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Relationship between LCM and GCD

If fx=x2+3x...

Question

If $f (x) = x^{2} + 3 x + 2$ , $q (x) = x^{2} - 4 x + 4$ whose L.C.M is $x^{2} - 4$ . Find their H.C.F.

(x + 1) (x - 2)

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(x - 1) (x - 2)

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(x + 1) (x + 2)

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

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Solution

The correct option is A $(x + 1) (x - 2)$
Given:
H.C.F $= ?$
L.C.M $= x^{2} - 4$
$f (x) = x^{2} + 3 x + 2$
$q (x) = x^{2} - 4 x + 4$
Factor of $x^{2} + 3 x + 2 = (x + 1) (x + 2)$
Factor of $x^{2} - 4 x + 4 = (x - 2)^{2}$
Using the formula,
$f (x) \times q (x) = H . C . F \times L . C . M$
$H . C . F = \frac{(x + 1) (x + 2) (x - 2)^{2}}{(x + 2) (x - 2)}$
$= (x + 1) (x - 2)$
Therefore, H.C.F is $(x + 1) (x - 2)$

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