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y^{2}\left(y-9\right)+y-9
Do the grouping y^{3}-9y^{2}+y-9=\left(y^{3}-9y^{2}\right)+\left(y-9\right), and factor out y^{2} in y^{3}-9y^{2}.
\left(y-9\right)\left(y^{2}+1\right)
Factor out common term y-9 by using distributive property. Polynomial y^{2}+1 is not factored since it does not have any rational roots.