Polynomials and Equations

CONTENT

CONTENT Preface 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 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 Chapter Three Notes on the study of equations in ancient civilizations vii 1 4 8 14 17 27 33 38 41 44 53 3.1 Ancient Egyptian and Babylonian algebra 55 3.2 Ancient Chinese algebra 57 3.3 Ancient Greek algebra 61 3.4 The modern notations 63 Chapter Four Linear, quadratic and cubic equations 4.1 Terminology 65 4.2 Linear and quadratic equations 67 4.3 Cubic equations 69 4.4 Equations of higher degree 79 Chapter Five Roots and coefficients 5.1 Basic relations 81 5.2 Integral roots 92 5.3 Rational roots 98 5.4 Reciprocal equations 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 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 Chapter Ten Approximation to real roots 10.1 Newton-Raphson method 10.2 Qin-Horner method Appendix Two theorems on separation of roots Numerical answers to exercises Index vi 113 116 118 125 132 134 140 142 145 149 153 155 160 167 171 180 182 187 192 207 217 231 ...