# Fundamental theorem of algebra

Topic:
Algebra
A classic proof for the fundamental theorem of algebra (that is, each non-constant polynomial has a root among the complex numbers) uses topology.
It is to prove that has a root in the complex plane, that is, a is required (on the left) for which (on the right) goes through the origin (X). The circles on the left are always transformed to a closed curve on the right which wind around the origin, if is sufficiently large. On the other hand, if , the point on the left is transformed to a point on the right. Between these two extremal situations the right curve continuously changes. This results in a position when the right curve obviously crosses the origin. You can try some activities out:
1. Change the radius of the circle to find an image of the circle which crosses the origin on the right.
3. Show the image of the circle to immediately know that a radius will be fine or not.
4. Trace the images of the circles to have a general view about the map .
5. Change to different polynomials. Learn how the higher order polynomials behave.
6. You can also try non-polynomial functions. For example, is an interesting option.