Shall we use Newton's Sum?

Algebra Level 5

Consider the cubic equation, \( 2x^3 - 7x^2 + 10x - 6 = 0 \).

The roots of the above cubic equation are \( \alpha , \beta , \gamma \). If,

\[ \displaystyle\sum_{r=1}^{10} [ \alpha^{r} + \beta^{r} + \gamma^{r} ] = \dfrac{m}{n}, \] where \(m\) and \(n\) and coprime positive integers,
then find \( m + n \).

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