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This problem trains for: AMC-10, SAT-II, AMC-12.
Which of the following cannot be the number of positive real roots of a degree 5 polynomial with real coefficients?
Since a polynomial with real coefficients has an even number of complex roots, it follows that the total number of real roots of a polynomial of degree 5 is odd.
If the number of real roots is odd, then the number of positive real roots cannot equal the number of non-positive real roots (since in this case the total number of real roots would be even.)