by calculating $(x^{2010}-f_{2009}x-f_{2008})''=2010*2009x^{2008}$ we know that $x^{2010}-f_{2009}x-f_{2008}$ is a convex function hence it has no more than $2$ zeroes on $R$ we can confirm that $x=\frac{1+\sqrt5}{2}$ and $\frac{1-\sqrt5}{2}$ are solutions,so they are the only roots.