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X² + x + 1 = 0

⇒ x² + x + 1 -x = 0 - x

⇒x² + 1 = -x

⇒(x² + 1)/x = -x/x

⇒ x + 1/x = -1

Using a³+b³ = (a+b)³ - 3ab(a+b)

x³ + 1/x³ = (x+1/x)³ - 3×x×1/x×(x+1/x)

Thus (x³ + 1/x³)³ = [(x+1/x)³ - 3×x×1/x×(x+1/x)]³

=[ (-1)³ - 3×(x×1/x)×(-1) ]³

=[ (-1)³ - 3×(1)×(-1) ]³

=[ -1 -3×(-1) ]³

=[ -1 + 3 ]³

=[2]³

=8

⇒ x² + x + 1 -x = 0 - x

⇒x² + 1 = -x

⇒(x² + 1)/x = -x/x

⇒ x + 1/x = -1

Using a³+b³ = (a+b)³ - 3ab(a+b)

x³ + 1/x³ = (x+1/x)³ - 3×x×1/x×(x+1/x)

Thus (x³ + 1/x³)³ = [(x+1/x)³ - 3×x×1/x×(x+1/x)]³

=[ (-1)³ - 3×(x×1/x)×(-1) ]³

=[ (-1)³ - 3×(1)×(-1) ]³

=[ -1 -3×(-1) ]³

=[ -1 + 3 ]³

=[2]³

=8

### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

X² + x + 1 = 0

multiply with (x-1) both sides:

(x-1)(x² + x + 1) = (x-1) 0

x³ - 1 = 0

x³ = 1

x³ + 1/x³ = 1 + 1/1 = 2

Answer is 2³ = 8

multiply with (x-1) both sides:

(x-1)(x² + x + 1) = (x-1) 0

x³ - 1 = 0

x³ = 1

x³ + 1/x³ = 1 + 1/1 = 2

Answer is 2³ = 8