Third Degree Polynomial Roots Calculator

A third degree polynomial, also known as a cubic polynomial, is a mathematical expression of the form ax³ + bx² + cx + d = 0. This calculator finds the real and complex roots of such equations using Cardano's formula.

What is a Third Degree Polynomial?

A third degree polynomial is a polynomial equation where the highest power of the variable is three. The general form is:

ax³ + bx² + cx + d = 0

Where:

a, b, c, and d are coefficients
a ≠ 0 (since it's a third degree polynomial)
x is the variable we're solving for

Third degree polynomials can have one real root and two complex conjugate roots, or three real roots (which may be identical).

How to Use the Calculator

Enter the coefficients a, b, c, and d of your polynomial equation
Click "Calculate Roots" to find the solutions
Review the results and interpretation
Use the chart to visualize the polynomial function

Note: For complex roots, the calculator will display them in the form of a + bi, where i is the imaginary unit (√-1).

The Formula

The roots of a cubic equation are found using Cardano's formula:

For equation ax³ + bx² + cx + d = 0, the roots can be found using:

Let Δ₀ = b² - 3ac

Δ₁ = 2b³ - 9abc + 27a²d

C = ∛[(Δ₁ ± √(Δ₁² - 4Δ₀³))/2]

Roots: x = [-(b + C + Δ₀/C)/(3a)]

This formula provides all roots, both real and complex, of the cubic equation.

Worked Example

Let's solve x³ - 6x² + 11x - 6 = 0:

Identify coefficients: a=1, b=-6, c=11, d=-6
Calculate Δ₀ = (-6)² - 3(1)(11) = 36 - 33 = 3
Calculate Δ₁ = 2(-6)³ - 9(1)(-6)(11) + 27(1)²(-6) = -432 + 594 - 162 = 0
C = ∛[0 ± √(0 - 4(3)³)]/2 = ∛[0 ± √(-108)]/2 = ∛[±6i√3]/2 = ±i√3
Roots: x = [6 ± i√3 ± 3]/3

This gives three roots: 2, 2, and 2 (a triple root).

Interpreting Results

When using the calculator, consider these points:

Real roots are points where the polynomial crosses the x-axis
Complex roots come in conjugate pairs (a + bi and a - bi)
Multiple roots (like x³ - 3x² + 3x - 1 = 0) have repeated solutions
The discriminant (Δ₀) helps determine the nature of the roots

For practical applications, real roots are typically more useful than complex ones.

FAQ

What if my polynomial has only one real root?

If the discriminant (Δ₀) is positive, there are three distinct real roots. If Δ₀ is zero, there's a multiple root and two other real roots. If Δ₀ is negative, there's one real root and two complex conjugate roots.

Can this calculator handle complex coefficients?

No, this calculator only handles real coefficients. For complex coefficients, you would need a more advanced mathematical tool.

What's the difference between a cubic and quadratic equation?

A cubic equation has the highest power of x as 3, while a quadratic equation has it as 2. Cubic equations can have up to three roots, while quadratics have up to two.

How accurate are the results?

The calculator uses precise mathematical formulas and JavaScript's floating-point arithmetic. For most practical purposes, the results are accurate to about 15 decimal places.