Polynomials

1 Fundamentals.- 1.1 The Anatomy of a Polynomial of a Single Variable.- 1.2 Quadratic Polynomials.- 1.3 Complex Numbers.- 1.4 Equations of Low Degree.- 1.5 Polynomials of Several Variables.- 1.6 Basic Number Theory and Modular Arithmetic.- 1.7 Rings and Fields.- 1.8 Problems on Quadratics.- 1.9 Other Problems.- Hints.- 2 Evaluation, Division, and Expansion.- 2.1 Horner’s Method.- 2.2 Division of Polynomials.- 2.3 The Derivative.- 2.4 Graphing Polynomials.- 2.5 Problems.- Hints.- 3 Factors and Zeros.- 3.1 Irreducible Polynomials.- 3.2 Strategies for Factoring Polynomials over Z.- 3.3 Finding Integer and Rational Roots: Newton’s Method of Divisors.- 3.4 Locating Integer Roots: Modular Arithmetic.- 3.5 Roots of Unity.- 3.6 Rational Functions.- 3.7 Problems on Factorization.- 3.8 Other Problems.- Hints.- 4 Equations.- 4.1 Simultaneous Equations in Two or Three Unknowns.- 4.2 Surd Equations.- 4.3 Solving Special Polynomial Equations.- 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.- 4.7 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.- 5.2 Tests for Real Zeros.- 5.3 Location of Complex Roots.- 5.4 Problems.- Hints.- 6 Symmetric Functions of the Zeros.- 6.1 Interpreting the Coefficients of a Polynomial.- 6.2 The Discriminant.- 6.3 Sums of the Powers of the Roots.- 6.4 Problems.- Hints.- 7 Approximations and Inequalities.- 7.1 Interpolation and Extrapolation.- 7.2 Approximation on an Interval.- 7.3 Inequalities.- 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.