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Relationship between LCM and GCD

Find the H.C....

Question

Find the H.C.F and L.C.M of the polynomial whose $p (x) = x^{2} - 5 x + 6$ and $q (x) = x^{2} + 5 x + 4$ .

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Solution

Given:
$p (x) = x^{2} - 5 x + 6$
$q (x) = x^{2} + 5 x + 4$
Factor of $x^{2} + 5 x + 4 = (x + 1) (x + 4)$
Factor of $x^{2} - 5 x + 6 = (x - 2) (x - 3)$
So, H.C.F of $p (x)$ and $q (x) =$ $(x + 1)$
Using the formula,
$p (x) \times q (x) =$ H.C.F $\times$ L.C.M
$L . C . M = \frac{(x + 1) (x + 4) (x - 2) (x - 3) (x + 1)}{x + 1)}$
$= (x + 1) (x + 4) (x - 2) (x - 3)$
Therefore, L.C.M is $(x + 1) (x + 4) (x - 2) (x - 3)$ .

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