Degrees of a Polynomial

A polynomial is an expression that has multiple terms in it. The root of the word is poly-meaning many and nomial-meaning numbers. So, polynomial literally means many numbers. A polynomial’s name is based on the number of terms that the expression has. Use the following table to identify the name of a polynomial based on the number of terms it has.

Number of Terms Name Example
1 Monomial x
2 Binomial 2x – 3
3 Trinomial 2x3 – 3x + 1
4+ Polynomial 4x5 +3x3 -6x- 3

Once a polynomial has four or more terms it is just called a polynomial. The reason for this is that there are properties associated with monomials, binomials, and trinomials that do not exist for polynomials beyond a trinomial.

The degree of a polynomial tells us quite a bit about the features of that polynomial. To identify the degree of the polynomial you look at the exponents for each of the terms, and the polynomial’s degree is the highest exponent. The following table gives the name of various degrees of a polynomial.

Highest Power Name Example
0 Constant 4
1 Linear x
2 Quadratic x2
3 Cubic x3
4 Quartic x4
5 Quintic x5
6 Sextic x6
7 Septic x7
8 Octic x8
9 Nonic x9
10 Decic x10

Take the following polynomial:

Polynomial - 1.png

This polynomial has three terms: 3x6, 2x, -1, making it a trinomial. Each of the three terms has a power associated with it. The 3×6‘s power is 6, the 2x‘s power is 1, and the -1′s power is 0. Since 6 is the highest power that makes it a sextic. So, this term is a sextic trinomial.

To identify the following polynomial:

Polynomial - 2.png

It has two terms. 4×9, and -6×3, making it a binomial. This term has two powers, and 3. is the highest of the two terms so it is a Nonic Binomial.