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Math Help - Complete factorisation of irreducible polynomial over Q

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    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?
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    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.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post

    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.
    Last edited by ThePerfectHacker; March 22nd 2009 at 07:05 PM. Reason: I fixed NCA's mistake
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