Reciprocated Insane Roots

Algebra Level 5

Let \(a,b,c,d,e\) be the roots to the polynomial \(f(x) = x^5 + 9x^4 + 7x^3 + 16x^2 + 6x + 7 \)

The value of \( \frac {1}{a^2} + \frac {1}{b^2} + \frac {1}{c^2} + \frac {1}{d^2} + \frac {1}{e^2} \) can be written in the form of \( - \frac {p}{q} \) where \(p,q\) are coprime positive integers.

What is the value of \(p-q\)?

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