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?
You first use rational roots test. After you look for rational roots you see that $\displaystyle x=1$ is a zero and so dividing by $\displaystyle x-1$ we get $\displaystyle (x-1)(x^4 + 4x^3 + 6x^2+7x+7)$. 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 $\displaystyle x^4 + 4x^3+6x^2+7x+7 = (x^2+ax+b)(x^2+cx+d)$ is impossible for any integers $\displaystyle a,b,c,d$ by expanding out this side and comparing coefficients. Same idea for the second polynomial.