Complex Analysis - Rouches Theorem

Question

Using Rouches Theorem, Determine the number of roots of the equation $z^4-6z^3+9z^2-24z+20=0$ inside the circle $|z|=2$. My problem here is that i cannot find a single dominant term? Can i choose multiple terms for my $f(z)$? If i say chose $f(z)=-6z^3+24z$ then how many roots are there?

Answer 1

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Using the rational root theorem, it's easy to see that $1$ and $5$ are roots of $z^4-6z^3+9z^2-24z+20$. Furthermore, $z^4-6z^3+9z^2-24z+20=(z-1)(z-5)(z^2+4)$. Therefore, the roots of your polynomial are $1$, $5$, and $\pm2i$. So, there's one and only one root inside the circle $|z|=2$.