# 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 .