Calculator
Example data table
Example polynomial: x³ − 6x² + 11x − 6 (roots: 1, 2, 3).
| a | b | c | d | x | f(x) |
|---|---|---|---|---|---|
| 1 | -6 | 11 | -6 | 0 | -6 |
| 1 | -6 | 11 | -6 | 1 | 0 |
| 1 | -6 | 11 | -6 | 2 | 0 |
| 1 | -6 | 11 | -6 | 3 | 0 |
| 1 | -6 | 11 | -6 | 4 | 6 |
Formula used
- Cubic polynomial: f(x) = ax³ + bx² + cx + d
- Derivative: f'(x) = 3ax² + 2bx + c
- Second derivative: f''(x) = 6ax + 2b
- Antiderivative: ∫f(x)dx = (a/4)x⁴ + (b/3)x³ + (c/2)x² + dx + K
- Inflection point: solve f''(x)=0 → x = −b/(3a)
- Turning points: solve f'(x)=0 (a quadratic equation)
- Discriminant (standard): Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d²
- Root solving (Cardano via depressed cubic): let x = t − b/(3a) giving t³ + pt + q = 0, then compute roots from p and q.
This page uses numeric approximations for roots and curve features. Increase decimals for more precise rounding.
How to use this calculator
- Enter coefficients a, b, c, d for ax³ + bx² + cx + d.
- Choose x₀ to evaluate the function and derivative.
- Set x₁ and x₂ to compute the definite integral.
- Adjust xmin, xmax, points for the value table range.
- Pick the decimals level, then press Calculate.
- Use the export buttons to download CSV or PDF.
FAQs
1) What makes a polynomial “cubic”?
A cubic polynomial has highest power three, like ax³ + bx² + cx + d, with a ≠ 0. It can have up to three real roots and always has one inflection point.
2) Why must coefficient a be non‑zero?
If a = 0, the x³ term disappears, so the expression is no longer cubic. The curve’s shape and root behavior change to quadratic, linear, or constant.
3) Are the roots exact or approximate?
They are numeric approximations computed from a standard cubic method. Use a higher decimals setting to reduce rounding error in the displayed roots, factor form, and table values.
4) What does the discriminant tell me?
The discriminant indicates root structure. Typically, a positive value implies three distinct real roots, zero implies repeated roots, and a negative value implies one real root with a complex conjugate pair.
5) How are turning points found?
Turning points occur where the derivative is zero. This calculator solves f'(x)=0, which is a quadratic equation, then evaluates f(x) and uses f''(x) to classify maxima or minima.
6) What is the inflection point in a cubic?
A cubic always has one inflection point where concavity changes. It is found by solving f''(x)=0, giving x = −b/(3a), then computing y = f(x).
7) What does the definite integral represent?
The definite integral from x₁ to x₂ gives the signed area under the curve. It can be negative if the function is below the x‑axis over part of the interval.
8) How do I export my results?
After calculation, use Download CSV for spreadsheets or Download PDF for sharing or printing. Exports include coefficients, evaluations, roots, integral result, and the generated value table.