Problem 36 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If all the roots of \(z^{3}+a z^{2}+b z+c=0\) are of unit modulus, then (A) \(|a| \leq 3\) (B) \(|b|>3\) (C) \(|c| \leq 3\) (D) None of these

Step-by-Step Solution

Answer

Option (C): \(|c| \leq 3\) is correct.

Step 1: Understanding Roots of Unit Modulus

If the roots of the polynomial \(z^3 + az^2 + bz + c = 0\) are of unit modulus, it means each root \(z_i\) satisfies \(|z_i| = 1\). Therefore, the roots can typically be represented as \(e^{i\theta_1}, e^{i\theta_2}, e^{i\theta_3}\), where \(\theta_1, \theta_2, \theta_3\) are real numbers.

Step 2: Using Vieta's Formulas

According to Vieta's formulas for a cubic polynomial \(z^3 + az^2 + bz + c = 0\), the sum of the roots \(\alpha + \beta + \gamma = -a\), the sum of the product of roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = b\), and the product of the roots \( \alpha\beta\gamma = -c\).

Step 3: Analyzing Product of Roots

Since \(|\alpha| = |\beta| = |\gamma| = 1\), it follows that \(|\alpha \beta \gamma| = 1\). Therefore, \(|-c| = 1\) which implies \(|c| = 1\). Thus, \(|c| \leq 3\) is satisfied.

Step 4: Conclusion

Only one of the options clearly satisfies the given condition when the roots of the polynomial are of unit modulus. Since the modulus of \(c\) is equal to 1, option \((C) |c| \leq 3\) is correct.

Key Concepts

Unit Modulus

In complex numbers, the unit modulus refers to a situation where the magnitude or absolute value of a complex number is equal to 1. This is an important concept in many areas of mathematics, particularly when dealing with roots of polynomials. For a complex number written in the form \( z = a + bi \), the modulus is represented as \( |z| = \sqrt{a^2 + b^2} \). When we say a complex number has unit modulus, it means \(|z| = 1\).

Typically, when a polynomial's roots have unit modulus, each root can be expressed in the form \( e^{i\theta} \). This shows that these complex numbers lie on the unit circle in the complex plane. This property is exploited when solving polynomial problems, as it simplifies equations and facilitates the use of other mathematical tools like Vieta's Formulas.

• Roots \( z_i \) of unit modulus correspond to \(|z_i| = 1\).
• These roots are expressed in the form \( e^{i\theta} \), representing their position on the unit circle.
• This concept is crucial for analyzing and simplifying polynomial equations.

Vieta's Formulas

Vieta's formulas provide insightful relationships between the coefficients of a polynomial and its roots. They are especially useful in identifying certain properties of the polynomial without solving it entirely. For a cubic polynomial like \( z^3 + az^2 + bz + c = 0 \), Vieta's formulas give us three key equations:

• The sum of the roots: \( \alpha + \beta + \gamma = -a \).
• The sum of the products of the roots taken two at a time: \( \alpha\beta + \beta\gamma + \gamma\alpha = b \).
• The product of the roots: \( \alpha\beta\gamma = -c \).

In the context of unit modulus roots, Vieta’s formulas are a powerful tool. For example, if each root has unit modulus, we know \( |\alpha| = |\beta| = |\gamma| = 1 \). Therefore, the product of the roots would also have a modulus of 1, i.e., \( |alpha\beta\gamma| = 1 \). This directly leads to insights about the coefficient \(c\) because we then have \( |-c| = 1 \), suggesting \( |c| = 1 \), which is crucial for determining if \( |c| \leq 3 \). This illustrates the way Vieta’s formulas can aid in understanding the relationships in polynomial equations.

Cubic Polynomial Roots

Understanding the roots of cubic polynomials is fundamental in algebra. A cubic polynomial is expressed in the form \( z^3 + az^2 + bz + c = 0 \), where its solutions are termed the roots. Each root can be a real number or a complex number. In our case with roots of unit modulus, they are complex and lie on the unit circle of the complex plane.

When dealing with cubic polynomials, it is important to remember that the nature of the roots strongly influences the properties of the coefficients of the polynomial. If all the roots of a cubic polynomial have unit modulus, they can be expressed as \( e^{i\theta_1}, e^{i\theta_2}, e^{i\theta_3} \).

• This expression links directly to the modulus and the circle.
• Roots on the unit circle imply certain symmetrical properties and help derive relations among coefficients.
• Such roots also tell us about the polynomial's behavior and its graphical representation.

In the scenario where the roots have unit modulus, this directly affects the calculation and verification of the polynomial's coefficients, as seen with \(|c|\). This knowledge simplifies the determination of which properties apply or are satisfied, such as the specific condition \( |c| \leq 3 \) in the original exercise.