Here is another equation where I am stuck to find the exact number of real roots: $\displaystyle x^5+x^3-2x+1=0$. Descarte's rule says it has $\displaystyle 0$ or $\displaystyle 2$ +ve real roots and $\displaystyle 1$ -ve real root. How do I know (apart from Wolframalpha, of course, which says it has no +ve root) how many +ve real roots does this equation have? I see $\displaystyle f(0),f(1),f(2)$ all are $\displaystyle >0$. But I want something more general to rely on.

Thanks