Mock AIME I 2012 Problems/Problem 4
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 .
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 .