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
![$p(x)=(x^2+x+1)(x^6-x^5+2x^3-x^2+1)$](http://latex.artofproblemsolving.com/b/c/a/bca9abdcec507b7fddd9ec50cd3abf6621bbe4c8.png)
Using the Rational Root Theorem on the second polynomial gives us that are possible roots. Only
is a possible root. Dividing through gives us
![$p(x)=(x+1)(x^2+x+1)(x^5-2x^4+2x^3-x+1)$](http://latex.artofproblemsolving.com/0/0/5/005304c9ac9f680c3b3c8435b326d311cd641a2b.png)
Note that can be factored into the product of a cubic and a quadratic. Let the product be
![$x^5-2x^4+2x^3-x+1=(x^2+ax+b)(x^3+cx^2+dx+e)$](http://latex.artofproblemsolving.com/0/4/8/048769cdb4559eb34824a05468d6fbab1d2805c9.png)
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
![$p(x)=(x+1)(x^2+x+1)(x^2-x+1)(x^3-x^2+1)$](http://latex.artofproblemsolving.com/b/7/7/b775c04682ef984d925dc218ed2e9c63ec3939cc.png)
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.