Polynomials are simple to define objects that are both a useful tool for solving
mathematical problems, and spawn interesting research in their own right. This talk will be a brief tour through some of the interconnected ideas in the areas of number theory, geometry, and dynamical systems as seen from the point of view of properties of polynomials. Along the way, we will uncover a new relation between two well-known numbers: the golden mean, and Lehmer's number.