Roots of a Rational Cubic

Algebra Level 5

How many ordered triples of rational numbers (a,b,c) (a, b, c) are there such that the cubic polynomial f(x)=x3+ax2+bx+c f(x) = x^3 + ax^2 + bx + c has roots a,b a, b and cc?

Details and assumptions

The polynomial is allowed to have repeated roots. The polynomial (x1)2(x+2) (x-1)^2 (x+2) has 3 roots which are 1,1,21, 1, -2, while the polynomial (x1)(x+2) (x-1)(x+2) has 2 roots which are 1,21, -2.

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