Roots of a Rational Cubic

Algebra Level 5

How many ordered triples of rational numbers \( (a, b, c)\) are there such that the cubic polynomial \( f(x) = x^3 + ax^2 + bx + c\) has roots \( a, b\) and \(c\)?

Details and assumptions

The polynomial is allowed to have repeated roots. The polynomial \( (x-1)^2 (x+2)\) has 3 roots which are \(1, 1, -2\), while the polynomial \( (x-1)(x+2)\) has 2 roots which are \(1, -2\).

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