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

$$f(x) = x^7+x^6-x^4-x^3-x^2+x-1.$$

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 $g(x)$ 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

$$f(x) = x^7+x^6-x^4-x^3-x^2+x-1,$$

and compute the new Polynomial

$$g(x)=-x^7+x^6-x^4+x^3-x^2-x-1.$$

There are four Sign changes, so there are a maximum of four Negative Roots.

See also Bound, Sturm Function

References

Anderson, B.; Jackson, J.; and Sitharam, M. ``Descartes' Rule of Signs Revisited.'' Amer. Math. Monthly 105, 447-451, 1998.

Hall, H. S. and Knight, S. R. Higher Algebra: A Sequel to Elementary Algebra for Schools. London: Macmillan, pp. 459-460, 1950.

Struik, D. J. (Ed.). A Source Book in Mathematics 1200-1800. Princeton, NJ: Princeton University Press, pp. 89-93, 1986.