Polynomials (Paperback)

'1 Fundamentals.- 1.1 The Anatomy of a Polynomial of a Single Variable.- 1.1.5 Multiplication by detached coefficients.- 1.1.19 Even and odd polynomials.- E.1 Square of a polynomial.- E.2 Sets with equal polynomial-value sums.- E.3 Polynomials as generating functions.- 1.2 Quadratic Polynomials.- 1.2.1 Quadratic formula.- 1.2.4 Theory of the quadratic.- 1.2.14 Cauchy-Schwarz inequality.- 1.2.17 Arithmetic-geometric mean inequality.- 1.2.18 Approximation of quadratic irrational by a rational.- E.4 Graphical solution of the quadratic.- E.5 Polynomials, some of whose values are squares.- 1.3 Complex Numbers.- 1.3.8 De Moivre's theorem.- 1.3.10 Square root of a complex number.- 1.3.15 Tchebychef polynomials.- E.6 Commuting polynomials.- 1.4 Equations of Low Degree.- 1.4.4 Cardan's method for cubic.- 1.4.11 Descartes' method for quartic.- 1.4.12 Ferrari's method for quartic.- 1.4.13 Reciprocal equations.- E.7 The reciprocal equation substitution.- 1.5 Polynomials of Several Variables.- 1.5.2 Criterion for homogeneity.- 1.5.5 Elementary symmetric polynomials of 2 variables.- 1.5.8 Elementary symmetric polynomials of 3 variables.- 1.5.9 Arithmetic-geometric mean inequality for 3 numbers.- 1.5.10 Polynomials with n variables.- E.8 Polynomials in each variable separately.- E.9 The range of a polynomial.- E.10 Diophantine equations.- 1.6 Basic Number Theory and Modular Arithmetic.- 1.6.1 Euclidean algorithm.- 1.6.5 Modular arithmetic.- 1.6.6 Linear congruence.- E.11 Length of Euclidean algorithm.- E.12 The congruence a? ? b (mod m).- E.13 Polynomials with prime values.- E.14 Polynomials whose positive values are.- Fibonacci numbers.- 1.7 Rings and Fields.- 1.7.6 Zm.- E.15 Irreducible polynomials of low degree modulo p.- 1.8 Problems on Quadratics.- 1.9 Other Problems.- Hints.- 2 Evaluation, Division, and Expansion.- 2.1 Horner's Method.- 2.1.8-9 Use of Horner's method for Taylor expansion.- E.16 Number of multiplications for cn.- E.17 A Horner's approach to the binomial expansion.- E.18 Factorial powers and summations.- 2.2 Division of Polynomials.- 2.2.2 Factor Theorem.- 2.2.4 Number of zeros cannot exceed degree of polynomial.- 2.2.7 Long division of polynomials; quotient and remainder.- 2.2.9 Division Theorem.- 2.2.12 Factor Theorem for two variables.- 2.2.15 Gauss' Theorem on symmetric functions.- E.19 Chromatic polynomials.- E.20 The greatest common divisor of two polynomials.- E.21 The remainder for special polynomial divisors.- 2.3 The Derivative.- 2.3.4 Definition of derivative.- 2.3.5 Properties of the derivative.- 2.3.9 Taylor's Theorem.- 2.3.15 Multiplicity of zeros.- E.22 Higher order derivatives of the composition of two functions.- E.23 Partial derivatives.- E.24 Homogeneous polynomials.- E.25 Cauchy-Riemann conditions.- E.26 The Legendre equation.- 2.4 Graphing Polynomials.- 2.4.6 Symmetry of cubic graph.- E.27 Intersection of graph of polynomial with lines.- E.28 Rolle's Theorem.- 2.5 Problems.- Hints.- 3 Factors and Zeros.- 3.1 Irreducible Polynomials.- 3.1.3 Irreducibility of linear polynomials.- 3.1.10 Irreducibility over Q related to irreducibility over Z.- 3.1.12 Eisenstein criterion.- 3.2 Strategies for Factoring Polynomials over Z.- 3.2.6 Undetermined coefficients.- E.29 t2 ? t + a as a divisor of tn + 1 + b.- E.30 The sequence un(t).- 3.3 Finding Integer and Rational Roots: Newton's Method of Divisors.- 3.3.5 Newton's Method of Divisors.- E.31 Rational roots of nt2 + (n + 1)t ? (n + 2).- 3.4 Locating Integer Roots: Modular Arithmetic.- 3.4.7 Chinese Remainder Theorem.- E.32 Little Fermat Theorem.- E.33 Hensel's Lemma.- 3.5 Roots of Unity.- 3.5.1 Roots of unity.- 3.5.7 Primitive roots of unity.- 3.5.9 Cyclotomic polynomials.- 3.5.18 Quadratic residue.- 3.5.19 Sicherman dice.- E.34 Degree of the cyclotomic polynomials.- E.35 Irreducibility of the cyclotomic polynomials.- E.36 Coefficients of the cyclotomic polynomials