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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.
gives isolating intervals for real roots of a single polynomial.
Details
![](Files/RootIntervals.en/details_1.png)
- 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
Get isolating intervals, together with a list of which polynomial has which root:
![](Files/RootIntervals.en/I_1.png)
https://wolfram.com/xid/05fuof7dac-le271
![](Files/RootIntervals.en/O_1.png)
The isolating intervals are always specified by exact rationals:
![](Files/RootIntervals.en/I_2.png)
https://wolfram.com/xid/05fuof7dac-c9r06z
![](Files/RootIntervals.en/O_2.png)
Scope (6)Survey of the scope of standard use cases
Isolate the real roots of a polynomial:
![](Files/RootIntervals.en/I_3.png)
https://wolfram.com/xid/05fuof7dac-jyy5i
![](Files/RootIntervals.en/O_3.png)
Isolate the real roots of a list of polynomials:
![](Files/RootIntervals.en/I_4.png)
https://wolfram.com/xid/05fuof7dac-jzdn21
![](Files/RootIntervals.en/O_4.png)
Isolate the complex roots of a polynomial:
![](Files/RootIntervals.en/I_5.png)
https://wolfram.com/xid/05fuof7dac-lu2g3i
![](Files/RootIntervals.en/O_5.png)
Isolate the complex roots of a list of polynomials:
![](Files/RootIntervals.en/I_6.png)
https://wolfram.com/xid/05fuof7dac-4jscs
![](Files/RootIntervals.en/O_6.png)
Polynomials may have multiple roots; pairs of polynomials may have common roots:
![](Files/RootIntervals.en/I_7.png)
https://wolfram.com/xid/05fuof7dac-c1dk5o
![](Files/RootIntervals.en/O_7.png)
![](Files/RootIntervals.en/I_8.png)
https://wolfram.com/xid/05fuof7dac-2x5j8
![](Files/RootIntervals.en/O_8.png)
Isolating intervals of rational roots may be single points:
![](Files/RootIntervals.en/I_9.png)
https://wolfram.com/xid/05fuof7dac-jb1noo
![](Files/RootIntervals.en/O_9.png)
Applications (1)Sample problems that can be solved with this function
Find numeric approximations of real roots of a polynomial:
![](Files/RootIntervals.en/I_10.png)
https://wolfram.com/xid/05fuof7dac-bq4wo4
![](Files/RootIntervals.en/I_11.png)
https://wolfram.com/xid/05fuof7dac-cn71sp
![](Files/RootIntervals.en/O_10.png)
![](Files/RootIntervals.en/I_12.png)
https://wolfram.com/xid/05fuof7dac-dafgmq
![](Files/RootIntervals.en/O_11.png)
Reduce uses a similar approach, but factoring the polynomial for Root objects takes time:
![](Files/RootIntervals.en/I_13.png)
https://wolfram.com/xid/05fuof7dac-det4oo
![](Files/RootIntervals.en/O_12.png)
Compute approximations of the Root objects:
![](Files/RootIntervals.en/I_14.png)
https://wolfram.com/xid/05fuof7dac-g7yqbf
![](Files/RootIntervals.en/O_13.png)
Properties & Relations (1)Properties of the function, and connections to other functions
Find real and complex roots of polynomials:
![](Files/RootIntervals.en/I_15.png)
https://wolfram.com/xid/05fuof7dac-cikx4x
Isolate the real roots; multiple roots are indicated in the second part of the output:
![](Files/RootIntervals.en/I_16.png)
https://wolfram.com/xid/05fuof7dac-c4ai2m
![](Files/RootIntervals.en/O_14.png)
Use CountRoots to count the real roots; multiple roots are counted with multiplicities:
![](Files/RootIntervals.en/I_17.png)
https://wolfram.com/xid/05fuof7dac-cze7ae
![](Files/RootIntervals.en/O_15.png)
Use Reduce to find the real roots; multiple roots are given once:
![](Files/RootIntervals.en/I_18.png)
https://wolfram.com/xid/05fuof7dac-drxqyf
![](Files/RootIntervals.en/O_16.png)
Isolate the complex roots; multiple roots are indicated in the second part of the output:
![](Files/RootIntervals.en/I_19.png)
https://wolfram.com/xid/05fuof7dac-d3hwjd
![](Files/RootIntervals.en/O_17.png)
Use Reduce to find the complex roots; multiple roots are given once:
![](Files/RootIntervals.en/I_20.png)
https://wolfram.com/xid/05fuof7dac-b2a4v
![](Files/RootIntervals.en/O_18.png)
Use Solve to find the complex roots with multiplicities:
![](Files/RootIntervals.en/I_21.png)
https://wolfram.com/xid/05fuof7dac-cwhf2m
![](Files/RootIntervals.en/O_19.png)
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: 17-February-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: 17-February-2025
]}