Polynomials

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.- E.37 Little Fermat Theorem generalized.- 3.6 Rational Functions.- 3.6.4–6 Partial fractions.- E.38 Principal parts and residues.- 3.7 Problems on Factorization.- 3.8 Other Problems.- Hints.- 4 Equations.- 4.1 Simultaneous Equations in Two or Three Unknowns.- 4.1.2 Two linear homogeneous equations in three unknowns.- 4.1.7 Use of symmetric functions of zeros.- 4.2 Surd Equations.- 4.3 Solving Special Polynomial Equations.- 4.3.6 Surd conjugate.- 4.3.6–10 Field extensions.- E.39 Solving by radicals.- E.40 Constructions using ruler and compasses.- 4.4 The Fundamental Theorem of Algebra: Intersecting Curves.- 4.5 The Fundamental Theorem: Functions of a Complex Variable.- 4.6 Consequences of the Fundamental Theorem.- 4.6.1 Decomposition into linear factors.- 4.6.2 C as an algebraically closed field.- 4.6.3 Factorization over R.- 4.6.7 Uniqueness of factorization.- 4.6.8 Uniqueness of polynomial of degree n taking n + 1 assigned values.- 4.6.10 A criterion for irreducibility over Z.- E.41 Zeros of the derivative: Gauss-Lucas Theorem.- 4.7 Problems on Equations in One Variable.- 4.8 Problems on Systems of Equations.- 4.9 Other Problems.- Hints.- 5 Approximation and Location of Zeros.- 5.1 Approximation of Roots.- 5.1.1 The method of bisection.- 5.1.3 Linear interpolation.- 5.1.4 Horner’s Method.- 5.1.5 Newton’s Method.- 5.1.9 Successive approximation to a fixpoint.- E.42 Convergence of Newton approximations.- E.43 Newton’s Method according to Newton.- E.44 Newton’s Method and Hensel’s Lemma.- E.45 Continued fractions: Lagrange’s method of approximation.- E.46 Continued fractions: another approach for quadratics.- 5.2 Tests for Real Zeros.- 5.2.7 Descartes’ Rule of Signs.- 5.2.12 A bound on the real zeros.- 5.2.15 Rolle’s Theorem.- 5.2.17–20 Theorem of Fourier-Budan.- E.47 Proving the Fourier-Budan Theorem.- E.48 Sturm’s Theorem.- E.49 Oscillating populations.- 5.3 Location of Complex Roots.- 5.3.3 Cauchy’s estimate.- 5.3.8 Schur-Cohn criterion.- 5.3.9 Stable polynomials.- 5.3.10 Routh-Hurwitz criterion for a cubic.- 5.3.11 Nyquist diagram.- 5.3.13 Rouche’s Theorem.- E.50 Recursion relations.- 5.4 Problems.- Hints.- 6 Symmetric Functions of the Zeros.- 6.1 Interpreting the Coefficients of a Polynomial.- 6.1.9 Condition for real cubic to have real zeros.- 6.1.12 The zeros of a quartic expressed in terms of those of its resolvent sextic.- 6.2 The Discriminant.- 6.2.5 Discriminant of a cubic.- E.51 The discriminant of tn ? 1.- 6.3 Sums of the Powers of the Roots.- 6.3.6 The recursion formula.- E.52 Series approach for sum of powers of zeros.- E.53 A recursion relation.- E.54 Sum of the first nkth powers.- 6.4 Problems.- Hints.- 7 Approximations and Inequalities.- 7.1 Interpolation and Extrapolation.- 7.1.5 Lagrange polynomial.- 7.1.7–13 Finite differences.- 7.1.16 Factorial powers.- E.55 Building up a polynomial.- E.56 Propagation of error.- E.57 Summing by differences.- E.58 The absolute value function.- 7.2 Approximation on an Interval.- 7.2.1 Least squares.- 7.2.2 Alternation.- 7.2.5 Bernstein polynomials.- E.59 Taylor approximation.- E.60 Comparison of methods for approximating square roots.- 7.3 Inequalities.- 7.3.5–6 Generalizations of the AGM inequality.- 7.3.8 Bernoulli inequality.- 7.3.10 Newton’s inequalities.- E.61 AGM inequality for five variables.- 7.4 Problems on Inequalities.- 7.5 Other Problems.- Hints.- 8 Miscellaneous Problems.- E.62 Zeros of z?1[(1 + z)n ? 1 ? zn].- E.63 Two trigonometric products.- E.64 Polynomials all of whose derivatives have integer zeros.- E.65 Polynomials with equally spaced zeros.- E.66 Composition of polynomials of several variables.- E.67 The Mandelbrot set.- E.68 Sums of two squares.- E.69 Quaternions.- Hint.- Answers to Exercises and Solutions to Problems.- Notes on Explorations.- Further Reading.