Newtons polynomial

Algebra Level 5

The polynomial \(x^3+Mx-N \) has roots \(a,b\) and \(c\), where \(M \) and \(N\) are constant integers satisfying the inequality \(1<|M| <|N| \).

Given that \(a^5 +b^5+c^5=90\) and \( a^6 + b^6 + c^6 = 227\), find the value of \(M^5 + N^5\).

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