If `(x + 6)` is a factor of `f (x) = x ^(3) + 3x ^(2) + 4x + P,` then find the value of P.

Options

A. 42
B. 132
C. 192
D. 82

Answer

To find the value of \( P \) such that \( (x + 6) \) is a factor of \( f(x) = x^3 + 3x^2 + 4x + P \), we can use the Factor Theorem. According to the Factor Theorem, if \( (x + 6) \) is a factor of \( f(x) \), then \( f(-6) = 0 \).

### Step-by-step Solution:

1. **Substitute \( x = -6 \) into \( f(x) \)**:
\[ f(-6) = (-6)^3 + 3(-6)^2 + 4(-6) + P \]

2. **Calculate each term**:
- Calculate \( (-6)^3 \):
\[ (-6)^3 = -216 \]
- Calculate \( 3(-6)^2 \):
\[ 3(-6)^2 = 3 \times 36 = 108 \]
- Calculate \( 4(-6) \):
\[ 4(-6) = -24 \]

3. **Combine the results**:
\[ f(-6) = -216 + 108 - 24 + P \]

4. **Simplify the expression**:
\[ f(-6) = -216 + 108 - 24 + P = -132 + P \]

5. **Set the equation to zero** (since \( f(-6) = 0 \)):
\[ -132 + P = 0 \]

6. **Solve for \( P \)**:
\[ P = 132 \]

### Final Answer:
The value of \( P \) is \( 132 \).

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