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I know that this is not the right place for questions like that, but I need someone that explain me step-by-step how can I resolve this exercise (I've exam in the next days):

Write as products of irreducible factors the polynomial $f=x^3-3x^2+x-3 \in\mathbb{R}[x]$ and $f=x^3-\overline{3}x^2+x-\overline{3} \in\mathbb{Z}_5[x]$.

Thank you for helping.

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    Use factor by grouping on each one, and notice that while $x^2+1$ doesn't factor in $\mathbb{R}$, it does factor in $\mathbb{Z}_5$.2012-03-19
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    Some answers are mentioning the rational root test, but I would advise you generally use that after you have tried grouping. If a polynomial has $k$ terms, and $d$ is a divisor of $k$, then I would try grouping $d$ terms at a time and see if a common polynomial falls out. In this case, $k=4$, and there was only one choice for $d$: $d=2$. For one thing, this is easier to do - there are fewer cases to check. For another, sometimes this yields a factorization when there is no root.2012-03-19

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