Find the remainder when the polynomial `p(x)=x^(3)-3x^(2)+4x+50` is divided by (x+3).

Options

A. `-13`
B. `-14`
C. `-15`
D. `-16`

Explanation

To find the remainder when the polynomial \( p(x) = x^3 - 3x^2 + 4x + 50 \) is divided by \( x + 3 \), we can use the Remainder Theorem. According to the theorem, the remainder of the division of a polynomial \( p(x) \) by \( x - c \) is \( p(c) \). In this case, we need to evaluate \( p(-3) \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
- The polynomial is \( p(x) = x^3 - 3x^2 + 4x + 50 \).
- We are dividing by \( x + 3 \), which can be rewritten as \( x - (-3) \).

2. **Apply the Remainder Theorem**:
- According to the Remainder Theorem, the remainder when \( p(x) \) is divided by \( x + 3 \) is \( p(-3) \).

3. **Calculate \( p(-3) \)**:
- Substitute \( -3 \) into the polynomial:
\[ p(-3) = (-3)^3 - 3(-3)^2 + 4(-3) + 50 \]

4. **Evaluate each term**:
- Calculate \( (-3)^3 = -27 \).
- Calculate \( -3(-3)^2 = -3 \times 9 = -27 \).
- Calculate \( 4(-3) = -12 \).
- Now, substitute these values back into the equation:
\[ p(-3) = -27 - 27 - 12 + 50 \]

5. **Combine the values**:
- Combine the terms:
\[ p(-3) = -27 - 27 = -54 \]
\[ -54 - 12 = -66 \]
\[ -66 + 50 = -16 \]

6. **Conclusion**:
- The remainder when \( p(x) \) is divided by \( x + 3 \) is \( -16 \).

### Final Answer:
The remainder is \( -16 \).