What is the L.C.M of the polynomial, if $f (x) = x^{2} - 3 x + 1$ and $g (x) = x^{2} - 1$ whose H.C.F is $(x - 1)^{2}$ ?

\frac{1}{(x + 2) (x + 1)}

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\frac{1}{(x - 2) (x - 1)}

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\frac{(x - 2) 1}{(x - 2) (x + 1)}

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\frac{1}{(x - 2) (x + 1)}

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Solution

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

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