Using binomial coefficients to find sum of roots

In summary, the conversation discusses finding the sum of the roots, both real and non-real, of the equation x^{2001}+\left(\frac 12-x\right)^{2001}=0, assuming there are no multiple roots. Three solutions are presented, with one suggesting the use of Vieta's formula. However, there is a mistake in the reasoning and the correct answer is 1,000,500. The error is explained and corrected by adjusting the leading coefficient in the factorization.
  • #1
JC2000
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Homework Statement



>Find the sum of the roots, real and non-real, of the equation [tex]x^{2001}+\left(\frac 12-x\right)^{2001}=0[/tex], given that there are no multiple roots.

While trying to solve the above problem (AIME 2001, Problem 3), I came across three solutions on [tex][AoPS](https://artofproblemsolving.com/wiki/index.php?title=2001_AIME_I_Problems/Problem_3#See_also)[/tex]. But, I wonder if it could be solved as follows :

> Let the roots be [tex]P_1,P_2,...P_{2000}[/tex].
> The polynomial can be expressed as a product of factors as follows :
> [tex](1/2)(x-P_1)(x-P_2)...(x-P_{2000}) = 0[/tex].
> The above expression is the same as [tex]x^{2001}+\left(\frac 12-x\right)^{2001}=0[/tex].

>Thus, [tex]x^{2001}+\left(\frac 12-x\right)^{2001} = (1/2)(x-P_1)(x-P_2)...(x-P_{2000})[/tex]

> Here the coefficient of [tex]x^{1999}[/tex] on the RHS should represent ## \sum\limits_{i=1}^{2000}P_i*(-1/2)##.

> On the LHS the corresponding term would be the term with [tex]x^{1999}[/tex] and thus the coefficient of this term on the LHS should also be the required sum.

> On the LHS the coefficient of the [tex]x^{1999}[/tex] term is -##{2001}\choose{2})## into ##(1/2)^2## which represent the sum of the roots.

I have the following questions regarding the above :

1. Are there any inconsistencies in the reasoning?
2. The answers do not match, which seems to suggest so. (The answer through the methods on AoPS, using Vieta's is 500)
3. Is there a way of arriving at the answer without using Vieta's formula and by expressing the polynomial as a product of factors and then using binomial coefficients as attempted above?
 
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  • #2
JC2000 said:
> Here the coefficient of [tex]x^{1999}[/tex] on the RHS should represent [tex]\sum\limits_{i=1}^{2000}P_i[/tex].

That's not correct. It should be ##- \frac{1}{2} \sum\limits_{i=1}^{2000}P_i##, shouldn't it?
 
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  • #3
stevendaryl said:
That's not correct. It should be ##- \frac{1}{2} \sum\limits_{i=1}^{2000}P_i##, shouldn't it?

Oh yes! After considering that, ##\sum\limits_{i=1}^{2000}P_i## should be 1,000,500! Thank you!
 
  • #4
Thanks for your time! The error in this solution was explained to me elsewhere. It seems that the leading coefficient for the factorization on RHS should actually be 2001/2 instead of 1/2. Making that changes gives the correct answer.
 

Related to Using binomial coefficients to find sum of roots

What are binomial coefficients?

Binomial coefficients are the numerical coefficients that appear in the expansion of a binomial expression, such as (a+b)^n. They represent the number of ways to choose a certain number of items from a larger set.

How are binomial coefficients used to find the sum of roots?

Binomial coefficients can be used in conjunction with the Fundamental Theorem of Algebra to find the sum of the roots of a polynomial equation. The sum of the roots can be calculated by taking the negative of the coefficient of the second-to-last term and dividing it by the coefficient of the leading term.

Is it necessary to use binomial coefficients to find the sum of roots?

No, there are other methods for finding the sum of roots, such as using Vieta's formulas or solving the equation directly by factoring. However, using binomial coefficients can be a useful and efficient method for finding the sum of roots in certain cases.

What are some real-life applications of using binomial coefficients to find the sum of roots?

Binomial coefficients and the sum of roots can be applied in various fields such as engineering, statistics, and economics. For example, in engineering, binomial coefficients can be used to determine the maximum load that a structure can withstand by finding the sum of the roots of the equation representing the structure's stability.

What is the relationship between binomial coefficients and Pascal's triangle?

Pascal's triangle is a triangular arrangement of numbers where each number is the sum of the two numbers above it. The numbers in each row of Pascal's triangle represent the binomial coefficients of the corresponding expansion of (a+b)^n. This relationship can be used to quickly determine binomial coefficients and their properties.

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