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If [TEX]n \geq 3[/TEX], prove that [TEX]x^{2^n} + x + 1[/TEX] is reducible over [TEX]\mathbb{Z}_2[/TEX].

Not sure how to go about this. I was thinking it might involve induction.

For [TEX]n=3[/TEX], we have

[TEX]x^8+x+1=(x^2+x+1)(x^6+x^5+x^3+x^2+1)[/TEX], but I can't find any pattern to help with the induction.

Thanks in advance!

Not sure how to go about this. I was thinking it might involve induction.

For [TEX]n=3[/TEX], we have

[TEX]x^8+x+1=(x^2+x+1)(x^6+x^5+x^3+x^2+1)[/TEX], but I can't find any pattern to help with the induction.

Thanks in advance!

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