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6th-8th Grade Math: Practice & Review55 chapters | 469 lessons

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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.

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.

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, 4*x*, 3*x*^2, 8*y* and 14*s*^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.

7*x*^6*y*^4*z*

*a*^3*b*^2*c*

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

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.

3*x*^2 + 4*x* + 9

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

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.

10*x*^8*y*^3 + 8*x*^7 - 2*x*^5*y*^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.

We can actually say that a monomial is a special kind of polynomial with just one term.

Our polynomials and monomials also have a degree connected with them. The **degree of a term** is the sum of the exponents.

Because our monomial has just one term, the degree of the monomial is just that of the term. If our term or monomial has just one variable, then the exponent of that variable is the degree. If our variable doesn't have an exponent written, the degree is 1, so 9*x* has a degree of 1. If there is no variable, the degree is 0. So, *10* has a degree of 0, since there is no variable. If our term has more than one variable, then the degree is the sum of the exponents of the variables. So, 8*x*^2*y* has a degree of 3, 2 from the *x* and 1 from the *y.*

The **degree of a polynomial** is the highest degree of its terms. Each term of a polynomial has its own degree. The largest degree of these terms is the degree of the polynomial. So, 3*x*^2 + 4*x* + 9 has a degree of 2 because that is the highest degree of all the terms. The polynomial 10*x*^8*y*^3 + 8*x*^7 - 2*x*^5*y*^2 + *xy* has a degree of 11 because that is the highest degree of any of the terms.

Let's review what we've learned. **Coefficients** are your numbers multiplied with an unknown value. **Variables** are your letters representing an unknown value.

**Monomials** are a combination of a coefficient and variables that is one term. A term includes just a coefficient multiplied by a variable or variables.

**Polynomials** are more than one monomial connected together via addition or subtraction. Polynomials are written ordered so that the variable with the highest exponent is written first, and then the other terms are written in decreasing order according to their exponents.

The **degree of a term** is the sum of the exponents. In the case when a term has no variable, the degree is 0, whereas the degree is 1 when the variable has no exponent. The **degree of a polynomial** is the highest degree of its terms.

When you finish this video on monomials and polynomials, you could be ready to:

- Compare coefficients and variables
- Write monomials and polynomials
- Find the degree of a term and the degree of a polynomial

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6th-8th Grade Math: Practice & Review55 chapters | 469 lessons

- What Are Monomials & Polynomials? 5:55
- How to Multiply & Divide Monomials 5:14
- Square & Cube Roots of Monomials 4:59
- How to Evaluate a Polynomial in Function Notation 8:22
- Understanding Basic Polynomial Graphs 9:15
- Basic Transformations of Polynomial Graphs 7:37
- How to Add, Subtract and Multiply Polynomials 6:53
- Pascal's Triangle: Definition and Use with Polynomials 7:26
- The Binomial Theorem: Defining Expressions 13:35
- How to Divide Polynomials with Long Division 8:05
- How to Use Synthetic Division to Divide Polynomials 6:51
- Dividing Polynomials with Long and Synthetic Division: Practice Problems 10:11
- Operations with Polynomials in Several Variables 6:09
- Go to 6th-8th Grade Algebra: Monomials & Polynomials

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