Francisco's Cubic Roots

Algebra Level 5

Let $r_1, r_2, r_3$ be the roots of polynomial $P(x) = x^3 + 3x + 1$ . Evaluate the product $\displaystyle \prod_{k=1}^3 (r_k^2 + r_k + 1).$

This problem is posed by Francisco R.

Details and assumptions

For those unfamiliar with the product notation,

$\displaystyle \prod_{k=1}^3 (r_k^2 + r_k + 1) = (r_1 ^2 + r_1 + 1) ( r_2 ^ 2 + r_2 + 1 ) ( r_ 3 ^2 + r_3 + 1 ) .$