Possible values of $a+b^2+c^3$? If $a, b$ and $c$ are rational numbers satisfying the equation $x^3+ax^2+bx+c=0$ then find the possible values of $a+b^2+c^3$.
I have found one of the possible values as 0 using some vieta's rules and substituting $c$ for $x$.  But can't find other possible values. The question specifically asks $3$ possible values.

Let $f(x)=x^3+ax^2+bx+c$. We are given that $f(a)=f(b)=f(c)=0$. Therefore
\begin{align*}
a+b+c & = -a\\
ab+bc+ca & =b\\
abc & = -c.
\end{align*}
The last equation suggests either $c=0$ or $ab=-1$. 
If $c=0$, then from the first two equations we get $2a+b=0$ and $b(a-1)=0$. This gives us two possibilities $a=1, b=-2$ OR $a=0,b=0$. Consequently the possible values for $a+b^2+c^3=5 \text{ or } 0$.
Now consider the case $ab=-1$. Can you proceed from here?
