Polynomial Division:
When dividing a polynomial by another polynomial, the remainder theorem can be used to determine the remainder. The remainder theorem states that if a polynomial f(x) is divided by a divisor d(x), the remainder is equal to f(a), where a is the root of the divisor.

Given Polynomials:
The polynomial p(x) = x^3 - x^2 - x + 1
The divisor q(x) = 2x - 1

Dividing p(x) by q(x):
To find the remainder when p(x) is divided by q(x), we will use polynomial long division. Here are the steps:

1. Arrange the terms of p(x) and q(x) in descending order of their exponents.
p(x) = x^3 - x^2 - x + 1
q(x) = 2x - 1

2. Divide the first term of p(x) by the first term of q(x) to obtain the first term of the quotient.
x^3 / (2x) = (1/2) * x^2

3. Multiply the divisor q(x) by the first term of the quotient obtained in step 2, and subtract it from p(x).
(1/2) * x^2 * (2x - 1) = x^3 - (1/2) * x^2
(x^3 - x^2 - x + 1) - (x^3 - (1/2) * x^2) = (1/2) * x^2 - x + 1

4. Repeat steps 2 and 3 with the new polynomial obtained in step 3.
Now we have the new polynomial (1/2) * x^2 - x + 1 and the same divisor 2x - 1.

(1/2) * x^2 / (2x) = (1/4) * x

(1/4) * x * (2x - 1) = (1/2) * x^2 - (1/4) * x
((1/2) * x^2 - x + 1) - ((1/2) * x^2 - (1/4) * x) = (3/4) * x + 1

5. Repeat steps 2 and 3 with the new polynomial obtained in step 4.
Now we have the new polynomial (3/4) * x + 1 and the same divisor 2x - 1.

(3/4) * x / (2x) = (3/8)

(3/8) * (2x - 1) = (3/4) * x - (3/8)
((3/4) * x + 1) - ((3/4) * x - (3/8)) = (11/8)

Remainder:
After the polynomial long division, the remainder obtained is (11/8).

Conclusion:
When the polynomial p(x) = x^3 - x^2 - x + 1 is divided by the divisor 2x - 1, the remainder is (11/8).