Answer
LCM =$x(x-1)^{2}(x+1)(x^{2}+x+1)$
Work Step by Step
Step 1:
Factor each polynomial completely.
$x^{3}-x=x(x^{2}-1)=$
... recognize a difference of squares
$=x(x-1)(x+1)$
$x^{3}-2x^{2}+x=x(x^{2}-2x+1)$
... recognize perfect square
$=x(x-1)^{2}$
$ x^{3}-1=\qquad$... a difference of cubes,
$=(x-1)(x^{2}+x+1)$
Step 2:
The LCM is the product of each of these factors raised to a power equal to the greatest number of times that the factor occurs in the polynomials.
LCM =$x(x-1)^{2}(x+1)(x^{2}+x+1)$
