Methods of Solving Nonstandard Problems

This book can be considered a sequel of sorts to the author’s previous Methods of Solving Complex Geometry Problems (CGP). Like CGP, this is a problem book, but now the subject of the problems is not geometry but equations. The primary focus is on polynomial and trigonometric equations, but other kinds (including differential equations) make an appearance as well. So do interesting mathematical topics like basic number theory, combinatorics, and symmetric functions. To give a sense of the kind of equations that appear in the book, here is a very small sample:

• \(\sqrt{1+x+x^2}+\sqrt{1-x+x^2}=2\)
• \( ((x+2)^2 + x^2)^3 = 8x^4-(x+2)^2\)
• \(x^2+x = 1111111122222222\)
• \(\cos 3x + \sin 2x - \sin 4x=0\)
• \(\sqrt{-\cos x} = \sqrt{-1+2\sin^2x}\)

And these are some of the simpler equations to write down; there are some that have so many superscripts, subscripts, and/or square root signs within square root signs that I couldn’t begin to try and reproduce them here.

In addition to equations, the book contains problems that would not be out of place in an elementary number theory course (e.g., prove that \(n^4+4\) is never prime if \(n>1\); find all integers \(x\) and \(y\) such that \(xy=x+y\)) or a basic combinatorics course (there are several problems, for example, involving binomial coefficients). Other topics that are covered here include symmetric polynomials and cubic and quartic polynomials, including discussions of Cardano, Viète and Ferrari. (The insolvability of the quintic is mentioned briefly.)

Now, I must confess that solving complicated-looking equations is not a big thrill for me; confronted with a ghastly-looking equation, my first inclination is not to solve it but to ignore it and hope it goes away. For this reason, and also because I actually do teach several courses in geometry, I have to say that I found CGP more to my taste than this book. However, the quality of a book does not depend on how much I like the subject matter; we must look to see what the author intended to do, and whether she succeeded in doing it.

The author’s goals are clear: she wants students to feel comfortable about (and even enjoy) mathematics, and she wants to provide them a range of techniques for thinking about equations- a sense of direction in solving problems such as are in the text, rather than freezing in fear if one is encountered. By virtue of clear, friendly writing (including use of the first person, as in “I remember when I was in 9th grade…”) and copious examples, these goals have been met.

Like most reviewers, I found a few nits to pick, but these are relatively minor and few in number. First, the book seems to be a bit unclear about just what kind of background is assumed on the part of the reader. In view of the fact that it starts with definitions of terms like “function”, “domain” and “range”, one would think that practically no background at all is assumed. However, on page 9 we encounter a reference to the derivative, a term which the author does not define; derivatives show up on a number of occasions later in the text as well, as do differential equations. So some background in calculus seems to be assumed, at least for parts of the book; it seems odd, then, that the author would spend time on preliminary issues like function domains and continuity.

Second, there were times when I thought that the author, in her zeal to write clearly and informally, was a little too informal and crossed the line into imprecision. This occurred in both definitions and problem solutions. For example, in discussing monotonic functions, the author first writes: “A function is [a] monotonic function if its first derivative is continuous and does not change sign.” This is immediately followed by the sentence: “Monotonic function is a function which is either entirely nonincreasing or nondecreasing.” Which of these is the definition? Does the author expect a student to believe that they are equivalent statements? A student may find this confusing, and come away with the (obviously incorrect) belief that a monotonic function must be continuously differentiable.

Likewise, problem 38 on page 156 of the book asks the reader to prove that “if \(x^2+ax+b=0\) has a rational root, then it is an integer.” The statement of this problem does not specify the assumption that \(a\) and \(b\) are integers, and also uses the word “root” where I would say “solution”. (I think that polynomials have roots and equations have solutions.) Then, after working the problem, the author writes “the given equation cannot have rational roots, only integer roots.” Of course, I know what the author meant, and I assume most students will too, but since every integer is a rational number, sentences like this make me wince.

There are also occasional typos, one of which turned out to be kind of amusing. In defining the trigonometric functions, the author writes: “Then (by Embry, Calculus, and Linear Algebra) we define its first coordinate as \(\cos t\) and the second coordinate as \(\sin(t)\).” The trivial typo of putting the \(t\) in parentheses in one case and not the other didn’t bother me at all, but I did spend a few minutes puzzling over what this mysterious “Embry” was that was teaming up with calculus and linear algebra to help define the trigonometric functions. Finally, I googled “embry calculus linear algebra” and discovered that what the author meant was the book Calculus and Linear Algebra by Embry et al.

I don’t think, however, that these issues seriously detract from this book’s value to its intended audience. Let me end this review, therefore, by explicitly identifying just who that audience might be. Certainly students with an interest in problem-solving (particularly students who like to take Olympiad-style contests) will find suggestions, techniques and examples here that will serve them well. Students who are studying, or want to brush up on, basic algebra or trigonometry should also find material of interest here. Instructors of basic algebra, trigonometry, calculus, and even number theory or combinatorics may also find some interesting problems in this book to present to this class either in lecture or as homework.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.