Consider the sequence of polynomials given by
Pn+2(x)=xPn+1(x)-Pn(x)
where
P0(x)=0 and P1(x)=1
(i) Show that every root of P3 is a root of P6.
(ii) Show that every root of P4 is a root of P8.
(ii) Show that every root of P5 is a root of P10.
You can do this by finding the polynomials and then finding their roots
(maybe using a computer), but try to find another way to get this result
without finding the roots of the polynomials.
One of the skills of a research mathematician is making conjectures about
results that no-one has thought of and that turn out to be provable.
In this problem there is a conjecture about a general result which you may
be able to make quite easily although the proof is well beyond the scope of
school mathematics. Go on learning mathematics and in a few years you will be
able to prove it.