Example 6 Find all cubic polynomials $f(x)=x^{3}+a x^{2}+b x+c$ with rational roots $a, b, c$.
(IMO - 26 Shortlist)
Solve the system of equations given by $\left\{\begin{array}{l}a+b-c=-a, \\ a b+b c+c a=b, \\ a b c=-c .\end{array}\right.$
If $c=0$, then from (1) and (2) we get $a+b=-a, a b=b$.
Thus, $\left\{\begin{array}{l}a=0 \\ b=0\end{array}\right.$ or $\left\{\begin{array}{l}a=1, \\ b=-2 .\end{array}\right.$
If $c \neq 0$, then from (3) we have $a b=-1$, and from (1) we have $c=-2 a-b$. Substituting into (2) gives $(b+1)\left(b^{3}-2 b+2\right)=0$, and since $b^{2}-2 b+2=0$ has no rational roots, we have $b=-1$.

The required polynomials are $f_{1}(x)=x^{3}+x^{2}-2 x, f_{2}(x)=x^{3}+x^{2}-x-1 .\left(f_{3}(x)=0\right.$ is not valid, discard it.)