`p(x)=4(-x^(3)+11x+12)-6(x-c)`
In the polynomial function p(x) defined above c is a constant. If p(x) is divisible by x, what is the value of c?

Options

A. `-8`
B. `-6`
C. 0
D. 6

Answer

To solve the problem, we need to find the value of \( c \) such that the polynomial \( p(x) \) is divisible by \( x \). This means that when we substitute \( x = 0 \) into \( p(x) \), the result should be zero.

### Step-by-Step Solution:

1. **Write down the polynomial**:
\[ p(x) = 4(-x^3 + 11x + 12) - 6(x - c) \]

2. **Distribute the terms**:
- First, distribute \( 4 \) in the first part:
\[ p(x) = 4(-x^3) + 4(11x) + 4(12) - 6(x - c) \]
- This simplifies to:
\[ p(x) = -4x^3 + 44x + 48 - 6x + 6c \]

3. **Combine like terms**:
- Combine the \( x \) terms:
\[ p(x) = -4x^3 + (44x - 6x) + 48 + 6c \]
- This simplifies to:
\[ p(x) = -4x^3 + 38x + 48 + 6c \]

4. **Set \( p(0) = 0 \)**:
- Since \( p(x) \) is divisible by \( x \), we need:
\[ p(0) = 0 \]
- Substitute \( x = 0 \) into \( p(x) \):
\[ p(0) = -4(0)^3 + 38(0) + 48 + 6c = 48 + 6c \]
- Set this equal to zero:
\[ 48 + 6c = 0 \]

5. **Solve for \( c \)**:
- Rearranging gives:
\[ 6c = -48 \]
- Dividing both sides by 6:
\[ c = \frac{-48}{6} = -8 \]

### Conclusion:
The value of \( c \) is \( -8 \).