Example 7. Determine the number of roots of the equation

$$
z^{6}-6 z+10=0
$$

inside the circle $|z|<1$.
Solution. Let, for example, $f(z)=10$ and $\varphi(z)=z^{6}-6 z$. On the circle $|z|=1$ we have

$$
|f(z)|=10, \quad|\varphi(z)|=\left|z^{6}-6 z\right| \leqslant\left|z^{6}\right|+6|z|=7
$$

Thus, in all points of the circle $|z|=1$, the inequality $|f(z)|>|\varphi(z)|$ holds. The function $f(z)=10$ has no zeros inside the circle $|z|<1$, and therefore, by Rouché's theorem, the function $z^{6}-6 z+10$ also has no zeros.

## Problems for Independent Solution

Using Rouché's theorem, find the number of roots of the given equations in the specified domains:

441. $z^{4}-3 z^{3}-1=0, \quad|z|<2$.
442. $z^{3}+z+1=0, \quad|z|<\frac{1}{2}$.
443. $z^{5}+z^{2}+1=0, \quad|z|<2$.
444. $z^{8}+6 z+10=0,|z|<1$.
445. $27 z^{11}-18 z+10=0,|z|<1$.
446. $z^{8}-6 z^{6}-z^{3}+2=0,|z|<1$.