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x^{2}\left(x-9\right)+9\left(x-9\right)
Do the grouping x^{3}-9x^{2}+9x-81=\left(x^{3}-9x^{2}\right)+\left(9x-81\right), and factor out x^{2} in the first and 9 in the second group.
\left(x-9\right)\left(x^{2}+9\right)
Factor out common term x-9 by using distributive property. Polynomial x^{2}+9 is not factored since it does not have any rational roots.