Problem 181 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The roots of the equation \(z^{4}-z^{3}+z^{2}-z+1=0\) are \(\cos \left(\frac{p \pi}{5}\right)+i \sin \left(\frac{p \pi}{5}\right)\) where \(p=\) (A) \(1,3,5,7,9\) (B) \(1,3,7,9\) (C) \(3,5,7,9\) (D) None of these

Step-by-Step Solution

Answer

(B) 1, 3, 7, 9

Step 1: Identify the Polynomial

The given equation is a polynomial of degree 4: \( z^4 - z^3 + z^2 - z + 1 = 0 \). It is expressed as a real polynomial with complex roots.

Step 2: Roots as Complex Numbers

According to the problem, the roots are of the form \( \cos \left( \frac{p \pi}{5} \right) + i \sin \left( \frac{p \pi}{5} \right) \), which corresponds to points on the unit circle in the complex plane. This form corresponds to Euler's formula for the complex exponential: \( e^{i \theta} = \cos(\theta) + i \sin(\theta) \).

Step 3: Use the Roots of Unity

The expression \( z^4 - z^3 + z^2 - z + 1 = 0 \) can be rewritten as \( \frac{z^5 - 1}{z - 1} = 0 \), which implies roots at the fifth roots of unity, excluding \( z = 1 \). These roots have the form \( e^{i \frac{2k\pi}{5}} \) for \( k = 1, 2, 3, 4 \).

Step 4: Determine the Angles Corresponding to Roots

The fifth roots of unity are given by \( \cos \left( \frac{2k\pi}{5} \right) + i \sin \left( \frac{2k\pi}{5} \right) \). For the valid roots \( k = 1, 2, 3, 4 \), this translates to angles \( \frac{2 \cdot 1\pi}{5}, \frac{2 \cdot 2\pi}{5}, \frac{2 \cdot 3\pi}{5}, \frac{2 \cdot 4\pi}{5} \), which simplify to \( \frac{2\pi}{5}, \frac{4\pi}{5}, \frac{6\pi}{5}, \frac{8\pi}{5} \).

Step 5: Identify Corresponding Values of p

For each angle \( \frac{2\pi}{5}, \frac{4\pi}{5}, \frac{6\pi}{5}, \frac{8\pi}{5} \), they can be interpreted as \( \frac{p\pi}{5} \) where \( p = 2, 4, 6, 8 \). Since \( p \) corresponds to positions in the unit circle's rotation, using odd symmetry, these translate to the odd integer sequence with a wrap-around (modulo 10): \( p = 1, 3, 7, 9 \).

Step 6: Choose the Correct Option

Given the values of \( p \) calculated: \( 1, 3, 7, 9 \), the correct answer matches with option (B) \( 1, 3, 7, 9 \).

Key Concepts

Roots of Unity

In the realm of complex numbers, the concept of roots of unity refers to the solutions of the equation \(z^n = 1\). These solutions are especially fascinating because they lie evenly on the unit circle in the complex plane, forming a regular \(n\)-gon. In our exercise, the polynomial equation \(z^4 - z^3 + z^2 - z + 1 = 0\) can be transformed using roots of unity. By rewriting the equation as \(\frac{z^5 - 1}{z - 1} = 0\), it reveals that the roots, apart from \(z=1\), are actually the fifth roots of unity.

• The roots of unity are expressed as \(e^{i\frac{2k\pi}{5}}\), where \(k = 0, 1, 2, \, \ldots \, , n-1\).
• In this equation, we exclude \(k=0\) since \(z-1\) is in the denominator.

For \(k = 1, 2, 3, 4\), these roots correspond to angles on the complex plane that lead to symmetric, equidistant points. They have a direct relationship with the periodic nature and symmetry of complex numbers, forming a pivotal part of algebraic operations in complex analysis.

Polynomial Equations

Polynomial equations are expressions consisting of variables and coefficients, which involve operations of addition, subtraction, multiplication, and non-negative integer powers. The polynomial \(z^4 - z^3 + z^2 - z + 1 = 0\) in our exercise is of degree 4, which means it can have up to four roots. However, the choice of transformation using roots of unity reduces the complexity of finding these roots by relating them to those of a simpler equation, \(z^5 - 1 = 0\).

• Such polynomials generate complex conjugate pairs when roots are not real.
• By manipulating input equations, we can simplify the process of finding solutions.

This use of factorization and symmetries is a powerful method in algebra, allowing us to solve complex polynomial equations effectively by making use of properties of complex numbers. The entire process reflects how algebra and geometry combine beautifully to expose solutions that might initially seem intimidating.

Euler's Formula

Euler's formula is a profound relation in mathematics expressed as \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\). This elegant equation links trigonometry with complex exponentials, allowing us to describe complex numbers using angles and magnitudes. In our exercise, the roots presented as \(\cos\left(\frac{p\pi}{5}\right) + i \sin\left(\frac{p\pi}{5}\right)\) are derived from Euler's formula, highlighting the roots' position on the unit circle.

• This representation helps visualize complex numbers as rotations and scalings in the plane.
• Through Euler's formula, multiplication of complex numbers translates into addition of angles.

By understanding that any complex number can be expressed in polar form \(re^{i\theta}\), where \(r\) is the modulus and \(\theta\) is the argument, we gain insights into transformations and symmetry in complex number systems. Euler's formula not only simplifies calculations involving complex numbers but also unveils the geometrical nature underlying these fascinating entities.