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I'm asked questions like to find the zeros, and multplicities of the $\mathbb{C}$-polynomials

$$f(z) =z^6 +4z^2 - 1 \hspace{10mm} , |z| < 1$$ or worse yet $$f(z)=z^{87}+36z^{57}+71z^4+z^3-z+1 \hspace{10mm}, 1<|z|<2$$

Surely factoring these things isn't the right approach...and I can't seem to think of a way to find them otherwise? We've been looking at some nice things we can do if we know what the zeros/singularities are, but as for finding them.. I don't know.

Thanks for any push in the right direction!

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A Cauchy integral can tell you the number of zeroes in e.g. the disc $|z|<1$, also their sum, the sum of their squares etc. Since I expect there to be way less than 87 roots in the domain in question, this should give you a linear or maybe quadratic. –  Hagen von Eitzen Nov 20 '12 at 15:58
    
Man how does everyone here know so much. You didn't even blink solving that. –  Chloe.H Nov 20 '12 at 15:59
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$4|z|^2>|z^6-1|$ for $|z|=1$, so Rouche Theorem tells you how many they are. Also, for the second one, $71|z|^4 > |z^{87}+..|$ on $|z| = 1$ and $|z|^{87} >...$ on $|z| =2$. –  N. S. Nov 20 '12 at 16:07
    
You can find multiple factors of $p$ by computing the greatest common divisor of $p$ and $p'$ ... if there are any. –  Hans Engler Nov 20 '12 at 16:07
    
@Chloe.H You're absolutely right: you shouldn't, and most likely can't, factorise them. The Abel–Ruffini theorem tells us that there is no general algebraic solution to polynomial equations of degree five or higher. –  Fly by Night Nov 20 '12 at 16:14

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