Cubic polynomials (third-degree equations) appear in various fields, including physics, engineering, and economics. Finding their roots manually can be complex, but a Roots of cubic polynomial calculator simplifies the process.
Roots of Cubic Polynomial Calculator
Enter the coefficients of the cubic polynomial:
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What Is a Cubic Polynomial?
A cubic polynomial is a third-degree equation of the form:
ax³ + bx² + cx + d = 0
Where:
- a, b, c, d are coefficients (with a ≠ 0)
- x is the variable
- The equation has three roots (real or complex)
Types of Roots in Cubic Equations
- Three Real Roots (e.g., x³ – 6x² + 11x – 6 = 0 → x = 1, 2, 3)
- One Real and Two Complex Roots (e.g., x³ + x = 0 → x = 0, ±i)
- A Repeated Real Root (e.g., x³ – 3x² + 3x – 1 = 0 → x = 1, 1, 1)
How to Find Roots of a Cubic Polynomial
Method 1: Factorization
Example: x³ – 6x² + 11x – 6 = 0
- Factorize: (x-1)(x-2)(x-3) = 0
- Roots: x = 1, 2, 3
Method 2: Cardano’s Formula
For x³ + px² + qx + r = 0, substitute x = y – p/3 to eliminate the quadratic term. Then solve using:
- Discriminant (Δ) = q² + 4p³r
- If Δ > 0: 1 real + 2 complex roots
- If Δ = 0: 3 real roots (at least two identical)
- If Δ < 0: 3 distinct real roots
This method is complex, so using a calculator is recommended.
Why Use a Cubic Polynomial Roots Calculator?
✅ Saves time – No manual calculations
✅ Handles complex roots – Displays results in a + bi form
✅ Step-by-step solutions – Some calculators show solving steps
✅ Error-free results – Avoids human calculation mistakes
How to Use Our Free Cubic Roots Calculator
Our Cubic Polynomial Roots Calculator is simple and accurate. Here’s how to use it:
Step 1: Enter Coefficients
Input values for a, b, c, d (e.g., for 2x³ – 4x² + 2x – 1 = 0, enter a=2, b=-4, c=2, d=-1).
Step 2: Click “Calculate Roots”
The calculator processes the equation using numerical methods.
Step 3: View Results
- Real roots (e.g., x = 1.3247)
- Complex roots (e.g., x = -0.1623 ± 0.5i)
Frequently Asked Questions (FAQ)
Q1: Can a cubic equation have no real roots?
No, a cubic equation always has at least one real root (due to the Intermediate Value Theorem).
Q2: What if the discriminant is zero?
It means two or three roots are identical (e.g., (x-1)³ = 0 → x = 1, 1, 1).
Q3: How accurate is the calculator?
Our tool provides 6 decimal places of precision, ensuring reliable results.
Q4: Can I use this for higher-degree polynomials?
No, this is only for cubic (degree 3) equations. For quartics, try a quartic solver.