Introduction

The solution of general quadratic equations becomes possible, in terms of simple square roots, once one has access to the machinery of complex numbers. The question naturally arises as to whether it is possible to solve higher-order equations in the same way. In fact, we must be careful to pose this question properly. We might be interested in whether we need to extend the number system still further. For example, if we write down a cubic equation with coefficients that are complex numbers, can we find all the roots in terms of complex numbers? We can ask similar questions for higher-order polynomial equations. The investigation of the solution of cubic and quartic equations is a topic that used to be popular in basic courses on complex numbers, but has become less fashionable recently, probably because of the extensive manipulations that are required. Armed with Mathematica, however, such manipulations become routine, and we can revisit some of the classic developments in algebra quite straightforwardly. These topics have become so unfashionable, in fact, that the author received some suggestions from readers of early drafts of this book that this material should be, if not removed altogether, relocated to an appendix! I have left this material here quite deliberately, having found numerous applications for the solutions of cubics, at least, in applied mathematics. You may feel free to skip this part of the material if you have no interest in cubics and higher order systems.