Find the roots of the cubic equation x^3 + x^2 - x - 1 = 0.

The roots of the cubic equation x^3 + x^2 - x - 1 = 0 are -1, 1, and -1/2.

To find the roots of a cubic equation, we can use the Rational Root Theorem to test possible rational roots. The Rational Root Theorem states that if a polynomial has integer coefficients, then any rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

In this case, the constant term is -1 and the leading coefficient is 1, so the possible rational roots are ±1 and ±1/1. We can test these roots by plugging them into the equation and seeing if they make it equal to 0.

Testing x = 1, we get:
1^3 + 1^2 - 1 - 1 = 0
So x = 1 is a root.

Testing x = -1, we get:
(-1)^3 + (-1)^2 - (-1) - 1 = 0
So x = -1 is a root.

Testing x = 1/2, we get:
(1/2)^3 + (1/2)^2 - (1/2) - 1 = -1/8
So x = 1/2 is not a root.

Testing x = -1/2, we get:
(-1/2)^3 + (-1/2)^2 - (-1/2) - 1 = 0
So x = -1/2 is a root.

Now that we have found three roots, we can factor the equation as:
(x - 1)(x + 1)(x + 1/2) = 0

Simplifying, we get:
x^3 + x^2 - x - 1 = (x - 1)(x + 1)(x + 1/2)

So the roots of the equation are -1, 1, and -1/2.