Polynomial

A polynomial in one variable $x$ has the form $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where each $a_i$ is a constant (the coefficient) and $n$ is a non-negative integer. The largest exponent with a non-zero coefficient is the polynomial's degree.

Polynomials are closed under addition, subtraction, and multiplication — but not division (which produces rational expressions). Special cases by degree: degree 0 is a constant, degree 1 is linear, degree 2 is quadratic, degree 3 is cubic.

Polynomials underpin calculus (differentiation/integration of polynomials is mechanical), numerical analysis (interpolation, approximation), and algebra (factoring theorems). The Fundamental Theorem of Algebra guarantees a degree-$n$ polynomial has exactly $n$ complex roots counted with multiplicity.