5-54 Find all cubic polynomials with rational roots $a, b, c$
$$
x^{3}+a x^{2}+b x+c .
$$
[Solution] According to Vieta's formulas, $a, b, c$ should satisfy
(1) $a+b+c=-a$,
(2) $a b+b c+c a=b$,
(3) $a b c=-c$.

If $a=0$ (or $b=0$), from (3) we get $c=0$, and from (1) we get $b=0$ (or $a=0$). Therefore, we can assume $a \neq 0, b \neq 0$.

If $c=0$, from (2) we get $a=1$, and from (1) we get $b=-2$. Clearly, 1, $-2,0$ are the roots of the equation $x^{3}+x^{2}-2 x=0$.

If $c \neq 0$, then from (3), $b=-\frac{1}{a}$, and by eliminating $c$ from (1) and (2), we get $b^{4}+b^{3}-2 b^{2}+2=0$. The only rational root of this equation is $b=-1$, and $a=1, c=-1$. In fact, the equation
$$
x^{3}+x^{2}-x-1=0
$$

has $1,-1,-1$ as its roots.