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Monomial: Definition, Examples & Factors

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  • 0:01 What Is a Monomial?
  • 0:57 Identifying Monomials
  • 1:49 Simplification
  • 3:25 Factoring Monomials
  • 4:57 Lesson Sumamry
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Instructor
Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

A monomial is a polynomial with only one term. In this lesson, you will learn how to identify and factor monomials, then take a quiz to cement your understanding.

What Is a Monomial?

What's the first math problem you ever solved in your life? It probably happened so long ago that you can't actually remember, though a good guess would be 1 + 1 = 2. Not only is this likely to be the first math problem you ever solved, but it's also the first time you ever worked with monomials. Here you're adding two monomials, 1 and 1, together to get a third monomial, 2. While this is the first time you might have worked with monomials, they're not often called by that name until you start taking algebra classes.

In our 1 + 1 = 2 example, the monomials are simply numbers, but monomials can get more complicated than that. The mathematical definition of a monomial is that it is a polynomial with only one term. The word 'monomial' comes from Latin, mono meaning one and mial meaning term. Each term in a polynomial is separated by addition or subtraction signs.

polynomial terms

Identifying Monomials

A monomial can be a constant (number), a variable (letter), or the product of one or more constants and variables. It's important to note that the variables of a monomial cannot have a negative or fractional exponent.

non-monomials

There are two basic rules for monomials:

1. When you multiply a monomial by a constant, you get another monomial.

monomial rule1

2. When you multiply a monomial by another monomial, you still get a monomial in return.

monomial rule2

To make sure you have a good grasp of what monomials look like, lets look at a few examples of monomials as constants and variables.

Identifying Monomials Through Simplification

Another thing you need to know about identifying monomials is that sometimes you can have a mathematical expression that appears not to be a monomial but can be turned into one by simplifying it. Let's look at the following two expressions:

monomial simplification examples

Our first expression is a binomial that consists of two monomials. However, monomials can be added or subtracted if they have like terms. This means that they have the same variables with identical exponents. The two monomials of our first expression have the like term of x^2, so we can add them together.

monomial simplification example1

The second expression appears not to be a monomial because a variable in the denominator of a fraction is the same as having that variable raised to a negative exponent.

negative exponent formula

Remember, a monomial cannot have variables with negative exponents. Luckily, since the numerator and denominator have the same variable (a), we can divide them.

monomial simplification example2

You need to be careful when dividing two monomials though; you can end up with answers that are not monomials.

division non-monomial examples

Factoring Monomials

One more important thing you should learn how to do with monomials is factor them. When we factor a monomial, we break it down into its prime factors. A monomial can consist of both constants and variables. To do the prime factorization of a monomial, you find the prime factors of each constant and variable separately. Let's look at an example monomial:

factoring example part1

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Additional Activities

Further Monomial Examples

In the video lesson, we learned that a monomial is a polynomial that has only one term. This means that there is no addition or subtraction (since terms are separated by addition and subtraction) and that we could have a constant, a variable, or a product of constants and variables. Any exponents in a monomial are positive whole number exponents - since that is the type of exponents polynomials have as well. To factor a monomial, find the prime factorization of any constant and rewrite any variables to a power as the variable multiplied by itself that number of times. Knowing how to factor a monomial can help us factor polynomials in the future - by finding the greatest common factor of each term.

Examples

Examine the following four problems and determine if they are monomials. If they are monomials, factor the monomial fully, as shown in the video lesson. If they are not monomials, explain how you know they are not.


Solutions

This is a monomial.


This is not a monomial because the exponent on the x is negative.


This is a monomial.


4. This is not a monomial because we have more than one term.

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