Find the remainder when the expression `x^(3)+x^(2)+x+1` is divided by `x+1`.

Options

A. 3
B. 5
C. 2
D. 0

Answer

To find the remainder when the polynomial \( P(x) = x^3 + x^2 + x + 1 \) is divided by \( x + 1 \), we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial \( P(x) \) by \( x - c \) is equal to \( P(c) \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
- We have the polynomial \( P(x) = x^3 + x^2 + x + 1 \).
- We are dividing by \( x + 1 \), which can be rewritten as \( x - (-1) \). Here, \( c = -1 \).

2. **Apply the Remainder Theorem**:
- According to the Remainder Theorem, we need to evaluate \( P(-1) \).

3. **Calculate \( P(-1) \)**:
- Substitute \( -1 \) into the polynomial:
\[ P(-1) = (-1)^3 + (-1)^2 + (-1) + 1 \]
- Simplifying this:
\[ P(-1) = -1 + 1 - 1 + 1 \]
- Combine the terms:
\[ P(-1) = 0 \]

4. **Conclusion**:
- The remainder when \( x^3 + x^2 + x + 1 \) is divided by \( x + 1 \) is \( 0 \).

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

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