If $x^{2} + p x + 1$ is a 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 C $a^{2} - c^{2} = a b$
As $x^{2} + p x + 1$ is a factor of $a x^{3} + b x + c = 0$
Then $a x^{3} + b x + c = (x^{2} + p x + 1) (a x + c)$
The other factor is of first degree whose coefficients are chosen keeping in view the coefficient of $x^{3}$ and constant term in cubic,
Comparing the coefficient of $x^{2}$
$a p + c = 0 ∴ p = - \frac{c}{a}$ ...(1)
And comparing the coefficients of $x$
$a + c p = b$ (2)
From (1) and (2), we have
$a + c (\frac{- c}{a}) = b$
$\Rightarrow a^{2} - c^{2} = a b .$
Hence, option 'C' is correct.

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