Category:Descartes's Rule of Signs

This category contains pages concerning Descartes's Rule of Signs:

Let $\map f x$ be a polynomial equation over the real numbers:

$a_n x^n + a_{n - 1} x^{n - 1} + \cdots + a_1 x + a_0 = 0$

where $a_j \in \R$.

Let $s_n$ be the number of variations in sign of $\map f x$.

Let $p_n$ be the number of positive real roots of $\map f x$ (counted with multiplicity).

Then:

$\forall n \in \Z_{>0}: s_n - p_n$ is a nonnegative even integer.

That is:

for every polynomial of degree $1$ or higher, the number of sign changes less than the number of positive real roots will be a nonnegative even integer.