MATH 455                      HINTS AND REMARKS FOR HOMEWORK ASSIGNED FROM SECTION 3.1
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#9. First note that since z_o is a zero of order d, the polynomial p(z) may be written as
    p(z) = (z - z_o) Q(z), where the quotient polynomial Q has the following characteristics:

    a) Q has order n - d;
    b) Q(z_o) is NOT zero;
    c) Hence Q(z) = a_m z^m + ... + a_1 z + a_0, where m = n - d and a_m is not = 0.

    Now, take the modulus of p: |p(z) - z_o| = |z - z_p| |Q(z)|. We need to find UNIFORM bounds for Q(z)
    when z is restricted to an appropriate neighborhood of z_o. In order to establish the existence of
    these uniform upper and lower bounds we first note that Q is a polynomial. Hence it is an entire
    function and in particular a CONTINUOUS function throughout the complex plane.

    UPPER BOUND: This is a consequence of the fact that any continuous function is uniformly bounded
    on a compact set.

    LOWER BOUND: This is the more complex issue. What is involved here is the fact (which we will prove
    formally in a later chapter) that zeros of analytic functions are isolated. Namely, how do we know 
    that Q(z) does not have a sequence of zeros converging to z_o when Q(z_o) is not equal to 0? This is 
    a consequence of the fact that Q is a CONTINUOUS function. Refer to your notes for a definition of
    continuity that I gave you a while back and write a complete proof.

    Further Elaboration: In essence you will have to prove that since Q is continuous and not zero at
    z_o, Q is in fact not zero in an appropriate neighborhood of z_o.

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#12.  Consider the function h(z) := R(z) - r(z). Assume that R and r agree at the points z_1, ...,
      z_(m+n+1) and no other point. Now note that R and r are two non-constant function and further
      note that h is also a rational function. Think about the zeros of h and apply the fundamental
      thm of algebra.
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