Polynomials and Equations

PREFACE

PREFACE Like its predecessor Fundamental Ooncepts ofMathematics (HKUP, 1988) and its successor Vectors, Matrices and Geometry (to be published), the present volume Polynomials and Equations is primarily a textbook for students of the Sixth Form. It contains the necessary materials for the preparation of the different public examinations of this level in Hong Kong. Moreover, this book also includes parts of the more advanced theory of equations (in Chapters 6, 8, 9 and 10) that are not required in these examinations but are of sufficient importance to serious students of mathematics. Hence it may also, serve as a reference book for undergraduate students. The first two chapters present the algebra of the domain of polynomials with real coefficients and include a proof of the unique factorization theorem which is an importa.nt item in the undergraduate algebra syllabus but is usually not required in the Sixth Form examinations . For the benefit of the interested readers, notes are taken, at appropriate places, of polynomials with other coefficients and in more than one indeterminates. Chapters Three to Five form a self-contained unit on elementary theory of equations. A brief outline of history is given in Chapter Three. This is probably a novelty in a textbook of this level and the section on Chinese mathematics may have a special appeal for students in Hong Kong. Here again Section 4.3 on Cardano's method is some additional material which is not required in the Sixth Form examinations. In the remaining five chapters of the book polynomials are treated as functions of a real variable. Chapter Seven on derivatives should be relevant to the Sixth Form examination syllabuses. While the derivative of a polynomial is defined here in purely algebraic terms, it is shown to coincide with the analytic notion of the derivative of a differentiable function given in terms of limit. Taylor's expansion is used extensively in the classification of multiple roots. Undergraduate students may find Chapter Eight a useful revision of the most Preface important concept of continuous function. The results of these two chapters find applications in the separation of roots and in the approximation to roots in the theory of equations presented in the last two chapters of this book. My former students and friends Miss I.A.C. Mok and Mr. S.N. Suen have provided the book with an excellent set of exercises without which this book would be incomplete and inadequate. My colleagues Dr. M.K. Siu and Dr. K.M. Tsang have been very generous with their suggestions and comments during the preparation of the main text. To them all I would like to express my gratitude. Last but not least I would like to thank Mrs. Annie Cheung for setting the whole text in the present form on AMS-Tex, and Mr. E.T.B. Lau for the line drawings. viii K.T. Leung November 1991 ...