Cubics

Algebra Level 4

Let a,b,ca, b, c be the roots of the cubic x3+3x2+5x+7x^3 + 3x^2 + 5x + 7. Given that PP is a cubic polynomial such that P(a)=b+c,P(b)=c+a,P(c)=a+b,P(a) = b + c, P(b) = c + a, P(c) = a + b, and P(a+b+c)=16P(a + b + c) = -16, find P(0)P(0).


This problem is from the OMO.
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