A method of determining the maximum number of positive and negative real roots of a polynomial.
For positive roots, start with the sign of the coefficient
of the lowest (or highest) power. Count the number of sign changes as you proceed from the lowest to the highest power
(ignoring powers which do not appear). Then is the maximum number of positive roots. Furthermore, the number of allowable roots
.... For example, consider the polynomial
Since there are three sign changes, there are a maximum
of three possible positive roots.
For negative roots, starting with a polynomial , write a new polynomial with the signs
of all odd powers reversed,
while leaving the signs of the even powers unchanged. Then proceed as before to count the number
of sign changes . Then is the maximum number of negative roots. For example, consider the polynomial
and compute the new polynomial
In this example, there are four sign changes, so there are
a maximum of four negative roots.
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ReferencesAnderson, B.; Jackson, J.; and Sitharam, M. "Descartes' Rule of Signs Revisited." Amer. Math. Monthly 105, 447-451, 1998.Grabiner,
D. J. "Descartes' Rule of Signs: Another Construction." Amer. Math.
Monthly 106, 854-855, 1999.Hall, H. S. and Knight, S. R.
Algebra: A Sequel to Elementary Algebra for Schools. London: Macmillan, pp. 459-460,
1950.Henrici, P. "Sign Changes. The Rule of Descartes." §6.2
and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal
Mapping-Location of Zeros. New York: Wiley, pp. 439-443, 1988.Itenberg,
U. and Roy, M. F. "Multivariate Descartes' Rule." Beiträge
Algebra Geom. 37, 337-346, 1996.Struik, D. J. (Ed.).
Source Book in Mathematics 1200-1800. Princeton, NJ: Princeton University
Press, pp. 89-93, 1986.
Referenced on Wolfram|AlphaDescartes' Sign Rule
Cite this as:
Weisstein, Eric W. "Descartes' Sign Rule."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DescartesSignRule.html