The binomial expansion Revision Notes for AQA GCSE Further Maths

The binomial expansion

Introduction to binomial expansion

When we want to expand expressions like $(a + b)^n$ where $n$ is a positive integer, we use the binomial expansion. This mathematical technique allows us to write out the full expanded form without having to multiply the brackets repeatedly.

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The key insight is that when you expand $(a + b)^n$ , you get exactly n + 1 terms in the final answer. Each term follows a specific pattern, and the coefficients (the numbers in front of each term) can be found using Pascal's triangle.

Understanding the basic pattern

Let's start by looking at what happens when we expand some simple cases:

$(a + b)^2 = a^2 + 2ab + b^2$
$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
$(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$

Notice that each term has the form Pa^r b^s, where $P$ is a coefficient and the powers $r$ and $s$ are non-negative integers whose sum equals $n$ . This means r + s = n for every term in the expansion.

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Critical Pattern Recognition: The sum of the powers in each term always equals the original power $n$ . This is your main error-checking tool when expanding binomial expressions.

Pascal's triangle

The coefficients in binomial expansions follow a beautiful pattern called Pascal's triangle. This triangular arrangement of numbers gives us all the coefficients we need.

Each row of Pascal's triangle corresponds to the expansion of $(a + b)^n$ for different values of $n$ :

Row 0: coefficients for $(a + b)^0 = 1$
Row 1: coefficients for $(a + b)^1 = 1, 1$
Row 2: coefficients for $(a + b)^2 = 1, 2, 1$
Row 3: coefficients for $(a + b)^3 = 1, 3, 3, 1$
Row 4: coefficients for $(a + b)^4 = 1, 4, 6, 4, 1$

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Construction Rule for Pascal's Triangle: The triangle has a simple construction rule: each number is the sum of the two numbers directly above it. The edges are always 1, and every row starts and ends with 1.

Working with substitutions

To expand expressions like $(2x + 3y)^5$ or $(3 - w)^4$ , we substitute our specific values into the general pattern $(a + b)^n$ .

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Worked Example: Expanding $(2x + 3y)^5$

For this expansion, we replace $a$ with $2x$ and $b$ with $3y$ in the general form.

Using row 5 of Pascal's triangle $(1, 5, 10, 10, 5, 1)$ , we get:

$(2x + 3y)^5 = 1(2x)^5(3y)^0 + 5(2x)^4(3y)^1 + 10(2x)^3(3y)^2 + 10(2x)^2(3y)^3 + 5(2x)^1(3y)^4 + 1(2x)^0(3y)^5$

Working out each term: $= 1 \times 32x^5 \times 1 + 5 \times 16x^4 \times 3y + 10 \times 8x^3 \times 9y^2 + 10 \times 4x^2 \times 27y^3 + 5 \times 2x \times 81y^4 + 1 \times 1 \times 243y^5$

$= 32x^5 + 240x^4y + 720x^3y^2 + 1080x^2y^3 + 810xy^4 + 243y^5$

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Worked Example: Expanding $(3 - w)^4$

Here we replace $a$ with $3$ and $b$ with $-w$ . Notice the negative sign!

Using row 4 of Pascal's triangle $(1, 4, 6, 4, 1)$ :

$(3 - w)^4 = 1(3)^4(-w)^0 + 4(3)^3(-w)^1 + 6(3)^2(-w)^2 + 4(3)^1(-w)^3 + 1(3)^0(-w)^4$

$= 1 \times 81 \times 1 + 4 \times 27 \times (-w) + 6 \times 9 \times w^2 + 4 \times 3 \times (-w^3) + 1 \times 1 \times w^4$

$= 81 - 108w + 54w^2 - 12w^3 + w^4$

Finding specific terms

Sometimes you only need certain terms from an expansion rather than the whole thing.

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Worked Example: First three terms of $(2 + 5x)^6$

Row 6 of Pascal's triangle starts: $1, 6, 15, 20, 15, 6, 1$ We only need the first three coefficients: $1, 6, 15$

The first three terms are: $1 \times 2^6 \times (5x)^0 + 6 \times 2^5 \times (5x)^1 + 15 \times 2^4 \times (5x)^2$

$= 1 \times 64 \times 1 + 6 \times 32 \times 5x + 15 \times 16 \times 25x^2$

$= 64 + 960x + 6000x^2$

Important exam tips

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Essential Exam Strategies:

Check your indices: After expanding any binomial expression, always verify that the sum of the powers in each term equals $n$ . This is a crucial error-checking method.
Building Pascal's triangle: If you need a row that you haven't memorised, remember that each number equals the sum of the two numbers above it. Every row starts and ends with 1.
Sign patterns: When dealing with expressions involving subtraction like $(a - b)^n$ , be extra careful with alternating positive and negative signs. The pattern depends on whether the power is even or odd.
Ascending vs descending powers: Pay attention to whether the question asks for terms in ascending or descending powers of a variable, as this affects the order you write your answer.

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Key Points to Remember:

The expansion of $(a + b)^n$ always has exactly n + 1 terms
Each term has the form $Pa^r b^s$ where r + s = n
Pascal's triangle provides all the coefficients you need
Always check that the sum of indices in each term equals $n$
Be careful with negative signs when expanding expressions like $(a - b)^n$

The binomial expansion (AQA GCSE Further Maths): Revision Notes