Classifications

The most important classification of polynomials is based of the number of distinct variables. A polynomial in one variable is called a univariate polynomial, a polynomial in more than one variable 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 variables at all. It is possible to further classify multivariate polynomials as bivariate, trivariate etc., according to the number of variables, but this is rarely done; it is more common for instance to say simply "polynomials in x, y, and z". A (usually mulitvariate) polynomial is called homogeneous of degree n if all its terms have degree n.
Univariate polynomials have many properties not shared by multivariate polynomials. For instance, the terms of a univariate polynomial are completely ordered by their degree, and it is conventional to always write them in order of decreasing degree. A univariate polynomial in x of degree n then takes the general form