Geometric interpretation of identity in complex analysis.

Asked Apr 12 '23 at 00:04

Active Apr 12 '23 at 00:33

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Show that if $1, z_1, z_2, \ldots,z_n$ are the $(n + 1)$th roots of $1$, then $$(z-z_1)(z-z_2)\cdots (z-z_n)=1+z+z^2+\ldots +z^n.$$

Make a graphic interpretation of the situation.

I already did the main result, which was to prove that identity (factor theorem, mainly what I used), but I don't understand how to give an interpretation to that, someone could explain that part to me.

edited Apr 12 '23 at 00:33

Gary

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asked Apr 12 '23 at 00:04

James A.

Draw the nth roots of a unit circle. z-z_n is the vector from a point outside to the point on circle – Clemens Bartholdy Apr 12 '23 at 00:35
@Tryst, I don't see how that connects with $1+z+z^2+\cdots+z^n$ (but I also don't see anything better). – Gerry Myerson Apr 12 '23 at 00:49
The equality for $z=1$ is Simple prove that product of the diagonals of a polygon = N. – dxiv Apr 12 '23 at 01:18
What about something like this? – The Art Of Repetition Apr 12 '23 at 02:22

Geometric interpretation of identity in complex analysis.

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