Polynomials and equations

Primarily a textbook to prepare Sixth Form students for public examinations in Hong Kong, this book is also useful as a reference for undergraduate students since it contains some advanced theory of equations beyond the sixth form level. Cover......Page 1 Title......Page 4 CONTENT......Page 6 Preface......Page 8 1.1 Terminology......Page 10 1.2 Polynomial functions......Page 13 1.3 The domain R[x]......Page 17 1.4 Other polynomial domains......Page 23 1.5 The remainder theorem......Page 26 1.6 Interpolation......Page 36 2.1 Divisibility......Page 42 2.2 Divisibility in other polynomial domains......Page 47 2.3 LCM and HCF......Page 50 2.4 Euclidean algorithm......Page 53 2.5 Unique factorization theorem......Page 62 3.1 Ancient Egyptian and Babylonian algebra......Page 64 3.2 Ancient Chinese algebra......Page 66 3.3 Ancient Greek algebra......Page 70 3.4 The modern notations......Page 72 4.1 Terminology......Page 74 4.2 Linear and quadratic equations......Page 76 4.3 Cubic equations......Page 78 4.4 Equations of higher degree......Page 88 5.1 Basic relations......Page 90 5.2 Integral roots......Page 101 5.3 Rational roots......Page 107 5.4 Reciprocal equations......Page 112 6.1 The leading term......Page 122 6.2 The constant term......Page 125 6.3 Other bounds of real roots......Page 127 7.1 Differentiation......Page 134 7.2 Taylor's formula......Page 141 7.3 Multiple roots......Page 143 7.4 Tangent......Page 149 7.5 Maximum and minimum......Page 151 7.6 Bend points and inflexion points......Page 154 8.1 Continuity......Page 158 8.2 Convergence......Page 162 8.3 Bolzano's theorem......Page 164 8.4 Rolle's theorem......Page 169 9.1 The Sturm sequence......Page 176 9.2 Sturm's theorem......Page 180 9.3 Fourier's theorem......Page 189 9.4 Descartes' rule of signs......Page 191 10.1 Newton-Raphson method......Page 196 10.2 Qin-Horner method......Page 201 Appendix. Two theorems on separation of roots......Page 216 Numerical answers to exercises......Page 226 Index......Page 240