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

Options

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

Explanation

To find the remainder when \( x^{51} + 51 \) is divided by \( x + 1 \), we can use the Remainder Theorem. Here’s a step-by-step solution:

### Step 1: Identify the polynomial and the divisor
The polynomial we are working with is:
\[ f(x) = x^{51} + 51 \]
We are dividing this polynomial by:
\[ x + 1 \]

### Step 2: Apply the Remainder Theorem
According to the Remainder Theorem, if a polynomial \( f(x) \) is divided by \( x + a \), the remainder is \( f(-a) \). In our case, \( a = 1 \), so we need to find \( f(-1) \).

### Step 3: Substitute -1 into the polynomial
Now, we calculate \( f(-1) \):
\[ f(-1) = (-1)^{51} + 51 \]

### Step 4: Evaluate the power of -1
Since 51 is an odd number:
\[ (-1)^{51} = -1 \]
Thus, we can substitute this back into our equation:
\[ f(-1) = -1 + 51 \]

### Step 5: Simplify the expression
Now, simplify the expression:
\[ f(-1) = 50 \]

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

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