Answer:
Use the Rational Root Theorem.
Explanation:
Any rational roots will be factors of the ratio of the constant (=p(0)) to the leading coefficient of the polynomial p(x). In the general case, that ratio is a rational number and the roots have numerator that is a factor of its numerator, and a denominator that is a factor of its denominator.
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To see how this works, consider the polynomial with rational roots b/a and d/c. Factors of it will be ...
p(x) = (ax -b)(cx -d)( other factors if p(x) is of higher degree )
The leading coefficient here is ac; the constant term is bd. The rational root theorem says any rational roots are factors of (bd)/(ac), which b/a and d/c are.