Question

if x^2 +x+1=0 what is the value of (x^3+1/x^3)^3

Answer 1

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

Another way to find the answer,
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