`x^51 + 16` is divided by (x +1), the remainder obtained is ?

Options

A. 1
B. 2
C. 0
D. 15

Answer

To find the remainder when \( x^{51} + 16 \) is divided by \( x + 1 \), we can use the Remainder Theorem. The Remainder Theorem states that 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 \( f(x) = x^{51} + 16 \).
- The divisor is \( x + 1 \), which can be rewritten as \( x - (-1) \). Thus, we will evaluate \( f(-1) \).

2. **Substitute \( x = -1 \) into the polynomial**:
- We need to calculate \( f(-1) \):
\[ f(-1) = (-1)^{51} + 16 \]

3. **Evaluate \( (-1)^{51} \)**:
- Since 51 is an odd number, \( (-1)^{51} = -1 \).

4. **Calculate \( f(-1) \)**:
- Now substitute back into the equation:
\[ f(-1) = -1 + 16 \]
- Simplifying this gives:
\[ f(-1) = 15 \]

5. **Conclusion**:
- Therefore, the remainder when \( x^{51} + 16 \) is divided by \( x + 1 \) is **15**.

### Final Answer:
The remainder is **15**.