Binomial

A binomial is a polynomial with exactly two terms separated by addition or subtraction. Examples: $x + 3$, $2x^2 - 5$, $a - b$.

The binomial theorem expands $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$. The coefficients $\binom{n}{k}$ are entries of Pascal's triangle.

Special cases used constantly: $(a + b)^2 = a^2 + 2ab + b^2$, $(a + b)(a - b) = a^2 - b^2$ (difference of squares), $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.

Beyond algebra, binomials appear in probability (binomial distribution), combinatorics (binomial coefficients), and calculus (binomial series).