All mathematical competitions include at least one problem on polynomials. The problems in polynomials are charming since they require considerable ingenuity for their solution. Usually, Algebra texts on competition mathematics do not delve deeper on all aspects of polynomials. Books devoted to polynomials exclusively are rarer. In this context, this book, Awesome Polynomials for Mathematics Competitions, the second in a three volume series (the first one being ”117 Polynomial problems”), is a welcome addition. Just like its predecessor, the book is well written, presents the topic in a lucid way and contains several interesting problems. One welcome feature of this book is that it discusses various strategies for problem solving and these strategies are presented in display boxes throughout the book.
The book has 8 chapters. The first chapter discusses the relation between a polynomial and its reciprocal polynomial. In the second chapter complex numbers and their use in polynomial problems are presented. This chapter is named ”Complex Numbers and Polynomials, Part I”. There is no Part II in this volume – perhaps we will see the advanced techniques on complex numbers in the next volume. The chapter contains several interesting problems. A non-intuitive result is Corollary 2.15 – If the coefficients of a polynomial are positive real numbers in ascending order, then all the roots lie outside or on the unit circle.
The next three chapters are devoted to finding all polynomials satisfying given conditions. This type of problems appear often in the national olympiads.
These three chapters discuss such problems comprehensively listing out several strategies for their solution. Two powerful and useful uniqueness lemmas are proved. Composition of polynomials, P(Q(x)), is discussed in detail. After a thorough study of these three chapters, one can gain an in depth understanding of solution strategies of these types of problems.
The next chapter discusses Lagrange’s Interpolation Formula. This formula is an essential tool in solving certain types of polynomial problems. A novel feature of this book is the use of (x d). Any polynomial can be written in terms of such polynomials but this technique is not that well-known. This book showcases several nice applications of this technique.
The seventh chapter discusses Newton’s identities. This chapter is a beautiful one with several charming problems. Three proofs are given for Newton’s identities – combinatorial, algebraic and using calculus. This is the chapter I enjoyed the most due to the several beautiful solutions.
The final chapter lists additional problems.
The inclusion of a brief alphabetical index is a welcome feature of this book. Since the book is fairly long containing 530 pages, the presence of an index helps one to locate the key results quickly.
The book contains more than 400 problems with full solutions. For many problems, more than one solution is given – for instance, for Example 4.24, four solutions are presented and each one teaches something new! The exposition of the theory is very comprehensive and clear. Many problems are original and other problems have been taken from team selection texts and international competitions. On the whole, this is an excellent book (just like the previous one from the same authors) for competition contestants. Looking forward to the next volume in the series.