Question

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