PolynomialsThe book extends the high school curriculum and provides a backdrop for later study in calculus, modern algebra, numerical analysis, and complex variable theory. Exercises introduce many techniques and topics in the theory of equations, such as evolution and factorization of polynomials, solution of equations, interpolation, approximation, and congruences. The theory is not treated formally, but rather illustrated through examples. Over 300 problems drawn from journals, contests, and examinations test understanding, ingenuity, and skill. Each chapter ends with a list of hints; there are answers to many of the exercises and solutions to all of the problems. In addition, 69 "explorations" invite the reader to investigate research problems and related topics. |
Contents
III | 1 |
IV | 9 |
V | 14 |
VI | 18 |
VII | 25 |
VIII | 31 |
IX | 38 |
X | 40 |
XXXII | 150 |
XXXIII | 154 |
XXXIV | 155 |
XXXV | 159 |
XXXVI | 160 |
XXXVII | 163 |
XXXIX | 174 |
XL | 183 |
XI | 43 |
XII | 46 |
XIII | 50 |
XIV | 57 |
XV | 66 |
XVI | 74 |
XVII | 78 |
XVIII | 81 |
XIX | 83 |
XX | 93 |
XXI | 98 |
XXII | 104 |
XXIII | 110 |
XXIV | 114 |
XXV | 117 |
XXVI | 120 |
XXVII | 125 |
XXVIII | 128 |
XXIX | 130 |
XXX | 142 |
XXXI | 146 |
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Common terms and phrases
a₁ Algebra algorithm Amer approximation at² complex numbers complex plane compute congruence constant cubic equation cyclotomic polynomials Deduce derivative determine discriminant divides Eisenstein Criterion equal Exercise factors greatest common divisor Hence induction integer integral domain interval irreducible Lagrange polynomial left side Let p(t Math Mathematics modulo monic Monthly multiple nonnegative nonzero obtain pair points polynomial equation polynomial of degree polynomial p(t positive integer possible prime problem Prove quadratic equation quadratic polynomial quartic quartic equation rational real numbers real roots real zeros reciprocal polynomial result follows Rolle's Theorem root of unity satisfy sequence Show solution Solve square Suppose symmetric functions t₁ Theorem values vanish variables Verify whence yields zero of p(t