Problem 1 - Complex Analysis

Question

The zeros \(e_{1}, e_{2}\) and \(e_{3}\) of the polynomial \(4 X^{3}-g_{2} X-g_{3}\) are all real, iff \(g_{2}, g_{3}\) are real, and the discriminant \(\Delta=g_{2}^{3}-27 g_{3}^{2}\) is non-negative.

Step-by-Step Solution

Answer

The roots are real if \(g_2\) and \(g_3\) are real, and \(g_2^3 \geq 27g_3^2\).

Step 1: Understanding the Polynomial

The polynomial given is a cubic polynomial of the form \(4X^3 - g_2X - g_3\), where \(g_2\) and \(g_3\) are coefficients that we need to consider. Here, \(e_1, e_2, e_3\) are the zeros or roots of the polynomial and we need to determine the conditions under which all roots are real.

Step 2: Identifying Conditions for Real Roots

A cubic polynomial of the form \(aX^3 + bX^2 + cX + d\) (here \(a=4, b=0, c=-g_2, d=-g_3\)) can have all real roots if its discriminant is non-negative. The discriminant \(\Delta\) for this cubic is given by \(g_2^3 - 27g_3^2\). We are asked to determine when this discriminant is non-negative and \(g_2, g_3\) are real.

Step 3: Analyzing the Discriminant

Given the formula for the discriminant of the polynomial \(\Delta = g_2^3 - 27g_3^2\), we conclude that for the polynomial to have all roots real, the condition \(\Delta \geq 0\) must hold. This implies \(g_2^3 \geq 27g_3^2\), meaning the value on the left must be greater or equal to the value on the right.

Step 4: Conclusively Defining the Realness condition

Hence, the zeros \(e_1, e_2, e_3\) of the polynomial \(4X^3 - g_2X - g_3\) are real if \(g_2\) and \(g_3\) are themselves real numbers and the calculated discriminant \(g_2^3 - 27g_3^2\) is non-negative. Explicitly, when \(g_2\), \(g_3\) are real and \(g_2^3 \geq 27g_3^2\).

Key Concepts

Real Roots Condition

To determine if a cubic polynomial has all real roots, we need to consider both the nature of its coefficients and its discriminant value. A key condition is that the coefficients, in this case represented by \(g_2\) and \(g_3\), must be real numbers. This is fundamental, as having real coefficients sets the stage for finding real roots.
However, real coefficients alone are not enough. For a cubic polynomial expressed as \(4X^3 - g_2X - g_3\) to have all roots real, the discriminant must also be greater than or equal to zero.

• Discriminant \(\Delta\) determines the nature of the roots.
• All three roots are real if the discriminant is non-negative.

This means that calculating the discriminant and ensuring it meets this condition is crucial.

Polynomial Zeros

In polynomial mathematics, the zeros (or roots) of a polynomial are the values of \(X\) that satisfy the equation \(4X^3 - g_2X - g_3 = 0\). Understanding these zeros is critical because they represent the solutions of the polynomial equation.

• The zeros can tell us about the shape of the graph.
• For cubic polynomials, having all real zeros affects how the graph crosses the X-axis.

Each zero \(e_1, e_2, e_3\) provides specific values where the polynomial becomes zero. Real zeros indicate actual intercept points on the graph along the X-axis, showing tangible solutions.

Discriminant Formula

The discriminant of a cubic polynomial is a specific formula that helps us know about the nature of its roots. For the polynomial \(4X^3 - g_2X - g_3\), the discriminant is given as \(\Delta = g_2^3 - 27g_3^2\).

• If \(\Delta > 0\), all roots are real and distinct.
• If \(\Delta = 0\), the polynomial has a multiple root, but all roots are still real.
• If \(\Delta < 0\), the polynomial has one real root and two complex conjugate roots.

The discriminant formula is a straightforward check to predict the type of roots without solving the polynomial fully, simplifying our analysis significantly.