Descartes's Rule of Signs/Examples/Arbitrary Example 1

Example of Use of Descartes's Rule of Signs

$x^5 + x^4 - 2 x^3 + x^2 - 1 = 0$

This has three variations in sign:

from $x^4$ to $-2 x^3$, where it goes from positive to negative

from $-2 x^3$ to $x^2$, where it goes from negative to positive

from $x^2$ to $-1$, where it goes from positive to negative.

Hence $\map f x$ has no more than $3$ positive real roots.

Replacing $x$ with $-x$ in $\map f x$ gives us the polynomial equation $\map {f'} x$:

$-x^5 + x^4 + 2 x^3 + x^2 - 1 = 0$

from $-x^5$ to $x^4$, where it goes from negative to positive

from $x^2$ to $-1$, where it goes from positive to negative.

Hence $\map f x$ has no more than $2$ negative real roots.