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Awesome Polynomials for Mathematics Competitions

Titu Andreescu, Navid Safaei, and Alessandro Ventullo
XYZ Press
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Andrzej Sokolowski
, on
The book Awesome Polynomials for Mathematics Competitions by T. Andreescu, N. Safaei, and A. Ventullo aims to be a source of problems on polynomial functions for advanced mathematical competitions. It comprises eight chapters and offers about three hundred problems with detailed solutions. The volume merges the concepts of algebra and the authors' teaching experience; therefore, it can serve not only mathematics competitions but also support the teaching of mathematics at any level. Furthermore, frequent references to mathematical Olympiads from countries such as Italy, Bulgaria, Poland, the USA, and South Korea stand for a high quality of the book content.
Among traditional operations on polynomials, the book introduces reciprocal polynomial functions and offers unconventional approaches to finding their roots. Complex numbers and their support for expressing the solutions of polynomial equations concisely and trigonometric representations of complex numbers and the De Moivre theorem make the book content applicable to advanced precalculus courses. A subsequent part of the book is dedicated to finding algebraic expressions for polynomial functions given by various conditions; for example, by using symmetry, the First and the Second Uniqueness Lemma, or an infinite sequence of roots.
The book encompasses numerous mathematical ideas that polynomial functions can engage in, and as such, it generates exciting yet intellectually challenging and rewarding algebraic thought processes. However, two suggestions emerged that could perhaps be considered for the future reference: 
  • The authors did not offer supporting visual (graphical) representations of the abstract ideas. While including graphs is unnecessary, it is seen that including such could attract readers who do not necessarily consider mathematics as a major area of study.
  • While the solutions to suggested problems are mathematically rigorous and detailed, there seems to be a need for sample problems that would provide opportunities for exploring the problems' complexity and prompting the readers to create solutions that, while being mathematically sound, are creative.
As the book has the potential to be expanded to other volumes, including such methods of presenting or solving the problems would perhaps make the book attractive to a broader audience. Even in the current form, the volume can serve as a source not only for math competitions but can enhance any undergraduate and graduate math courses that deal with the analysis of polynomial functions. A continuation of the book series is anticipated, and I am sure it will be very much appreciated.


Andrzej Sokolowski, Ph.D., authored several books and research papers on developing students' mathematical reasoning and integrating this reasoning to improve their understanding of science.