Could anyone provide me with a proof for Newton's Sums?(adsbygoogle = window.adsbygoogle || []).push({});

So far, I've gotten as far as showing for a polynomial P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}with roots x_{1}, x_{2},..., x_{n}that

P(x_{1})+P(x_{2})+P(x_{3})+...+P(x_{n})=0

and so,

a_{n}S_{n}+a_{n-1}S_{n-1}+...+a_{1}S_{1}+na_{0}=0

and,

a_{n}S_{n+k}+a_{n-1}S_{n+k-1}+...+a_{1}S_{k+1}+a_{0}S_{k}=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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# Proof for Newton's Sums

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