How Does Cyclic Product Change for Polynomial Roots?

  • MHB
  • Thread starter anemone
  • Start date
In summary, "POTW #334" refers to a specific math problem involving evaluating a product of fractions with polynomial expressions as the numerator and denominator. The notation \prod_{\text{cyclic}} means to take the product in a cyclic order over all the roots of the polynomial. There are various approaches to solving this problem, including using Vieta's formulas or the fundamental theorem of algebra. The significance of finding the value of this product for polynomial roots is that it can have applications in understanding polynomial functions and certain equations. There are resources and tools, such as online resources and math software programs, that can assist in solving this problem, as well as seeking help from a math tutor or joining a math forum or study group.
  • #1
anemone
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MHB
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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 \(\displaystyle \prod_{\text{cyclic}}\frac{a-1}{a+1}\).

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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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  • #2
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,

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

FAQ: How Does Cyclic Product Change for Polynomial Roots?

1. What is "POTW #334: Evaluating \prod_{\text{cyclic}}\frac{a-1}{a+1} for Polynomial Roots"?

"POTW #334" stands for "Problem of the Week #334" and refers to a specific math problem posted on a website or forum. The problem itself involves evaluating a product of fractions, where the numerator and denominator are both polynomial expressions, for certain values of the variable "a".

2. What does the notation \prod_{\text{cyclic}} mean in this problem?

The symbol \prod is a mathematical notation for a product, similar to how \sum represents a sum. The subscript "cyclic" means that the product should be taken in a cyclic order, meaning the terms should be multiplied in a specific repeating pattern. In this case, it means that the product should be taken over all the roots of the polynomial in a specific order.

3. How do I approach solving this problem?

There are several approaches you can take to solving this problem, but one possible way is to use Vieta's formulas, which relate the coefficients and roots of a polynomial. You can also use the fundamental theorem of algebra to find the roots of the polynomial and then plug them into the given expression.

4. What is the significance of evaluating this product for polynomial roots?

Finding the value of this product can be useful in various mathematical applications, such as determining the behavior of a polynomial function or solving certain types of equations. It can also help in understanding the relationship between the roots and coefficients of a polynomial.

5. Are there any resources or tools that can help with solving this problem?

Yes, there are many online resources and math software programs that can assist with solving this problem. Some examples include WolframAlpha, Symbolab, and Desmos. Additionally, seeking help from a math tutor or joining a math forum or study group can also be beneficial in solving this problem.

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