Which one of the following statements is correct?

Options

A. If `x^6 + 1` is divided by `x + 1`, then the remainder is `-2`
B. If `x^6 + 1` is divided by `x - 1`, then the remainder is 2
C. If `x^6 + 1` is divided by `x + 1` , then the remainder is 1
D. If `x^6 + 1` is divided by `x - 1`, then the remainder is -1

Answer

To determine which statement is correct among the given options, we will use the Remainder Theorem. According to this theorem, if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). Conversely, if it is divided by \( x + a \), the remainder is \( f(-a) \).

### Step-by-Step Solution:

1. **Identify the Polynomial and the Divisor**:
We are given the polynomial \( f(x) = x^6 + 1 \) and we need to check the divisibility by \( x + 1 \) (which means we will use \( -1 \) for our calculations).

2. **Apply the Remainder Theorem**:
Since we are dividing by \( x + 1 \), we will find the remainder by evaluating \( f(-1) \):
\[ f(-1) = (-1)^6 + 1 \]

3. **Calculate \( f(-1) \)**:
\[ (-1)^6 = 1 \quad \text{(since any even power of -1 is 1)} \]
Therefore,
\[ f(-1) = 1 + 1 = 2 \]

4. **Check the Options**:
Now we will check the options provided to see which one matches our calculated remainder of 2.

- **Option A**: Remainder is -2 (Incorrect, as we found 2)
- **Option B**: Remainder is 2 (Correct, matches our calculation)
- **Option C**: Remainder is 1 (Incorrect, as we found 2)
- **Option D**: Remainder is -1 (Incorrect, as we found 2)

5. **Conclusion**:
The correct statement is **Option B**, which states that the remainder is 2.