Exercise 15.  Find all four roots of  ,  and use them to demonstrate that    can be factored into two quadratics with real coefficients.  

Solution 15 .

See text and/or instructor's solution manual.

Start with  ,  then use  ,       and   .

Given  .  Then    where  ,       and   .

The primitive fourth root of unity is  ,  
and the fourth roots of unity are  .  

The principal fourth root is  

Thus, the fourth roots of    are   which can be written as  

            

                      for  .

Now use the roots as linear factors in conjugate pairs to get  .

Use   and   to form   .

Use   and   to form   .

Then  .

We are done.   

Aside.  We can let Mathematica double check our work.

We are done.   

The fourth roots of    can be calculated in the ordinary way.

We are done.   

Remark.  The traditional answers for the fourth roots of    are

            for  .

We are done.   

Aside.  We can let Mathematica double check our work.

The principal value is

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell