RootIntervals
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.
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
open allclose allBasic Examples (2)Summary of the most common use cases
Scope (6)Survey of the scope of standard use cases
Isolate the real roots of a polynomial:

https://wolfram.com/xid/05fuof7dac-jyy5i

Isolate the real roots of a list of polynomials:

https://wolfram.com/xid/05fuof7dac-jzdn21

Isolate the complex roots of a polynomial:

https://wolfram.com/xid/05fuof7dac-lu2g3i

Isolate the complex roots of a list of polynomials:

https://wolfram.com/xid/05fuof7dac-4jscs

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

https://wolfram.com/xid/05fuof7dac-c1dk5o


https://wolfram.com/xid/05fuof7dac-2x5j8

Isolating intervals of rational roots may be single points:

https://wolfram.com/xid/05fuof7dac-jb1noo

Applications (1)Sample problems that can be solved with this function
Find numeric approximations of real roots of a polynomial:

https://wolfram.com/xid/05fuof7dac-bq4wo4

https://wolfram.com/xid/05fuof7dac-cn71sp


https://wolfram.com/xid/05fuof7dac-dafgmq

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

https://wolfram.com/xid/05fuof7dac-det4oo

Compute approximations of the Root objects:

https://wolfram.com/xid/05fuof7dac-g7yqbf

Properties & Relations (1)Properties of the function, and connections to other functions
Find real and complex roots of polynomials:

https://wolfram.com/xid/05fuof7dac-cikx4x
Isolate the real roots; multiple roots are indicated in the second part of the output:

https://wolfram.com/xid/05fuof7dac-c4ai2m

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

https://wolfram.com/xid/05fuof7dac-cze7ae

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

https://wolfram.com/xid/05fuof7dac-drxqyf

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

https://wolfram.com/xid/05fuof7dac-d3hwjd

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

https://wolfram.com/xid/05fuof7dac-b2a4v

Use Solve to find the complex roots with multiplicities:

https://wolfram.com/xid/05fuof7dac-cwhf2m

Wolfram Research (2007), RootIntervals, Wolfram Language function, https://reference.wolfram.com/language/ref/RootIntervals.html.
Text
Wolfram Research (2007), RootIntervals, Wolfram Language function, https://reference.wolfram.com/language/ref/RootIntervals.html.
Wolfram Research (2007), RootIntervals, Wolfram Language function, https://reference.wolfram.com/language/ref/RootIntervals.html.
CMS
Wolfram Language. 2007. "RootIntervals." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RootIntervals.html.
Wolfram Language. 2007. "RootIntervals." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RootIntervals.html.
APA
Wolfram Language. (2007). RootIntervals. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RootIntervals.html
Wolfram Language. (2007). RootIntervals. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RootIntervals.html
BibTeX
@misc{reference.wolfram_2025_rootintervals, author="Wolfram Research", title="{RootIntervals}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RootIntervals.html}", note=[Accessed: 09-May-2025
]}
BibLaTeX
@online{reference.wolfram_2025_rootintervals, organization={Wolfram Research}, title={RootIntervals}, year={2007}, url={https://reference.wolfram.com/language/ref/RootIntervals.html}, note=[Accessed: 09-May-2025
]}