# Counting and Isolating Polynomial Roots

## Counting Roots of Polynomials

CountRoots[poly,x] | give the number of real roots of the polynomial poly in x |

CountRoots[poly,{x,a,b}] | give the number of roots of the polynomial poly in x with |

Counting roots of polynomials.

CountRoots accepts polynomials with Gaussian rational coefficients. The root count includes multiplicities.

This gives the number of real roots of

.

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This counts the roots of

in the closed interval

.

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The roots of

in the vertical axis segment between

and

consist of a triple root at

and a single root at

.

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This counts 17

-degree roots of unity in the closed unit square.

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The coefficients of the polynomial can be Gaussian rationals.

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## Isolating Intervals

A set , where is or , is an *isolating set* for a root of a polynomial if is the only root of in . Isolating roots of a polynomial means finding disjoint isolating sets for all the roots of the polynomial.

RootIntervals[{poly_{1},poly_{2},...}] | give a list of disjoint isolating intervals for the real roots of any of the , together with a list of which polynomials actually have each successive root |

RootIntervals[poly] | give disjoint isolating intervals for real roots of a single polynomial |

RootIntervals[polys,Complexes] | give disjoint isolating intervals or rectangles for complex roots of polys |

IsolatingInterval[a] | give an isolating interval for the algebraic number a |

IsolatingInterval[a,dx] | give an isolating interval of width at most dx |

Functions for isolating roots of polynomials.

RootIntervals accepts polynomials with rational number coefficients.

For a real root the returned isolating interval is a pair of rational numbers , such that either or . For a nonreal root the isolating rectangle returned is a pair of Gaussian rational numbers , such that and either or .

Here are isolating intervals for the real roots of

.

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The second list shows which interval contains a root of which polynomial.

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This gives isolating intervals for all complex roots of

.

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Here are isolating intervals for the third- and fourth-degree roots of unity. The second interval contains a root common to both polynomials.

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Here is an isolating interval for a root of a polynomial of degree seven.

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This gives an isolating interval of width at most

.

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All numbers in the interval have the first 10 decimal digits in common.

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