simplify $2(3x + 4) + 5(x - 1)$

$2(3x + 4) = 6x + 8$ — $2 \times 3x = 6x$ and $2 \times 4 = 8$. [M1]
$5(x - 1) = 5x - 5$ — $5 \times x = 5x$ and $5 \times -1 = -5$. [M1]
$6x + 8 + 5x - 5$ — Place all four terms in one line before collecting.
$6x + 5x = 11x$ — Add the coefficients: $6 + 5 = 11$.
$8 - 5 = 3$ — Subtract: $8 - 5 = 3$.
$11x + 3$ — Fully simplified. [A1]

Answer: $11x + 3$

Worked example: Expand and Simplify: Subtracting a Bracket

Expand and simplify $4(2x + 3) - 3(x + 5)$

$4(2x + 3) = 8x + 12$ — $4 \times 2x = 8x$ and $4 \times 3 = 12$. [M1]
$-3(x + 5) = -3x - 15$ — The minus sign in front means the outside term is $-3$. So $-3 \times x = -3x$ and $-3 \times 5 = -15$. [M1]
$8x + 12 - 3x - 15$ — Both expansions written out in full before collecting.
$8x - 3x = 5x$ — Subtract the coefficients: $8 - 3 = 5$.
$12 - 15 = -3$ — Subtract: $12 - 15 = -3$.
$5x - 3$ — Fully simplified. [A1]

Answer: $5x - 3$

6. Exam Technique and Common Mistakes

The most reliable way to lose marks on these questions is to let sign errors creep in. Before you write your final answer, re-read every term and check whether the sign in front of it is correct.

On Paper 1 (non-calculator), all arithmetic must be shown clearly. Write each expansion on its own line, then collect — never try to do both stages in your head at once.

Sources and content standards

Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-14.

AQA Level 2 Certificate in Further Mathematics (8365) specification How we check content Report a correction — include this lesson URL

Expanding Brackets and Collecting Like Terms Revision Notes | GCSE AQA Higher Further Maths