Descartes's Rule of Signs/Original Statement/Negative Roots

Theorem

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 $\map {f'} x$ be the polynomial equation over the real numbers derived from $\map f x$ by replacing $x$ with $-x$ throughout.

The number of negative real roots of $\map f x$ cannot be greater than the number of variations in sign of $\map {f'} x$ (although it may be fewer).

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Descartes's rule of signs
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Descartes's rule of signs