TABLE OF CONTENTS

1.1 Greatest Common Factor

 

Prerequisite knowledge and skills:
  • a working knowledge and understanding of the prime factorization of a composite number
  • a working knowledge and understanding of the greatest common factor of two  numbers
  • a working knowledge and understanding of the rules of exponents
  • a working knowledge and understanding of negative and fractional exponents

 

 

Terms to know:

Factor

Factoring

Greatest Common Factor

Integer

Product

Term

 

Skill Prep: Finding the GCF ;    

 Checkpoint GCF 1

 

In the prep exercise, we reviewed how to find the greatest common factor of a set of polynomial terms:

 

 

When given any number of terms with the same variable to different powers, the GCF is the term with the lowest exponent.

 

 

We can extend this notion of the GCF to include expressions with negative and fractional exponents. When given expressions with such exponents, we follow the same procedure as we did when dealing with positive integer exponents: we factor out the term with the lower exponent.

 

Example 1. Find the term to be factored out: .

 

The term to be factored out is the term with the lower power:

 

 

 

Example 2. Find the term to be factored out:  

 

The term to be factored out is the term with the lower power:

Since , the term to be factored out is .

 

Checkpoint GCF 2

 

II. Factoring out the GCF

 

Now we look at factoring out the GCF, once we have found it.

 

Example 3. Write in factored form: .

 

The GCF is the term with the lower power:

 

 

The term with the lower power is x2.

So =  

Divide each term by the GCF to obtain the other factor:

 

Multiply the two factors together:

 

 

Check 1: We can use a table of values to support our answer.

 

Check 2: We can use graphs to support our answer.

 

Example 4. Write in factored form: .

 

The term with the lower power is  

So  

 

Divide each term by the term with the lower power to obtain the other factor:

 

 

Multiply the two factors together:

 or  

 

Check 1: Are the tables of values equivalent?

 

Check 2: Do the graphical representations support our answer?

 

Example 5. Write in factored form: .

 

Find the term with the lower power:

Since term with the lower power is  

Divide all the terms by the term with the lower power to obtain the other factor:

 

 

Multiply the two factors togeyther

 

 

Check 1:Are the tables of values equivalent?

 

Check 2: Are the graphs equivalent?

 

     Checkpoint GCF 3

 

 

To factor an expression with the same variable to different powers:

1. Factor out the term with the lowest power;

2. Divide all terms by this factor to obtain the other factor;

3. Multiply the two factors together.

 

 

 

More worked examples

 

Homework problems

 

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