Calculate (1/x1)+(1/x2)+(1/x3), where x1,x2,x3 are the solutions of the eq. x^3 -3x +1=0

You need to
remember that Vieta's formulas provide the relations between coefficients of equation and its
roots.

The problem provides an polynomial equation of third order,
+1=0

You should use Vieta's formulas
such that:

  (coefficient of term that contains  
is 0, since this term is missing) =>

-1

You need to evaluate the sum , hence you should
bring these terms to a common denominator   such that:

1/x_2 + 1/x_3 = (x_2*x_3 + x_1*x_3 + x_1*x_2)/x_1*x_2*x_3

You need to
substitute -1 for   and -3 for   such that:

3

Hence, evaluating the sum    yields
.