Maths353 views·Updated 20 Jul 2026·3 pages

Mastering the Binomial Expansion

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Binomial Expansion Basics

Ever wondered how to expand something like $x+2$ ^8 without multiplying it out eight times? Binomial expansion is your answer, and it works for three main types of expressions.

The binomial theorem gives us a shortcut to expand any binomial raised to a power. Instead of multiplying $a+b$$a+b$$a+b$ ... repeatedly, we can use patterns to write out the answer directly.

Pascal's triangle is brilliant for finding coefficients. Each row gives you the numbers you need - just start with 1 at the top, then each number below is the sum of the two numbers above it. Row 4 gives you 1, 4, 6, 4, 1 for expanding something to the power of 4.

When expanding $a+b$ ^n, the powers of 'a' start at n and decrease by 1 each term, whilst the powers of 'b' start at 0 and increase by 1 each term. The coefficients come straight from Pascal's triangle.

Quick tip: The row number in Pascal's triangle matches the power you're expanding to!

of 3

Working Through Examples

Let's see binomial expansion in action with some proper examples that show you exactly how it works.

For $x+2$ ^6, you'll use row 6 of Pascal's triangle: 1, 6, 15, 20, 15, 6, 1. Each term follows the pattern: coefficient × x^(decreasing power) × 2^(increasing power). So you get x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64.

The tricky bit comes with negative terms like $2x - 3y$ ^4. Remember that $-3y$ raised to odd powers stays negative, but even powers become positive. This gives you 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4.

Setting up a table can really help you keep track of all the calculations. List your coefficients, then work out each term systematically - don't try to do it all in your head!

Watch out: When you've got constants multiplying your variables (like the 2 in 2x), you need to raise the entire term to the power, not just the variable.

of 3

Binomial Coefficients and Combinations

When you need to expand something massive like $a+b$ ^20, Pascal's triangle becomes impractical. That's where binomial coefficients using combinations come to the rescue.

Factorial notation $like 3! = 3×2×1$ helps us calculate how many ways we can choose objects. The formula for choosing r objects from n objects is nCr = n!/ $r!(n-r)!$ , and this is exactly what gives us our binomial coefficients.

In any binomial expansion $a+b$ ^n, the coefficient of the term a^r × b^ $n-r$ is nCr. This works because you're essentially choosing which r brackets contribute the 'a' term, whilst the remaining $n-r$ brackets contribute the 'b' term.

Your calculator probably has an nCr button, which makes finding specific terms in large expansions dead easy. No more drawing out massive triangles or doing hundreds of multiplications!

Exam tip: You can find any specific term in a binomial expansion without working out the whole thing - just use nCr for the coefficient you need.

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Maths353 views·Updated 20 Jul 2026·3 pages

Mastering the Binomial Expansion

Jess@jess_eqxp

Add to Google sources

Binomial expansion is a clever way to expand expressions like (a+b)^n without doing loads of tedious multiplication. It's built on patterns you can spot and techniques that'll save you ages in your A-level maths exams.

of 3

Binomial Expansion Basics

Ever wondered how to expand something like $x+2$ ^8 without multiplying it out eight times? Binomial expansion is your answer, and it works for three main types of expressions.

The binomial theorem gives us a shortcut to expand any binomial raised to a power. Instead of multiplying $a+b$$a+b$$a+b$ ... repeatedly, we can use patterns to write out the answer directly.

Quick tip: The row number in Pascal's triangle matches the power you're expanding to!

of 3

Working Through Examples

Let's see binomial expansion in action with some proper examples that show you exactly how it works.

Setting up a table can really help you keep track of all the calculations. List your coefficients, then work out each term systematically - don't try to do it all in your head!

Watch out: When you've got constants multiplying your variables (like the 2 in 2x), you need to raise the entire term to the power, not just the variable.

of 3

Binomial Coefficients and Combinations

When you need to expand something massive like $a+b$ ^20, Pascal's triangle becomes impractical. That's where binomial coefficients using combinations come to the rescue.

Your calculator probably has an nCr button, which makes finding specific terms in large expansions dead easy. No more drawing out massive triangles or doing hundreds of multiplications!

Exam tip: You can find any specific term in a binomial expansion without working out the whole thing - just use nCr for the coefficient you need.

We thought you’d never ask...

You can download the app from Google Play Store and Apple App Store.

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Students love us — and so will you.

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Binomial Expansion Basics

Working Through Examples

Binomial Coefficients and Combinations

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