Basic Algebra: One Step at a Time.  Pages 165-172:   46, 47, 48, 49

Dr. Robert J. Rapalje

Seminole Community College

Sanford, FL  32773

p. 171:   43-58.    Notice that these exercises are actually a series of exercises in which the First times First must be  , and the LAST times LAST must be.  Obviously, there is only one way to get the First times First to be !  It must be  times .  To make the LAST times LAST come out to, you can use either  or .  These can be determined by trial and error as illustrated by the following exercises.

46.    

SOLUTION:  Notice that this is a trinomial since it has three terms.  Since the coefficient of  is NOT 1, and it can’t be factored out as a common factor, this is what I call an “Advanced Trinomial Factoring” problem.  Begin by remembering that this is actually “undoing” a  F OI L  (that is, F L OI ).   

In this case, the First times First is obviously 5x times x.  In the factored form the signs will BOTH be positive.               

The Last times Last must be two numbers whose product is.  This would be either or .  It can’t be , because the middle term is an odd number.   The correct combination must be . 

However, because of the 5x times  x, there are two ways to write the combination:

To decide which is the correct combination, you must do the OUTER times OUTER and the INNER times INNER, and see which one gives the sum of 13x.    

In the case of  , the OUTER times OUTER is 40x, and the INNER times INNER is 1x.  This obviously does not add up to 13x. 

In the case of  , the OUTER times OUTER is 5x, and the INNER times INNER is 8x, which DOES add up to 13x.  This is the correct combination. 

Final answer:   .

47.     

SOLUTION:  Notice that this is a trinomial since it has three terms.  Since the coefficient of  is NOT 1, and it can’t be factored out as a common factor, this is what I call an “Advanced Trinomial Factoring” problem.  Begin by remembering that this is actually “undoing” a  F OI L  (that is, F L OI ).   

In this case, the First times First is obviously 5x times x.   In the factored form, the signs will both be negative.                

The Last times Last must be two numbers whose product is.  This would be either or .  Since the middle term is an even number, the correct combination might be (but not necessarily!) .   It’s a good place to start.  However, because of the 5x times  x, there are two ways to write the combination:

To decide which is the correct combination, you must do the OUTER times OUTER and the INNER times INNER, and see which one gives the sum of -14x.    

In the case of  , the OUTER times OUTER is -20x, and the INNER times INNER is -2x.  This obviously does not add up to -14x. 

In the case of  , the OUTER times OUTER is -10x, and the INNER times INNER is -4x, which DOES add up to -14x.  This is the correct combination. 

Final answer:   .

48.    

SOLUTION:  Notice that this is a trinomial since it has three terms.  In this case, the First times First is obviously 5x times x.   In the factored form, the signs will both be negative.   

The Last times Last must be two numbers whose product is.  This would be either or .  It can’t be , because the middle term is an odd number.   The correct combination must be . 

However, because of the 5x times  x, there are two ways to write the combination:

To decide which is the correct combination, you must do the OUTER times OUTER and the INNER times INNER, and see which one gives the sum of 41x.    

In the case of  , the OUTER times OUTER is -40x, and the INNER times INNER is -1x.  This DOES add up to  -41x, so this is the correct combination. 

Final answer:   .

49.    

SOLUTION:  Notice that this is a trinomial since it has three terms.  In this case, the First times First is obviously 5x times x.   In the factored form, the signs will be opposite, and the middle terms must be subtracted to get +3x.            

The Last times Last must be two numbers whose product is.  This would be either or .  It can’t be , because the middle term is an odd number.   The correct combination must be . 

However, because of the 5x times  x, there are two ways to write the combination:

To decide which is the correct combination, you must do the OUTER times OUTER and the INNER times INNER, and see which one gives the difference of +3x.    

In the case of  , the OUTER times OUTER is  40x, and the INNER times INNER is 1x.  This does not subtract to give you 3x, so this is not the correct combination. 

In the case of  , the OUTER times OUTER is  5x, and the INNER times INNER is 8x.  This DOES subtract to give you 3x, so this IS the correct combination. 

In order to get the +3x, you need to have +8x and -5x.  Therefore, the correct combination is

Final answer:   .