Find common factor of the quadratic polynomials $3 x^{2} - x - 10$ and $2 x^{2} - x - 6$ .

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Solution

$p (x) = 3 x^{2} - x - 10$
$q (x) = 2 x^{2} - x - 6$
Let $(x - k)$ be a common factor of $p (x)$ and $q (x)$ .
Therefore, $p (k) = q (k) = 0$
Therefore,
$3 k^{2} - k - 10 = 2 k^{2} - k - 6$
$k^{2} = 4$
$k$ can be $\pm 2$ but if $k = - 2$ ,
$p (- 2) = 12 + 2 - 10 = 4 \neq 0$
Hence $k = 2$
Therefore, required common factor is $(x - 2)$

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