How can I prove Newton's Sums?

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Discussion Overview

The discussion revolves around seeking a proof for Newton's Sums, a concept in polynomial theory. Participants explore the relationships between the roots of polynomials and their symmetric sums, with a focus on deriving the sums from polynomial equations.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • One participant presents an initial approach by showing that the polynomial evaluated at its roots sums to zero, leading to a relationship involving symmetric sums.
  • Another participant seeks clarification on the notation used, specifically what Sn, Sn-1, and Supn+k represent.
  • A participant provides links to external resources that may contain relevant information or proofs related to Newton's Sums.
  • Further clarification is given on the definition of Sm as the sum of the m-th powers of the roots of the polynomial.

Areas of Agreement / Disagreement

The discussion does not reach a consensus, as participants are still clarifying terms and exploring different aspects of the proof without a definitive agreement on the approach to take.

Contextual Notes

There are unresolved questions regarding the notation and definitions used in the discussion, which may affect the clarity of the proof being sought.

ForMyThunder
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Could anyone provide me with a proof for Newton's Sums?

So far, I've gotten as far as showing for a polynomial P(x)=anxn+an-1xn-1+...+a1x+a0 with roots x1, x2,..., xn that

P(x1)+P(x2)+P(x3)+...+P(xn)=0

and so,

anSn+an-1Sn-1+...+a1S1+na0=0

and,

anSn+k+an-1Sn+k-1+...+a1Sk+1+a0Sk=0

but I don't know if I'm heading in the right direction and I can't seem to go anywhere that seems to lead towards Newton's sums. Any suggestions will be much appreciated. Thanks.
 
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What are Sn, Sn-1, Supn+ k, etc.?
 
ForMyThunder: http://www.mathlinks.ro/viewtopic.php?t=213867

HoI: http://www.artofproblemsolving.com/Wiki/index.php/Newton_sums
 
Last edited by a moderator:
HallsofIvy said:
What are Sn, Sn-1, Supn+ k, etc.?

Sm = x1m+x2m+x3m+...xn-1m+xnm
 
Oh, thanks. :)
 

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