Given:
The function f(x) = 0 has eight distinct real solutions.
f(4x) = f(4-x).

To find:
The sum of all eight solutions of f(x) = 0.

Explanation:
To find the sum of all eight solutions of f(x) = 0, we first need to understand the given function and its properties.

Understanding the given function:
The function f(x) = 0 represents a horizontal line on the x-axis. Since the function is equal to zero for all values of x, it does not depend on the value of x and remains constant.

Property: f(4x) = f(4-x)
The given function satisfies the property f(4x) = f(4-x). This property tells us that the value of the function at 4x is equal to the value of the function at 4-x. In other words, the function is symmetric about the line x = 2.

Implications of the symmetric property:
1. The function has a total of eight distinct real solutions. Since the function is symmetric about x = 2, it means that for every solution x, there is another solution (4 - x) on the other side of the line x = 2. This results in a total of eight distinct solutions.

2. The sum of each pair of symmetric solutions is equal to 4. For example, if one solution is x = 1, the other solution will be (4 - 1) = 3. The sum of these two solutions is 1 + 3 = 4. This property holds true for all pairs of symmetric solutions.

Calculating the sum of all eight solutions:
To calculate the sum of all eight solutions, we can pair them up and find the sum of each pair. Since there are four pairs of symmetric solutions, we can calculate the sum of one pair and multiply it by four to get the total sum.

Let's assume the eight solutions of f(x) = 0 are a, b, c, d, e, f, g, and h.

Sum of the first pair: a + (4 - a) = 4
Sum of the second pair: b + (4 - b) = 4
Sum of the third pair: c + (4 - c) = 4
Sum of the fourth pair: d + (4 - d) = 4

Total sum of all eight solutions = 4 + 4 + 4 + 4 = 16

Therefore, the sum of all eight solutions of f(x) = 0 is 16.

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