Inspired by Issac Newton

Algebra Level 4

k=120(α3k+β3k+ξ3k)\large \sum _{ k=1 }^{ 20 }{ \left( { \alpha }^{ 3k }+{ \beta }^{ 3k }+{ \xi }^{ 3k } \right) }

{f(1)=9f(1)=7f(3)=19 \begin{cases} {f\left( -1 \right) =-9} \\ {f\left( 1 \right) =-7 } \\ {f\left( 3 \right) =19} \end{cases}

If f(x)f\left( x \right) is monic cubic polynomial having roots α,β,ξ\alpha ,\beta ,\xi . Then evaluate topmost expression modulo 17.

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