Thread: roots of a 5-degree polynomial

Here is another equation where I am stuck to find the exact number of real roots: . Descarte's rule says it has or +ve real roots and -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 all are . But I want something more general to rely on.

Thanks

First note that every odd degree polynomial has at least one real root.



So the function is increasing on



and has a max at



and a min at



Note that



So the function has one zero in



and then is never negative again so it has only one real zero.