TABLE OF CONTENTS

1.2. When the GCF is a binomial.

Prerequisite knowledge and skills:

         a working knowledge and understanding of the greatest common factor of two terms in     a polynomial

         a working knowledge and understanding of the rules of exponents

        a working knowledge and understanding of negative and fractional exponents

Terms to know: 

·         Binomial

·         Factor

·         Factoring

·         Greatest Common Factor

·         Monomial

·         Term

Sometimes the GCF is a binomial.  

 Example 1.    Factor out the GCF:    

Think of this expression as having two big terms:          

The binomial  is a factor in both terms.  How many time is it in common to both?

The GCF is the binomial with the lower power:   

 Divide all the terms by the GCF:                                

Multiply the two factors together:                                   

Check 1:  Are the tables of values equivalent?

Check 2:  Are the graphical representations identical?

Example 2.    Factor out the GCF:     

This expression, though longer, still has only two terms:

The binomials  are factors in both terms.  How many time are they common to both?

The GCF consists of each common factor to the lower power: 

The GCF is                                                                                            

Divide each term by the GCF and cancel fractions = to 1:   

Multiply the factors together:                                                                      

 Check 1:  Are the tables of values equivalent?

 Check 2:  Are the graphical representations identical?

      Checkpoint Binomial GCF 1

We can extend this idea to include binomials taken to negative or fractional powers as we did in section 1.1 with monomial common factors.

Example 3.    Factor out the term with the lower power:     

As before, we have two terms:                                              

The factor  is common to both.  How many times is it in common?

We take out the binomial with the lower power:                                         

Divide each term by this factor:                                                                        

  Multiply the factors together:                    

Check 1:  Are the tables of values equivalent?

Check 2:  Are the graphical representations identical?  Where is the expression undefined?    Where is the x-intercept?

      Checkpoint Binomial GCF 2

    Example 4.    Factor out the term with the lowest power:     

     The factor  is common to both terms.  How many times is it in common?

      We take out the binomial with the lower power:                           

      Divide both terms by this factor:

         Multiply the factors together:    

            Check 1:  Are the tables of values equivalent?

            Check 2:  Are the graphical representations identical?  Where is the    

                          expression undefined?  Where is the x-intercept?

      Checkpoint Binomial GCF 3

Example 5.    Factor out the GCF:                      

          The factors 4, x, and  are common to both terms.  How many times      

          are they in common to both terms?

          The GCF consists of each common factor to the lower power:    

          Divide each term by this GCF:

         Multiply the factors together:     

The checks are left for you.        

      Checkpoint Binomial GCF 4