find the sum of squares of all roots of the equation. `x^(8) - x^(7) + x^(6) -x^(5) + x^(4) -x^(3) +x^(2)-x + 1 =0`

Explanation

Given equation is `x^(8)-x^(7)+x^(6)-x^(5)+x^(4)-x^(3)+x^(2)-x+1`
`=0.`
Multiplying by `x+1`, we get
`x^(9)+1=0`
`impliesx^(9)=(-1)`
Hence, the roots are `9^(th)` roots of -1 excluding -1.
Let `alpha_(1),alpha_(2),alpha_(3),...,alpha_(8)` be the complex roots, then
`(-1)^(2)+alpha_(1)^(2)+alpha_(2)^(2)+...+alpha_(8)^(2)=0`
`impliesalpha_(1)^(2)+alpha_(2)^(2)+...+alpha_(8)^(2)=-1`