Find the remainder when `x^2018 + x^2017 + x^2016 + … + x^2 + x + 1` is divided by x+1.

Options

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

Answer

To find the remainder when the polynomial \( P(x) = x^{2018} + x^{2017} + x^{2016} + \ldots + x^2 + x + 1 \) is divided by \( x + 1 \), we can use the Remainder Theorem.

### Step-by-Step Solution:

1. **Identify the polynomial**:
We have the polynomial \( P(x) = x^{2018} + x^{2017} + x^{2016} + \ldots + x^2 + x + 1 \).

2. **Apply the Remainder Theorem**:
According to the Remainder Theorem, the remainder of \( P(x) \) when divided by \( x + a \) is \( P(-a) \). Here, \( a = 1 \), so we need to find \( P(-1) \).

3. **Substitute -1 into the polynomial**:
\[ P(-1) = (-1)^{2018} + (-1)^{2017} + (-1)^{2016} + \ldots + (-1)^2 + (-1) + 1 \]

4. **Evaluate each term**:
- The terms with even powers of \(-1\) will be \(1\) (since \((-1)^{\text{even}} = 1\)).
- The terms with odd powers of \(-1\) will be \(-1\) (since \((-1)^{\text{odd}} = -1\)).

5. **Count the number of terms**:
- There are \(2018 + 1 = 2019\) terms in total (from \(x^{2018}\) down to \(x^0\)).
- Among these, there are \(1009\) even powers (from \(0\) to \(2018\)) and \(1009\) odd powers (from \(1\) to \(2017\)).

6. **Calculate the total**:
- Contribution from even powers: \(1009 \times 1 = 1009\)
- Contribution from odd powers: \(1009 \times (-1) = -1009\)

7. **Combine the contributions**:
\[ P(-1) = 1009 - 1009 = 0 \]

8. **Conclusion**:
The remainder when \( P(x) \) is divided by \( x + 1 \) is \(0\).

### Final Answer:
The remainder is \(0\).