Exercise 7.   Let  .  

7 (b).   Show that there are three zeros in  .  

Solution 7 (b).

Solution.   Let  .  Then for    we have

          

Since    has three 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).

                    Three zeros of    lie in the disk .  

Aside.  We can let Mathematica double check our work.

However, the zeros of    have a complicated algebraic representation.  

So we might prefer 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