Question

If $ x^2 + px + 1 $ is a factor of $ ax^3 + bx + c $ , then

• A. $ a^2 + c^2 = ab $
• B. $ a^2 + c^2 = -ab $
• C. $ a^2 - c^2 = ab $
• D. None of these

Answer 1

Answer: (C)

Since, $x^{2}+p x+1$ is a factor of $a x^{3}+b x+c$, so the remainder will be zero.
Now, $a x^{3}+b x+c=\left(x^{2}+p x+1\right)(a x-a p)+x\left(b-a+a p^{2}\right)+(c+a p) $
$\Rightarrow x\left(b-a+a p^{2}\right)+(c+a p)=0$
$\Rightarrow b-a+a p^{2}=0 \text { and } c+a p=0$
On putting $p=-\frac{c}{a}$ in $b-a+a p^{2}=0$, we get
$b-a+a \cdot\left(\frac{c^{2}}{a^{2}}\right)=0$
$ \Rightarrow a b-a^{2}+c^{2} =0 $
$ \Rightarrow a^{2}-c^{2} =a b$