# Sum of roots of any equation

Is the sum of the roots of any equation of one term (e.g., $x^n=c$) always equal to zero? I found this out by having a look at the roots of any equations that have been solved till now. Does it hold true only for $n>1?$

$1.$ How to prove it mathematically

$2.$ What insight or intuition does it provide, if any, in the realm of abstract thinking or mathematics itself.

• What about $x^2-x-2=(x-2)(x+1)$? Commented Apr 8, 2017 at 8:55
• Sounds unbelievable. Can you show us three of your equations that have this property ? (Notice that by Vieta's relations, your conjecture is clearly false.)
– user65203
Commented Apr 8, 2017 at 9:01
• @ResearchEngineer: Leila gave the answer for this case.
– user65203
Commented Apr 8, 2017 at 9:23
• @Leila Too much editing. Every time you edit the question it bumps to the front page. Your last edit is imo pointless and I'm rolling it back. Other than that, two or three times you edited something that had already existed on a previous edit, so you should have edited everything at once instead. Anyway, thank you for editing questions because not nearly enough users here do. Commented Apr 8, 2017 at 21:10
• I hope that I'm not offending you. To clarify, most of the edits you made were improvements, except the last one that baffles me, seems maybe an OCD thing that you wanted the parentheses to be math mode, but they shouldn't be. The only thing I really wanted to tell you was to be careful of edits because they do bump the question. And it may be annoying for both the OP and other users on the page who keep having to refresh. Commented Apr 8, 2017 at 21:12

Consider $x-1=0$.

It has one root $x=1$ and it's not zero.

There is a well-known theorem $($Vieta's theorem$)$ says the sum of root of polynomial

$$a_nx^n+a_{n-1}x^{n-1}+...a_1x+a_0$$

is equals to

$$-\frac{a_{n-1}}{a_n}$$ and not necessary zero. Probably all of your equations had $a_{n-1}=0$

For your modified version, The $($complex$)$ roots of $x^n=c$ has an interesting form. They all are located on a circle and they sum to zero by symmetry for $n>1$. See the wikipedia page for more details.

P.S: I've just found this video about roots of unity. I think you will probably like it:

• No It does not hold. See the edited answer. Commented Apr 8, 2017 at 8:56
• @ResearchEngineer See the linked page for details. Commented Apr 8, 2017 at 9:02
• @ResearchEngineer Your welcome. I recommend reading the "proof" part of wiki page. It has a simple proof.... Commented Apr 8, 2017 at 9:16
• @ResearchEngineer The circle will be scaled to the $r=\sqrt[7]{3}$. The circle itself is somehow an Intuition. Commented Apr 8, 2017 at 9:36
• @ResearchEngineer Your welcome dear researcher. Commented Apr 8, 2017 at 9:38

The sum of the roots of any equation of one term (e.g., $x^n=c$) is always zero?

Yes if $n>1$, no if $n=1$. @Leila already gave a more general answer for polynomials, but for this specific form, it is more readily seen. Define $\omega\equiv e^{2\pi i/n}$. Then the sum of roots of $x^n=c$ is

$$\sqrt[n]{c}\left(1+\omega+\omega^2+\cdots+\omega^{n-1}\right)=\begin{cases}c&\text{if}\; n=1\\\sqrt[n]{c}\frac{\omega^n-1}{\omega-1}=0&\text{if}\;n>1\end{cases}.$$

What insight or intuition does it provide, if any, in the realm of abstract thinking or mathematics itself.

The roots of unity can be thought of as weights placed around the perimeter of a circular disc. When there is only 1 weight ($n=1$) then the weight can never be balanced around the center. When there is more than 1 weight ($n>1$) then by equally spacing the weights, as are the $n^\text{th}$ roots of unity, the center of gravity is always the center of the disc$-$in other words they sum to 0.

• I like how this answer pointing out all the roots of $x^n - c$ are $\sqrt[n]{c}$ $\omega^k$ where $k = 0, 1, \dots n - 1$. Commented Apr 8, 2017 at 13:45

Let the polynomial $P(x)$ with roots $r_k$ such that $$\sum_{k=1}^n r_k=0.$$

Then $P(x-1)$ is also a polynomial, and it is such that the sum of its roots is $n$, which refutes the claim.

• If this downvote is for irrelevance to the question, please undo it and downvote the OP, which was silently modified.
– user65203
Commented Apr 8, 2017 at 10:42
• @YvesDaoust I am not that guy, but in general, I think you can edit the question to make it clear that OP has changed the question. Commented Apr 8, 2017 at 13:39
• @ResearchEngineer "OP" means "original poster", which is you in this case. As a side note, I see people marked the edits they made which changes the question with EDITED in 1st April: blablabla to avoid running into situation like this. Commented Apr 8, 2017 at 14:19
• @YvesDaoust I made a typo, I meant editing the answer not the question. Commented Apr 8, 2017 at 14:29
• @ResearchEngineer Sorry, maybe I have confused you. I mean you can edit the question to include: This question has been modified, with FOO changed into BAR., so that people know you have changed the question. Commented Apr 8, 2017 at 15:38

Since you asked for some insight in the realm of mathematics it worth noting that even though the term "roots of any equation" is much more general than "root of polynomials" (which is still more general than x^n=c) most of simple equations that brings to mind has some symmetric properties. One of the most common of these properties is that the sum of their roots are zero. Indeed, it's possible to be even/odd function which their roots are conjugated. Consider sin(x)=0 or cos(x)=1/2.

So an philosophical question may be arose. Is it something metaphysical in the nature or simply our mind have been attracted to this symmetries. Why you just consider x^n=c instead of x^2+x=0 ?