I was working on proofs by induction and had to produce numerous proofs about the properties the Fibonacci sequence as exercises.
It was pointed out to me while proving that $ F_n = \frac 1{\sqrt 5} \left(\phi^n-\left(-\frac 1\phi\right)^n\right)= F_{n-1}+F_{n-2}$, that the roots of the second degree polynomial $$y = x^2-x-1$$ are the golden ratio $\phi = \frac {1+\sqrt 5}2 = \lim_{n \to \infty} \frac {F_{n+1}}{F_n}$ and its negative inverse $\left(-\frac 1\phi\right)$.
I never realized that before and now I wonder whether there is something special about the polynomial $y=x^2-x-1$ appart from the fact that its positive root is the golden ratio $\phi$?