5. Find all cubic polynomials $x^{3}+a x^{2}+b x+c$ with rational roots $a, b, c$.
(1985 Turkish Competition Problem)
5. By Fengda's theorem, $a, b, c$ should satisfy $a+b+c=-a$, (1) $ab+bc+ca=b$, (2) $abc=-c$. (3) 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 $x^{3}+x^{2}-2x=0$. If $c \neq 0$, then from (3), $b=-\frac{1}{a}$, and by eliminating $c$ from (1) and (2), we get $b^{4}+b^{3}-2b^{2}+2=0$, the only rational root of this equation is $b=-1$, and hence $a=1, c=-1$. In fact, the equation $x^{3}+x^{2}-x-1=0$ has $1, -1, -1$ as its roots.