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The concept of "roots of unity" is a fascinating area in complex analysis. These are complex numbers that equate to 1 when raised to a specific power. Let's take the "nth" roots of unity as an example. These numbers are solutions to the equation \(z^n = 1\). They are evenly distributed on the unit circle in the complex plane. Each root can be represented in exponential form as \( z_k = e^{2\pi i k / n} \), where \( k \) is an integer ranging from 0 to \( n-1 \).

Visualize the unit circle as a clock. If \( n \) is 12, each root corresponds to a number on the clock face, starting at \( z_0 = 1 \) (akin to 12 on the clock) and proceeding round clockwise. These roots have a neat property: if you sum them all, the result is zero (except for the root at \( z_0 = 1 \)). This symmetry is vital in simplifying many mathematical equations related to polynomials and series.

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ratio. Recognizing when you have a geometric series can greatly simplify the sum of the series. For instance, \(1 + z + z^2 + \cdots + z^{n-1}\) is an example of such a series with the first term \(a = 1\) and the common ratio \(r = z\).

The formula for summing the first \(n\) terms of a geometric series is \(S = \frac{a(r^n - 1)}{r - 1}\). When \(z^n = 1\) (since these terms are the roots of unity), the sum simplifies to \(\frac{1 - 1}{z - 1} = 0\), unless \(z = 1\). However, when addressing the polynomial roots excluding 1, this sum becomes relevant as a way of tying back to the product of the polynomial roots.

Understanding the roots of a polynomial is crucial when solving equations. For a polynomial such as \((z - z_1)(z - z_2)\cdots(z - z_{n-1})\), the roots are \(z_1, z_2, \ldots , z_{n-1}\). These roots are strategically selected to exclude \(z_0 = 1\), focusing on the non-trivial solutions. The polynomial equation \(z^n - 1 = (z-1)(z - z_1)(z - z_2)\cdots(z - z_{n-1})\) brings in every nth root of unity, incorporating symmetry naturally.

Dividing \(z^n - 1\) by \(z - 1\) (since \(z eq 1\)) derives \((z - z_1)(z - z_2)\cdots(z - z_{n-1})\). What is intriguing here is how such roots relate to the geometric series sum \(1 + z + z^2 + \cdots + z^{n-1}\), showcasing the intricate symmetry and interconnection between polynomial roots and these series.