Binomial Theorem
The Binomial Theorem expands (a + b)ⁿ for a positive integer n. Examiners use it for the general term, the middle term, a term independent of a variable, and simple sums of binomial coefficients.
The expansion
(a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ for r from 0 to n. The general term is T_{r+1} = nCr aⁿ⁻ʳ bʳ; the +1 matters because counting starts at r = 0, and the expansion has n + 1 terms. If n is even there is a single middle term T_{n/2 + 1}. If n is odd there are two middle terms, T_{(n+1)/2} and T_{(n+3)/2}. Symmetry gives nCr = nC_{n−r}. Setting a = b = 1 gives C₀ + C₁ + ... + Cₙ = 2ⁿ; setting a = 1, b = −1 gives the alternating sum equal to 0.
Worked example
Find the term independent of x in (2x² − 1/x)⁶. The general term is 6Cr (2x²)⁶⁻ʳ (−1/x)ʳ = 6Cr · 2⁶⁻ʳ · (−1)ʳ · x^{12−3r}. For the term to be free of x, 12 − 3r = 0, so r = 4. The coefficient is 6C₄ · 2² · (−1)⁴ = 15 × 4 = 60.
Traps
- Labelling the general term with r instead of r + 1.
- Dropping the sign when the second term is negative and r is odd.
- Forgetting the power of the numerical coefficient, here 2⁶⁻ʳ.
- Reporting one middle term when n is odd and there should be two.