# Vieta's is the easy part

**Number Theory**Level 5

A **quartic** polynomial has the very interesting property that the coefficient of \(x^{ n }\) for \(n=0,1,2,3,4\) is given by \(\displaystyle \sum _{ j=1 }^{ n+1 }{ \sum _{ k=1 }^{ 5 }{ \frac { \phi (k^{ j }) }{ \phi (k) } } }\).

If the sum of the roots of this polynomial is \(X\), and the product of the roots of this polynomial is \(Y\), calculate \(Y-X\).

If the answer is in the form \(\dfrac { p }{ q }\) where \(p\) and \(q\) are co-prime positive integers, submit your answer as \(p+q+145\).

**Notation**: \(\phi (k)\) denotes the Euler's totient function.

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