Most people learn in linear algebra that its possible to calculate the eigenvalues of a matrix by finding the roots of its characteristic polynomial. However, this method is actually very slow, and while its easy to remember and its possible for a person to use this method by hand, there are many better techniques available (which do not rely on factoring a polynomial).
So I was wondering, why on earth is it actually important to have techniques available to solve polynomial equations? (to be specific, I mean solving over $\mathbb{C}$)
I actually used to be fairly interested in how to do it, and I know a lot of the different methods that people use. I was just thinking about it though, and I'm actually not sure what sort of applications there are for those techniques.