Find the LCM of the following polynomials
$a^{2} - 3 a + 2 and a^{2} + 2 a - 8$ whose GCD is $(a - 2)$

(a + 1) (a + 2) (a - 4)

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(a - 1) (a + 2) (a + 4)

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(a - 1) (a - 2) (a - 4)

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(a - 1) (a - 2) (a + 4)

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Solution

The correct option is D $(a - 1) (a - 2) (a + 4)$
Let
$p (x) = a^{2} - 3 a + 2 = a^{2} - 2 a - a + 2 = (a - 1) (a - 2)$

and $g (x) = a^{2} + 2 a - 8 = a^{2} + 4 a - 2 a - 8 = (a + 4) (a - 2)$

We know that
product of polynomials $= L C M \times G C D$
$\Rightarrow p (x) \times q (x) = L C M \times G C D \Rightarrow L C M = \frac{p (x) \times g (x)}{G C D} \Rightarrow L C M = \frac{(a - 1) (a - 2) (a + 4) (a - 2)}{a - 2} = (a - 1) (a - 2) (a + 4)$

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