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Calcul Racines Polynome Degré 3

Reviewed by Calculator Editorial Team

This calculator finds all roots of a cubic polynomial equation of the form x³ + ax² + bx + c = 0. It uses the cubic formula to determine the real and complex roots of the equation.

How to Use This Calculator

To calculate the roots of a cubic polynomial:

  1. Enter the coefficients a, b, and c from your polynomial equation x³ + ax² + bx + c = 0.
  2. Click the "Calculate" button.
  3. Review the results, which will show all real and complex roots.
  4. Use the chart to visualize the polynomial and its roots.

The calculator will display the roots in a clear format, with real roots shown as decimal numbers and complex roots shown in the form x + yi.

The Cubic Formula

The cubic formula is used to find the roots of a cubic polynomial equation. For an equation of the form:

x³ + ax² + bx + c = 0

The roots can be found using the following steps:

  1. Calculate the discriminant Δ = 18abc - 4a³c + a²b² - 4b³ - 27c².
  2. If Δ > 0, there are three distinct real roots.
  3. If Δ = 0, there are multiple roots (either three real roots with at least two equal, or one real root and two complex conjugate roots).
  4. If Δ < 0, there is one real root and two complex conjugate roots.

The exact formulas for the roots are complex and involve cube roots of complex numbers. The calculator implements these formulas to provide accurate results.

Worked Example

Let's solve the equation x³ - 6x² + 11x - 6 = 0.

Using the calculator:

  1. Enter a = -6, b = 11, c = -6.
  2. Click "Calculate".
  3. The calculator will display the roots: 1, 2, and 3.

This shows that the polynomial can be factored as (x-1)(x-2)(x-3) = 0.

Interpreting the Results

The calculator provides the roots of the cubic polynomial. Each root represents a solution to the equation x³ + ax² + bx + c = 0.

For real roots, you can substitute them back into the original polynomial to verify the solution. For complex roots, they come in conjugate pairs and represent points where the polynomial crosses the complex plane.

Note: Complex roots are mathematically valid solutions to the equation, even though they don't appear on the real number line.

Frequently Asked Questions

What is a cubic polynomial?
A cubic polynomial is a polynomial of degree 3, which means the highest power of x is 3. It has the general form x³ + ax² + bx + c = 0.
How many roots can a cubic polynomial have?
A cubic polynomial always has three roots, counting multiplicities. These can be all real, two real and one complex, or one real and two complex (which come in conjugate pairs).
What is the discriminant of a cubic polynomial?
The discriminant Δ determines the nature of the roots. If Δ > 0, there are three distinct real roots. If Δ = 0, there are multiple roots. If Δ < 0, there is one real root and two complex roots.
Can this calculator handle complex roots?
Yes, the calculator can find and display complex roots in the form x + yi, where i is the imaginary unit.
What if my polynomial has a leading coefficient other than 1?
You can divide the entire equation by the leading coefficient to convert it to the form x³ + ax² + bx + c = 0 before using this calculator.