The H.C.F of $a^{4} + 3 a^{3} + 5 a^{2} + 26 a + 56$ and $a^{4} + 2 a^{3} - 4 a^{2} - a + 28$ is $a^{2} + 5 a + 7$ . Find their L.C.M.

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Solution

$L C M (x, y) = \frac{x \times y}{H C F (x, y)}$
$∴ L C M = \frac{(a^{4} + 3 a^{3} + 5 a^{2} + 26 a + 56) \times (a^{4} + 2 a^{3} - 4 a^{2} - a + 28)}{(a^{2} + 5 a + 7)}$
$L C M = \frac{(a^{2} + 5 a + 7) (a^{2} - 2 a + 8) (a^{4} + 2 a^{3} + 4 a^{2} - a + 28)}{(a^{2} + 5 a + 7)} = (a^{2} - 2 a + 8) (a^{4} + 2 a^{3} - 4 a^{2} - a + 28)$

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