polynomial

Formally,
a polynomial is a finite sum of monomials:

p = \sum_{k = 0}^{n} p_{k} x^{k} = p_{0} + p_{1} x + p_{2} x^{2} + \dots + p_{n} x^{n}

where $p_{i}$ are constants, called the coefficients of the polynomial,
and $x$ is the indeterminate;
the highest exponent with a nonzero coefficient is the degree $n$ .

The word "indeterminate" is used to distinguish that $x$ represents no particular value, though any value may be substituted for it via a function, specifically a polynomial function.

The roots of a polynomial are the zeroes of the polynomial function. By the Fundamental Theorem of Algebra, there are $n$ such roots (counting multiplicity).

A polynomial in one indeterminate is in standard form when the exponents of the terms decrease from left to right. The coefficient of the first term of a polynomial written in standard form is called the leading coefficient.

Polynomials of degree zero are constant polynomials, or simply constant.
Polynomials of degree one are linear polynomials.
Polynomials of degree two are quadratic polynomials.
Polynomials of degree three are cubic polynomials.

Polynomials can be classified by the number of terms with nonzero coefficients;

a one-term polynomial is called a monomial,
a two-term polynomial is called a binomial,
a three-term polynomial is called a trinomial.

The polynomial $0$ , which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero; it is explicitly left undefined, or defined as negative. The zero polynomial is unique in that it is the only polynomial in one indeterminate that has an infinite number of roots.

Two terms with the same indeterminates raised to the same powers are called "similar terms" or "like terms", and they can be combined using the distributive property into a single term.