Calculator input
Enter coefficients for a third degree polynomial in the form ax³ + bx² + cx + d.
Formula used
General cubic: ax3 + bx2 + cx + d = 0, where a ≠ 0.
Depressed cubic substitution: x = t - b / 3a.
Depressed form: t3 + pt + q = 0.
p: (3ac - b2) / 3a2.
q: (27a2d - 9abc + 2b3) / 27a3.
Cardano check: Δ = (q / 2)2 + (p / 3)3.
If Δ is positive, one real root and two complex roots appear. If Δ is zero, repeated roots appear. If Δ is negative, three real roots appear. The factor form is built from these roots.
How to use this calculator
- Write your expression as ax³ + bx² + cx + d.
- Enter a, b, c, and d in the matching fields.
- Choose graph range, table sample count, and decimal precision.
- Press the submit button to show the factor form above the form.
- Review roots, Cardano values, turning points, and the graph.
- Use CSV for spreadsheets or PDF for a printable report.
Example data table
This example uses x3 - 6x2 + 11x - 6.
| a | b | c | d | Factor form | Roots |
|---|---|---|---|---|---|
| 1 | -6 | 11 | -6 | (x - 1)(x - 2)(x - 3) | 1, 2, 3 |
| 2 | -4 | -22 | 24 | 2(x - 1)(x - 3)(x + 4) | 1, 3, -4 |
| 1 | 0 | 1 | 1 | One real factor and one quadratic factor | One real, two complex |
Understanding cubic factorization
Why cubic factors matter
Cubic polynomials appear in algebra, physics, economics, and design problems. A factored form gives more meaning than an expanded form. It shows where the graph crosses the horizontal axis. It also shows repeated roots, turning behavior, and useful symmetry. This calculator converts coefficient data into roots, factors, steps, and a visual graph.
How the method works
The tool begins with the standard form ax³ + bx² + cx + d. It first confirms that a is not zero. Then it normalizes the polynomial. The substitution x = t - b / 3a removes the squared term. This creates a depressed cubic. Cardano values p, q, and Δ then guide the root type. A positive Δ means one real root. A zero Δ means repeated roots. A negative Δ means three real roots.
Exact checks and numeric support
When coefficients look like small integers, the calculator also tests rational root candidates. This helps with classroom examples. It can find roots such as 1, -2, or 3/2 quickly. For harder cubics, numeric Cardano roots are used. Complex roots are paired into a real quadratic factor, so the final answer remains readable.
Using the graph and exports
The graph helps you check the answer visually. Real roots should match x-axis crossings. Turning points explain where the curve rises or falls. The generated table gives coordinate samples across your chosen range. Export the CSV when you need spreadsheet data. Export the PDF when you need a clean report for notes, lessons, or records.
Common checks
After finding factors, multiply them back in order. The expanded result should match the original coefficients. Check the constant term first. It often reveals sign errors. Next, check the leading coefficient. Finally, substitute one root into the original expression. A correct root should return a value near zero.
Practical advice
Use enough decimal places when coefficients are not integers. Increase the sample count for a smoother graph. Widen the graph range when roots are far from zero. Always compare the factor form with the original polynomial. Expanding the factors should return the same coefficients. That check protects against entry mistakes and rounding confusion.
FAQs
What is a third degree polynomial?
It is a polynomial where the highest power of the variable is three. Its usual form is ax³ + bx² + cx + d, with a not equal to zero.
Can every cubic be factored?
Every cubic has three roots when complex roots are allowed. Over real numbers, it has at least one real linear factor and may also include a quadratic factor.
Why does the calculator show complex roots?
Some cubics have only one real root. The other two roots are complex conjugates. They combine into a real quadratic factor in the factor form.
What does the discriminant mean?
The cubic discriminant helps describe root behavior. Positive, zero, and negative values point to different combinations of distinct, repeated, real, or complex roots.
What is Cardano's method?
Cardano's method solves cubic equations by converting them into a depressed cubic. It then uses p, q, and Δ to calculate the roots.
Can I use decimal coefficients?
Yes. Decimal coefficients are supported. Rational root scanning is strongest for small integer coefficients, but the numeric root method still works with decimals.
Why is coefficient a required?
The coefficient a controls the cubic term. If a is zero, the expression becomes quadratic or lower, so it is not a third degree polynomial.
What do CSV and PDF downloads include?
The CSV includes summary values and generated graph table points. The PDF includes the polynomial, factor form, roots, and key Cardano values.