Polynomials and Equations

Cover

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Title Page, Copyright

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CONTENT

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PREFACE

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pp. vii-viii

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...

CHAPTER ONE: POLYNOMIALS

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pp. 1-32

The study of polynomials constitutes a major component of the mathematics course in secondary school. There polynomials first appear in connection with equations where the main concern is the evaluation of roots. Later they are treated as functions; as such we examine their derivatives, their integrals and their maxima and...

CHAPTER TWO: FACTORIZATION OF POLYNOMIALS

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pp. 33-54

A comparison between the number system Z and the polynomial domain R[ x] will show that they are very similar as far as formal properties of addition and multiplication are concerned. In fact for most calculations that were carried out in the last chapter, we almost could have operated with polynomials as if they were integers. We...

CHAPTER THREE: NOTES ON THE STUDY OF EQUATIONSIN ANCIENT CIVILIZATIONS

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pp. 55-64

Equations are among the topics of mathematics that have been studied extensively for thousands of years. As equations will be the main subject for the rest of the present course, we shall begin here with a brief description of a small selection of results obtained by mathematicians in the antiquity...

CHAPTER FOUR: LINEAR, QUADRATIC AND CUBIC EQUATIONS

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pp. 65-80

A polynomial g(x} = bmxm bm_Ixm - 1 ... bix bo defines a polynomial function g(x} : R - R which maps every real number c of the domain to the real number g(c} of the range. The evaluation of g(x} at x = c is a very staight-forward matter and there are simple methods of calculation by which the correct value of g( c} can be obtained. We...

CHAPTER FIVE: ROOTS AND COEFFICIENTS

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pp. 81-112

We remarked in the last chapter that for an equation of degree higher than four we do not possess a general method of solution and that the roots of such equations may not be obtained by root extractions and rational operations on the coefficients. Naturally this does not mean that we shall henceforth neglect the study of equations of higher...

CHAPTER SIX: BOUNDS OF REAL ROOTS

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pp. 113-124

Let given be an equation f(x) = anxn an_lxn- 1 ... ao = 0 where the coefficients have numerical real values. In our attempt to find the real roots of f(x) = 0, it would be very advantageous if we knew the range of values in which they might occur. To put it in another way, we wish to obtain for the search of the real roots of...

CHAPTER SEVEN: THE DERIVATIVE

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pp. 125-148

Up to the last chapter, only purely algebraic properties of polynomials are used in our study of equations. Beginning with this chapter, we shall put more emphasis on the functional aspect of the polynomial and examine in detail the change of the value of a polynomial corresponding to a minute increase or diminution of the variable...

CHAPTER EIGHT: POLYNOMIALS AS CONTINUOUS FUNCTIONS

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pp. 149-166

In the last chapter we treat polynomials as differentiable functions and study their derivatives and Taylor's expansions. As each differentiable function is also continuous, polynomials are continuous functions. In this chapter we shall first introduce the general concept of continuous function and prove that polynomial functions are...

CHAPTER NINE: SEPARATION OF REAL ROOTS

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pp. 167-186

The method of separation of roots based on Rolle's theorem of the last chapter has one major disadvantage in that the roots of the equation f'{x) = 0 have to be found before the roots of f{x) = 0 can be isolated. Now if deg f{x) = n, then deg /,(x) = n - 1. For...

CHAPTER TEN: APPROXIMATION TO REAL ROOTS

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pp. 187-206

We recall that given an equation of degree less than five, the exact values of its roots can be written as expressions that involve only rational operations and root extractions on the coefficients. It is also known that such expressions of roots are not generally available for an equation of higher degree. Therefore, for such equations, we shall...

APPENDIX: TWO THEOREMS ON SEPARATION OF ROOTS

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pp. 207-216

NUMERICAL ANSWERS TO EXERCISES

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pp. 217-230

INDEX

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pp. 231-232