Newton's Formula for Power Sums of Zeros of Polynomial

Newton's Formula for Power Sums of Zeros
2  1  2  -2 
1  2  2  2 
-2  3  -6  4 
3  4  12  -26 
      58 
      -88 
      54 
      174 
      -752 
      1702 
      -2466 
      1204 
      5718 
      -22678 
Let z1, z2, z3, z4 be the zeros of x4 + 2x3 + x2 - 2x + 3. Then
z16 + z26 + z36 + z46 = - 88.

Reference: H. S. Wilf, Mathematics for the Physical Sciences, pp. 100-102.

 

Note on Newton's Relations

 Let tex2html_wrap_inline5335 be the sum of the nth powers of all of the roots of a polynomial equation

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then

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where tex2html_wrap_inline5343 . This can be seen trivially for a 3rd degree polynomial

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so

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