Zeros of a polynomial

Apr 2010
6
0
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
 

chiph588@

MHF Hall of Honor
Sep 2008
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Champaign, Illinois
Your argument does show there are \(\displaystyle 3 \) zeros for \(\displaystyle |z|<2 \).