The G.C.D of $x^{3} + x^{2} - x - 1$ and $x^{2} - 1$ is

x^{2} - 1

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x + 1

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x^{3} - 1

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x - 1

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Solution

The correct option is C $x^{2} - 1$
Let $p (x) = x^{3} + x^{2} - x - 1$ and $q (x) = x^{2} - 1$
$p (x) = x^{3} + x^{2} - x - 1$
$= x^{2} (x + 1) - 1 (x + 1)$
$= (x^{2} - 1) (x + 1)$
$= (x - 1) (x + 1) (x + 1)$
and
$q (x) = x^{2} - 1$
$= (x + 1) (x - 1)$
$∴$ G.C.D of $p (x)$ and $q (x)$ = $(x + 1) (x - 1) = x^{2} - 1$
Option A is correct.

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