Monday, 20 October 2014

polynomials - Factor x8x in BbbZ[x] and in BbbZ2[x]




Factor x8x in Z[x] and in Z2[x]



Here what I get is x8x=x(x71)=x(x1)(1+x+x2++x6) now what next? Help in both the cases in Z[x] and in Z2[x]



Edit: I think (1+x+x2++x6) is cyclotomic polynomial for p=7 so it is irred over Z. Now the problem remains for Z2[x]


Answer



After my edit I finally got the answer it was under my nose the polynomial 1++x6 does not have zeros at 0 and 1 so it can't be factored as a multiple of a one degree polynomial and one other as a whole.



Edit: But as Lubin mentioned in the comment 1++x6=(1+x+x3)(1+x2+x3)



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