# Sum of roots of a number

• May 27th 2010, 12:18 PM
GeoC
Sum of roots of a number
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 $\lambda$, such that $\lambda _k=\left | \lambda \right |^{1/n}exp^{2\pi ik/n} ...... k=0,1,2....n$ are the nth roots of $\lambda$, what is the sum of the roots,

i.e. $\sum_{k=0}^{n-1}\lambda _k = ?$.

In other words, is there a closed form, and general solution for the sum of roots, i.e. $\left | \lambda \right |^{1/n}\sum_{k=0}^{n-1}exp^{2\pi ik/n}$
• May 27th 2010, 12:35 PM
Bruno J.
Yes, it's $0$, unless $n=1$!

Hint : the sum of the roots of a polynomial $f(z)$ is (up to sign) the coefficient of $z$, and the roots of a complex number $\lambda$ are the roots of $f(z)=z^n-\lambda$.
• May 27th 2010, 03:44 PM
GeoC
So, are you saying the sum of the roots of $\lambda^{1/n}$ is 0 for n not equal 1, and, for n=1, the sum must be simply $\lambda$? Thanks.. looking at the $\sum$, 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
Bruno J.
Quote:

Originally Posted by GeoC
So, are you saying the sum of the roots of $\lambda^{1/n}$ is 0 for n not equal 1, and, for n=1, the sum must be simply $\lambda$?

Yes!

Here's another way to prove it : if $n>1$, let $\omega = e^{2\pi i / n}$ and $S=1+\omega + \dots + \omega^{n-1}$. Then, since $\omega^n=1$, we have $\omega S = \omega + \dots + \omega^n = S$. Since $\omega \neq 1$, we must have $S=0$.
• May 28th 2010, 11:58 AM
wonderboy1953
Comment for Bruno J
• May 28th 2010, 12:53 PM
chisigma
In general if we have a polynomial of degree $n$ in $z$ it can be written as...

$p(z) = z^{n} + a_{n-1} z^{n-1} + \dots + a_{1} z + a_{0} = \prod_{k=0}^{n-1} (z-z_{k})$ (1)

... where the $z_{k}$, $k=0,1,\dots , n-1$ are the roots of the polynomial. If the roots are all distinct then from (1) is easy to derive that is...

$\sum_{k=0}^{n-1} z_{k} = - a_{n-1}$ (2)

If $p(z) = z^{n} - \lambda$ with $\lambda>0$ is...

$\sum_{k=0}^{n-1} z_{k} = \sum_{k=0}^{n-1} \lambda^{\frac{1}{n}} e^{2 \pi i \frac{k}{n}} =0$ (3)

Kind regards

$\chi$ $\sigma$
• May 28th 2010, 02:17 PM
Bruno J.
Quote:

Originally Posted by wonderboy1953