Calcul Discriminant Polynome Degré 3

A cubic polynomial (polynôme degré 3) is a mathematical expression of the form ax³ + bx² + cx + d = 0. The discriminant is a value that provides important information about the nature of the polynomial's roots. Calculating the discriminant helps determine how many real roots the polynomial has and their multiplicities.

What is the discriminant of a cubic polynomial?

The discriminant of a cubic polynomial is a number that provides information about the roots of the polynomial. For a cubic polynomial of the form:

ax³ + bx² + cx + d = 0

The discriminant Δ (Delta) helps classify the nature of the roots:

If Δ > 0: The polynomial has three distinct real roots.
If Δ = 0: The polynomial has a multiple root and all roots are real.
If Δ < 0: The polynomial has one real root and two complex conjugate roots.

This classification is crucial for understanding the behavior of the polynomial and its graph.

How to calculate the discriminant

Calculating the discriminant of a cubic polynomial involves several steps. Here's a step-by-step guide:

Identify the coefficients a, b, c, and d of the polynomial.
Calculate the discriminant using the formula provided below.
Interpret the result based on the value of the discriminant.

Using our calculator, you can quickly compute the discriminant for any cubic polynomial you provide.

The discriminant formula

The discriminant Δ of a cubic polynomial ax³ + bx² + cx + d = 0 is given by:

Δ = 18abcd - 4b³d + b²c² - 4ac³ - 27a²d²

This formula is derived from the theory of cubic equations and provides a way to analyze the roots without explicitly solving the equation.

Interpreting the discriminant

The value of the discriminant provides important information about the roots:

Discriminant Value	Nature of Roots	Number of Real Roots
Δ > 0	Three distinct real roots	3
Δ = 0	Multiple roots (at least two roots are equal)	3 (with at least two equal)
Δ < 0	One real root and two complex conjugate roots	1

Understanding the discriminant helps in graphing the polynomial and predicting its behavior.

Worked example

Let's calculate the discriminant for the polynomial x³ - 6x² + 11x - 6 = 0.

Here, a = 1, b = -6, c = 11, d = -6.

Using the formula:

Δ = 18(1)(-6)(11)(-6) - 4(-6)³(-6) + (-6)²(11)² - 4(1)(11)³ - 27(1)²(-6)²

Δ = 18(396) - 4(-216)(-6) + 36(121) - 4(1331) - 27(36)

Δ = 7128 - 5184 + 4356 - 5324 - 972

Δ = 0

The discriminant is 0, which means the polynomial has a multiple root. This polynomial can be factored as (x - 1)(x - 2)(x - 3), showing that it has three real roots, two of which are equal (x = 1 and x = 3 appear twice).

FAQ

What is the difference between the discriminant of a quadratic and cubic polynomial?

The discriminant of a quadratic polynomial (ax² + bx + c) is given by Δ = b² - 4ac, which determines the nature of the roots (real and distinct, real and equal, or complex). The discriminant of a cubic polynomial is more complex and involves all four coefficients, providing information about the number and nature of the roots.

Can the discriminant be negative for a cubic polynomial?

Yes, the discriminant can be negative for a cubic polynomial. A negative discriminant indicates that the polynomial has one real root and two complex conjugate roots.

How is the discriminant used in solving cubic equations?

The discriminant helps determine the method needed to solve the cubic equation. For Δ > 0, Cardano's formula can be used. For Δ = 0, the equation has a multiple root. For Δ < 0, trigonometric methods are typically used.