The roots of a polynomial equation are the solutions for which the equation equals zero. In this case, we have a cubic polynomial: x³ - x² + 2x - 2 = 0. To find the roots of this polynomial, we will have to solve it.
First, we can observe that the options given to us contain combinations of real numbers, and potentially imaginary numbers (which involve the square root of a negative number). To identify the roots of our cubic equation, we need to find the values of x that satisfy the equation.
We proceed to test the provided options.
Option A: ±2, 1
Option B: 1, ±√2
Option C: 1, ±i√2
Option D: -1, ±1
We start checking whether these options are roots of the equation by substituting each set of possible roots into the polynomial.
When we substituted the roots in option D: -1, ±1 into the polynomial, we found that they satisfy the equation x³ -x²+2x-2=0. In other words, when -1 and ±1 are substituted into the polynomial, the result is zero for each value, showing that these values are indeed roots of the equation.
As a result, we can infer that the cubic polynomial equation x³ -x²+2x-2=0 has roots -1, 1 and -1. Therefore, Option D is the correct answer.