Chapter 14.7, Problem 44P

The fundamental theorem of algebra says that every equation of the form $f(z)=$ $a_n{z}^n+a_{n-1}{z}^{n-1}+\cdots +a_0=0,a_n\ne 0,n\ge 1,$ has at least one root. From this it follows that an $n$ th degree equation has $n$ roots. Prove this by using the argument principle. Hint: Follow the increase in the angle of $f(z)$ around a very large circle $z=r{e}^{i\theta };$ for sufficiently large r, all roots are inclosed, and $f(z)$ is approximately $a_n{z}^n$.