What do we see? To the left is a circle centred at the origin with radius [math]r=|A|[/math] where [i]A[/i] lies on the x-axis. The point, [i]z[/i], lies on this circle. We think of [i]z[/i] as a complex number. Indeed, [i]z[/i] can be [b]any[/b] complex number. Here [math]r=0.2[/math]. To the right there appears to be another circle, but this is not so. The closed path is the locus of [math]w=p(z)[/math] where [i]p[/i] is some polynomial with complex coefficients. Here [math]p(z)=z^2+z+1[/math].[br]Check the following:[br]Rotating [i]z[/i] through one revolution about the circle causes [i]w[/i] to rotate one revolution about its locus. As the radius of the circle diminishes ([math]r=|A| \rightarrow 0[/math]), the locus of [i]w[/i] approximates a circle of radius [i]r[/i] centred at 1. As this radius increases, the locus of [i]w[/i] behaves strangely! Try [i]A[/i] = 0.4, 0.6, 0.8 and 1. Describe what you see.

Now zoom out, so you can see the whole picture. Notice that the locus passes through the origin. Why is this? Now try [i]A[/i] = 2, 3. Zoom out again. Describe what you see. Notice that as [i]z[/i] rotates through a revolution, [i]w[/i] does so twice. We say that the locus has [b]winding number[/b] 2 (about the origin). Try [i]A[/i] = 5, 10. Zoom out and describe. What about 50?[br]Now explore all of this for [math]p(z)=z^n+z+1[/math] with [i]n[/i] = 3, 4, 5, ... (To do this, just right click on [i]w[/i] and edit Object Properties.) Try any polynomial, [i]p[/i], of your choice, including ones with imaginary coefficients, and, in particular, with imaginary constant coefficients.[br]No matter what polynomial, [i]p[/i], we choose, we see that for sufficiently small [i]r[/i], the locus of [i]w[/i] is almost a (small) circle centred at [math]z_0[/math], the constant coefficient in [math]p(z)[/math]. On the other hand, for sufficiently large [i]r[/i], the locus is almost a (large) circle (with winding number [i]n[/i]) centred at the origin. (What is the radius of each of these circles?) If [math]z_0\neq 0[/math], then the locus must, for some value of [i]r[/i], pass through the origin. Thus [math]w=p(z)=0[/math] for some [i]z[/i], proving the [b]Fundamental Theorem of Algebra[/b]. Of course, the word 'must' needs further scrutiny and relies on the fact that the locus of [i]w[/i] varies [b]continuously[/b] with [i]r[/i].