Binomial expansion formula: (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ
The binomial expansion formula is (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, with general term T(r+1) = nCr aⁿ⁻ʳ bʳ.
What each letter means
- \(n\) the power (a positive whole number)
- \(\binom{n}{r}\) the binomial coefficient nCr
- \(r\) which term: T₁ has r = 0
- \(a,\ b\) the two terms in the bracket, signs included
When to use it
To expand a bracket raised to a power without multiplying it out, or to find one term or coefficient (for example the x³ term) straight away.
AA HL also extends the expansion to rational n with |x| < 1.
Worked example
Find the coefficient of \(x^3\) in the expansion of \((2+x)^7\).
- The \(x^3\) term has \(r=3\): \(\binom73\,2^{4}x^3\)
- \(\binom73=35\), \(2^4=16\)
Answer: \(35\times16=560\)
Common mistake
Forgetting to raise the number with x to the power: in (1 + 3x)⁵ the r = 2 term has (3x)² = 9x², not 3x².
On your course
| Course | In the exam |
|---|---|
| AA SL | In the IB formula booklet |
| AA HL | In the IB formula booklet |
From our own IB Maths formula sheets, in our words. The IB gives its booklet to schools, so we checked against our own copy: your teacher has the official one. Official: IB DP Mathematics page.
Practise and revise
- Practise AA SL binomial theorem questions
- Practise AA HL counting and binomial questions
- Revise Binomial expansion (AA SL)
- Revise Binomial theorem (AA HL)
- Print AA SL one-page formula sheet
Questions
What is the binomial expansion formula?
The binomial expansion formula is (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, with general term T(r+1) = nCr aⁿ⁻ʳ bʳ. n: the power (a positive whole number); nCr: the binomial coefficient nCr; r: which term: T₁ has r = 0; a, b: the two terms in the bracket, signs included.
Is the binomial expansion given in the exam?
AA SL: in the IB formula booklet. AA HL: in the IB formula booklet. This comes from our own IB 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 IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.