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 is a zero and so dividing by we get . 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 is impossible for any integers by expanding out this side and comparing coefficients. Same idea for the second polynomial.