3.02  Polynomial Expressions and Equations

College Algebra: One Step at a Time,  Pages 352 - 360:   #19, 26, 29, 32, 53, 58, 59, 60

Dr. Robert J. Rapalje

Seminole Community College

Sanford, FL  32773

19.  Find all real solutions for  .

Solution:    Start by setting the equation equal to zero, and arrange the terms in descending powers of x.

To solve this by algebraic methods, notice that it factors by grouping.  Group the first two and the last two terms.

Factor the common factor of :

Factor the difference of two squares:

Solve for x:                      There are three real solutions!

If you happen to have a graphing calculator with the program POLYSMLT, this works well on this exercise.

26.  Find all real solutions for  .

Solution:    Start by trying to factor the left side of the equation.  Take out a factor of . 

This gives you a solution of  and a trinomial  that does NOT factor.  You can use either the quadratic formula or the method of completing the square to solve this trinomial equation.  Because the coefficient of x is even, completing the square is probably easier.

Add 6 to each side, and prepare to complete the square:

Take half of the which is , and square which gives you , and add  to each side:

Take the square root of each side of the equation.  Don’t forget the on the right side of the equation!

Subtract  from each side: 

Final Answer:   There are three real solutions:  

29.  Find all solutions, both real and complex, for  .

Solution:    Start by setting the equation equal to zero, and factor the resulting trinomial. 

Set each factor equal to zero, and solve for x:

Take the square root of each side.  Remember that  , and don’t forget the signs.

Final Answer:   There are four solutions:  

32.  Find all solutions, both real and complex, for .

Solution:    Start factoring the trinomial. 

Set each factor equal to zero, and solve for x:

Take the square root of each side.  Remember that  , and don’t forget the signs.

Final Answer:   There are four solutions:  

53.  Find the equation whose roots are .

Solution:    In most exercises, you are given an equation and asked to solve it.  Now, the process is reversed.  You are given the solutions to an equation, and you are asked to find the equation.  What you have to do in this exercise is to reverse the steps of a completing the square problem.  If you need to see a completing the square problem, see #26, which is solved on this webpage. 

                                      Beginning with these two roots:   

                                     Add +3 to each side:                      

                                     Next, square both sides:         

                                     Remember that ,   

                                     Add +16 to each side       

Final Answer:  Be sure to answer in the form of an equation!

You might want to check this by completing the square or by calculator methods (POLYSMLT).

58.  Find the equation whose roots are        .

Solution:                  First look at the root:                                   

                                     This came from factor:                              

                                     Now, look at the last two roots:               

                                     Subtract 3 from each side:                    

                                     Next, square both sides:                 

Final Answer:       Be sure to answer in the form of an equation!

59.  Find the equation whose roots are .

Solution:    First look at the roots:                        

                       These came from factors:              

                       Which multiplied out becomes              

                        Now, look at the last two roots:               

                        Subtract 4 from each side:                   

                        Next, square both sides:                  

Final Answer:  Be sure to answer in the form of an equation!

60.  Find the equation whose roots are .

Solution:    First look at the roots:                       

                    These came from factors:              

                    Which multiplied out becomes         

                     Now, look at the last two roots:          

                     Add +3 to each side:                    

                     Next, square both sides:           

Final Answer:  Be sure to answer in the form of an equation!

While some instructors may want this product  multiplied out, it seems a tedious task for College Algebra students.  This form of the answer is good enough for me!!  It is not difficult to multiply a trinomial times a trinomial if that form of the answer is preferred.