If $(x^{2} + p x + 1)$ is factor of $(a x^{3} + b x + c)$ , then

a^{2} + c^{2} = - a b

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a^{2} - c^{2} = - a b

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a^{2} - c^{2} = a b

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none of these

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Solution

The correct option is D $a^{2} - c^{2} = a b$
$x^{2} + p x + 1$ is a factor of $a x^{3} + b x + c$
Let other root be $(m x + n)$
Hence, $m = a$ and $n = c$
$(x^{2} + p x + 1) (a x + c) = (a x^{3} + b x + c)$
Now let us compare the terms of $x$ and $x^{2}$ on both sides,
$c + a p = 0$ ........(1)
$p c + a = b$ ......(2)
$p = - \frac{c}{a}$
Eliminate $p$ from eq.(2)
$a^{2} - c^{2} = a b$

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