Find the number of zeroes of a function $z^3+z^2 +z+1$ lying inside $|z|=2$ by Rouche's theorem?

Question

This function have mamy possible values. Which one we should select?

Answer 1

Hint For $|z|=2$ you have.

$$|z^3| > |z^2+z+1|$$

Apply the Rouche theorem.

Answer 2

If $\lvert z\rvert=2$, then\begin{align}\lvert z^2+z+1\rvert&\leqslant\lvert z\rvert^2+\lvert z\rvert+1\\&\leqslant4+2+1\\&<8\\&=\lvert z\rvert^3.\end{align}So, by Rouché's theorem, your function has as many zeros in the given region as $z^3$, which has $3$ zeros there.

Answer 3

On $ |z| =2 $, $|z^3|>|z^2+z+1|$.

Hence by Rouche's Theorem, the number of zeros of $z^3+z^2+z+1$ is equal to the number of zeros of $z^3$ inside $|z|=2$.

The zeros of $z^3$ are just 0, and all three of them lie inside $|z|=2$, hence the number of zeros of $z^3+z^2+z+1$ is equal to three.