Binomial Theorem
The Binomial Theorem domain tests expansion of (a + b)ⁿ, isolating a particular term or coefficient, and the infinite binomial series for a rational index.
Core rules
(a + b)ⁿ = Σ nCr a^(n−r) b^r. The general term is T_(r+1) = nCr a^(n−r) b^r, so the term you want is the (r + 1)th, not the rth. If n is even there is one middle term, T_(n/2+1); if n is odd there are two, T_((n+1)/2) and T_((n+3)/2). Also Σ nCr = 2ⁿ, Σ (−1)^r nCr = 0, and Σ (nCr)² = 2nCn.
For a rational index, (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + …, valid only when |x| < 1; this series is infinite and never terminates.
Worked example
Find the term independent of x in (x² + 2/x)⁹. Here T_(r+1) = 9Cr (x²)^(9−r) (2/x)^r = 9Cr 2^r x^(18−3r). The power of x vanishes when 18 − 3r = 0, that is r = 6. The term is 9C6 × 2⁶ = 84 × 64 = 5376.
Traps
Using r instead of r + 1; dropping the constant factor 2^r when only the binomial coefficient is recorded; using the rational form while |x| ≥ 1; assuming a single middle term when n is odd.