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Complex variable ( Rouche's Theorem)

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I need some help showing a step-by-step calculation to this question: (better formula representation in attachment):
Astronomer Cal Sagan Discovered from smoke filled telescopic lenses, a planet with billions upon billions of moons call that number N moons, including multiplicity, which he termed "fuzzy" moons. He came up with a 2-dimensional experimentally based model of this with |z|≤1 representing the planet and the location of moons given by the roots of Z^n + a_(N+1) Z^(N-1)..........+ a_1Z + a_0 = 0 in the Complex-plane with |a_0|>1 + |a_(N-1)| + ......... + |a_1| and 2^N> |a_(N-1)|2^(N-1) + |a_(N-2)|2^(N-2) + ......+|a|2 + |a_0|.
He couldn't quite figure out why all of his moons lied in this region. Show using Rouche's Theorem that all the N moons lie in 1 <|Z|>2

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Solution Summary

This solution uses Rouche's Theorem to find the location of roots of a complex polynomial with step-by-step calculations.

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