# RootIntervals

RootIntervals[{poly1,poly2,}]

gives a list of isolating intervals for the real roots of any of the polyi, together with a list of which polynomials actually have each successive root.

RootIntervals[poly]

gives isolating intervals for real roots of a single polynomial.

RootIntervals[polys,Complexes]

gives bounding rectangles for complex roots.

# Details • The coefficients of poly must be integers or rationals.
• An isolating interval for a root of a polynomial poly is an interval where the only root of poly contained in the interval is .
• If a root is real, the isolating interval is an open real interval, or a point. If a root is not real, the isolating interval is an open rectangle, disjoint from the real axis.
• Multiple roots give multiple entries in the second list generated by RootIntervals.

# Examples

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## Basic Examples(2)

Get isolating intervals, together with a list of which polynomial has which root:

The isolating intervals are always specified by exact rationals:

## Scope(6)

Isolate the real roots of a polynomial:

Isolate the real roots of a list of polynomials:

Isolate the complex roots of a polynomial:

Isolate the complex roots of a list of polynomials:

Polynomials may have multiple roots; pairs of polynomials may have common roots:

Isolating intervals of rational roots may be single points:

## Applications(1)

Find numeric approximations of real roots of a polynomial:

Find isolating intervals:

Find root approximations:

Reduce uses a similar approach, but factoring the polynomial for Root objects takes time:

Compute approximations of the Root objects:

## Properties & Relations(1)

Find real and complex roots of polynomials:

Isolate the real roots; multiple roots are indicated in the second part of the output:

Use CountRoots to count the real roots; multiple roots are counted with multiplicities:

Use Reduce to find the real roots; multiple roots are given once:

Isolate the complex roots; multiple roots are indicated in the second part of the output:

Use Reduce to find the complex roots; multiple roots are given once:

Use Solve to find the complex roots with multiplicities: