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Math Help - Zeros of a polynomial

  1. #1
    Newbie
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    Zeros of a polynomial

    Hi,

    I'm working on the following problem where I need to find the zeros using Rouche's theorem:

    z^4 + 3z^3 + 6 inside |z| = 2

    So far, I have the following:

    I let f(z) = 3z^3 and g(z) = z^4 + 6.

    I have |z^4 + 6| < |z|^4 + 6 = 22 < |f(z)| = 24.

    so, |f(z)| > |g(z)| and according to the theorem, f(z) and f(z) + g(z) have the name number of zeros, counting multiplicities inside |z| = 2.

    Is the correct answer f(z) has 3 zeros and f(z) + g(z) also have 3 zeros inside z = |2| = 2 ? Or is it the correct answer that there are no zeros?

    Thanks a lot
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  2. #2
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
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    Champaign, Illinois
    Posts
    1,163
    Your argument does show there are  3 zeros for  |z|<2 .
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