Descartes' Sign Rule
A method of determining the maximum number of Positive and Negative Real Roots of aPolynomial.
For Positive Roots, start with the Sign of the Coefficient of the lowest (or highest) Power. Countthe number of Sign changes 
as you proceed from the lowest to the highest Power (ignoringPowers 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 theSigns of all Odd Powers reversed, while leaving the Signs of theEven 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.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.
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