RootIntervals
✖
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
![](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
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
]}