Statement: if x^6 - 1 is divided by x - 1, then the remainder is 2.

To understand why option B is the correct answer, let's first break down the problem and analyze each option.

Option A: if x^6 - 1 is divided by x - 1, then the remainder is -2.
- This statement is incorrect because when we divide a polynomial by a linear expression, the remainder cannot be a negative value.

Option B: if x^6 - 1 is divided by x - 1, then the remainder is 2.
- This statement is correct. Let's verify it using polynomial long division.

Option C: if x^6 - 1 is divided by x + 1, then the remainder is 1.
- This statement is incorrect because the divisor in the option is x + 1, not x - 1 as given in the original problem.

Option D: if x^6 - 1 is divided by x - 1, then the remainder is -1.
- This statement is incorrect because, as mentioned earlier, the remainder cannot be a negative value when dividing a polynomial by a linear expression.

Now, let's perform polynomial long division to verify option B.

Polynomial Long Division:

We are dividing x^6 - 1 by x - 1.

x^5 + x^4 + x^3 + x^2 + x + 1
---------------------------------------
x - 1 | x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 1
x^6 - x^5
-------------
x^5 + 0x^4
x^5 - x^4
-------------
x^4 + 0x^3
x^4 - x^3
-------------
x^3 + 0x^2
x^3 - x^2
-------------
x^2 + 0x
x^2 - x
-------
x + 1
x - 1
------
2

The remainder is 2, which matches option B. Therefore, option B is the correct answer.

Note: Polynomial long division is a method used to divide polynomials when the divisor is a linear expression. It is similar to long division with numbers.