Find the LCM of polynomials $P (y) = (y^{3} - 27) (y^{2} - 3 y + 2)$ and $Q (y) = (y^{2} + 3 y + 9) (y^{2} - 5 y + 6)$ .

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Solution

We have
$P (y) = (y^{3} - 27) (y^{2} - 3 y + 2) = (y - 3) (y^{2} + 3 y + 9) (y - 2) (y - 1)$ ,
$Q (y) = (y^{2} + 3 y + 9) (y^{2} - 5 y + 6) = (y^{2} + 3 y + 9) (y - 3) (y - 2)$ .
We observe that $y - 3, y - 2, y - 1$ and $y^{2} + 3 y + 9$ are the irreducible factors each having maximum exponent equal to $1$ . Hence their product
$(y - 3) (y - 2) (y^{2} + 3 y + 9)$
is the LCM of given polynomials.

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