A problem by Brian Yao

Level pending

Let \(N\) be the sum of the real roots of the polynomial function

\(P(x)=x^{6}+2x^{5}+x^{4}-7x^{3}-18x^{2}-9x-18\)

\(\frac{1}{N}\) can be expressed in the form \(a^{\frac{4}{3}}+b^{\frac{1}{3}}+c\), where \(a\), \(b\), and \(c\) are positive integers. Find \(a+b+c\). Do not make any unwarranted assumptions about the values of \(a\), \(b\), and \(c\).

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