Find the greatest common factor of two or more expressionsFactor the greatest common factor from a polynomialFactor by grouping

### Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

You are watching: How to find gcf with variables and exponents   Find the GCF of 48 and 80.

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Find the GCF of 18 and 40.

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We summarize the steps we use to find the GCF below.

Find the Greatest Common Factor (GCF) of two expressions.
Factor each coefficient into primes. Write all variables with exponents in expanded form.List all factors—matching common factors in a column. In each column, circle the common factors.Bring down the common factors that all expressions share.Multiply the factors.

In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.

Find the greatest common factor of Multiply the factors. and Multiply the factors. and Multiply the factors. , .

Solution
 Find the GCF of 5a and 5. Rewrite each term as a product using the GCF. Check by mulitplying the factors to get the orginal polynomial. Rewrite each term as a product using the GCF. Check by mulitplying the factors. Rewrite each term as a product using the GCF. Check by mulitplying. Rewrite each term. Check. Factor the GCF.  Rewrite each term. Check.  Check. Rewrite each term using the GCF. Check. Check on your own by multiplying.

Factor: .

Factor: .

### Factor by Grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.

(Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.)

How to Factor by Grouping

Factor: .

Solution
Factor: .

Factor: .

Factor by grouping.
Group terms with common factors.Factor out the common factor in each group.Factor the common factor from the expression.Check by multiplying the factors.

Factor: .

Solution

Factor: .

Factor: .

Access these online resources for additional instruction and practice with greatest common factors (GFCs) and factoring by grouping.

### Key Concepts

Finding the Greatest Common Factor (GCF): To find the GCF of two expressions:Factor each coefficient into primes. Write all variables with exponents in expanded form.List all factors—matching common factors in a column. In each column, circle the common factors.Bring down the common factors that all expressions share.Factor the Greatest Common Factor from a Polynomial: To factor a greatest common factor from a polynomial:Find the GCF of all the terms of the polynomial.Rewrite each term as a product using the GCF.Use the ‘reverse’ Distributive Property to factor the expression.Factor by Grouping: To factor a polynomial with 4 four or more termsGroup terms with common factors.Factor out the common factor in each group.Factor the common factor from the expression.
Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

8, 18

2

24, 40

72, 162

18

150, 275

10a, 50

10

5b, 30

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

Factor by Grouping

In the following exercises, factor by grouping.

Mixed Practice

In the following exercises, factor.

Everyday Math

Area of a rectangle The area of a rectangle with length 6 less than the width is given by the expression , where width. Factor the greatest common factor from the polynomial.

Height of a baseball The height of a baseball t seconds after it is hit is given by the expression . Factor the greatest common factor from the polynomial.

Writing Exercises

The greatest common factor of 36 and 60 is 12. Explain what this means.

What is the GCF of ? Write a general rule that tells you how to find the GCF of .

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no – I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.

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### Glossary

factoringFactoring is splitting a product into factors; in other words, it is the reverse process of multiplying.greatest common factorThe greatest common factor is the largest expression that is a factor of two or more expressions is the greatest common factor (GCF).
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