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Finding the Greatest Common Factor of Monomials

Instructor: Yuanxin (Amy) Yang Alcocer

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

After reading this lesson, you'll know the steps you can take to make finding the greatest common factor easier when you are working with monomials with various variables and exponents.

Definition

First, some definitions. In this lesson, we'll be working with monomials and greatest common factors. What are these two things? A monomial is a group of variables and numbers all multiplied together. All your variables and numbers multiplied together make a term. When you have a mathematical expression made up of just one term, you have a monomial. In other words, a monomial is a mathematical expression made up of just one term. The greatest common factor is the largest factor that can divide into all your numbers or monomials.

The process of finding the greatest common factor for monomials is very similar to that for finding the greatest common factor between two numbers. Let's look how you can go about finding the greatest common factor between two numbers.

For example, the greatest common factor between 30 and 36 is 6 because it is the largest factor that both numbers have in common, thus making it the largest factor that can divide into both numbers. To find this greatest common factor, you separate each of your numbers into as many factors as you can, and then you find which factors both numbers have in common. All the factors that both numbers have in common will be your greatest common factor. So, separating your numbers into its factors, you get 2, 3, and 5 for 30. For the number 36, you get the factors 3, 3, 2, and 2. The factors that both numbers have in common are 2 and 3. As with all factors, they are always multiplied together. So, the greatest common factor between 30 and 36 is 2 * 3 = 6.

Now, let's look at the process when you are working with monomials.

Two Monomials

We'll start with just two monomials.

2 monomials
GCF monomials

We'll first break up each monomial into its factors. To do this, you separate each monomial into its factors as you would any other number. When you go about separating your variables, your exponent tells you how many variables make up the factors for that variable. For example, if you see an x to the third power, this tells you that your factors are three x's: x, x, and x. So separating your monomials into their respective factors, you get 2, x, y, y, z, z, z, z, z, and z for the first one 2xy^2z^6. For the 4xy^3 monomial, your factors are 2, 2, x, y, y, and y. Now, to find your greatest common factor, you combine all the factors that both monomials have in common. Looking through the factors for each, you see that both monomials have these factors in common: 2, x, y, and y. Multiplying all these together, you get a greatest common factor of 2xy^2.

Three Monomials

Now, let's try finding the greatest common factor when you have three monomials.

3 monomials
GCF monomials

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