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The following rules can sometimes be helpful. Say o is an odd number and e is an even number. We can sometimes eliminate some possibilities given by the Rational Root Theorem, and save on unnecessary applications of Horner's Scheme.
e+e=e (That is, adding two even numbers gives an even number)e+o=o (Adding an even and an odd number gives an odd number)o+o=eo*e=e;o*o=oon=o (n is a natural number)en=eFor a polynomial, if it has rational roots, p/q, (p and q are integers) then there are three possibilities in terms of odd and even:
p is even, and q is oddp is odd and q is evenp is odd and q is oddThe option p is even and q is even is not present when the equation is normalised, because cases where 2p/2q might exist become p/q, because the common factor cancels. That is, p and q are relatively prime, they share no common factors.
If testing an equation with these rules produces a contracictory result, then that combination of odd and even is impossible. We are sure such roots do not exist. If the results are logical, then it means that combination of odd and even is not illogical, but it does not prove that such roots exist. One cannot use logic to prove existence. 
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