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I barely remember that I could identify them by attempts. I would get the coefficient, see what numbers can divide it and put them into the Ruffini's rule.

But I don't even know what such method is called.

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    Do you still need the algorithm of how to find these or your question is answered?2011-02-07

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The OP was after the Rational root theorem.

(Added so the question could be officially answered.)