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Use the rational root theorem
Use Descartes rules of signs to find how many roots you need, in order to know when to stop.
Then use synthetic division to check your answer (If the remainder is 0, then it is a root.)
If the number of roots found by descartes rule does not match the degree you might have to factor. OR! You can use synthetic division on the orginal polynomial by the roots, and the resulting equation will possibly give you a solvable equation (such as x^2+5, then \pm 5i)