This is a two-faced book, and that’s a good thing. One face is a set of enrichment materials for bright high school students. The other face is a fairly comprehensive textbook on algebraic properties of polynomials. The main narrative requires only high-school math and no calculus, but some of the more advanced investigations use Rolle’s Theorem, Taylor expansions, and Rouché’s Theorem from complex analysis (although only for polynomial functions)
Looking on its enrichment face, the book does very little exposition but guides the student to discovery through problem sequences. The problems are graded by difficulty and maturity level, with the more routine ones (that make up most of the book) being labeled Exercises. The more difficult problems (taken from a variety of sources, including the Putnam competition and the problem sections of several magazines) are labeled Problems. Finally some open-ended investigations are labeled Explorations. Solutions (usually brief) are given for all Exercises and Problems, and hints and references are given for the Explorations.
On the textbook face, the book covers everything that would traditionally be considered the Theory of Equations, including solution of equations, approximation of roots, factorization, irreducibility, Hensel’s Lemma (for congruences mod pn), ruler and compass constructions, and methods of solution of cubics and quartics by radicals (there’s no Galois theory, so the book doesn’t cover arbitrary degrees). In the topic of isolation of real zeroes it covers not only the familiar Descartes Rule of Signs but also the more sensitive Fourier–Budan and Sturm tests, and it shows ways of bounding the absolute value of complex roots. The book is weak on analytic properties of polynomials, and has nothing about orthogonal polynomials, but does cover numerical approximation to roots and has a chapter on inequalities and approximation by polynomials.
The present book is an excellent introduction to the subject for anyone, from high schooler to professional. A much more advanced and comprehensive but concise book that covers all these topics and more is Prasolov’s Polynomials.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.