If the Coefficients of the Polynomial

(1) |

Let the Roots of the polynomial

(2) |

(3) | |||

(4) | |||

(5) |

These can be derived by writing

(6) |

Any Polynomial can be numerically factored, although different Algorithms have different strengths and weaknesses.

If there are no Negative Roots of a Polynomial (as can be determined by Descartes' Sign
Rule), then the Greatest Lower Bound is 0. Otherwise, write out the Coefficients, let ,
and compute the next line. Now, if any Coefficients are 0, set them to minus the sign of the next
higher Coefficient, starting with the second highest order Coefficient. If all the signs alternate, is the
greatest lower bound. If not, then subtract 1 from , and compute another line. For example, consider the Polynomial

(7) |

0 | 2 | 2 | 1 | ||

2 | 0 | 8 | |||

-- | 2 | 8 | |||

2 | 7 | ||||

2 | 5 | 35 |

so the greatest lower bound is .

If there are no Positive Roots of a Polynomial (as can be determined by Descartes' Sign Rule),
the Least Upper Bound is 0. Otherwise, write out the Coefficients of the
Polynomials, including zeros as necessary. Let . On the line below, write the highest order
Coefficient. Starting with the second-highest Coefficient, add times the number just written to the original
second Coefficient, and write it below the second Coefficient. Continue through order zero. If all the
Coefficients are Nonnegative, the least upper bound is . If not, add one to and repeat
the process again. For example, take the Polynomial

(8) |

0 | 2 | 1 | |||

1 | 2 | 1 | |||

2 | 2 | 3 | |||

3 | 2 | 5 | 8 | 25 | 68 |

so the Least Upper Bound is 3.

© 1996-9

1999-05-25