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

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

x^3 - 3x^2 + 4x - 3

x + 1 | x^4 - 2x^3 + x^2 - 2x + 1

-x^4 - x^3

-------------

-3x^3 + x^2

-(-3x^3 - 3x^2)

--------------

4x^2 - 2x

-(4x^2 + 4x)

------------

-6x + 1

-(-6x - 6)

--------

7

Therefore, the remainder when x^4 - 2x^3 + x^2 - 2x + 1 is divided by x + 1 is 7.

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