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Is the rational zero test (rational root theorem) always true?

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Well it is the RATIONAL root theorem. If a polynomial fails the test, then it is certain that there aren't rational roots. If it passes the test, it doesn't guarantee the presence of rational roots - it merely gives you the possibilities.
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It works for rational zeros. If your equation has one rational zero, then synthetic division would give you the depressed quadratic eqn which can be solved using the quadratic formula.
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Thus the name...
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