In the following case, use factor theorem to find whether $g (x)$ is a factor of the polynomial $p (x)$ or not.
$p (x) = 2 x^{4} + 3 x^{3} - 2 x^{2} - 9 x - 12 g (x) = x^{2} - 3$

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Solution

$g (x) = x^{2} - 3$
$⟹ g (x) = (x + \sqrt{3}) (x - \sqrt{3})$
Now,
$p (x) = 2 x^{4} + 3 x^{3} - 2 x^{2} - 9 x - 12$
$p (\sqrt{3}) = 2 (\sqrt{3})^{4} + 3 (\sqrt{3})^{3} - 2 (\sqrt{3})^{2} - 9 (\sqrt{3}) - 12$
$= 2 \times 9 + 3 \times 3 \sqrt{3} - 2 \times 3 - 9 \sqrt{3} - 12$
$= 18 + 9 \sqrt{3} - 18 - 9 \sqrt{3} = 0$
$∴ (x - \sqrt{3})$ is a factor of $p (x)$

$p (- \sqrt{3}) = 2 (- \sqrt{3})^{4} + 3 (- \sqrt{3})^{3} - 2 (- \sqrt{3})^{2} - 9 (- \sqrt{3}) - 12$
$= 2 \times 9 + 3 \times - 3 \sqrt{3} - 2 \times 3 + 9 \sqrt{3} - 12$
$= 18 - 9 \sqrt{3} - 18 + 9 \sqrt{3} = 0$
$∴ (x + \sqrt{3})$ is a factor $p (x)$

$∵$ both are the facors of $p (x), g (x)$ is a factor of the polynomial $p (x)$

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