MHB How Does Cyclic Product Change for Polynomial Roots?

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The discussion revolves around evaluating the cyclic product \(\prod_{\text{cyclic}}\frac{a-1}{a+1}\) for the roots \(a, b, c\) of the polynomial \(P(x) = x^3 - 2007x + 2002\). Participants share their approaches and solutions, with several members successfully providing correct answers. The solution by greg1313 is highlighted as a notable contribution. The thread encourages engagement with the Problem of the Week (POTW) format and recognizes members for their participation. Overall, the focus is on the mathematical evaluation of the cyclic product related to polynomial roots.
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Here is this week's POTW:

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If $a,\,b$ and $c$ are roots of the polynomial $P(x) = x^3 - 2007x + 2002$, evaluate $$\prod_{\text{cyclic}}\frac{a-1}{a+1}$$.

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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!
 
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Congratulations to the following members for their correct solution!(Cool)

1. castor28
2. greg1313
3. lfdahl
4. kaliprasad

Solution from greg1313:
Using Vieta's formulas,

$$\prod_{\text{cyclic}}\frac{a-1}{a+1}=\frac{(a-1)(b-1)(c-1)}{(a+1)(b+1)(c+1)}=\frac{abc-ab-ac-bc+a+b+c-1}{abc+ab+ac+bc+a+b+c+1}=\frac{-2002+2007+0-1}{-2002-2007+0+1}=\boxed{-\frac{1}{1002}}$$
 
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