Final answer:
C) x=2x=2. The correct answer to the equation x3 - 2x2 + x - 2 = 0 is C, where x equals 2. Other provided options do not satisfy the equation when substituted for x, and thus we can confirm that x = 2 is the correct solution.
Step-by-step explanation:
To solve for x in the equation x3 - 2x2 + x - 2 = 0, we can approach this polynomial equation by looking for rational roots that satisfy the equation. One way to do this is to apply the Rational Root Theorem, but since the options are given, we can check each option by substituting the value of x into the equation to see if it satisfies the equation, making it a root.
- For option A (x = 1): 13 - 2(1)2 + 1 - 2 = 0 | 1 - 2 + 1 - 2 = -2, which is not 0, so x = 1 is not a root.
- For option B (x = -1): (-1)3 - 2(-1)2 + (-1) - 2 = 0 | -1 - 2 - 1 - 2 = -6, which is not 0, so x = -1 is not a root.
- For option C (x = 2): 23 - 2(2)2 + 2 - 2 = 0 | 8 - 8 + 2 - 2 = 0, which is 0, so x = 2 is a root of the equation.
- For option D (x = -2): (-2)3 - 2(-2)2 + (-2) - 2 = 0 | -8 - 8 - 2 - 2 = -20, which is not 0, so x = -2 is not a root.
Thus, the correct answer is option C, where x equals 2. Other methods such as factoring by grouping, synthetic division, or the use of the quadratic formula are also applicable if the polynomial can be reduced to a quadratic equation.