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 n as you proceed from the lowest to the highest power (ignoring powers which do not appear). Then n is the maximum number of positive roots. Furthermore, the number of allowable roots is n, n-2, n-4, .... 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 f(x), write a new polynomial f(-x) 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 n. Then n 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.

See also

Bound, Root, Sturm Function

Explore with Wolfram|Alpha


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.

Subject classifications