Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents.
Formula
a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0
Definition
A polynomial is an expression built by adding or subtracting terms that each have a variable raised to a whole-number exponent; in one variable $x$, it has the general form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where the $a_i$ are real coefficients, the exponents are non-negative integers, and the degree is the highest exponent present. Polynomials cannot have negative exponents, fractional exponents, or variables in a denominator, and they are closed under addition, subtraction, and multiplication. Formally, the set of polynomials over a field $F$ in variable $x$, denoted $F[x]$, forms a commutative ring (in fact a principal ideal domain); by the Fundamental Theorem of Algebra, every polynomial of degree $n$ over the complex numbers has exactly $n$ roots (counted with multiplicity), and polynomial rings are the algebraic setting for Galois theory, which classifies which polynomial equations can be solved by radicals.
Example
$3x^2 + 5x - 2$ is a polynomial with three terms, and $x^4 - 7$ is another, both using only whole-number exponents ($0, 1, 2, 3, \ldots$). $4x^3 - 2x^2 + x - 9$ is a degree-$3$ polynomial, and adding $(2x^2 + 3)$ to $(4x^2 - x)$ gives $6x^2 - x + 3$, still a polynomial. $\mathbb{Z}[x]$ (polynomials with integer coefficients) is a unique factorization domain but not a field; Gauss's Lemma states that a primitive polynomial in $\mathbb{Z}[x]$ is irreducible in $\mathbb{Z}[x]$ if and only if it is irreducible in $\mathbb{Q}[x]$.
Key Insight
Polynomials are the building blocks of algebra: most equations studied in algebra involve them, and these restrictions on exponents are exactly what separates a polynomial from a rational or radical expression. The idea generalizes to multivariate polynomial rings $F[x_1,\ldots,x_n]$, the central objects of algebraic geometry, where varieties are zero sets of polynomial ideals, connecting algebra directly to geometry.