Polynomials are classified according to two attributes -- number of terms and degree.

A monomial is an expression with a single term. It is a real
number, a variable, or the product of real numbers and variables. For
example,
4
,
3*x*
^{2}
, and
15*xy*
^{3}
are all monomials, but
4*x*
^{2} + *x*
,
(3 + *y*)^{2}
, and
12 - *z*
are not monomials.

A polynomial is a monomial or the sum or difference of monomials.
4*x*
^{3} +3*y* + 3*x*
^{2} + *z*
,
-12*zy*
, and
15 - *x*
^{2}
are all polynomials.

Polynomials are classified according to their number of terms.
4*x*
^{3} +3*y* + 3*x*
^{2}
has
three terms,
-12*zy*
has 1 term, and
15 - *x*
^{2}
has two terms. As already mentioned,
a polynomial with 1 term is a monomial. A polynomial with two terms is
a binomial, and a polynomial with three terms is a trinomial.

The degree of a monomial is the sum of the exponents of its
variables. For example,
12*x*
^{3}
has degree 3,
*x*
^{2}
*y*
^{5}
has degree
2 + 5 = 7
, and
11*xy*
has degree
1 + 1 = 2
.

A polynomial can be arranged in ascending order, in which the
degree of each term is at least as large as the degree of the
preceding term, or in descending order, in which the degree of
each term is no larger than the degree of the preceding term. The
polynomial
3 + 12*x* - *xy* + 7*x*
^{2}
*y* + *y*
^{5} -12*x*
^{3}
*y*
^{3}
is written in
ascending order, while the same polynomial expressed as
-12*x*
^{3}
*y*
^{3} + *y*
^{5} +7*x*
^{2}
*y* - *xy* + 12*x* + 3
is written in descending order.

Mathematicians generally write polynomials in descending order. The coefficient of the first term of a polynomial written in descending order is known as the leading coefficient.

The degree of a polynomial is the largest of the degrees of its monomial terms.