What Are Polynomials?

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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 an Equation

x2 - 7x - 6 

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

Term Numerical Coefficient


8x2 3x -2 Polynomial
8x-3 7y -2 NOT a Polynomial The exponent is negative.
9x2 8x -2/3 NOT a Polynomial Cannot have division.
7xy Monomial

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 that is actually referred to as a monomial.

One thing you will do when solving polynomials is combined like terms.

  • Like terms: 6x 3x - 3x
  • NOT like terms: 6xy 2x - 4

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

  • 5x
  • 2 2x2 - 3


  • 10x4 - 3
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Your Citation
Russell, Deb. "What Are Polynomials?" ThoughtCo, Apr. 5, 2023, thoughtco.com/what-are-polynomials-understanding-polynomials-2311946. Russell, Deb. (2023, April 5). What Are Polynomials? Retrieved from https://www.thoughtco.com/what-are-polynomials-understanding-polynomials-2311946 Russell, Deb. "What Are Polynomials?" ThoughtCo. https://www.thoughtco.com/what-are-polynomials-understanding-polynomials-2311946 (accessed June 8, 2023).