If `x^(21)+101` is divided by x+1 then the remainder is

Options

A. `-1`
B. `102`
C. `0`
D. `100`

Answer

To find the remainder when \( x^{21} + 101 \) is divided by \( x + 1 \), we can use the Remainder Theorem. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Identify the Polynomial**:
We have the polynomial \( P(x) = x^{21} + 101 \).

2. **Determine the Value for Remainder Theorem**:
According to the Remainder Theorem, if we are dividing by \( x + 1 \), we need to find \( P(-1) \) because \( x + 1 = 0 \) gives \( x = -1 \).

3. **Substitute \( -1 \) into the Polynomial**:
We substitute \( x = -1 \) into \( P(x) \):
\[ P(-1) = (-1)^{21} + 101 \]

4. **Calculate \( (-1)^{21} \)**:
Since 21 is an odd number, \( (-1)^{21} = -1 \).

5. **Add 101 to the Result**:
Now, we compute:
\[ P(-1) = -1 + 101 \]

6. **Simplify the Expression**:
\[ P(-1) = 100 \]

7. **Conclusion**:
Therefore, the remainder when \( x^{21} + 101 \) is divided by \( x + 1 \) is \( 100 \).

### Final Answer:
The remainder is \( 100 \).

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