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 power
(ignoring powers which do not appear). Then maximum number of positive roots . Furthermore, the number of allowable roots
is polynomial
(1)
Since there are three sign changes, there are a maximum
of three possible positive roots .
For negative roots , starting with a polynomial polynomial 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 maximum 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 .
See also Bound ,
Root ,
Sturm Function
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References 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. 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. 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. Referenced on Wolfram|Alpha Descartes' Sign Rule
Cite this as:
Weisstein, Eric W. "Descartes' Sign Rule."
From MathWorld https://mathworld.wolfram.com/DescartesSignRule.html
Subject classifications
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
, 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
