# Thread: Complete factorisation of irreducible polynomial over Q

1. ## Complete factorisation of irreducible polynomial over Q

For each polynomial below give the factorisation into Q - irreducible factors:
i) x^5 + 3x^4 + 2x^3 + x^2 - 7
ii) x^5 +10x^4 +13x^3 -25x^2 -68x -60

How do you this kind of question?

2. Originally Posted by knguyen2005
For each polynomial below give the factorisation into Q - irreducible factors:
i) x^5 + 3x^4 + 2x^3 + x^2 - 7
You first use rational roots test. After you look for rational roots you see that $x=1$ is a zero and so dividing by $x-1$ we get $(x-1)(x^4 + 4x^3 + 6x^2+7x+7)$. Now move on over to the second factor. Notice it has no zeros. But it does not necessarily make it irreducible. You need to show that $x^4 + 4x^3+6x^2+7x+7 = (x^2+ax+b)(x^2+cx+d)$ is impossible for any integers $a,b,c,d$ by expanding out this side and comparing coefficients. Same idea for the second polynomial.

3. Originally Posted by ThePerfectHacker

You need to show that $x^4 + 4x^3+6x^2+7x+7 = (x^2+ax+b)(x^2+cx+d)$ is impossible for any integers $a,b,c,d$ by expanding out this side and comparing coefficients. Same idea for the second polynomial.
alternatively, if you let $f(x)=x^4 + 4x^3+6x^2+7x+7,$ then $f(x-1)=x^4 + 3x + 3.$ now Eisenstein's criterion proves that $f(x)$ is irreducible.