We will take a detailed look at some of the mathematics underlying the most fundamental problem of algebra -- solving polynomial equations. A review of quadratic equations and the quadratic formula will motivate a closer look at rational and real numbers. The study of the solution of cubic equations discovered by the great Italian mathematicians of the sixteenth century will focus our attention on complex numbers and lead to some trigonometric issues. After solving quartic equations by similar methods, we will consider issues arising in trying to solve polynomial equations of arbitrary degree.