Maybe the roots are rational? I'll try that

Algebra Level 4

Let \(p,q\) and \(r\) be the roots of \(f(x)=x^3+3x+8\). Suppose a monic cubic \(g(x)\) has the roots

\[\frac{p^2+q^2}{r^2}, \frac{p^2+r^2}{q^2}, \frac{q^2+r^2}{p^2}\]

The value of \(g(-1)\) is \(\dfrac{a}{b}\) for relatively prime \(a\) and \(b\). Find \(a+b\).

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