The L.C.M of $x^{2} - 4$ and $x^{2} + 5 x + 6$ is $(x - 2)^{2}$ . Find the H.C.F.

\frac{(x + 2) (x + 3)}{(x - 2)^{2}}

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\frac{(x + 2) (x + 3)}{x - 2}

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\frac{(x + 2)^{2} (x + 3)}{x - 2}

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\frac{(x - 2)^{2} (x - 3)}{x - 2}

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Solution

The correct option is C $\frac{(x + 2)^{2} (x + 3)}{x - 2}$
Given:
$f (x) = x^{2} - 4$
$g (x) = x^{2} + 5 x + 6$
L.C.M $=$ $(x - 2)^{2}$
Factor of $x^{2} + 5 x + 6 = (x + 2) (x + 3)$
Factor of $x^{2} - 4 = (x + 2) (x - 2)$
Using the formula,
$f (x) \times g (x) = H . C . F \times L . C . M$
H.C.F $=$ $\frac{(x + 2) (x + 3) (x + 2) (x - 2)}{(x - 2)^{2}}$
H.C.F $=$ $\frac{(x + 2)^{2} (x + 3)}{x - 2}$

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