A Level Maths: General Binomial Expansion Topic Summary and Resources
Video Lessons
Watch alongside the worksheet for the full lesson experience, then test your understanding with the lesson questions.
Revision Notes
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Exam Questions
Past-paper-style questions organised by topic, with full mark schemes.
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Drawn from OCR and Edexcel past papers but designed to be useful for students of all UK exam boards — including AQA and OCR MEI — unless a sheet is explicitly board-specific.
Before You Start This Topic
It will help if you are confident with the following:
- Binomial Expansion — the positive integer case is the direct prerequisite
- Rational Expressions — partial fractions are needed for expanding harder rational functions
- Indices — needed for handling negative and fractional powers
A Level Maths general binomial expansion extends Binomial Expansion from positive integer powers to ANY rational power $n$. This gives an infinite series rather than a finite sum, which lets you generate polynomial approximations to functions like $(1 + x)^{1/2}$, $(1 – 2x)^{-3}$, and many others.
You expand $(1 + x)^n$ for rational $n$ using the formula: $1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \ldots$ You apply this to expressions of the form $(a + bx)^n$ by first factorising out $a^n$ to leave $(1 + (b/a)x)^n$. The expansion is only valid for $|bx/a| < 1$ — you must always state the range of validity. You combine the general binomial expansion with Rational Expressions (decomposing into partial fractions first, then expanding each piece) to expand more complex rational functions. Common applications include approximating square roots, cube roots, and reciprocals; you check accuracy by comparing your truncated series to known values.
General binomial expansion is part of the Pure Maths strand of A Level Maths for AQA, Edexcel, OCR, and OCR MEI students.
Watch out for…
A few things to be careful with: the expansion is INFINITE for non-integer $n$, so you only ever write the first few terms (usually as many as the question asks for); the validity condition $|bx/a| < 1$ must be stated and applied — substituting a value of $x$ outside this range gives a meaningless result; the formula starts with $x^0 = 1$ — do not forget the leading $1$; and after factorising $a$ out, do NOT forget to multiply the whole expansion by $a^n$ at the end.