Calculate the remainder when x^5 - 4x^4 + 6x^3 - 4x^2 + x - 1 is divided by x - 1.

The remainder when x^5 - 4x^4 + 6x^3 - 4x^2 + x - 1 is divided by x - 1 is -4.

To find the remainder when a polynomial is divided by a linear factor, we can use the Remainder Theorem. This states that if we divide a polynomial f(x) by x - a, the remainder is f(a).

In this case, we want to find the remainder when x^5 - 4x^4 + 6x^3 - 4x^2 + x - 1 is divided by x - 1. Using the Remainder Theorem, we can substitute x = 1 into the polynomial to find the remainder:

1^5 - 4(1)^4 + 6(1)^3 - 4(1)^2 + 1 - 1 = -4

Therefore, the remainder when x^5 - 4x^4 + 6x^3 - 4x^2 + x - 1 is divided by x - 1 is -4.