If and are the distinct roots of the equation x^2 - x + 1 = 0 , then the value of ²⁰⁰ + ²⁰⁶ + 2 is equal to

Step-by-step solution

Given equation: x^2 - x + 1 = 0 Multiplying both sides by (x + 1) , we get: (x + 1)(x^2 - x + 1) = 0 x^3 + 1 = 0 x^3 = -1 Since and are roots of x^2 - x + 1 = 0 , they satisfy the equation x^3 = -1 . Therefore, ^3 = -1 and ^3 = -1 . Now, evaluating the given expression ²⁰⁰ + ²⁰⁶ + 2 : ²⁰⁰ = ( ^3)⁶⁶ ^2 = (-1)⁶⁶ ^2 = ^2 ²⁰⁶ = ( ^3)⁶⁸ ^2 = (-1)⁶⁸ ^2 = ^2 Substituting these values, the expression becomes ^2 + ^2 + 2 . From the given quadratic equation, the sum of roots + = 1 and the product of roots = 1 . Using the ide