I want to prove that $x^7-120x^5+1875x^3+12500x+5$ has exactly 5 real roots. I have been able to show (by plugging in values and using the intermediate value theorem) that there are at least 5, but I don't know how to prove that there are exactly 5.
By the way, I am trying to prove this as a lemma to show that the polynomial has $S_7$ as its Galois group. See Is there an irreducible, degree 7 polynomial in $\mathbb{Q}[x]$ which has exactly two of its roots in $\mathbb{C-R}$?
There are two sign changes in the coefficients, so there are either 2 or 0 positive roots, by Descartes's rule of signs. Replacing $x$ by $-x$ gives $-x^7+()x^5-()x^3-()x+()$, where all the $()$s are positive, which has 3 sign changes, so there are either 3 or 1 negative roots. Hence there are at most 5 real roots.