How to Multiply & Divide Monomials

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  • 0:05 What is a Monomial?
  • 1:07 Multiplying Monomials
  • 2:29 Dividing Monomials
  • 4:33 Lesson Summary
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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.

What Is a Monomial?

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 5x 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 328x 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.

Multiplying Monomials

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 5x. Likewise, if you were to multiply 6x by 4, the resulting answer is 24x. You can even do this with decimals: 0.5 * 8x comes out to 4x. 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 3x by 4x, you'd end up with 12x^2.

Dividing Monomials

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, 4x divided by 2 is 2x. Wait a minute, what if it isn't so clear cut? What if you were dividing 3x by 2? In that case, your coefficient becomes 3/2, leaving you with an answer of 3/2x.

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