Example 7 Given that the function $y=f(x)$ is an even function, and its graph intersects the $x$-axis at four points, find the sum of all real roots of the equation $f(x)=0$.

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Let $x_{0}$ be a root of the equation $f(x)=0$, then $f\left(x_{0}\right)=0$, and $f\left(-x_{0}\right)=f\left(x_{0}\right)$, so $-x_{0}$ is also a root of the equation $f(x)=0$. Therefore, the sum of all real roots of the equation $f(x)=0$ is 0.

Key Points for Solving the Problem For an odd or even function $y=f(x)$, the non-zero real roots of the equation $f(x)=0$ always appear in pairs.