Relation of coefficient of Cubic polynomial If $a.b $ & $c$ are rational number satisfying the equation $x^3+ax^2+bx+c=0$, then which of the following can also be true .
A) $a+b^2+c^3=0$
B) $a+b^2+c^3=5$
C) $a+b^2+c^3=1$
D) None of these
If it is a cubic function the equation has at least one real root. So for any value of a, b & c we have one real root, hence i am confused. The question i presume is cotrect

Since no relation between $a, b$ and $c$ need to hold in order to satisfy $$x^3+ax^2+bx+c=0 \tag{i}$$ as there always exist some real $x$ which is one of the roots of $\text{(i)}$, I'm assuming that $a, b$ and $c$ satisfy $\text{(i)}$ for all $x \in \mathbb R$
Thus, for $x=0$, equation $\text{(i)}$ becomes
$0+0+0+c=0 \implies c=0$
Also,
$1+a+b+c=0$ and $ -1+a-b+c=0$ for $x=1$ and $x=-1$ in $\text{(i)}$ respectively.
Solving them will give you $a=0,b=-1,c=0$
But for $x=2$,we get $8+4a+2b+c=0 \implies 8-2=0$ which is a contradiction, so there do not exists any rational numbers $a,b,c$ that can satisfy $\text{(i)}$ for all $x \in \mathbb R$
