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

Options

A. `-16`
B. `-12`
C. `-20`
D. `-10`

Answer

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

Here, since we are dividing by \( x + 3 \), we set \( c = -3 \).

### Step-by-step Solution:

1. **Identify the polynomial and the value of c**:
\[ f(x) = x^3 - 3x^2 + 4x + 50 \]
We need to evaluate \( f(-3) \).

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

3. **Calculate each term**:
- Calculate \( (-3)^3 \):
\[ (-3)^3 = -27 \]
- Calculate \( -3(-3)^2 \):
\[ (-3)^2 = 9 \quad \Rightarrow \quad -3 \times 9 = -27 \]
- Calculate \( 4(-3) \):
\[ 4 \times (-3) = -12 \]

4. **Combine all the terms**:
\[ f(-3) = -27 - 27 - 12 + 50 \]

5. **Simplify the expression**:
- First, combine \( -27 - 27 \):
\[ -27 - 27 = -54 \]
- Now, combine \( -54 - 12 \):
\[ -54 - 12 = -66 \]
- Finally, add \( 50 \):
\[ -66 + 50 = -16 \]

6. **Conclusion**:
The remainder when \( f(x) \) is divided by \( x + 3 \) is:
\[ \text{Remainder} = -16 \]

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