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What Are Monomials & Polynomials?

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  • 0:05 Coefficients and Variables
  • 0:46 Monomials
  • 1:44 Polynomials
  • 3:48 The Degree
  • 5:07 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how coefficients and variables combine to form monomials and polynomials. You'll also find out how the exponent determines the degree of polynomials.

Coefficients and Variables

In this lesson, we will talk about how coefficients and variables combine together to form different things. Your coefficients are your numbers multiplied with an unknown value. Your variables are your letters representing an unknown value.

So, why do you need to know this information? This information is important because understanding these combinations will help you to solve your math problems more easily. Once you know what you are working with and the kind of combination that you have, you are in a better position to use the proper techniques to solve that particular combination. However, we're not going to discuss these problem-solving techniques in this lesson.

Let's take a look at a couple of possible combinations: monomials and polynomials.

Monomials

The first we will look at are monomials. Monomials are a combination of a coefficient and variables that is just one term. A term includes just a coefficient multiplied by a variable or variables. You will not have anything else being added to it, subtracted from it, or divided from it.

For example, 4x, 3x^2, 8y and 14s^8 are all monomials. It is okay for your variables to include an exponent. This exponent simply tells you how many of that variable you have. So, the x^2 tells you that you have two x's. The s^8 tells you that you have eight s's.

It is also possible to have a monomial with several variables like these.

7x^6y^4z

a^3b^2c

If we see a monomial with just variables, the coefficient in this case is a 1.

Polynomials

Now, let's talk about polynomials. Polynomials are more than one monomial connected together via addition or subtraction. Each monomial is referred to as a term of the polynomial.

We have a particular way of writing polynomials. The standard form of writing polynomials involves ordering the terms so that the variable with the highest exponent is written first, and then the other terms are written in decreasing order based on the exponent. Many times, the terms of a polynomial will have the same variable included in them. These are all examples of polynomials written in standard form.

3x^2 + 4x + 9

10t^4 + 7t^2 - 91t

Notice how each of these polynomials all share the same variable in each term. In the first polynomial, you might have noticed the single 9 without a variable. Because all the other terms have an x as a variable, we can actually say that this 9 also has an x variable. However, this x variable connected with the 9 has an exponent of 0. So, what happens when our exponent is 0? Our value is 1. Therefore, x^0 is equal to 1. What is 9 times 1? 9. That is why we won't write out the variable if our variable has an exponent of 0.

It is very possible for a polynomial to consist of more than one variable. In this case, we choose a variable and use that to order our variables starting out with the term that includes the variable to the highest exponent.

10x^8y^3 + 8x^7 - 2x^5y^2 + xy

In this polynomial, we chose the x variable to order our terms by. We started out with the term that includes the variable to the highest exponent. In this case, the x in our first term has an exponent of 8. The next term's x variable has an exponent of 7, and then our x variable exponent keeps decreasing after that in order.

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