Binomial
AlgebraA 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 terms that are added or subtracted. "Bi" means two.
Example
$x + 3$, $2y - 7$, $x^2 + 4$, and $5a - 3b$ are all binomials. Each has exactly two unlike terms.
Key Insight
"Bi" means two. A bicycle has two wheels. A binomial has two terms.
Definition
A binomial is a polynomial consisting of exactly two unlike terms, connected by addition or subtraction. Binomials can be multiplied using the distributive property or the FOIL method. Special binomial products include the difference of squares $(a+b)(a-b) = a^2 - b^2$ and the perfect square $(a+b)^2 = a^2 + 2ab + b^2$.
Example
$(x + 5)(x - 3) = x^2 - 3x + 5x - 15 = x^2 + 2x - 15$ using FOIL or distribution.
Key Insight
Many factoring techniques are specifically designed to break polynomials back into binomials. Recognizing binomial patterns speeds up both multiplication and factoring.
Definition
A binomial is a two-term polynomial. The Binomial Theorem gives the 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
$(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$, with coefficients from Pascal's Triangle row $4$: $1, 4, 6, 4, 1$.
Key Insight
Binomial coefficients $\binom{n}{k}$ count the ways to choose $k$ items from $n$, connecting algebra to combinatorics. The Binomial Theorem appears in probability (binomial distribution), calculus (Taylor series), and number theory.