Note that I am talking about rational roots not rational coefficients. I know that Galois theory can tell you but I want to know if knowing whether all the roots of a polynomial are rational can also tell you. Note also that the roots and their properties are usually unknown, but I'm talking about when you only know that they are rational but don't know their values.
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By definition, a polynomial is solvable if and only if its roots can be expressed using rational numbers, addition/subtraction, multiplication/division, and radicals (squareroot symbols, cuberoot symbols, etc.) If the roots are rational, they certainly can be expressed in the above form.
You can always find all the rational roots of a polynomial using the rational root theorem. See http://en.wikipedia.org/wiki/Rational_root_theorem