If `'x^(13) + 1'` is divided by 'x-1' , the remainder is :

Options

A. 1
B. `-1`
C. 0
D. 2

Answer

To find the remainder when \( x^{13} + 1 \) is divided by \( x - 1 \), we can use the Remainder Theorem. According to the theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is equal to \( f(c) \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
- The polynomial is \( f(x) = x^{13} + 1 \).
- The divisor is \( x - 1 \).

2. **Apply the Remainder Theorem**:
- According to the Remainder Theorem, we need to evaluate \( f(1) \) because we are dividing by \( x - 1 \) (where \( c = 1 \)).

3. **Substitute \( x = 1 \) into the polynomial**:
\[ f(1) = 1^{13} + 1 \]

4. **Calculate \( f(1) \)**:
- Since \( 1^{13} = 1 \), we have:
\[ f(1) = 1 + 1 = 2 \]

5. **Conclusion**:
- The remainder when \( x^{13} + 1 \) is divided by \( x - 1 \) is \( 2 \).

### Final Answer:
The remainder is \( 2 \).