Descartes' sign rule is a method of bounding the numbers of positive and negative real roots of a nonzero polynomial with real coefficients, counting roots with multiplicity.
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 an upper bound for the 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 an upper bound for the 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.
If 
is a real-rooted polynomial with real coefficients, both bounds are attained. The number of
positive roots equals the number
of sign changes in the nonzero coefficients of 
, and the number of negative roots equals the corresponding count for 
, with multiplicity included
in both counts. Roots at zero are counted separately. In
general, the sign counts alone do not establish real-rootedness.
For example, 
has two sign changes but no real
roots.
