Polynomials of degree three
Sticking with polynomials for a bit more, here is a plot of a polynomial of degree three:
\(f(z) = (z-2)(z+i)(z+2-i),\quad z \in \mathbb{C}(0,3)\)
The zeros can be located in this case perhaps even more accurately. All three is of multiplicity one. (According to the fundamental theorem of algebra all n roots of a polynomial with complex coefficients of degree n can be found on the complex plane with multiplicities taken into account.)
See the same polynomial below with another coloring:
\(f(z) = (z-2)(z+i)(z+2-i),\quad z \in \mathbb{C}(0,3)\)
Observe the three roots of multiplicity one on this picture too.
Now let us examine yet another polynomial of degree three:
\(f(z) = (z-2)(z+1)^2,\quad z \in \mathbb{C}(0,3)\)
This possesses a zero of multiplicity two and a zero of multiplicity one.