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Method of Finding HCF

Find the HCF ...

Question

Find the HCF of $p (x) = (x^{3} - 27) (x^{2} - 3 x + 2)$ and $q (x) = (x^{2} + 3 x + 9) (x^{2} - 5 x + 6)$ .

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Solution

Given : $p (x) = (x^{3} - 27) (x^{2} - 3 x + 2)$ and $q (x) = (x^{2} + 3 x + 9) (x^{2} - 5 x + 6)$ .
Now, $p (x) = (x^{3} - 27) (x^{2} - 3 x + 2)$
$= (x^{3} - 3^{3}) (x^{2} - 2 x - x + 2)$
$= (x - 3) (x^{2} + 3 x + 9) (x - 2) (x - 1)$ ........ [Using formula of $(a^{3} - b^{3})$ ]
Also, $q (x) = (x^{2} + 3 x + 9) (x^{2} - 5 x + 6)$
$= (x^{2} + 3 x + 9) (x^{2} - 3 x - 2 x + 6)$
$= (x^{2} + 3 x + 9) (x - 2) (x - 3)$
The common factors in $p (x)$ and $q (x)$ are $(x - 3), (x - 2)$ and $(x^{2} + 3 x + 9)$
Thus, the HCF of $p (x)$ and $q (x)$ is $(x - 2) (x - 3) (x^{2} + 3 x + 9)$

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