If $x^{2} + p x + 1$ is a factor of the expression $a x^{3} + b x + c$ , 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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Solution

The correct option is A $a^{2} - c^{2} = a b$
Given that $x^{2} + p x + 1$ is a factor of $a x^{3} + b x + c$ .
Let $a x^{3} + b x + c = (x^{2} + p x + 1) (a x + λ)$ , where $λ$ is a constant.
Then equating the coefficients of line powers of $x$ on both sides, we get
$0 = a p + λ, b = p λ + a, c = λ$
$\Rightarrow p = - \frac{λ}{a} = - \frac{c}{a}$
Hence,
$b = (- \frac{c}{a}) c + a$
$\Rightarrow a b = a^{2} - c^{2}$

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