Mock AIME I 2012 Problems/Problem 4
Problem
Consider the polynomial . Let
and
. The product of the roots of
can be expressed in the form
where
and
are relatively prime positive integers. Find the remainder when
is divided by
.
Solution
Let be the leading coefficient of
and let
be the constant coefficient of
. Therefore, we would like to find
in reduced form.
It is easy to see that we have the following recursive relations: .
Notice that . It is quickly deduced that
. Now let us evaluate
.
Notice that from some computations. Note that
. Therefore
, so
. So then it suffices to evaluate
.
Note that , so
, since
. Therefore we have that
, so our answer is
.