# How to find the number of roots using Rouche theorem?

Find the number of roots $f(z)=z^{10}+10z+9$ in $D(0,1)$. I want to find $g(z)$ s.t. $|f(z)-g(z)|<|g(z)|$, but I cannot. Any hint is appreciated.

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## 1 Answer

First, we factor by $z+1$ to get $f(z)=(z+1)(z^9-z^8+z^7+\dots-z^2+z+9)$. Let $F(z):=z^9-z^8+z^7+\dots-z^2+z+9$ and $G(z)=9$. Then for $F$ of modulus strictly smaller than $1$, $|F(z)-G(z)|\leqslant 9|z| \lt |G(z)|$. thus for each positive $\delta$, we can find the number of zeros of $f$ on $B(0,1-\delta)$.

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@Sanchez You can post your alternative approach as an answer. – Davide Giraudo Nov 30 '12 at 23:13
actually let me delete the last comment, as I hope that OP can figure it out himself. – user27126 Nov 30 '12 at 23:14
Thank you very much, @Davide. I think you are right and there is no root inside the region. Only when we remove $(x+1)$, things get clearer. Also, thank Sanchez's comments. – Sam Nov 30 '12 at 23:20
How about $z=-1$ again? – Ma Ming May 3 '13 at 13:56
Also, in your last inequality, what if $z=-1$? – The Substitute Feb 9 '15 at 10:47