E.J. Barbeau. Polynomials. With 36 Illustrations. Springer

Transcription

1 E.J. Barbeau Polynomials With 36 Illustrations Springer

2 Contents Preface Acknowledgment of Problem Sources vii xiii 1 Fundamentals 1 /l.l The Anatomy of a Polynomial of a Single Variable Multiplication by detached coefficients Even and odd polynomials E.I Square of a polynomial E.2 Sets with equal polynomial-value sums E.3 Polynomials as generating functions ^1.2 Quadratic Polynomials Quadratic formula Theory of the quadratic Cauchy-Schwarz inequality Arithmetic-geometric mean inequality Approximation of quadratic irrational by a rational E.4 Graphical solution of the quadratic E.5 Polynomials, some of whose values are squares j(l.3 Complex Numbers De Moivre's theorem Square root of a complex number ; Tchebychef polynomials E.6 Commuting polynomials 1.4 Equations of Low Degree Cardan's method for cubic Descartes' method for quartic Ferrari's method for quartic Reciprocal equations E.7 The reciprocal equation substitution 1.5 Polynomials of Several Variables Criterion for homogeneity Elementary symmetric polynomials of 2 variables Elementary symmetric polynomials of 3 variables Arithmetic-geometric mean inequality for 3 numbers Polynomials with n variables E.8 Polynomials in each variable separately

3 xviii Contents E.9 The range of a polynomial E.10 Diophantine equations 1.6 Basic Number Theory and Modular Arithmetic Euclidean algorithm Modular arithmetic Linear congruence E.ll Length of Euclidean algorithm E.12 The congruence ax = b (mod m) E.13 Polynomials with prime values E.14 Polynomials whose positive values are Fibonacci numbers 1.7 Rings and Fields Z m E.15 Irreducible polynomials of low degree modulo p 1.8 Problems on Quadratics Other Problems 43 Hints 46 2 Evaluation, Division, and Expansion Horner's Method Use of Horner's method for Taylor expansion E.16 Number of multiplications for c n E.17 A Horner's approach to the binomial expansion E.18 Factorial powers and summations 2.2 Division of Polynomials Factor Theorem Number of zeros cannot exceed degree of polynomial Long division of polynomials; quotient and remainder Division Theorem Factor Theorem for two variables 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 Definition of derivative Properties of the derivative Taylor's Theorem Multiplicity of zeros E.22 Higher order derivatives of the composition of two functions E.23 Partial derivatives E.24 Homogeneous polynomials

4 Contents xix E.25 Cauchy-Riemann conditions E.26 The Legendre equation 2.4 Graphing Polynomials Symmetry of cubic graph E.27 Intersection of graph of polynomial with lines E.28 Rolle's Theorem 2.5 Problems 78 Hints 81 3 Factors and Zeros Irreducible Polynomials Irreducibility of linear polynomials Irreducibility over Q related to irreducibility over Z Eisenstein criterion 3.2 Strategies for Factoring Polynomials over Z Undetermined coefficients E.29 t 2 - t + a as a divisor of t n +1 + b E.30 The sequence u n (t) 3.3 Finding Integer and Rational Roots: Newton's Method of Divisors Newton's Method of Divisors E.31 Rational roots of nt 2 + (n + l)t - (n + 2) 3.4 Locating Integer Roots: Modular Arithmetic Chinese Remainder Theorem E.32 Little Fermat Theorem E.33 Hensel's Lemma 3.5 Roots of Unity Roots of unity Primitive roots of unity Cyclotomic polynomials Quadratic residue 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 Partial fractions E.38 Principal parts and residues 3.7 Problems on Factorization Other Problems 117 Hints 120

5 xx Contents 4 Equations Simultaneous Equations in Two or Three Unknowns Two linear homogeneous equations in three unknowns Use of symmetric functions of zeros 4.2 Surd Equations Solving Special Polynomial Equations Surd conjugate Field extensions E.39 Solving by radicals E.40 Constructions using ruler and compasses 4.4 The Fundamental Theorem of Algebra: Intersecting Curves The Fundamental Theorem: Functions of a Complex Variable Consequences of the Fundamental Theorem Decomposition into linear factors C as an algebraically closed field Factorization over R Uniqueness of factorization Uniqueness of polynomial of degree n taking n + 1 assigned values A criterion for irreducibility over Z E.41 Zeros of the derivative: Gauss-Lucas Theorem 4.7 Problems on Equations in One Variable Problems on Systems of Equations Other Problems 159 Hints Approximation and Location of Zeros Approximation of Roots The method of bisection Linear interpolation Horner's Method Newton's Method 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

6 Contents xxi E.46 Continued fractions: another approach for quadratics 5.2 Tests for Real Zeros Descartes' Rule of Signs A bound on the real zeros Rolle's Theorem 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 Cauchy's estimate Schur-Cohn criterion Stable polynomials Routh-Hurwitz criterion for a cubic Nyquist diagram Rouche's Theorem E.50 Recursion relations 5.4 Problems 190 Hints Symmetric Functions of the Zeros Interpreting the Coefficients of a Polynomial Condition for real cubic to have real zeros The zeros of a quartic expressed in terms of those of its resolvent sextic 6.2 The Discriminant Discriminant of a cubic E.51 The discriminant of t n Sums of the Powers of the Roots The recursion formula E.52 Series approach for sum of powers of zeros E.53 A recursion relation E.54 Sum of the first n fcth powers 6.4 Problems 205 Hints Approximations and Inequalities Interpolation and Extrapolation Lagrange polynomial Finite differences

7 xxii E.55 E.56,E.57 E.58 E Hints Factorial powers Building up a polynomial Propagation of error Summing by differences The absolute value function Approximation on an Interval E.59 E.60 Inequalities Least squares Alternation Bernstein polynomials Taylor approximation Comparison of methods for approximating square roots Generalizations of the AGM inequality Bernoulli inequality Newton's inequalities AGM inequality for five variables Problems on Inequalities Other Problems Contents Miscellaneous Problems 236 E.62 E.63 E.64 E.65 E.66 E.67 E.68 E.69 Hints Zeros of*- 1 [(l + *) B -l-* n ] Two trigonometric products Polynomials all of whose derivatives have integer zeros Polynomials with equally spaced zeros Composition of polynomials of several variables The Mandelbrot set Sums of two squares Quaternions Answers to Exercises and Solutions to Problems Notes on Explorations Glossary Further Reading Index