Binomial Expansion Basics For A Level Maths

Hello A Level Pure Maths students! Binomial expansion is a topic you’ll encounter time and again on your practice exam papers, so it’s certainly worth investing the time and energy in developing a solid understanding of the underlying principles. 

We know that the long and complex-looking formulae can seem a little daunting, but don’t let this put you off! In this blog post, our in-house Maths expert Simon breaks down every aspect of this topic specifically to help out confused A Level students

Once you’ve read through the explanation a couple of times and tried the example question, make sure to keep building on your knowledge by exploring the more advanced aspects of the topic. You’ll find the links to the relevant revision notes (complete with diagrams and more worked examples) at the bottom of this post. 

 Getting started: the Binomial Theorem

We use this theorem to calculate each of the terms in the expansion. 

Now, we could expand (a+b)^{n} in a step-by-step method, however this would be time consuming and prone to error. 

Instead, we can use this theorem:

Using the Theorem in practice

When applying this theorem, it’s important to work slowly and carefully, using new lines for each step and always double-checking your answers. 

Remember that ! means factorial 

5! = 5x4x3x2x1 

The below example shows each of the terms of the expansion of  (3+2x)^{4}

Calculating nCr on a calculator

Although you may be able to work out the first lines of answers in your head, you’ll need a calculator for the majority.

Use the nCr button on your scientific calculator as below. 

Important points to remember

If asked to calculate a specific term in the expansion, Always go for the r value one lower than the required term (because ^{n}\textrm{C}r starts at r = 0). For example, to find the 4th term, you are looking for ^{n}\textrm{C}_{3} not ^{n}\textrm{C}_{4}

  • Look at the pattern
    • Start at ,^{n}\textrm{C}_{0} then ^{n}\textrm{C}_{1}, ^{n}\textrm{C}_{2} etc
    • Powers of a start at n and decrease by 1
    • Powers of b start at 0 and increase by 1
  • These are the shortcuts (but they hide the pattern): 
    • ^{n}\textrm{C}_{0} = ^{n}\textrm{C}_{n} = 1
    • {{}^{n}\textrm{C}}_{1} = n
    • (b)^{0}=(a)^{0}=1

Using Binomial Expansion to solve exam-style questions 

You could be asked to calculate the coefficient of a particular term in an expansion: 

Or to find the first few terms of the expansion:

Or to solve problems involving unknowns:

Worked example 

Ready for a worked example? Try this on your own before scrolling down to the answer. 

Find the first four terms, in ascending powers of x, in the expansion of (2-3x)^{6}

Did you get the correct answer? Ready to keep going? Find the full set of revision notes and many more worked examples here: 

General Binomial Expansion 

GBE: Subtleties 

Multiple GBEs

Approximating Values