The Binomial Theorem states (x+y)^n = sum_{r=0}^{n} C(n,r)*x^{n-r}*y^r for positive integer n, producing n+1 terms. The general term T_{r+1} = C(n,r)*x^{n-r}*y^r is the foundation of all binomial problems. The binomial coefficients C(n,r) = n!/(r!(n-r)!) satisfy symmetry C(n,r) = C(n,n-r), Pascal's rule C(n,r) = C(n-1,r-1) + C(n-1,r), and the sum identity C(n,0)+C(n,1)+...+C(n,n) = 2^n.

The middle term is T_{n/2+1} for even n (one term) or T_{(n+1)/2} and T_{(n+3)/2} for odd n (two terms). The greatest binomial coefficient is C(n, floor(n/2)). The numerically greatest term is found by the ratio method: T_{r+1}/T_r = (n-r+1)|y|/(r|x|), and the greatest term occurs at the transition point where this ratio crosses 1.

For finding specific coefficients, the general term is the workhorse: set the exponent of the desired variable to the target value, solve for r, and compute C(n,r) times any multiplicative constants. The term independent of x in expressions like (x^a + b/x^c)^n requires solving a(n-r) - cr = 0.

The generalized binomial theorem extends to any real exponent alpha: (1+x)^alpha = 1 + alpha*x + alpha(alpha-1)*x^2/2! + ... for |x| < 1. This infinite series is used for approximation: (1+x)^n is approximately 1+nx for small |x|.

Applications span three categories: (1) coefficient extraction and manipulation, (2) divisibility and remainder problems using expansions like (1+m)^n, and (3) summation of series involving binomial coefficients using differentiation, integration, or substitution of special values (x=1, -1, i, omega).