Greatest Common Factor with Variables Lesson

We previously learned that numbers being multiplied together are known as factors. The result of the multiplication operation is known as the product:
2 • 3 = 6
2, 3 » factors
6 » product
We can also reverse the process and go backward. This is known as factoring:
6 = 2 • 3
When an integer is a factor for each in a group of numbers, it is said to be a "common factor". The greatest common factor (GCF) is the largest integer that is a common factor for each number in a group.

GCF for a Group of Numbers

• Factor each number completely (write as the product of prime numbers)
• Create a list with each prime factor that is common to all numbers of the group
  • When a common factor has repeats or more than one factor, we list the smallest number of repeats between all factors
• Find the GCF as the product of all numbers on the list
  • If the list contains only one number, the GCF is just the number

Let's look at an example.
Example 1: Find the Greatest Common Factor (GCF).
45, 108, 270
Step 1) Factor each number:

Prime Factorization
45			3	3		5
108	2	2	3	3	3
270		2	3	3	3	5

Step 2) Create a list with each prime factor that is common to all numbers of the group. In this case, 3 is the only prime factor that is common to each number. The smallest number of repeats in any prime factorization is two. Therefore, our GCF will be two factors of 3 or 9.
GCF (45,108,270) = 9

GCF for a Group of Monomials

When we work with polynomials, we will often need to find the GCF for a group of monomials. When variables are involved, the process is the same for the number part. The variable part will include any variable that is common to all and the exponent on the variable is the smallest that occurs in the group. Let's take a look at a few examples.
Example 2: Find the Greatest Common Factor (GCF).
20x³, 30x²y, 100x²y³
Let's think about the number part first:
GCF (20, 30, 100) = 10
The variable part is pretty simple. We have the variable x in each monomial. We don't have the variable y in each monomial. Therefore, only x will be included in our GCF. The exponent on x will be the smallest exponent on x that occurs in the group. In this case, the smallest exponent on x is 2, therefore, x² will be the variable part of our GCF:
GCF (20x³, 30x²y, 100x²y³) = 10x²
Example 3: Find the Greatest Common Factor (GCF).
55xy²z, 110x²y⁵, 143x³y⁷z
Let's think about the number part first:
GCF (55, 110, 143) = 11
The variable part is pretty simple. We have the variables x and y in each monomial. We don't have the variable z in each monomial. Therefore, only x and y will be included in our GCF. The exponent on x will be the smallest exponent on x that occurs in the group. In this case, the smallest exponent on x is 1. The exponent on y will be the smallest exponent on y that occurs in the group. In this case, the smallest exponent on y is a 2, therefore, xy² will be the variable part of our GCF:
GCF (55xy²z, 110x²y⁵, 143x³y⁷z) = 11xy²

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