Descartes' Sign Rule

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.

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.