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

Answer

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

### Step-by-Step Solution:

1. **Identify the polynomial and divisor**:
- The polynomial is \( P(x) = x^3 - 3x^2 + x + 1 \).
- The divisor is \( x + 1 \), which can be rewritten as \( x - (-1) \). Thus, \( c = -1 \).

2. **Evaluate the polynomial at \( c = -1 \)**:
- We need to calculate \( P(-1) \):
\[ P(-1) = (-1)^3 - 3(-1)^2 + (-1) + 1 \]

3. **Calculate each term**:
- Calculate \( (-1)^3 = -1 \)
- Calculate \( -3(-1)^2 = -3 \cdot 1 = -3 \)
- Calculate \( (-1) = -1 \)
- The constant term is \( +1 \)

4. **Combine the results**:
\[ P(-1) = -1 - 3 - 1 + 1 \]
- Combine the terms:
\[ P(-1) = -1 - 3 - 1 + 1 = -4 \]

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

### Final Answer:
The remainder is \( -4 \).