If $a x^{3} + b x + c$ has a factor of the form $x^{2} + p x + 1$ , then $a^{2} - c^{2} =$

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a/b

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^{2}

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^{2}

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Solution

The correct option is A ab
By actual division, we get
$\frac{a x^{3} + b x + c}{x^{2} + p x + 1} = a x - a p + \frac{R (x)}{x^{2} + p x + 1}$
Remainder polynomial
$R (x) = (b - a + a p^{2}) x + c + a p$
If $a x^{3} + b x + c$ has a factor of the form $x^{2} + p x + 1$ , then R (x) must be identically zero,
if $b - a + a p^{2} = 0$ and $c + a p = 0$
Eliminating p between these equations, we get
$b - a + a {(- \frac{c}{a})}^{2} = 0$ or $a^{2} - c^{2} = a b$

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