Question

Die Nullstellen \(c_{1}, c_{2}\) und \(e_{3}\) des Polynoms \(4 X^{3}-g_{2} X-g_{3}\) sind genau dann reell, wenn \(g_{2}\) und \(g_{3}\) reell sind und die Diskriminante \(\Delta=g_{2}^{3}-27 g_{3}^{2}\) nicht negativ ist.

Short answer

The roots are real if \(g_2\) and \(g_3\) are real, and \(\Delta = g_2^3 - 27g_3^2 \geq 0\).

Step-by-step solution

Understanding the Problem

We are given a polynomial equation of the form \(4x^3 - g_2x - g_3\). We need to determine the conditions under which all roots \(c_1, c_2,\) and \(c_3\) are real numbers. This involves analyzing the coefficients \(g_2\) and \(g_3\) as well as the discriminant \(\Delta\).

Identifying Conditions for Real Roots

For the roots of a cubic polynomial \(a x^3 + b x^2 + c x + d\) to be real, all coefficients must be real, and the discriminant must be non-negative. Here, our polynomial has \(a = 4\), \(b = 0\), \(c = -g_2\), and \(d = -g_3\). Thus, \(g_2\) and \(g_3\) need to be real.

Using the Discriminant

The discriminant \(\Delta\) of the polynomial \(4x^3 - g_2x - g_3\) helps determine the nature of its roots. It is given by \(\Delta = g_2^3 - 27g_3^2\). For the roots to be real, \(\Delta\) must be non-negative (i.e., \(\Delta \geq 0\)).

Conclusion

All roots \(c_1, c_2,\) and \(c_3\) of the polynomial \(4x^3 - g_2x - g_3\) are real if \(g_2\) and \(g_3\) are real numbers and the discriminant \(\Delta = g_2^3 - 27g_3^2\) is non-negative.

Key concepts

Discriminant

In the context of cubic polynomials, the discriminant is a crucial component that helps us understand the nature of a polynomial's roots. For a cubic polynomial like \(4x^3 - g_2x - g_3\), the discriminant is expressed as \(\Delta = g_2^3 - 27g_3^2\). This equation plays a key role in determining whether the roots of the polynomial are real or complex.

Essential Points about the Discriminant:

• The discriminant \(\Delta\) is a single value calculated from the coefficients of the polynomial.
• If \(\Delta \geq 0\), the cubic polynomial has either three real roots or one real root and two complex conjugate roots.
• If \(\Delta < 0\), the polynomial has one real root and two non-real complex conjugate roots.

Understanding the discriminant thus provides a way to quickly assess the nature of all the roots without solving the polynomial explicitly.

Real Roots

Real roots are simply the points at which the polynomial intersects the x-axis in a graph. For the cubic polynomial \(4x^3 - g_2x - g_3\), having real roots means that these intersections occur for real values of \(x\).

Conditions for Real Roots:

• All coefficients of the polynomial must be real numbers.
• The discriminant \(\Delta = g_2^3 - 27g_3^2\) must be non-negative (i.e., \(\Delta \geq 0\)).

This set of conditions guarantees that the polynomial will not have any imaginary components in its roots. When these conditions are satisfied, you can have either:

• Three distinct real roots (if \(\Delta > 0\)).
• A triple real root or a combination of a real root and a repeated real root (when \(\Delta = 0\)).

Thus, real roots are important in determining the behavior of the polynomial and its graph in real-number space.

Polynomial Coefficients

Polynomial coefficients are the numbers that precede each term in a polynomial equation. In the cubic polynomial \(4x^3 - g_2x - g_3\), the coefficients are 4, \(-g_2\), and \(-g_3\).

Impact of Polynomial Coefficients:

• They influence the shape and position of the polynomial's graph.
• They determine the polynomial's discriminant, \(\Delta = g_2^3 - 27g_3^2\), which further informs us of the nature of the roots.
• In this context, the coefficients \(g_2\) and \(g_3\) need to be real for the roots to be real.

The coefficients define the major characteristics of the polynomial's behavior and are essential in solving for its roots. When dealing with questions of real roots, ensuring that the coefficients are real numbers is the first critical step.