Problem 74 - A Complete Resource Book in Mathematics for JEE Main

Question

All the roots of the equation \(a_{1} z^{3}+a_{2} z^{2}+a_{3} z+a_{4}=3\), where \(\left|a_{i}\right| \leq 1, i=1,2,3,4\) lie outside the circle with centre origin and radius (A) \(\frac{1}{3}\) (B) \(\frac{2}{3}\) (C) 1 (D) None of these

Step-by-Step Solution

Answer

(C) 1 - All roots lie outside a circle with radius 1.

Step 1: Rearrange the Equation

Let's rewrite the given equation into a polynomial form. Instead of having it equal to 3, move 3 to one side: \[ a_1 z^3 + a_2 z^2 + a_3 z + (a_4 - 3) = 0 \] This is now a standard cubic equation in terms of \(z\).

Step 2: Apply Rouche's Theorem Principle

According to Rouche's Theorem, for a polynomial of degree \(n\) like this, all roots will lie outside a circle if the magnitude of the constant term exceeds the sum of the magnitudes of the other coefficients at the particular distance. So we need to check if the absolute value of \(a_4 - 3\) is greater than \(|a_1| + |a_2| + |a_3|\).

Step 3: Calculate Magnitudes

Given \(|a_i| \leq 1\) for each \(i\), the maximum value of each could be 1. Calculate the maximum possible sum of these coefficients:\[ |a_1| + |a_2| + |a_3| \leq 1 + 1 + 1 = 3 \]Then, we find:\[ |a_4 - 3| > 3 \] implies \(|a_4|\) can be \( -1 \leq a_4 \leq 1 \), thus leading to \(|a_4 - 3| < |4|\). As \(|a_4 - 3| \leq 4| \), we want \(|a_4 - 3| > 3\).

Step 4: Determine Adequate Radius

If \(|a_4 - 3| > 3\), then a circle of radius greater than 1 is suitable since the roots are always greater. However, lesser radii such as \(\frac{1}{3}\) or \(\frac{2}{3}\) do not meet the condition based on the sums derived from prior steps; therefore, no root satisfies beyond explicit proof testing.

Key Concepts

Complex Analysis

Complex analysis is a branch of mathematics that investigates functions of complex numbers. It is rich in theory, dealing with complex planes and intriguing concepts like analytic functions and contour integrals.
One important tool in this field is **Rouche's Theorem**. It allows us to determine the number of zeros of a complex polynomial within a certain contour in the complex plane. Rouche's Theorem requires comparing a target polynomial to another "dominating" one. If one polynomial dominates the values on the boundary of a contour, the number of zeros inside matches that of the dominant polynomial.
This theorem is incredibly useful in confirming the locations of roots for complex polynomials. In our case, understanding where the roots of a particular cubic equation lie hinges on this theorem, focusing on the polynomial's magnitude along the contour of interest.

Cubic Equations

Cubic equations are algebraic expressions where the highest exponent of the variable is three. They have the general form:\[ ax^3 + bx^2 + cx + d = 0 \]These equations naturally extend from linear and quadratic equations, introducing additional complexity due to their degree. Solving cubic equations may require factoring, synthetic division, or intricate methods like Cardano's formula.
However, in the realm of **Complex Analysis**, it becomes crucial to evaluate the behavior and roots of cubic polynomials within the complex plane. Techniques often involve setting certain conditions, like those outlined in Rouche's Theorem, to determine the positions of these roots.
For equations like ours, knowing if roots exist inside or outside a specific radius around the origin involves careful manipulation and examinations of complex numbers and their magnitudes.

Polynomial Roots

Polynomial roots are fundamental solutions where the polynomial equals zero. For any polynomial, the sum of the exponents denotes a maximum number of roots it may have. These can be real or complex.
For our cubic equation, **Rouge’s Theorem** plays a pivotal role in locating these roots. The conditions set by the theorem (e.g., comparing magnitudes) help ensure that all roots lie outside a particular circle around the origin.
Checking the radius where roots reside involves ensuring the magnitude of differences (e.g., coefficients minus constants) surpasses the sum of other polynomials' components. Once these concepts are understood, determining all roots lying outside a circle informs us about the polynomial's nature and behavior, maintaining alignment with Rouche's established rules.
Understanding these basics aids students in grasping the underpinning aspects that control root placement, evaluation, and application within polynomial functions.