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How to Find the Greatest Common Factor of Expressions

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Factoring can be a great resource when solving math problems. Many times, it is very helpful to find the greatest common factor. In this lesson, we will learn how to find the greatest common factor of expressions.

The Greatest Common Factor

The variety of apples at the market can be mind boggling. Just look at all those apples! Granny Smith, Golden Delicious, Cortland, … What do these apples have in common? Besides the fact they are delicious, these apples all have stems and seeds and they grow in trees. Although math expressions do not grow in trees, we can still find what is common in math expressions.

Common apples

In factoring and simplifying expressions, we often use the greatest common factor (abbreviated GCF). We will work through finding the GCF by explaining how to deal with coefficients and exponents. Can you imagine if math expressions really did grow in trees?

What Is a Factor?

When multiplying things (typically, numbers and variables) together we get a result called a product. These things we multiplied to get that product are called factors. For example, when we multiply 2 x 6 to get 12, 2 and 6 are factors. When we multiply 2 x 4 to get 8, 2 and 4 are factors. As you can see, 2 is a factor of both 8 and 12, and we call factors that are shared by different numbers common factors.

The Making of an Expression

The expressions we will look at have two types of numbers: coefficients and exponents. A coefficient is the multiplying number in front of the expression. An exponent is also a number, but it is the power a variable is raised to. In the expression 2x3, the coefficient is 2 and the variable x is raised to the 3 power, shown by the exponent of 3.

Expression Example
Expression

Finding the GCF

Now we get to the core (no, not apple core) of this lesson: finding the GCF.

Expressions can also have common factors. The greatest of these common factors is the greatest common factor (GCF). We are now going to learn everything we need to know about finding the GCF of expressions by working together and finding the GCF of these three expressions:

30a2b 2

24a3b

42a5b3

GCF of Coefficients

What are the coefficients? Answer: 30, 24 and 42. For coefficients, we follow these three ideas in order: 1 - 'factor'; 2 - 'common'; and 3 - 'greatest'.

Finding the Factors of Coefficients

To find all the factors of a coefficient, we are looking for all numbers that can be multiplied together to equal the coefficient.

For the number 30, we can start our list of factors with 1 x 30. Then, we ask the question: 2 times 'what' equals 30? The answer: 15. How about the next number? Is 3 a factor? Well, 3 times 'what' equals 30? The answer: 10. Next number is 4. Is 4 a factor? Asking the question: 4 times 'what' equals 30, we answer: there isn't a number multiplying 4 giving us 30. Thus, 4 is not a factor. The next number is 5. Since 5 times 6 equals 30, 5 and 6 are factors of 30. The next number is 6. But we already have 6 as a factor, so we can stop. The list of factors for 30 are 1, 2, 3, 5, 6, 10, 15 and 30. Great!

We can do the same process with the number 24 to get a list of factors: 1, 2, 3, 4, 6, 8, 12 and 24. For the number 42, the factors are 1, 2, 3, 6, 7, 14, 21 and 42.

So far we have identified the factors of our coefficients:

• 30: 1, 2, 3, 5, 6, 10, 15, 30

• 24: 1, 2, 3, 4, 6, 8, 12, 24

• 42: 1, 2, 3, 6, 7, 14, 21, 42

We have looked at the F in GCF.

Identifying the Common Factors

Now we look at the C: the common factors. In the three lists of factors, which numbers appear in ALL three lists? These are the factors in common. We have 1, 2, 3 and 6.

Identifying the Greatest Common Factor

With this list of common factors, it is easy to pick the greatest one. This is the G in GCF. The GCF of the coefficients 30, 24 and 42 is 6. OK. Now, what about the exponents?


Finding the GCF
Factoring


GCF of Exponents

Let's look again at our three expressions:

30a2b 2

24a3b

42a5b3

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