Example 8. How many roots of the equation

$$
z^{4}-5 z+1=0
$$

lie in the annulus $1<|z|<2 ?$
Solution. Let $N$ be the number of roots of equation (4) in the ring $1<|\varphi(z)|$, since $|f(z)|=|-5 z|=5,|\varphi(z)|=\left|z^{4}+1\right| \leqslant$ $\left|z^{4}\right|+1=2$. The function $f(z)=-5 z$ has one root in the circle $|z|<1$, and thus $N_{1}=1$.

In the circle $|z|<2$, $|f(z)|>\left|\varphi(z)\right|$, since $|f(z)|=\left|z^{4}\right|=2^{4}=16,|\varphi(z)|=|1-5 z| \leqslant 1+5|z|=11$. The function $f(z)=z^{4}$ has four roots in the circle $|z|<2$, and therefore $N_{2}=4$.

The number of roots of equation (4) in the ring $1<|z|<2$ will be $N=4-1=3$.

## Problems for Independent Solution

Determine the number of roots of the given equations in the specified rings:

447. $4 z^{4}-29 z^{2}+25=0, \quad 2<|z|<3$.
448. $z^{7}-5 z^{4}+z^{2}-2=0,1<|z|<2$.
449. $z^{6}-8 z+10=0, \quad 1<|z|<3$.