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📘 Algebra • Lesson 7 of 22

Expanding Brackets

Expanding brackets means multiplying the number or expression outside the bracket by the correct term or terms inside the bracket.

✏️ Algebra 📘 Worked examples included 🧠 Step-by-step learning

Learning Objectives

Introduction

This lesson develops Expanding Brackets from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.

Key Notes

Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.

🎬 Distribute to every term

3(x+4)

The term outside multiplies every term inside.

🤔 Think About This...

A teacher is preparing prize packs for learners. Each prize pack contains:

There are 5 learners. Instead of writing the same thing five times, we can write:

5(2 books + 1 pencil)

Now multiply everything correctly.

5 × 2 books = 10 books
5 × 1 pencil = 5 pencils
5(2 books + 1 pencil) = 10 books + 5 pencils

This is exactly what we do when expanding brackets in Algebra.

📖 What Does "Expand" Mean?

To expand brackets means multiplying the number or expression outside the bracket by the correct term or terms inside the bracket.

Expanding brackets is simply multiplying correctly.

🧠 The StudyNest Thinking Method

Before expanding any brackets, always ask yourself:

👉 How many terms are inside the brackets?

A term is a part of an expression separated by a plus (+) or minus (−) sign.

x + 2

There are 2 terms.

2x

There is only 1 term because there is no plus or minus sign.

Do not count numbers and letters separately. 2x is one term.

📝 Worked Examples

Example 1

Expand:

3(x + 2)

🔍 StudyNest Question: How many terms are inside the brackets?

There are 2 terms.

  • x
  • 2

Multiply each term once.

3 × x = 3x
3 × 2 = 6
3(x + 2) = 3x + 6

Example 2

Expand:

5(2x)

🔍 StudyNest Question: How many terms are inside the brackets?

Only one term.

2x

Multiply the whole term.

5 × 2x = 10x
5(2x) = 10x
Do NOT do this:

5 × 2 = 10
5 × x = 5x ❌

Example 3

Expand:

2(3x + 5)

🔍 StudyNest Question: How many terms are inside the brackets?

There are 2 terms.

  • 3x
  • 5

Multiply each term once.

2 × 3x = 6x
2 × 5 = 10
2(3x + 5) = 6x + 10

Example 4

Expand:

7(y − 3)

Multiply the 7 by both terms.

7 × y = 7y
7 × (−3) = −21
7(y − 3) = 7y − 21

⭐ Expanding Two Brackets: The FOIL Method

So far, we have expanded expressions with one bracket.

We can also multiply two brackets, such as:

(x + 2)(x + 5)

Each term in the first bracket must multiply each term in the second bracket.

FOIL is a shortcut that helps us remember all four multiplications:

F – First
O – Outside
I – Inside
L – Last

Example: Expand (x + 2)(x + 5)

F
First terms

x × x = x²

O
Outside terms

x × 5 = 5x

I
Inside terms

2 × x = 2x

L
Last terms

2 × 5 = 10

Now write all four products:

x² + 5x + 2x + 10

Collect the like terms:

x² + 7x + 10
(x + 2)(x + 5) = x² + 7x + 10
Important:

FOIL is only a memory shortcut for multiplying two brackets containing two terms each.

The real rule is still: multiply every term in one bracket by every term in the other bracket.

🌍 Real-Life Example

A school is preparing prize packs. Each pack contains:

There are 8 prize packs.

8(3 pens + 2 pencils)

Multiply each item by 8.

24 pens + 16 pencils

Expanding brackets simply means multiplying every term in the bracket correctly.

⚠ Common Mistakes

Mistake 1

Forgetting to multiply the second term.

3(x + 5) ≠ 3x + 5 ❌
Mistake 2

Treating 2x as two separate terms.

Remember: 2x is ONE term.
Mistake 3

Forgetting that multiplying a negative number gives a negative answer.

Practice Questions

Expand the following expressions.

  1. 2(x + 7)
  2. 4(a + 3)
  3. 6(y − 2)
  4. 3(2m + 5)
  5. 9(p − 4)
  6. 5(4x)
  7. 8(2a − 1)
✅ Show Answers
  1. 2x + 14
  2. 4a + 12
  3. 6y − 12
  4. 6m + 15
  5. 9p − 36
  6. 20x
  7. 16a − 8

🧠 Did You Know?

Factories, supermarkets and warehouses use this same idea every day. If one box contains 5 bottles, then 200 boxes contain:

200 × 5 bottles

Expanding brackets is simply multiplying groups.

⭐ What You Can Do Now

🎉 Well Done!

You can now expand brackets correctly. This skill will help you simplify expressions, solve equations and understand factorisation.

Remember: First count the terms, then multiply each term once.

🧠 Let’s Think Together

Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.

📈 Your Progress

Lesson 7 of 22

Course Progress 44%

Exam Focus

Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.

Summary