What do polynomials look like in the complex plane? I have a hard time visualizing the fundamental theorem of algebra, which says that any polynomial has at least one zero, superficially I know this is true as every polynomial must have either an imaginary zero or real zero, but how do I visualize this in the complex plane?
For example if we have a real polynomial, we know that it is zero when it crosses the x axis this is because $y = 0$, however if $f(z) = 0$, then it must be the case that $f(z) = w = u+iv = 0+i0=0$ therefore every zero in $f(z)$ passes the origin? That does not make sense to me, what am I missing here?

See these:


*

*Visual Complex Functions by Wegert.

*Phase Plots of Complex Functions: A Journey in Illustration by Wegert and Semmler.

*The Fundamental Theorem of Algebra: A Visual Approach by Velleman.
Try an interactive demo at http://www.math.osu.edu/~fowler.291/phase/.
