Binomials

In algebra, a binomial is a polynomial with exactly two terms, typically written as A ± B, where A and B are monomials or polynomials. Examples include x + y and a^2 − b^3.

The most important result involving binomials is the binomial theorem, which gives the expansion of (x + y)^n for a nonnegative integer n. It states (x + y)^n = sum_{k=0}^n binom(n, k) x^{n−k} y^k, where binom(n, k) = n!/(k!(n−k)!). The binomial coefficients form Pascal's triangle and are symmetric: binom(n, k) = binom(n, n−k).

A generalized binomial theorem extends to any exponent α: (1 + x)^α = sum_{k=0}^∞ binom(α, k) x^k, valid for |x| < 1, with binom(α, k) = α(α−1)...(α−k+1)/k!.

Binomial coefficients have a combinatorial interpretation: binom(n, k) counts the number of k-element subsets of an n-element set. They appear in probability as the basis of the binomial distribution and in many combinatorial identities and series expansions.

In practice, binomials are used to expand expressions, factor polynomials, approximate powers, and reason about counts and probabilities. The term binomial comes from Latin bi- meaning two and nomen meaning name, reflecting the two-term structure.