The function `f (x)=0` has eight dinstict real solution and f also satisfy `f (4+x) =f (4-x).` The sum of all the eight solution of `f (x)=0` is :
(a). 12 (b). 32 (c). 16 (d). 15

Options

A. 12
B. 32
C. 16
D. 15

Answer

To solve the problem, we need to analyze the given function \( f(x) \) and its properties.

### Step-by-step Solution:

1. **Understanding the Function**:
We are given that \( f(x) = 0 \) has eight distinct real solutions. Additionally, the function satisfies the condition \( f(4 + x) = f(4 - x) \). This indicates that the function is symmetric about \( x = 4 \).

**Hint**: Recognize that symmetry in functions often leads to pairs of solutions that are equidistant from the line of symmetry.

2. **Identifying the Roots**:
Because the function is symmetric about \( x = 4 \), we can express the roots in pairs. Let’s denote the roots as follows:
- If \( r_1 \) is a root, then \( r_2 = 8 - r_1 \) is also a root (since \( 4 + k \) and \( 4 - k \) for some \( k \)).
- This means we can have pairs of roots: \( (4 - k_1, 4 + k_1), (4 - k_2, 4 + k_2), (4 - k_3, 4 + k_3), (4 - k_4, 4 + k_4) \).

**Hint**: For every root \( r \), there is a corresponding root \( 8 - r \) due to the symmetry.

3. **Calculating the Sum of the Roots**:
The eight distinct roots can be expressed as:
- \( 4 - k_1, 4 + k_1, 4 - k_2, 4 + k_2, 4 - k_3, 4 + k_3, 4 - k_4, 4 + k_4 \).
- The sum of these roots can be calculated as:
\[ (4 - k_1) + (4 + k_1) + (4 - k_2) + (4 + k_2) + (4 - k_3) + (4 + k_3) + (4 - k_4) + (4 + k_4) \]
- Simplifying this gives:
\[ 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 8 \times 4 = 32 \]

**Hint**: When summing symmetric pairs, the terms involving \( k \) will cancel out, simplifying the calculation.

4. **Conclusion**:
The sum of all the eight solutions of \( f(x) = 0 \) is \( 32 \).

**Final Answer**: The correct option is (b) 32.