Is there a theorem related to the sum of the roots of a positive number?
For example, for any integer n=1,2,3.....N, and positive number , such that are the nth roots of , what is the sum of the roots,
In other words, is there a closed form, and general solution for the sum of roots, i.e.
May 27th 2010, 12:35 PM
Yes, it's , unless !
Hint : the sum of the roots of a polynomial is (up to sign) the coefficient of , and the roots of a complex number are the roots of .
May 27th 2010, 03:44 PM
So, are you saying the sum of the roots of is 0 for n not equal 1, and, for n=1, the sum must be simply ? Thanks.. looking at the , I guess that makes sense.. I had long since forgotten, if I ever even knew it, that the sum of roots equals coefficient of Z in the polynomial.
May 27th 2010, 10:37 PM
Originally Posted by GeoC
So, are you saying the sum of the roots of is 0 for n not equal 1, and, for n=1, the sum must be simply ?
Here's another way to prove it : if , let and . Then, since , we have . Since , we must have .
May 28th 2010, 11:58 AM
Comment for Bruno J
I like your new Avatar.
May 28th 2010, 12:53 PM
In general if we have a polynomial of degree in it can be written as...
... where the , are the roots of the polynomial. If the roots are all distinct then from (1) is easy to derive that is...