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- Feb 14, 2012

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It's me again...I find this problem to be very interesting yet very difficult to me. I tried to approach it using the Vieta's formula, knowing the given function $p(x)$ has only one real root and 4 complex roots, where I let the 4 complex roots be $a\pm bi$ and $c\pm di$, but it failed me. I start to think this must be a problem that is not compatible to my level and thus I post it here hoping to find someone who will be interested with it and solve it for me...

Thanks in advance!

Problem:

Let \(\displaystyle p(x)=x^5+x^2+1\) have roots $r_1$, $r_2$, $r_3$, $r_4$, and $r_5$. Let \(\displaystyle q(x)=x^2-2\). Determine the product of \(\displaystyle q(r_1)\cdot q(r_2)\cdot q(r_3)\cdot q(r_4)\cdot q(r_5)\).