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

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Lesson Transcript

Instructor:
*Kevin Newton*

Kevin has edited encyclopedias, taught middle and high school history, and has a master's degree in Islamic law.

In this lesson, we're going to learn how to multiply and divide monomials. As you can imagine, this is a very important skill that much of algebra (and even more advanced math) depends upon.

One of the most confusing parts about starting to study algebra and higher mathematics is how in the world that a letter can be a number. By now, you've probably figured out that a **variable** is simply a letter that represents a number. You don't know what that number is and more often than not, it's your job to figure out what it is.

So, how can something like 5*x* be just one number? Well, that's what a **monomial** is. It's a number that has been reduced as far as it can be reduced and has only one term. Therefore, 9 is a monomial, but so is 328*x* to the 42nd power. They're both as simple as we can make them. The fact that one is an exponent doesn't matter as long as it's a whole positive number, and the fact that there is a coefficient, or the number in front of the variable, doesn't matter either.

Okay, that's all pretty easy, but now you've got your math teacher wanting you to multiply or divide these things! How do you do that? Luckily for you, that's what this lesson is going to focus on.

To multiply a monomial by a known number, simply multiply the coefficient by that number. Remember that *x* by itself has a coefficient of just 1. Remember that it is understood that the coefficient is being multiplied with the variable. Therefore, if you were to multiply *x* by 5, the resulting answer is 5*x*. Likewise, if you were to multiply 6*x* by 4, the resulting answer is 24*x*. You can even do this with decimals: 0.5 * 8*x* comes out to 4*x*. If you want, you can check these by substituting any number for *x*. You can even choose a completely irrational number like pi and the math still holds.

But, what if you're going to multiply a monomial by another variable or even another monomial. In that case, multiply the coefficients together to get the coefficient part of the answer, and then multiply the variables by each other. But wait, how do you multiply variables?

Think about it like this: if you multiply 3 by 3, you get 9. What's another way of saying that? If you guessed 3^2, you'd be right. Therefore, any variable multiplied by itself is simply itself squared. As a result, if we were to multiply 3*x* by 4*x*, you'd end up with 12x^2.

That's all fine and good for multiplying monomials, but what about dividing them? Luckily, that's just as easy. If you are just dividing by a known number, simply divide it into the coefficient. Therefore, 4*x* divided by 2 is 2*x*. Wait a minute, what if it isn't so clear cut? What if you were dividing 3*x* by 2? In that case, your coefficient becomes 3/2, leaving you with an answer of 3/2*x*.

Now, here's where things get tricky. Remember that you are multiplying the coefficient by the variable, but you leave the variable because you don't know what it is yet. It's important that you remember that the variable is over 1. Therefore, it's 3/2 * *x*/1. That means that you could simplify it further as 3*x*/2. Check with your teacher to see which format that he or she prefers.

But what about dividing by a variable? Luckily, that's just as easy. If you have a variable with a coefficient, carry over the variable, but then divide the variable by the other variable. If you are simply dividing *x* by *x*, that means the answer is 1. But, what about exponents? What if you had to divide *x*^3 by *x*^2?

Let's factor out the exponent for this one. *x*^3 is really *x* * *x* * *x*, right? So put that on top, then factor *x*^2 and put that on the bottom, which would be *x* * *x*. Two of the *x*'s divide out, meaning that you're left with one *x* on top. As a result, the answer is simply *x*.

If, by the way, you have a variable on the bottom, that just means it becomes part of the denominator. Keep this in mind when working with the coefficient. Remember that a whole number has a denominator of 1. Therefore, 3*x* divided by *x*^2 means you are left with an *x* on the bottom. Leave the 3 on top and you've got 3/*x*, which is your answer.

In this lesson, we learned how to **multiply and divide by monomials**. Remember that a **monomial** is simply a term that has been reduced as far as it can be with only one term.

To multiply a monomial by a known number, simply multiply the coefficient by the number. To multiply a monomial by a variable, simply multiply the variable by the other variable; this will often result in an exponent.

To divide a monomial by a known number, simply divide the coefficient by the number in question. Finally, to divide a monomial by a variable, do the exact opposite of what you did when you multiplied, which could mean factoring out some exponents.

- variable: a letter that represents a number
- monomial: a number that has been reduced as far as it can be reduced and has only one term

Upon reviewing this lesson, you should be able to accomplish the following:

- Multiply monomials
- Use factoring while dividing monomials

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

- What Are Monomials & Polynomials? 5:55
- How to Add & Subtract Monomials 5:29
- How to Multiply & Divide Monomials 5:14
- 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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