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22 Dec 2025

•

3 pages

Understanding Binomial Expansion

Lauren Stocker

@laurenstocker_123

Ever wondered why expanding brackets like $(a+b)^5$ creates such predictable... Show more

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Chapter 8,
Binomial Expansion
Pascal's Triangle
(a+b)⁰
(a+b)
(a+b)²
(a+b)³
(a+b)*
1
4
5
1
la + 1b
1a²+ 2ab + lb
ta³ + 3a²b
Examples:
1) Expa

Pascal's Triangle and Basic Binomial Expansion

Think of Pascal's triangle as your mathematical cheat sheet for expanding brackets. Each row gives you the coefficients for different powers - row 0 for $(a+b)^0$ , row 1 for $(a+b)^1$ , and so on.

The pattern is dead simple: start with 1, and each number below is the sum of the two numbers above it. For $(a+b)^4$ , you get coefficients 1, 4, 6, 4, 1, which create $a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$ .

When expanding expressions like $(x+2y)^3$ or $(2+3x)^4$ , you substitute your values for $a$ and $b$ , then apply the pattern. Remember to be extra careful with negative terms - $(1-2x)^3$ becomes $1 - 6x + 12x^2 - 8x^3$ because those minus signs create alternating positive and negative terms.

Top tip: Always double-check your arithmetic when dealing with coefficients and powers - small mistakes multiply quickly!

Finding Unknown Coefficients

Here's where binomial expansion gets really useful for problem-solving. When you're told that the coefficient of $x^2$ in $(2+ax)^3$ equals 54, you can work backwards to find the unknown value.

Expand the expression systematically: $(2+ax)^3 = 8 + 12ax + 6a^2x^2 + a^3x^3$ . The coefficient of $x^2$ is $6a^2$ , so you set $6a^2 = 54$ and solve to get $a = ±3$ .

This technique appears constantly in exam questions. You might need to find relationships between coefficients too - like when the coefficient of $x^2$ equals six times the coefficient of $x$ . Set up your equation carefully and solve step by step.

Exam hack: Write out the first few terms clearly before identifying the specific coefficient you need - rushing leads to silly errors.

Using Calculator Functions and nCr Notation

Your calculator becomes incredibly powerful once you master the nCr function. This "n choose r" notation calculates $^nC_r = \frac{n!}{r!(n-r)!}$ , which gives you the exact coefficient for any term in a binomial expansion.

For $(3-2x)^5$ , instead of drawing Pascal's triangle, use $^5C_0$ , $^5C_1$ , $^5C_2$ , etc. to get coefficients 1, 5, 10, 10, 5, 1. Then multiply by the appropriate powers of your terms: $^5C_2(3)^3(-2x)^2 = 10 × 27 × 4x^2 = 1080x^2$ .

This method is essential for larger powers where Pascal's triangle becomes impractical. Your calculator's factorial function (!) makes these calculations instant, so you can tackle expressions like $(2+\frac{x}{2})^5$ without breaking a sweat.

Calculator tip: Learn where the nCr and factorial functions are on your specific calculator - they're absolute lifesavers in exams.

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Maths

•

22 Dec 2025

•

3 pages

Understanding Binomial Expansion

Lauren Stocker

@laurenstocker_123

Ever wondered why expanding brackets like $(a+b)^5$ creates such predictable patterns? Binomial expansionis a brilliant mathematical shortcut that lets you expand any power of two terms without tedious multiplication. You'll discover how Pascal's triangle and combination notation make even... Show more

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Pascal's Triangle and Basic Binomial Expansion

Top tip: Always double-check your arithmetic when dealing with coefficients and powers - small mistakes multiply quickly!

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Finding Unknown Coefficients

Here's where binomial expansion gets really useful for problem-solving. When you're told that the coefficient of $x^2$ in $(2+ax)^3$ equals 54, you can work backwards to find the unknown value.

Expand the expression systematically: $(2+ax)^3 = 8 + 12ax + 6a^2x^2 + a^3x^3$ . The coefficient of $x^2$ is $6a^2$ , so you set $6a^2 = 54$ and solve to get $a = ±3$ .

Exam hack: Write out the first few terms clearly before identifying the specific coefficient you need - rushing leads to silly errors.

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Using Calculator Functions and nCr Notation

Calculator tip: Learn where the nCr and factorial functions are on your specific calculator - they're absolute lifesavers in exams.