Statement of Descartes Rule

Worked illustration. For $2x^7-3x^6-4x^5+5x^4+6x^3-7x+8$ (the $x^2$ term is absent, i.e. it has coefficient $0$), the sign of each nonzero coefficient in order (from $x^7$ down to the constant) is $+,-,-,+,+,-,+$. Reading consecutive nonzero signs (the zero $x^2$ coefficient is simply skipped), there are 4 changes of sign — occurring at $x^6$, $x^4$, $x^1$ and $x^0$ (labelling each change by the lower power of the pair that flips sign).

Two consequences, used constantly:

• $p$ can never exceed $s$: the number of positive roots is at most the number of sign changes.
• $s-p$ is even: if $s=4$, the actual number of positive roots is $4,2,$ or $0$ — never $3$ or $1$.

Negative roots, via $P(-x)$. A negative root of $P(x)$ is exactly a positive root of $P(-x)$ (substitute $x=-y$, $y>0$). So applying the theorem to $P(-x)$ instead: the number of negative roots of $P(x)$ is at most the number of sign changes in $P(-x)$'s coefficients, and the difference is again even. …