For example, to prove that the function $x^8+x−1=0$ has exactly two roots, we first prove that $f$ has at most $2$ real roots by using differentiation. $f′(x)=8x^7+1=0$, by using that the function has two roots and therefore it should have at least one point such that $f'(c)=0$ and then we use IVT to prove that it has exactly two roots.
However, the equation $x^4−6x^2−8x+1=0$, its derivative has $2$ roots but still, I should try to prove that it has exactly two roots. If its derivative had one solution, then I could first prove that it has two roots at most as I did, so how should we approach the problem?