Exercise 5.   Let  .  

5 (a).   Show that there are two zeros in  .  

Solution 5 (a).

Solution.   Let  .  Then for    we have

          

Since    has two roots in  ,  so does  ,  by Remark 8.5 to Rouché's Theorem.

We are done.   

Aside.  We can use Mathematica to verify the inequality mentioned above.

                    For    on the circle    we have the inequality
                    

We are really done.   

Aside.  We can use Mathematica to plot the zeros of  g(z).

                    Two zeros of    lie in the disk .  

Aside.  We can let Mathematica double check our work.

However, the zeros of    do not have a nice algebraic solution.  

So we must settle for numerical approximations:

The moduli of these roots are:  

                       

Remark.  A factorization of the polynomial using numerical approximations for the coefficients is  

                      

This solution is complements of the authors.

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