# 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 $$1, -2$$.

×