Problem 104 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the expression \((\omega-1)\left(\omega-\omega^{2}\right)\left(\omega-\omega^{3}\right)\) ... \(\left(\omega-\omega^{n-1}\right)\), where \(w\) is the \(n^{\text {th }}\) root of unity, is (A) \(n \omega^{n-1}\) (B) \(n \omega^{n}\) (C) \((n-1) \omega^{n}\) (D) \((n-1) \omega^{n-1}\)

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Understanding the Problem

We need to evaluate the expression \((\omega-1)(\omega-\omega^2)(\omega-\omega^3) \ldots (\omega-\omega^{n-1})\) where \(\omega\) is the \(n\)-th root of unity. This means \(\omega^n = 1\).

Step 2: Identifying Roots of Unity Properties

Since \(\omega\) is the \(n\)-th root of unity, the solutions to the equation \(x^n = 1\) are \(1, \omega, \omega^2, \ldots, \omega^{n-1}\). This implies that the polynomial \(x^n - 1\) can be factored as \((x-1)(x-\omega)(x-\omega^2)\ldots(x-\omega^{n-1})\).

Step 3: Simplication of the Expression

Notice that the expression \((\omega-1)(\omega-\omega^2)\ldots(\omega-\omega^{n-1})\) can be viewed as the value of the polynomial \((x-1)(x-\omega)(x-\omega^2)\ldots(x-\omega^{n-1})\) evaluated at \(x = \omega\).

Step 4: Evaluating the Polynomial at \(x = \omega\)

Substituting \(x = \omega\) in the polynomial \((x-1)(x-\omega)(x-\omega^2)\ldots(x-\omega^{n-1})\), notice that \((\omega - \omega) = 0\) for the second factor (\(\omega-\omega\)). Hence, the entire product becomes zero.

Step 5: Conclusion

Therefore, the value of the expression is zero for any \(n^{\text{th}}\) root of unity \(\omega\). None of the given options correctly describe this. Therefore, the answer must be 0.

Key Concepts

Polynomial Evaluation

Polynomial evaluation is the process by which the value of a polynomial is determined when a specific value is substituted for the variables in the polynomial. In this exercise, we evaluated a complex expression involving polynomials with roots of unity. Here,

1. The expression \((\omega-1)(\omega-\omega^2)(\omega-\omega^3) \ldots (\omega-\omega^{n-1})\) is essentially the polynomial \((x-1)(x-\omega)(x-\omega^2)\ldots(x-\omega^{n-1})\) evaluated at \(x = \omega\).
2. This substitution means replacing \(x\) with \(\omega\) in all terms.
3. The key factor \( (\omega - \omega) \) equals \(0\), and the product of any expression containing a factor of 0 is necessarily 0.

The systematic evaluation of each polynomial factor clearly shows why the expression as a whole evaluates to 0. Polynomial evaluation is a foundational concept useful for solving complex expressions by substituting values into polynomial equations and simplifying the results.

Product of Roots

The concept of the product of roots is about understanding the relationship between the roots of a polynomial and the polynomial itself. When dealing with the roots of unity, we are often interested in how factors of a polynomial combine and determine its overall value.

• Roots of unity are special because, for the polynomial \(x^n - 1\), they include all complex solutions to the equation \(x^n = 1\).
• This polynomial can be factored as \((x-1)(x-\omega)(x-\omega^2)\ldots(x-\omega^{n-1})\).
• Our exercise involved evaluating the product at the specific root \(x = \omega\), which contained all the terms except for \((x-\omega)\).

In practice, what makes this product significant is that even if it involves different roots like in this expression, it simplifies to showcase the nature of substitution yielding zero when one term equals zero. By understanding the product of roots concept, we can see that such expressions of polynomials can be broken down to demonstrate zero products due to the presence of nullifying factors.

Complex Numbers

Complex numbers are crucial for understanding advanced mathematical concepts including roots of unity and polynomial evaluation. The principle behind complex numbers lies in extending the real number system to include numbers that have a real part and an "imaginary" part (where the imaginary unit is \(i\), satisfying \(i^2 = -1\)). For roots of unity:

• Each \(n\)-th root of unity is a complex number expressed in the form \(\omega = e^{2\pi i k/n}\) for \(k = 0, 1, 2, \ldots, n-1\).
• The real and imaginary parts of these roots can be plotted on the complex plane, forming a regular polygon.
• This representation helps visualize how roots of unity relate to polynomial solutions, especially evaluating and simplifying expressions where real numbers alone are insufficient.

In this exercise, complex numbers allow us to evaluate differences like \((\omega - \omega^k)\) with mathematical precision, and observe how complex number properties lead to simplification processes. This understanding deepens comprehension of algebraic structures and operations involved in complex polynomial equations, especially as applied to evaluating roots of unity.