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
is ,
,
,
.... For example, consider the polynomial

(1)

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

(2)

and compute the new polynomial

(3)

In this example, there are four sign changes, so there are
a maximum of four negative roots .

See also Bound ,

Root ,

Sturm Function
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References Anderson, 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.
Higher
Algebra: A Sequel to Elementary Algebra for Schools. London: Macmillan, pp. 459-460,
1950. Henrici, P. "Sign Changes. The Rule of Descartes." §6.2
in Applied
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.).
A
Source Book in Mathematics 1200-1800. Princeton, NJ: Princeton University
Press, pp. 89-93, 1986. Referenced on Wolfram|Alpha Descartes' Sign Rule
Cite this as:
Weisstein, Eric W. "Descartes' Sign Rule."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/DescartesSignRule.html

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