Binomial expansion formula: (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + …
The binomial expansion formula is (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + ….
What each letter means
- \(n\) the power: a positive integer (the series stops) or any rational number (it goes on for ever)
- \(x\) for rational n, the series is valid only when |x| < 1
When to use it
To expand (a + bx)ⁿ: as a finite expansion when n is a positive integer, or as an approximation when n is negative or a fraction. For (a + bx)ⁿ take out aⁿ first: aⁿ(1 + (b/a)x)ⁿ.
Worked example
Expand \((1-2x)^{-\frac12}\) up to the term in \(x^2\), and state the values of \(x\) for which it is valid.
- \(n=-\tfrac12\), and \(-2x\) plays the part of \(x\)
- \(1+(-\tfrac12)(-2x)+\dfrac{(-\frac12)(-\frac32)}{2}(-2x)^2\)
Answer: \(1+x+\tfrac32x^2\), valid for \(|x|<\tfrac12\)
Common mistake
Forgetting the validity condition. (1 − 2x)^(−½) is valid only for |2x| < 1, that is |x| < ½.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) Binomial, \(n\in\mathbb N\) | In the Edexcel formulae booklet |
| Edexcel A Level (9MA0) Binomial, \(|x|<1\), \(n\in\mathbb Q\) | In the Edexcel formulae booklet |
| Edexcel International A Level Binomial, \(n\in\mathbb N\) | In Pearson’s IAL formulae booklet |
| Edexcel International A Level Binomial, \(|x|<1\), \(n\in\mathbb Q\) | In Pearson’s IAL formulae booklet |
| AQA or OCR A Level | Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page |
From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).
Practise and revise
- Practise A Level Maths practice
- Revise Sequences and series
- Revise IAL P2 binomial expansion
- Revise IAL P4 binomial expansion
- Print Edexcel A Level (9MA0) one-page formula sheet
Questions
What is the binomial expansion formula?
The binomial expansion formula is (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + …. n: the power: a positive integer (the series stops) or any rational number (it goes on for ever); x: for rational n, the series is valid only when |x| < 1.
Is the binomial expansion given in the exam?
Edexcel A Level (9MA0) (Binomial, n N): in the Edexcel formulae booklet. Edexcel International A Level (Binomial, n N): in Pearson’s IAL formulae booklet. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.
How do I find one term without expanding everything?
Use the general term T(r+1) = nCr aⁿ⁻ʳ bʳ and choose r so that the power of x is the one you want.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.