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 $m + n$ .