Find the LCM of $x^{4} + x$ and $x^{3} - x$ .

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Solution

The factors of these expressions are:
$x^{4} + x = x (x^{3} + 1) = x (x^{3} + 1^{3})$ [Use $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ ]
$= x (x + 1) (x^{2} - x + 1)$ ;
$x^{3} - x = x (x^{2} - 1) = x (x^{2} - 1^{2})$ [Use $a^{2} - b^{2} = (a + b) (a - b)$ ]
$= x (x + 1) (x - 1)$
the LCM of the above expression is $x (x - 1) (x^{2} - x + 1) (x + 1)$

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