Solution 1 (a).

See text and/or instructor's solution manual.

Answer.      has zeros of order  4   at   .  

Solution Method I.   Factorization reveals that  

                    ,

and it is evident that    has two zeros of order  4   at   .  

We are done.   

Solution Method II.   Consider   ,   here we have

                      

          and

                      

Thus  ,  and  .

Applying Definition 7.6 we can conclude that    has a zero of order  4  at the point  .  

Similarly,      has a zero of order  4  at the point  .  

We are really done.   

Solution Method III.   Consider    at the point  .

Expand  f(z) in its Taylor series for centered at    and get

                    ,

and it is evident that    has a zero of order  4  at the point  .  

Remark.  Horner's method could be used to divide    by    and obtain this result.

Similarly,     has a zero of order  4  at the point  .  

We are really really done.   

Aside.  We can let Mathematica double check our work.

We are really really really done.   

          The zeros    of order  , respectively.  

                              A contour plot for   .  

                                                                      Plots for   .  

This solution is complements of the authors.

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