The $H . C . F$ of the polynomials $p (x) = (x - 3) (x^{2} + x - 2)$ and $q (x) = x^{2} - 5 x + 6$ is $(x - 3),$ then there $L . C . M$ is

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$(x + 3) (x - 1) (x^{2} - 4)$

$(x - 3) (x - 1) (x^{2} - 4)$

$(x + 1) (x - 1) (x^{2} - 4)$

${(x + 1)}^{2} (x - 1) (x^{2} - 4)$

answer is C.

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Detailed Solution

$L . C . M = \frac{p r o d u c t o f t h e p o l y n o m i a l s}{H . C . F}$

$L . C . M = \frac{(x - 3) (x^{2} + x - 2) (x^{2} - 5 x + 6)}{(x - 3)}$

$= (x^{2} + x - 2) (x^{2} - 5 x + 6)$

$= (x^{2} + 2 x - x - 2) (x^{2} - 2 x - 3 x + 6)$

$= [x (x + 2) - 1 (x + 2)] [x (x - 2) - 3 (x - 2)]$

$= (x + 2) (x - 1) (x - 2) (x - 3)$

$= (x^{2} - 4) (x - 1) (x - 3)$

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