Descartes's Rule of Signs/Examples
Jump to navigation
Jump to search
Examples of Use of Descartes's Rule of Signs
Arbitrary Example
Consider the polynomial equation $\map f x$ over real numbers:
- $x^5 + x^4 - 2 x^3 + x^2 - 1 = 0$
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 $\map {f'} x$:
- $-x^5 + x^4 + 2 x^3 + x^2 - 1 = 0$
This has two variations in sign: