Find a L.C.M. [Least Common Multiple] of the following:
$x^{3} + y^{3}, x^{3} - y^{3}, x^{4} + x^{2} y^{2} + y^{4}$

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Solution

Given, $x^{3} + y^{3}, x^{3} - y^{3}, x^{4} + x^{2} y^{2} + y^{4}$
$\Rightarrow x^{3} + y^{3} = (x + y) (x^{2} - x y + x^{2})$
$\Rightarrow x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})$
$\Rightarrow x^{4} + x^{2} y^{2} + y^{4} = (x^{2} + y^{2})^{2} - x^{2} y^{2}$
$= (x^{2} + y^{2})^{2} - (x y)^{2}$
$= (x^{2} + y^{2} + x y) (x^{2} + y^{2} - x y)$
L.C.M. $= (x + y) (x^{2} - x y + y^{2})$
$(x - y) (x^{2} + x y + y^{2})$
$= (x^{3} + y^{3}) (x^{3} - y^{3})$
$= (x^{3})^{2} - y^{3})^{2} - x^{6} - y^{6}$

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