DeCarte's Rule of Signs

Pronunciation: /deɪˈkɑrtz rul ʌv saɪnz/ Explain

De Carte's rule of signs is a rule for determining the maximum number of positive roots of a polynomial.^[2] For any polynomial, the maximum number of positive roots is equal to the number of sign changes of the polynomial going from the term with the highest degree to the term with the least degree. If there are a maximum of n positive roots, the exact number of roots is one of n, n - 2, n - 4, …. Terms with a coefficient of 0 are ignored.

The maximum number of negative roots of the polynomial is equal to the number of sign changes for f(-x). This changes sign of the terms of odd degree. Each time a nonzero term changes sign from the previous nonzero term, it counts as a sign change. Any roots that are neither negative or positive are complex roots.

Examples

Terms	Sign Changes	Positive Roots	f(-x)	Sign Changes	Negative Roots	Complex Roots
3x2 + 2	0	0	3x2 + 2	0	0	2
x - 2	1	1	-x - 2	0	0	0
3x2 + 2x + 7	0	0	3x2 - 2x + 7	2	2 or 0	0 or 2
-x3 - x2 + 5	1	1	x3 - x2 + 5	2	2 or 0	0 or 2
2x4 - 3x2 + 2	2	2 or 0	2x4 - 3x2 + 2	2	2 or 0	0, 2, or 4
4x6 - 2x5 - 3x4 - x3 + 5x2 + 4x - 1 = 0	3	3 or 1	4x6 + 2x5 - 3x4 + x3 + 5x2 - 4x - 1 = 0	3	3 or 1	2, 4, or 6
Table 1

References

1. McAdams, David E.. All Math Words Dictionary, Decartes Rule of Signs. 2nd Classroom edition 20150108-4799968. pg 58. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Ron Larson, Robert P. Hostetler. Precalculus. 7th edition. pg 176. Brooks Cole. January 26, 2006. Last Accessed 7/3/2018. Buy the book

Cite this article as:

McAdams, David E. DeCarte's Rule of Signs. 4/20/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/decartesruleofsigns.html.

Revision History

4/20/2019: Updated equations and expressions to the new format (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/28/2018: Corrected spelling. (McAdams, David E.)
7/3/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
1/22/2010: Added "References". (McAdams, David E.)
1/22/2009: Changed equations from images to html. (McAdams, David E.)
12/4/2008: Initial version. (McAdams, David E.)