Find the HCF of $p (a) = 2 a^{2} - 7 a + 3$ and $q (a) = 2 a^{2} + 5 a - 3$ .

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Solution

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$p (a) = 2 a^{2} - 7 a + 3 = 2 a^{2} - 6 a + 3 = 2 a (a - 3) - 1 (a - 3) = (a - 3) (2 a - 1)$
and
$q (a) = 2 a^{2} + 5 a - 3 = 2 a^{2} + 6 a - a - 3 = 2 a (a + 3) - 1 (a + 3) = (a + 3) (2 a - 1)$
From this we conclude that the HCF of $p (a)$ and $q (a)$ is $2 a - 1$ .

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