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

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

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.

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.

© 1996-9

1999-05-24