# 2007 iTest Problems/Problem TB2

## Problem

Factor completely over integer coefficients the polynomial . Demonstrate that your factorization is complete.

## Solution

Note that . If and , then and . Therefore if and , then . Hence . Dividing through gives us

Using the Rational Root Theorem on the second polynomial gives us that are possible roots. Only is a possible root. Dividing through gives us

Note that can be factored into the product of a cubic and a quadratic. Let the product be

We would want the coefficients to be integers, hence we shall only look for integer solutions. The following equations must then be satisfied:

Since and are integers, is either or . Testing the first one gives

We must have that . Therefore , or . Solving for and gives . We don't need to test the other one.

Hence we have

For any of the factors of degree more than 1 to be factorable in the integers, they must have rational roots, since their degrees are less than 4. None of them have rational roots. Hence is completely factored.

### Alternate Solution

We write

The factorization of is trivial once we look at the exponents modulo ; since any root of satisfies , it follows that and the cubic factor comes as a result of polynomial division.

To prove that this is a complete factorization, it suffices to note that the factors of degree greater than have no rational roots (follows from the rational root theorem and some small cases).