If the polynomial $x^{6} + p x^{5} + q x^{4} - x^{2} - x - 3$ is divisible by $x^{4} - 1$ , then the value of $p^{2} + q^{2}$ is

1

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5

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10

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13

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Solution

The correct option is C $10$
The divisor is $x^{4} - 1 = (x - 1) (x + 1) (x^{2} + 1)$
By factor theorem, $f (1) = f (- 1) = 0$
Thus, $1 + p + q - 1 - 1 - 3 = 0$ $a n d$ $1 + q - 1 - 3 = p - 1$
i.e., $p + q = 4$ and $p - q = - 2$
Adding the two, $2 p = 2$ i.e. $p = 1$ and $∴$ $q = 3.$
$∴$ $p^{2} + q^{2} = 1 + 9 = 10$

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