Easy to think, Harder to solve

Algebra Level 5

\(f(x)\) is a monic quartic polynomial.

Such that

\(f(1)=-48\)

\(f(2)=0\)

\(f(3)=192\)

\(f(x)\) has real integer roots \(\alpha\) , \(\beta\) , \(\gamma\) and \(\delta\) and also integer coefficients .

Then find ,

[\(\left ( \alpha^{4} + \beta^{4} + \gamma^{4}+ \delta^{4}\right )\) + \(\left ( \alpha^{3} + \beta^{3} + \gamma^{3}+ \delta^{3}\right )\)] mod 11

This problem is original

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