## 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. |

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### 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 |

XXXVIII | 174 |

XXXIX | 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

AGM inequality Algebra algorithm approximation complex numbers complex plane compute congruence constant cubic equation cubic polynomial cyclotomic polynomials Deduce derivative determine discriminant divides divisible Eisenstein Criterion equal example Exercise expression formula given equation greatest common divisor Hence induction integer coefficients integral domain interval irreducible Lagrange polynomial leading coefficient left side Let p(t linear polynomial locus Math Mathematics method modulo monic Monthly negative Newton's Method nonnegative nonreal zeros nonzero obtain pair points polynomial equation polynomial of degree polynomial p(t polynomial t2 positive integer possible powers prime problem Prove quadratic equation quadratic polynomial quartic equation rational real numbers real polynomial real roots real zeros reciprocal polynomial result follows Rolle's Theorem root of unity satisfy sequence Show solution Solve square Suppose symmetric functions theory vanish variables Verify whence yields zero of p(t