Proof for Newton's Sums

1. Aug 6, 2008

ForMyThunder

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.

2. Aug 6, 2008

HallsofIvy

Staff Emeritus
What are Sn, Sn-1, Supn+ k, etc.?

3. Aug 6, 2008

4. Aug 6, 2008

ForMyThunder

Sm = x1m+x2m+x3m+...xn-1m+xnm

5. Aug 6, 2008

ForMyThunder

Oh, thanks. :)