Example 2 (1991 National High School League Question) Let the function $y=f(x)$ satisfy $f(3+x)=f(3-x)$ for all real numbers $x$, and the equation $f(x)=0$ has exactly 6 distinct real roots. Then the sum of these 6 real roots is $(\quad)$.
A. 18
B. 12
C. 9
D. 0
Solution: Choose A. Reason: If $3+\alpha$ is a root of $f(x)=0$, then by the given $f(3-\alpha)=f(3+\alpha)=$ 0, which means $3-\alpha$ is also a root. Therefore, we can assume that the six roots of the equation $f(x)=0$ are $3 \pm \alpha_{1}, 3 \pm \alpha_{2}, 3 \pm \alpha_{3}$. Thus, their sum equals 18.