A Return to Your Roots

Algebra Level 5

\[ \large P(x)=4x^{4}+\sum_{j=1}^4 (5-j)(x^{4+j}+x^{4-j}) \]

Let \(z_{1},z_{2},z_{3}, \ldots,z_{k}\) be the distinct roots of \(P(x)\), and let \(z_{n}=a_{n}+b_{n}i\) for \(n=1,2,3,\ldots,k\), where \(i=\sqrt{-1}\) and \(a_{n}\) and \(b_{n}\) are real numbers. Let

\[ \sum_{n=1}^k \left|b_{n}\right|=m+p\sqrt{q}\ \]

where \(m\), \(p\) and \(q\) are positive integers and \(q\) is not divisible by the square of any prime. Find \(m+p+q\).

Inspired by the 2003 AIME II Problem 15

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