Francisco's Cubic Roots

Algebra Level 5

Let r1,r2,r3 r_1, r_2, r_3 be the roots of polynomial P(x)=x3+3x+1 P(x) = x^3 + 3x + 1. Evaluate the product k=13(rk2+rk+1). \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,

k=13(rk2+rk+1)=(r12+r1+1)(r22+r2+1)(r32+r3+1). \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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