If `(x^(51)+51)` is divided by `( x+1)` then the remainder is

Options

A. `0`
B. `1`
C. `49`
D. `50`

Answer

To find the remainder when \( x^{51} + 51 \) 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 \( f(c) \).

### Step-by-step Solution:

1. **Identify the polynomial and divisor**:
The polynomial is \( f(x) = x^{51} + 51 \) and the divisor is \( g(x) = x + 1 \).

2. **Set the divisor equal to zero**:
To find the value of \( c \), we set \( g(x) = 0 \):
\[ x + 1 = 0 \implies x = -1 \]

3. **Substitute \( c \) into the polynomial**:
We need to evaluate \( f(-1) \):
\[ f(-1) = (-1)^{51} + 51 \]

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

5. **Add 51 to the result**:
Now we can substitute this back into the equation:
\[ f(-1) = -1 + 51 = 50 \]

6. **Conclusion**:
Therefore, the remainder when \( x^{51} + 51 \) is divided by \( x + 1 \) is:
\[ \text{Remainder} = 50 \]

### Final Answer:
The remainder is \( 50 \).

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