Cubics

Algebra Level 4

Let \(a, b, c\) be the roots of the cubic \(x^3 + 3x^2 + 5x + 7\). Given that \(P\) is a cubic polynomial such that \(P(a) = b + c, P(b) = c + a, P(c) = a + b, \) and \(P(a + b + c) = −16\), find \(P(0)\).


This problem is from the OMO.
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