Final answer:
The cubic equation x^3 - x^2 - x - 2 = 0 has three roots: -1, 1 + sqrt(3), and 1 - sqrt(3). By applying the Rational Root Theorem and using both polynomial and synthetic division, as well as the quadratic formula, we can find all roots of the equation.
Step-by-step explanation:
To find the roots of the equation x^3 - x^2 - x - 2 = 0, we'll start by looking for possible rational roots using the Rational Root Theorem, which suggests that any rational root, as a fraction in lowest terms p/q, is where p is a factor of the constant term and q is a factor of the leading coefficient.
For our equation, the constant term is -2 and the leading coefficient is 1, so the possible rational roots are ±2 and ±1. We will test these potential roots by substituting them into the equation and looking for a value that results in zero.
When testing these values, we find that x = -1 is a root since (-1)^3 - (-1)^2 - (-1) - 2 equals 0.
Now, we can perform polynomial division or use synthetic division to divide the original polynomial by (x + 1) to find the other roots.
The division gives us a quadratic polynomial x^2 - 2x - 2.
We now apply the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a), where a = 1, b = -2, and c = -2.
Solving this gives us two more roots, which are irrational.
After computation, the remaining roots are x = 1 + sqrt(3) and x = 1 - sqrt(3).
Therefore, the roots of the equation x^3 - x^2 - x - 2 = 0 are -1, 1 + sqrt(3), and 1 - sqrt(3).