Enter cubic coefficients and graph settings
The calculator accepts any real coefficients, graphs the polynomial, shows real and complex roots, and builds a practical factorization over the reals when possible.
Sample cubic expressions for testing
| Expression | Expected factorization | Notes |
|---|---|---|
| x3 - 6x2 + 11x - 6 | (x - 1)(x - 2)(x - 3) | Three distinct real roots. |
| x3 + 3x2 + 3x + 1 | (x + 1)(x + 1)(x + 1) | Triple repeated root. |
| 2x3 + x2 - 8x - 4 | (x + 1/2)(2x2 - 8) | One rational root, then a quadratic factor. |
| x3 + x + 1 | One real factor and one irreducible quadratic | No rational root appears. |
Methods behind the cubic factoring calculator
1. Standard cubic form
ax3 + bx2 + cx + d = 0, with a ≠ 0
2. Rational Root Theorem
When coefficients are integers, any rational root must be of the form p/q, where p divides d and q divides a.
3. Synthetic division
Once a root r is found, the cubic is divided by (x - r) to create a quadratic quotient for faster factoring.
4. Depressed cubic conversion
x = t - b / (3a)
This removes the squared term and gives t3 + pt + q = 0.
5. Discriminant
Δ = 18abcd - 4b3d + b2c2 - 4ac3 - 27a2d2
The discriminant reveals whether repeated roots or different real-root patterns are present.
6. Cardano fallback
If clean rational roots are unavailable, Cardano’s method is used to compute all real and complex roots numerically.
Simple workflow
Step 1
Enter values for a, b, c, and d from your cubic expression.
Step 2
Set the graph range and choose how many sample points you want plotted.
Step 3
Choose the display precision for roots, factors, and summary values.
Step 4
Press the calculate button to factor the cubic and generate the graph.
Step 5
Review the factors, roots, discriminant, turning points, and export buttons.
Common questions
1. What does this calculator factor?
It analyzes cubic polynomials written as ax³ + bx² + cx + d. The tool finds roots, checks rational candidates, forms practical factors, and graphs the expression.
2. Does it work with decimals?
Yes. Decimal coefficients are accepted. Rational-root testing only appears for integer coefficients, but the solver still computes numerical roots and a real-factor form.
3. Why are some factors approximate?
Some cubics have irrational or complex roots. In those cases, the calculator shows accurate decimal approximations instead of short integer or fraction factors.
4. What does the discriminant tell me?
The discriminant indicates root structure. It helps identify repeated roots, three real roots, or one real root with a complex conjugate pair.
5. What are turning points?
Turning points are where the cubic changes direction. They come from solving the derivative, and they help explain the curve’s shape on the graph.
6. Can I export my result?
Yes. After calculation, you can download a CSV summary or generate a PDF report containing the equation, factors, roots, and key values.
7. Why is a required to be non-zero?
If a equals zero, the expression is no longer cubic. It becomes quadratic or lower, so cubic factoring methods would not apply correctly.
8. Does the graph show complex roots?
No. The graph is drawn on the real x-y plane, so only real roots appear as x-axis intercept markers.