How Does Cyclic Product Change for Polynomial Roots?

[anemone]

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 \(\displaystyle \prod_{\text{cyclic}}\frac{a-1}{a+1}\).

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[anemone]

Congratulations to the following members for their correct solution!(Cool)

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

Solution from greg1313:

Spoiler

Using Vieta's formulas,

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