Mastering Expanding Brackets

From simple distribution to tackling double brackets with confidence.

📚 GCSE / A-Level Foundation 🎯 Difficulty: ⭐ ⏱️ Reading time: 15 minutes 📋 All Exam Boards

📚 Introduction

Expanding brackets is a fundamental skill in algebra. It's the process of removing brackets from an expression by multiplying everything inside the bracket by the term outside. It's like unlocking a puzzle to reveal a simpler expression inside.

🎯 Learning Objectives

Expand expressions with single brackets.
Expand the product of two brackets (double brackets) using the FOIL method.
Use the grid method as a visual alternative for expanding.
Simplify expressions by collecting like terms after expanding.

📝 Expanding Single Brackets

The rule is simple: multiply the term on the outside by every term on the inside. This is called the distributive law.

📋 Example 1: Expand $5(x + 3)$

Multiply 5 by $x$ and then multiply 5 by 3.

5(x+3) = (5 \times x) + (5 \times 3) = \boxed{5x + 15}

📋 Example 2: Expand $3x(2x - 7)$

Be careful with the signs and indices!

3x(2x - 7) = (3x \times 2x) + (3x \times -7) = \boxed{6x^2 - 21x}

🚀 Expanding Double Brackets (The FOIL Method)

When you have two brackets multiplied together, like $(a+b)(c+d)$, you need to multiply every term in the first bracket by every term in the second. A great way to remember this is **FOIL**.

📖 FOIL Method

First: Multiply the first terms in each bracket.
Outer: Multiply the outermost terms.
Inner: Multiply the innermost terms.
Last: Multiply the last terms in each bracket.

📋 Example 3: Expand $(x+4)(x+5)$

(x+4)(x+5) $$ $$ \text{F: } x \times x = x^2 $$ $$ \text{O: } x \times 5 = 5x $$ $$ \text{I: } 4 \times x = 4x $$ $$ \text{L: } 4 \times 5 = 20 $$ $$ \text{Combine them: } x^2 + 5x + 4x + 20 $$ $$ \text{Simplify by collecting like terms: } \boxed{x^2 + 9x + 20}

Visualising with the Grid Method

If FOIL feels a bit abstract, the grid method is a fantastic visual tool. It works just like a multiplication grid you might have used in primary school.

📋 Example 4: Expand $(x-3)(2x+1)$ using the grid method

Draw a 2x2 grid. Write the terms of the first bracket along the side and the terms of the second bracket along the top.

×	2x	+1
x	$2x^2$	$+x$
-3	$-6x$	$-3$

Now, add up all the terms from the results boxes and simplify:

2x^2 + x - 6x - 3 = \boxed{2x^2 - 5x - 3}

⚠️ Common Mistakes

🚫 Mistake 1: Forgetting the signs

What students do wrong: When expanding $(x-5)(x+2)$, they might write $x^2 + 2x + 5x + 10$.

Why it's wrong: The 5 is negative. You must multiply by $-5$, not 5.

How to avoid it: Always include the sign with the term. The inner multiplication is $(-5) \times x = -5x$. The last is $(-5) \times 2 = -10$. The correct expansion is $x^2 + 2x - 5x - 10 = x^2 - 3x - 10$.

🚫 Mistake 2: Only multiplying the first terms

What students do wrong: For $4(x+y)$, they write $4x+y$.

Why it's wrong: The distributive law says you must multiply the outer term by all inner terms.

How to avoid it: Draw arrows from the outer term to every inner term to remind yourself to multiply everything. $4(x+y) = 4x + 4y$.

💪 Practice Problems

🎯 Practice Questions

Expand and simplify the following expressions:

$7(2x - 3)$
$-4y(y^2 + 5)$
$(x+2)(x+6)$
$(x-8)(x+3)$
$(2x+1)(3x-4)$

🔍 Show Solutions

1. $14x - 21$

2. $-4y^3 - 20y$

3. $x^2 + 8x + 12$

4. $x^2 - 5x - 24$

5. $6x^2 - 5x - 4$

📋 Summary

🎯 Key Takeaways

Expanding brackets means multiplying out terms to remove the brackets.
For single brackets, multiply the outside term by every term inside.
For double brackets, use the FOIL or Grid method to ensure you multiply all term pairs.
Always simplify your final answer by collecting any like terms.