A polynomial in a single indeterminate can be written in the form

a_n x^n + a_{n-1}x^{n-1} + dotsb + a_2 x^2 + a_1 x + a_0,
where a_0, ldots, a_n are numbers, or more generally elements of a ring, and x is a symbol which is called an indeterminate or, for historical reasons, a variable. The symbol x does not represent any value, although the usual (commutative, distributive) laws valid for arithmetic operations also apply to it.

This can be expressed more concisely by using summation notation:

sum_{i=0}^n a_i x^i
That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a number—called the coefficient of the term[2]—and a finite number of indeterminates, raised to nonnegative integer powers. The exponent on an indeterminate in a term is called the degree of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest degree of any one term with nonzero coefficient. Since x = x1, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called respectively a constant term and a constant polynomial;[3] the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the zero polynomial (which has no term) is not defined.[4]

For example:

-5x^2y,
is a term. The coefficient is −5, the indeterminates are x and y, the degree of x is two, while the degree of y is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is 2 + 1 = 3.

Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:

underbrace{_,3x^2}_{begin{smallmatrix}mathrm{term}\mathrm{1}end{smallmatrix}} underbrace{-_,5x}_{begin{smallmatrix}mathrm{term}\mathrm{2}end{smallmatrix}} underbrace{+_,4}_{begin{smallmatrix}mathrm{term}\mathrm{3}end{smallmatrix}}.
It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.

Polynomials of small degree have been given specific names. A polynomial of degree zero is a constant polynomial or simply a constant. Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. For higher degrees the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, in x2 + 2x + 1 the term 2x is a linear term in a quadratic polynomial.

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. Rather the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞).[5] These conventions are useful when defining Euclidean division of polynomials. The zero polynomial is also unique in that it is the only polynomial having an infinite number of roots. In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all its terms have degree n. For example, x3y2 + 7x2y3 − 3x5 is homogeneous of degree 5. For more details, see homogeneous polynomial.

The commutative law of addition can be used to rearrange terms into any preferred order. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial in the example above is written in descending powers of x. The first term has coefficient 3, indeterminate x, and exponent 2. In the second term, the coefficient is −5. The third term is a constant. Since the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.[6]

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 law, into a single term whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0.[7] Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial,[8] a two-term polynomial is called a binomial, and a three-term polynomial is called a trinomial. The term "quadrinomial" is occasionally used for a four-term polynomial. A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance when working with univariate polynomials one does not exclude constant polynomials (which may result, for instance, from the subtraction of non-constant polynomials), although strictly speaking constant polynomials do not contain any indeterminates at all. It is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of indeterminates allowed. Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on. It is common, also, to say simply "polynomials in x, y, and z", listing the indeterminates allowed.

The evaluation of a polynomial consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions. For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using the Horner scheme:

(((dotsb((a_n x + a_{n-1})x + a_{n-2})x + dotsb + a_3)x + a_2)x + a_1)x + a_0.