Let's solve this problem step by step. We are given that the polynomial \(ax^3 + bx^2 + cx - 1\) has factors `(x - 1)`, `(x + 1)` and `(x - 2)`. We need to find the values of `a`, `b`, and `c`.

Since `(x - 1)`, `(x + 1)` and `(x - 2)` are factors of the polynomial, we can write the polynomial as: \(ax^3 + bx^2 + cx - 1 = k(x - 1)(x + 1)(x - 2)\), where `k` is a constant.

Expanding the right side of this equation, we get: \(k(x - 1)(x + 1)(x - 2) = k(x^2 - 1)(x - 2) = k(x^3 - x^2 - 2x + 2)\).

Comparing the coefficients of the left and right sides of this equation, we find that:

- `a = k`

- `b = -k`

- `c = -2k`

- `-1 = 2k`

Solving for `k`, we find that `k = -1/2`. Substituting this value into the equations for `a`, `b`, and `c`, we get:

- `a = k = -1/2`

- `b = -k = 1`

- `c = -2k = 1`

Therefore, the values of `a`, `b`, and `c` are **-1/2**, **1**, and **1/2**, respectively.