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}

.

(ii) 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.