What Are Polynomials?

Introduction to Polynomials

Noel Henderson / Getty Images
Noel Henderson / Getty Images

Polynomials are algebraic expressions that include real numbers and variables. Division and square roots cannot be involved in the variables. The variables can only include addition, subtraction and multiplication.

Polynomials contain more than one term. Polynomials are the sums of monomials.

A monomial has one term: 5y or -8x2 or 3.
A binomial has two terms: -3x2 2, or 9y - 2y2
A trinomial has 3 terms: -3x2 2 3x, or 9y - 2y2 y

The degree of the term is the exponent of the variable: 3x2 has a degree of 2.

When the variable does not have an exponent - always understand that there's a '1' e.g., 1x

Example of Polynomial in a Equation

x2 - 7x - 6 

(Each part is a term and x2 is referred to as the leading term.)

TermNumerical Coefficient




8x2 3x -2Polynomial           
8x-3 7y -2NOT a PolynomialThe exponent is negative.
9x2 8x -2/3NOT a PolynomialCannot have division.

Polynomials are usually written in decreasing order of terms. The largest term or the term with the highest exponent in the polynomial is usually written first. The first term in a polynomial is called a leading term. When a term contains an exponent, it tells you the degree of the term.

Here's an example of a three term polynomial:

6x2 - 4xy 2xy - This three term polynomial has a leading term to the second degree. It is called a second degree polynomial and often referred to as a trinomial.

9x5 - 2x 3x4 - 2 - This 4 term polynomial has a leading term to the fifth degree and a term to the fourth degree.

It is called a fifth degree polynomial.

3x3 - This is a one term algebraic expression which is actually referred to as a monomial.

One thing you will do when solving polynomials is combine like terms. This is also discussed in lesson 2 - Adding and Subtracting polynomials.

Like terms: 6x 3x - 3x

NOT like terms: 6xy 2x - 4

The first two terms are like and they can be combined:

5x2 2x2 - 3


10x4 - 3

Now you're ready to start adding polynomials.