If 1, a1, a2, ..., an-1 are the n roots of unity, then show that (1-a1)(1-a2)...(1-an-1) = n.

Q: Wait, why can we just divide by $(z-1)$? Isn't that undefined at $z=1$?

That is a brilliant observation! You are absolutely right that the expression is undefined at $z=1$. However, we are working with an algebraic identity that holds for all $z \neq 1$. Since both sides of the equation are polynomials (which are continuous functions), the limit as $z \to 1$ is perfectly well-defined and gives us the correct value for the product.

Q: How do I know there are exactly $n$ terms in the geometric series $1 + z + z^2 + \dots + z^{n-1}$?

Great question! Just count the powers of $z$. We start at $z^0$ (which is $1$) and go up to $z^{n-1}$. The number of terms is $(n-1) - 0 + 1$, which equals $n$. So, when we substitute $z=1$, we are summing $1$ exactly $n$ times, which gives us $n$.

Q: Is there a geometric way to see this without using the Factor Theorem?

Yes, there is! You can think of the product as the product of the lengths of the chords from the point $1$ to all other roots. While you can derive this using trigonometry and properties of chords, the algebraic approach using the Factor Theorem is the most elegant and standard method for JEE Advanced. It connects the geometry of the complex plane directly to the power of polynomial algebra.

Question

If $1, a_1, a_2, \dots, a_{n-1}$ are the $n$ roots of unity, then show that $(1-a_1)(1-a_2)(1-a_3)\dots(1-a_{n-1}) = n$.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The $n$-th roots of unity are defined by the equation $z^n - 1 = 0$. In the complex plane, these roots lie on the unit circle centered at the origin.

One of these roots is the real number $1$. The remaining $n-1$ roots, denoted as $a_1, a_2, \dots, a_{n-1}$, are spaced with perfect symmetry, dividing the circle into $n$ equal arcs.

Connecting these roots with straight lines forms a perfect regular $n$-gon. This geometric structure provides the foundation for our algebraic exploration.

### The Algebraic Foundation

We utilize the **Factor Theorem** to express the polynomial $z^n - 1$ in terms of its roots. Since $1, a_1, a_2, \dots, a_{n-1}$ are the roots, we can write:

$$z^n - 1 = (z-1)(z-a_1)(z-a_2)\dots(z-a_{n-1})$$

This equation reveals that the polynomial $z^n - 1$ is composed of these specific linear factors. Our objective is to determine the value of the product $(1-a_1)(1-a_2)\dots(1-a_{n-1})$.

### The Bridge: The Geometric Series

To isolate the product, we divide both sides of the factored equation by $(z-1)$:

$$\frac{z^n-1}{z-1} = (z-a_1)(z-a_2)\dots(z-a_{n-1})$$

The left-hand side is the sum of a finite geometric progression. We can rewrite this expression as:

$$\frac{z^n-1}{z-1} = 1 + z + z^2 + \dots + z^{n-1}$$

This transformation allows us to establish the following identity:

$$1 + z + z^2 + \dots + z^{n-1} = (z-a_1)(z-a_2)\dots(z-a_{n-1})$$

### The Final Leap

To find the product $(1-a_1)(1-a_2)\dots(1-a_{n-1})$, we evaluate the identity as $z 	o 1$.

On the left-hand side, as $z$ approaches $1$, each of the $n$ terms (from $z^0$ to $z^{n-1}$) approaches $1$. The sum becomes:

$$1 + 1 + \dots + 1 = n$$

On the right-hand side, substituting $z=1$ yields the product $(1-a_1)(1-a_2)\dots(1-a_{n-1})$. Therefore, we arrive at the elegant result:

$$(1-a_1)(1-a_2)\dots(1-a_{n-1}) = n$$

### A Final Reflection

This result demonstrates the harmony between geometry and algebra. By transitioning from a geometric representation on the complex plane to the algebraic power of the Factor Theorem, we uncover a simple, clean integer.

This derivation captures the essence of advanced mathematics: finding elegant truths hidden beneath complex structures. Keep visualizing these relationships to master the core principles of your studies.

Animated Solution for Mathematics - Complex Numbers: If $1, a_{1}, a_{2}, \dots, a_{n - 1}$ are the $n$ roots of unity, then show that $(1 - a_{1}) (1 - a_{2}) (1 - a_{3}) \dots (1 - a_{n - 1}) = n$ .

Visualized Solution

Roots of $z^{n} - 1 = 0$

Regular $n$ -gon on Unit Circle

Factor Theorem Application

Isolating the Target Expression

Sum of Geometric Progression

Equating the Forms

The Crucial Substitution

Geometric Interpretation: Chords

Evaluating the Left Hand Side

Final Result

The Sigma Insight: Cube Roots and nth Roots of Unity

Analyzing the Setup

The Algebraic Foundation

The Bridge

The Final Leap

A Final Reflection

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