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Poly Fibs

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Image of a Parrot
Consider the sequence of polynomials given by $P_{n+2}(x)=xP_{n+1}(x)-P_n(x)$ where $P_0(x)=0$ and $P_1(x)=1$

(i) Show that every root of $P_3$ is a root of $P_6$.

(ii) Show that every root of $P_4$ is a root of $P_8$.

(iii) Show that every root of $P_5$ is a root of $P_{10}$.

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.