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

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

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 are dividing x^4 - 3x^3 + 3x^2 - 3x + 1 by x + 1. Using long division, we get:

x^3 - 4x^2 + 7x - 8
x + 1 | x^4 - 3x^3 + 3x^2 - 3x + 1
-x^4 - x^3
--------------
-4x^3 + 3x^2
-4x^3 - 4x^2
--------------
7x^2 - 3x
7x^2 + 7x
--------
-8x + 1
-8x - 8
------
9

Therefore, the remainder is 9. However, we can check our answer using the Remainder Theorem. If we substitute -1 for x in the original polynomial, we get:

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

So the remainder is indeed 1.