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, 3x2, and 15xy3 are all monomials, but 4x2 + x, (3 + y)2, and 12 - z are not monomials.
A polynomial is a monomial or the sum or difference of monomials. 4x3 +3y + 3x2 + z, -12zy, and 15 - x2 are all polynomials.
Polynomials are classified according to their number of terms. 4x3 +3y + 3x2 has three terms, -12zy has 1 term, and 15 - x2 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, 12x3 has degree 3, x2y5 has degree 2 + 5 = 7, and 11xy 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 + 12x - xy + 7x2y + y5 -12x3y3 is written in ascending order, while the same polynomial expressed as -12x3y3 + y5 +7x2y - xy + 12x + 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.