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,

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

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