If (x+ 1/x)^2 = 3 then the value of x^206 + x^200 + x^90 + x^84 + x^18 + x^12 + x^6 + 1

x^206 + x^200 + x^90 + x^84 + x^18 + x^12 + x^6 + 1

= x^200(x^6+1) + x^84(x^6+1) + x^12(x^6+1) + (x^6+1)

= (x^6+1)(x^200 + x^ 84+ x^12 + 1)

x^2 + 1/x^2 + 2 = 3
so x^2 + 1/x^2 = 1
or x^4 + 1 = x^2
or x^4 - x^2 + 1 = 0
or x ^6 + 1 = 0 ( multiplying by x^2 + 1)

hence given sum = 0