Binomial
A binomial is a polynomial with exactly two unlike terms connected by addition or subtraction.
Formula
a + b \; \text{ or } \; a - b
Definition
A binomial is a polynomial with exactly two unlike terms, connected by addition or subtraction ("bi" means two). Binomials can be multiplied using the distributive property or the FOIL method, and special products recur often: the difference of squares $(a+b)(a-b) = a^2 - b^2$ and the perfect square $(a+b)^2 = a^2 + 2ab + b^2$. The general Binomial Theorem gives the full expansion $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$, where $\binom{n}{k} = n!/(k!(n-k)!)$ are binomial coefficients; Newton's generalization extends this to non-integer exponents as an infinite series, valid for $|b/a| < 1$.
Example
$x + 3$, $2y - 7$, $x^2 + 4$, and $5a - 3b$ are all binomials, each with exactly two unlike terms. $(x + 5)(x - 3) = x^2 - 3x + 5x - 15 = x^2 + 2x - 15$ using FOIL or distribution. $(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$, with coefficients from row $4$ of Pascal's Triangle: $1, 4, 6, 4, 1$.
Key Insight
"Bi" means two, the way a bicycle has two wheels, and many factoring techniques are specifically designed to break polynomials back down into binomials, so recognizing binomial patterns speeds up both multiplication and factoring. Binomial coefficients $\binom{n}{k}$ count the ways to choose $k$ items from $n$, connecting algebra to combinatorics, and the Binomial Theorem also appears in probability (the binomial distribution), calculus (Taylor series), and number theory.