1. ## Rouche's Theorem

Find the number of zeros of the following polynomial lying inside the unit circle;
f(z)=z^9 - 2z^6 + z^2 - 8z -2

I tried to use Rouche's Theorem
for differentiable f and g and all points inside s
if |f(z)-g(z)|<|f(z)| then
f and g have same zeros in s.
But which of the g(z) functions should I choose? -2z^6 or z^2 or -8z
how can ı determine this?

2. As far as i can remember you have to choose the dominating function within the region, i.e the one which grows the fastest.

Within the unit circle, the dominating part of your function is -8z, so try choosing that and see if that works, i may be wrong though.

3. Originally Posted by AcCeylan
But which of the g(z) functions should I choose? -2z^6 or z^2 or -8z how can ı determine this?

If $|z|=1$ then,

$|z^9 - 2z^6 + z^2 -2 |\leq |z^9|+|-2z^6|+|z^2|+|-2|=1+2+1+2=6<8=|-8z|$