If x^2 - x + 1 = 0 , then what is (x - 1 x )^2 + (x - 1 x )^4 + (x - 1 x )^8 equal to?

Step-by-step solution

Given x^2 - x + 1 = 0 Dividing the entire equation by x , we get: x - 1 + 1 x = 0 x + 1 x = 1 Using the algebraic identity (x - 1 x )^2 = (x + 1 x )^2 - 4 Substituting the value of x + 1 x : (x - 1 x )^2 = (1)^2 - 4 = -3 The given expression is (x - 1 x )^2 + (x - 1 x )^4 + (x - 1 x )^8 This can be written as (x - 1 x )^2 + ( (x - 1 x )^2 )^2 + ( (x - 1 x )^2 )^4 Substituting (x - 1 x )^2 = -3 into the expression: (-3) + (-3)^2 + (-3)^4 = -3 + 9 + 81 = 87 Answer: 87