Tanishq's roots

Algebra Level 4

Real numbers \(w\leq x\leq y\leq z\) satisfy

\[\begin{align} w+x+y+z&=12\\ wx+wy+wz+xy+xz+yz&=17\\ wxy+wxz+wyz+xyz&=-114\\ wxyz&=-216. \end{align}\]

Consider the polynomial \[g(u)= wu^3+xu^2+yu+z.\]

The equation \( g(u) = 0 \) has roots \(p, q, r \). The value of \( p^2 + q^2 + r^2 \) can be written as \( \frac{a}{b} \), where \(a\) and \(b\) are positive coprime integers. What is the value of \(a+b\)?

This problem is posed by Tanishq A.


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