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Question Number 29169 by abdo imad last updated on 04/Feb/18

factorize inside R[x] the polynomial p(x)= x^8 −1.

factorizeinsideR[x]thepolynomialp(x)=x8−1.

Commented by abdo imad last updated on 06/Feb/18

p(x)=(x^4 )^2 −1=(x^4 −1)(x^4 +1)=(x^2 −1)(x^2 +1)(x^4 +1) =(x−1)(x+1)(x^2 +1) (x^4 +1) but x^4 +1=(x^2 +1)^2 −2x^2 =(x^2 +1−(√)2x)(x^2 +1+(√)2x)so p(x)=(x−1)(x+1)(x^2 +1)(x^2 −(√2)x+1)(x^2 +(√2) x +1).

p(x)=(x4)2−1=(x4−1)(x4+1)=(x2−1)(x2+1)(x4+1)=(x−1)(x+1)(x2+1)(x4+1)butx4+1=(x2+1)2−2x2=(x2+1−2x)(x2+1+2x)sop(x)=(x−1)(x+1)(x2+1)(x2−2x+1)(x2+2x+1).

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