Complex Polynomials and roots

[BlakeJA@]

I was asked this question and it has stumped me for a week so i figured i'd come ask you all to see if anyone knows how to solve it.

 

let p(z) = z^n + iz^(n-1) - 10

 

if wi are the roots of this polynomial where i= 1,2,3,...,n

 

Compute:

 

Sum(wj) from j=1 to n

Product(wj) from j=1 to n

 

where sum is capital Sigma (summation)

and product is capital pi (product)

and the j is a subscript

and Z is a complex number

[Tartaglia]

Expanding

(z-w0)*(z-w1)*(z-w2)* ...*(z-wn-1) = z^n -(w0+w1+w2+w3 ....+wn-1)z^(n-1) + (-1)^n*(w0*w1*w2*... *wn-1)

 

with all other coefficients of z less than n-1 being zero.

 

equating coefficents

 

i = -(w0+w1+....+wn-1)

-10 = (-1)^n* (w0*w1*w2*...wn-1)

 

therefore sum of roots = -i, product = -10 if n is even or +10 if n is odd

[blakeja]

awesome, thanks so much i knew it was something simple that i was missing