If $ax^3+bx-c$ is divisible by $x^2+bx+c$ then $a,b,c$ are in what kind of progression? (arithmetic/geometric/etc)

Question

If a polynomial $$f(x)=ax^3+bx-c$$ is divisible by the polynomial $g(x)=x^2+bx+c$, then $a,b,c$ are in ...

$1.$ Arithmetic Progression
$2.$ Geometric Progression
$3.$ Harmonic Progression
$4.$ Arithmetic and Geometric Progression

I wrote $f(x)$ as $g(x)\cdot a(x-b)+x(b-ac+ab^2)+c(ab-1)$. Since it is divisible it means that $$x(b-ac+ab^2)+c(ab-1)=0$$ or $x=\frac{c(1-ab)}{b-ac+ab^2}$. I'm stuck here. What to do next? Any ideas?

Answer 1

You have:

$$(x^2+bx+c)(ax+m)=ax^3+bx-c$$

Since $cm=-c\implies m=-1$, then you get

$$ax^3+bx-c=ax^3+x^2(ab-1)+x(ac-b)-c$$

This leads to:

$$\begin{align}\begin{cases}ab=1\\ ac=2b\end{cases}\implies \begin{cases}a=a,a≠0\\b=\frac 1a\\ c=\frac {2}{a^2}\end{cases}\end{align}$$

I also stucked , what is the name of the following sequence?

$$a,\;\frac 1a\;,\frac {2}{a^2}$$

No option corresponds to this sequence.