If `x^2+x+1=0` then the value of `(x+1/x)^2+(x^2+1/(x^2))^2+...+(x^27+1/(x^27))^2` is

Explanation

`1 + omega + omega^2 = 0`

`1 + x + x^2 = 0 `

`omega^3 = 1 , 1 + omega + omega^2 = 0`

`(x + 1/x)^2 + ( x^2 + 1/x^2)^2 + ....... (x^27 + 1/x^27)^2`

when `x= omega`

`( omega + omega^2/omega^3)^2 + ( omega^2 + omega/omega^3) + (1+1)^2 + ( omega + omega^2/omega^6)^2 + (omega^2 + omega/omega^6)^2 + (1+1)^2`

= `(-1)^2 + (-1)^2 + (2)^2 + (-1)^2 + (-1)^2 + (2)^2 `

`= 9[(-1)^2 + (-1)^2 + 2^2]`

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