8.51 Let the function $y=f(x)$ satisfy for all real numbers $x$
$$
f(3+x)=f(3-x),
$$

and the equation $f(x)=0$ has exactly 6 distinct real roots, then the sum of these 6 roots is
(A) 18.
(B) 12.
(C) 9.
(D) 0.
(China High School Mathematics League, 1991)
[Solution] If $3+\alpha$ is a root of $f(x)=0$, then by the given condition,
$$
f(3-\alpha)=f(3+\alpha)=0 .
$$

This means that $3-\alpha$ is also a root of $f(x)=0$. Therefore, we can assume these 6 roots to be
$$
3 \pm \alpha_{1}, \quad 3 \pm \alpha_{2}, \quad 3 \pm \alpha_{3} .
$$

Thus, their sum is 18.
Hence, the correct choice is $(A)$.