Lesson Factoring Difference of Squares in Living Color

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This Lesson (Factoring Difference of Squares in Living Color) was created by by rapaljer(4551) About Me : View Source, Show
About rapaljer: Retired Professor of Mathematics from Seminole State College of Florida after 36 years.

This text was imported from http://www2.seminolestate.edu/rrapalje/Math%20in%20Living%20Color/Basic%20Living%20Color/Color%201205%20Difference%20of%20Squares%20Basic%20Alg.htm by its author.

2.05  Factoring Difference of Squares

Perfect Square Trinomials

Basic Algebra: One Step at a Time.  Pages 157-163:  67, 72, 74, 75, 77

Bittinger 5.4,  p. 332-333:  49, 63, 67, 69, 71, 81

Dr. Robert J. Rapalje

Seminole State College of Florida

Altamonte Springs Campus

                                                                            

To see Section 2.05, with detailed explanations, examples, exercises, and answers, click here!

 

p. 162:   67.           Notice that   and   are both perfect squares.

                                                  The First times First must be :      times

 

                     

 

                                                   The Last times Last must be 16:      4 times 4.

                    

 

                                                   Because the 16 is negative, use opposite signs.

                    

 

The factor is itself a difference of squares, and so it must be re-factored.  However, the factor is the SUM of squares.  It does not re-factor, and it must be left as it is in the final answer.

 

                   

                      Final Answer!!

 

 

p. 162:   72.      Notice that   and   are both perfect squares.

                                                  The First times First must be :     times

 

                          

 

                                                   The Last times Last must be  81:  9 times 9.

                        

 

                                                   Because the 81 is negative, use opposite signs.

                        

 

The factor is itself a difference of squares, and so it must be re-factored.  However, the factor is the SUM of squares.  It does not re-factor, and it must be left as it is in the final answer.

 

                     

                        Final Answer!!

 

p. 163:  74.         

                            Notice that First times First must be :   times                                          

                     

                                                The Last times Last must be  and the

                                                OI  term must add up to 13x2      (Try 9 • 4, both positive)                                           

                     

                      O term is 4x2, and the I term is  9x2, for a total of 13x2 ,   

 

The factors and are both sums of squares!  They cannot be NOT  re-factored so this is the final answer.

 

 p. 162:  75.         

                           Notice that First times First must be :   times                                          

                 

                                                The Last times Last must be  and the

                                                OI  term must add up to 13x2      (Try 9 • 4, both negative)                                                                                        

                 

                  O term is −4x2, and the I term is −9x2, for a total of −13x2   

 

The factors and are both difference of squares.  Each must be re-factored so this is NOT the final answer.

 

               

          Final Answer!!

 

 p. 163:  77.     

                   The  First times First must be :   times   .

             

                                               The Last times Last must be  and the OI  term

must add up to -29x2    (Try 25 • 4!!)        

           

           O term is −4x2, and the I term is −25x2,

                                                for a total of -29x2   

             

The factors and are each difference of squares.  Each must be re-factored.

             

            Final Answer!!

 

 

 

 

Bittinger p. 332-333:      49, 63, 67, 69, 71, 81

49.         In ANY factoring problem, the first step is to take out the common factor.  Remember FCFF:  Factor the Common Factor First!!  There is a common factor of 2 to all three terms.  It will help to take out the negative as well, so take out the –2.   This changes all the signs within the parentheses

           

      

                      

            Notice that this is a trinomial , but even more importantly, do you see the PERFECT SQUARES?  The First and Last terms are both PERFECT SQUARES!!  And the problem factors accordingly!

           

               

           Can it really be this easy??  Check the OUTER times OUTER (-20st) and the INNER times INNER (-20st), and see that it adds up to -40st !  Can you believe it ??

           The FINAL ANSWER can be written: 

63.         In ANY factoring problem, the first step is to take out the common factor.  Remember FCFF:  Factor the Common Factor First!!  Be sure to get ALL the common factors.  In this case, take out the 6.

                      

            What remains is a DIFFERENCE OF TWO SQUARES!!

           

             

           Can it really be this easy?? 

 

67.         In ANY factoring problem, the first step is to take out the common factor.  Remember FCFF:  Factor the Common Factor First!!  Be sure to get ALL the common factors.  In this case, take out the 2.

                      

            What remains is a DIFFERENCE OF TWO SQUARES !!

            

             

 

69.          In ANY factoring problem, the first step is to take out the common factor.  Remember FCFF:  Factor the Common Factor First!!  Notice that there is a common factor to both terms, and that common factor is 5.  What works even better than taking out the 5 is to take out the –5 .

           

                      

           

            Notice that this is a difference of squares which factors:

           

           

71.         In ANY factoring problem, the first step is to take out the common factor.  Remember FCFF:  Factor the Common Factor First!!  Be sure to get ALL the common factors.  In this case, take out the 5.

                      

            What remains is a DIFFERENCE OF TWO SQUARES!! 

             

 

81.         In ANY factoring problem, the first step is to take out the common factor.  Remember FCFF:  Factor the Common Factor First!!  There is a common factor of 3x to all three terms.  It will help to take out the negative as well, so take out the –3x.   This changes all the signs within the parentheses.  Note:  Be sure to get ALL the common factors, both the -3 and the x factors!

           

      

                      

            Notice that this is a trinomial which factors:

           

                 or    

 

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Dr. Robert J. Rapalje Altamonte Springs Campus
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