Polynomials And Equations

Polynomials and Equations POLYNOMIALS AND EQUATIONS K.T. Leung I.A.C. Mok S.N. Suen Hong Kong University Press * t~ *- 1.f ~ Xi iTd: Hong Kong University Press 139 Pokfulam Road, Hong Kong © Hong Kong University Press 1992 First published 1992 Reprinted 1993 ISBN 962 209 271 3 All rights reserved. No portion of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Printed in Hong Kong by Nordica Printing Co., Ltd. CONTENT Preface vii Chapter One Polynomials 1.1 Terminology 1.2 Polynomial functions 1.3 The domain R[x] 1.4 Other polynomial domains 1.5 The remainder theorem 1.6 Interpolation 1 4 8 14 27 Chapter Two Factorization of polynomials 2.1 Divisibility 2.2 Divisibility in other polynomial domains 2.3 LCM and HCF 2.4 Euclidean algorithm 2.5 Unique factorization theorem 33 38 41 44 53 Chapter Three 3.1 3.2 3.3 3.4 Notes on the study of equations in ancient civiliz a tions Ancient Egyptian and Babylonian algebra Ancient Chinese algebra Ancient Greek algebra The modern notations 17 Chapter Four 4.1 4.2 4.3 4.4 55 57 61 63 Linear, quadratic and cubic equations Terminology Linear and quadratic equations Cubic equations Equations of higher degree 65 67 69 79 Chapter Five Roots and coefficients 5.1 Basic relations 5.2 Integral roots 5.3 Rational roots 5.4 Reciprocal equations 81 92 98 103 Content Chapter Six Bounds of real roots 6.1 The leading term 6.2 The constant term 6.3 Other bounds of real roots Chapter Seven The derivative 7.1 Differentiation 7.2 Taylor's formula 7.3 Multiple roots 7.4 Tangent 7.5 Maximum and minimum 7.6 Bend points and inflexion points Chapter Eight Polynomials as continuous functions 8.1 Continuity 8.2 Convergence 8.3 Bolzano's theorem 8.4 Rolle's theorem 113 116 118 125 132 134 140 142 145 149 153 155 160 Chapter Nine Separation of real roots 9.1 The Sturm sequence 9.2 Sturm's theorem 9.3 Fourier's theorem 9.4 Descartes' rule of signs 167 171 Chapter Ten Approximation to real roots 10.1 Newton-Raphson method 10.2 Qin-Horner method 187 Appendix 207 Two theorems on separation of roots 180 182 192 Numerical answers to exercises 217 Index 231 vi PREFACE Like its predecessor Fundamental Ooncepts of Mathematics (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 th