This is what The Fundamental Theorem of Algebra looks like. Click on the above to get it to animate. The polynomial is given by f(z) = z³ – 4z² + 25z – 3000. Fix r>0, and plug in all values of z=x+iy into f such that x²+y²=r². This will give you a single snapshot of the above animation, which shows what happens as r increases. It looks as if there should be some value of r for which the curve crosses through 0, doesn’t it?

The animated .gif is so useful toward the explanation I have in mind that I may decide to write this next essay in .html form instead of as a .pdf. I’ve never tried to put LaTeX into html files before, so I don’t know exactly how well this is going to go. I’m also a bit annoyed that I can’t seem to get the animated .gif to autoplay, and I have to open a new window just for the image. Maybe it’s an annoying WordPress setting.