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x^{2}\left(x-3\right)+x-3
Do the grouping x^{3}-3x^{2}+x-3=\left(x^{3}-3x^{2}\right)+\left(x-3\right), and factor out x^{2} in x^{3}-3x^{2}.
\left(x-3\right)\left(x^{2}+1\right)
Factor out common term x-3 by using distributive property. Polynomial x^{2}+1 is not factored since it does not have any rational roots.