Q. If $x^{2} + ax + 1$ is a factor of $ax^{3} + bx + c$, then
A) $b + a + a^{2} = 0,\ a = c$
B) $b - a + a^{2} = 0,\ a = c$
C) $b + a - a^{2} = 0,\ a = 0$
D) None of these
Explanation
(d)
$x^{2} + ax + 1$ must divide $ax^{3} + bx + c$.
Now, $\frac{ax^{3} + bx + c}{x^{2} + ax + 1} = \frac{a(x - a) + \left( b - a + a^{3} \right)x + c + a^{2}}{x^{2} + ax + 1}$
Reminder must be zero
$\Rightarrow b - a + a^{3} = 0,\ a^{2} + c = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.