Possible values of $a+b^2+c^3$?

Question

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.

Can you please elaborate your method Raffaele – Rohan Shinde Sep 03 '17 at 13:23 — Rohan Shinde, Sep 03 '17 at 13:23

Anurag A · Accepted Answer · 2017-09-03T07:54:11.080

6

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?

edited Sep 03 '17 at 07:54

answered Sep 03 '17 at 07:48

Anurag A

41,067

I'll take it from here thanks – Rohan Shinde Sep 03 '17 at 08:26

Possible values of $a+b^2+c^3$?

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