MHB How Do You Evaluate the Product of g(x) at the Roots of a Quintic Polynomial?

[anemone]

Here is this week's POTW:

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Let $r_1,\,r_2,\,\cdots,r_5$ be the roots of $f(x)=x^5+x^2+1$ and $g(x)=x^2-2$. Evaluate $g(r_1)g(r_2)g(r_3)g(r_4)g(r_5)$.

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[anemone]

Congratulations to lfdahl for his correct solution (Cool) !

Spoiler

\[\coprod_{i=1}^{5}g(r_i) = \coprod_{i=1}^{5}(r_i^2-2) = \coprod_{i=1}^{5}(r_i-\sqrt{2})(r_i+\sqrt{2})=\coprod_{i=1}^{5}(r_i-\sqrt{2})\coprod_{i=1}^{5}(r_i+\sqrt{2})\\\\ = \left ( (-1)^5\coprod_{i=1}^{5}(\sqrt{2}-r_i) \right )\left ( (-1)^5\coprod_{i=1}^{5}(-\sqrt{2}-r_i) \right ) = f(\sqrt{2})f(-\sqrt{2}) =\left ( \left ( \sqrt{2} \right )^5+3 \right )\left ( -\left ( \sqrt{2} \right )^5+3 \right ) \\\\= -2^5+9 = -32 + 9 = -23.\]