If `3x^(2)+ax +4` is perfectly divisible by `x - 8`, then the value of a is:

Options

A. 24.5
B. `-25.5`
C. `-24.5`
D. 25.5

Answer

To solve the problem, we need to find the value of \( a \) such that the polynomial \( 3x^2 + ax + 4 \) is perfectly divisible by \( x - 8 \). This means that when we substitute \( x = 8 \) into the polynomial, the result should be zero.

### Step-by-Step Solution:

1. **Identify the Polynomial and Factor**:
We have the polynomial \( f(x) = 3x^2 + ax + 4 \) and we know it is divisible by \( x - 8 \).

2. **Set the Factor Equal to Zero**:
To find the value of \( x \) for which the polynomial is zero, we set the factor equal to zero:
\[ x - 8 = 0 \implies x = 8 \]

3. **Substitute \( x = 8 \) into the Polynomial**:
We substitute \( x = 8 \) into the polynomial:
\[ f(8) = 3(8^2) + a(8) + 4 \]

4. **Calculate \( 8^2 \)**:
Calculate \( 8^2 \):
\[ 8^2 = 64 \]

5. **Substitute Back into the Polynomial**:
Now substitute \( 64 \) back into the polynomial:
\[ f(8) = 3(64) + 8a + 4 \]

6. **Calculate \( 3 \times 64 \)**:
Calculate \( 3 \times 64 \):
\[ 3 \times 64 = 192 \]

7. **Combine the Terms**:
Now combine all the terms:
\[ f(8) = 192 + 8a + 4 = 196 + 8a \]

8. **Set the Polynomial Equal to Zero**:
Since \( f(8) \) must equal zero for \( x - 8 \) to be a factor:
\[ 196 + 8a = 0 \]

9. **Solve for \( a \)**:
Rearranging the equation gives:
\[ 8a = -196 \]
Now divide both sides by 8:
\[ a = \frac{-196}{8} = -24.5 \]

### Final Answer:
The value of \( a \) is \( -24.5 \).

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