If ` ( x + 4) ` is a factor of ` 3 x ^(2) + kx + 8` then what is the value of k ?

Options

A. 4
B. `-4`
C. `-14`
D. `14`

Answer

To find the value of \( k \) such that \( (x + 4) \) is a factor of \( 3x^2 + kx + 8 \), we can follow these steps:

### Step 1: Set up the equation
Since \( (x + 4) \) is a factor, we can set \( x + 4 = 0 \) which gives us \( x = -4 \). We will substitute \( x = -4 \) into the polynomial \( 3x^2 + kx + 8 \).

### Step 2: Substitute \( x = -4 \)
Substituting \( x = -4 \) into the polynomial:
\[ 3(-4)^2 + k(-4) + 8 = 0 \]

### Step 3: Calculate \( 3(-4)^2 \)
Calculating \( 3(-4)^2 \):
\[ 3 \times 16 = 48 \]
So the equation now becomes:
\[ 48 - 4k + 8 = 0 \]

### Step 4: Combine like terms
Combine \( 48 \) and \( 8 \):
\[ 56 - 4k = 0 \]

### Step 5: Solve for \( k \)
Now, isolate \( k \):
\[ -4k = -56 \]
Dividing both sides by \(-4\):
\[ k = \frac{56}{4} = 14 \]

### Conclusion
Thus, the value of \( k \) is \( 14 \).

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