If the HCF of $x^{2} + x - 12$ and ${2 x}^{2} - k x - 9$ is $x - k$ , find the value of $k$ .

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Solution

Let $p (x) = x^{2} + x - 12$ and $q (x) = 2 x^{2} - k x - 9$
Suppose, $h (x) = (x - k)$ be HCF of $p (x)$ and $q (x)$
Now, $p (x) = x^{2} + x - 12$
$= x^{2} + 4 x - 3 x - 12$
$= x (x + 4) - 3 (x + 4)$
$= (x + 4) (x - 3)$
So, either $k = - 4$ or $k = 3$ ...... $(i)$
Since, $(x - k)$ is HCF
$∴ q (x) = (x - k) g (x)$ for some factor $g (x)$ of $q (x)$
Take $x = k$
$⟹ 2 (k)^{2} - k \times k - 9 = 0$
$⟹ 2 k^{2} - k^{2} - 9 = 0$
$⟹ k^{2} = 9 ⟹ k = \pm 3$
From $(i)$ , we get
$k = 3$

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If the HCF of x2+x−12 and 2x2−kx−9 is x−k, find the value of k.

If the HCF of $x^{2} + x - 12$ and ${2 x}^{2} - k x - 9$ is $x - k$ , find the value of $k$ .