How can I prove Newton's Sums?

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In summary, the conversation discusses finding a proof for Newton's Sums, specifically in relation to showing for a polynomial with roots, and the use of variables such as Sn and Sm. The conversation also provides links for further information.
  • #1
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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  • #2
What are Sn, Sn-1, Supn+ k, etc.?
 
  • #3
ForMyThunder: http://www.mathlinks.ro/viewtopic.php?t=213867

HoI: http://www.artofproblemsolving.com/Wiki/index.php/Newton_sums
 
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  • #4
HallsofIvy said:
What are Sn, Sn-1, Supn+ k, etc.?

Sm = x1m+x2m+x3m+...xn-1m+xnm
 
  • #5
Oh, thanks. :)
 

1. What are Newton's Sums?

Newton's Sums, also known as Newton's Identities, are a set of equations used to express the elementary symmetric polynomials in terms of the coefficients of a polynomial equation.

2. Why are Newton's Sums important?

Newton's Sums provide a way to relate the coefficients of a polynomial equation to the roots of the equation, making it easier to solve for the roots or to find other important properties of the equation.

3. How do you use Newton's Sums?

To use Newton's Sums, you first need to know the coefficients of a polynomial equation and its roots. Then, you can plug these values into the equations to find the values of the elementary symmetric polynomials.

4. Are Newton's Sums only applicable to polynomials?

No, Newton's Sums can also be applied to power series, infinite series, and other mathematical functions that involve coefficients and roots.

5. Can Newton's Sums be proven?

Yes, there are several ways to prove Newton's Sums, including using mathematical induction, Vieta's formulas, and linear algebra techniques. These proofs rely on basic algebraic principles and properties of symmetric polynomials.

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