Calculator Input
Example Data Table
| Expression | Factored Form | Root Pattern | Note |
|---|---|---|---|
| x^3 - 6x^2 + 11x - 6 | (x - 1)(x - 2)(x - 3) | 1, 2, 3 | Three distinct real factors. |
| x^3 + 3x^2 + 3x + 1 | (x + 1)^3 | -1, -1, -1 | Triple repeated root. |
| 3x^3 - 12x | 3x(x - 2)(x + 2) | -2, 0, 2 | Common factor extracted first. |
| 2x^3 + 2x^2 - 16x - 16 | 2(x + 1)(x^2 - 8) | -1, ±2√2 | Mixed linear and irrational factors. |
Formula Used
ax^3 + bx^2 + cx + d = 0
ax^3 + bx^2 + cx + d = g · (Ax^3 + Bx^2 + Cx + D)
If p/q is a rational root, then p divides d and q divides a.
Once a root r is found, divide by (x - r) to reduce the cubic into a quadratic.
Δ = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2
A positive discriminant indicates three distinct real roots. Zero indicates repeated real roots. A negative value indicates one real root and two complex roots.
How to Use This Calculator
- Enter the four coefficients for the cubic expression.
- Choose the variable symbol and preferred decimal precision.
- Set the graph range and sample density.
- Click Factor Expression to generate the result.
- Review the factored form, roots, discriminant, turning points, and graph.
- Download the output as a CSV or PDF file if needed.
FAQs
1. What does this calculator factor?
It factors cubic expressions of the form ax^3 + bx^2 + cx + d. It also extracts a signed common factor and reports the resulting reduced expression.
2. Does it always produce exact integer factors?
No. Some cubic expressions have irrational or complex roots. In those cases, the calculator shows a valid real-factor form or a numerical factorization where needed.
3. Why are rational root candidates shown?
They help you test likely rational roots efficiently. The list comes from the Rational Root Theorem using divisors of the constant term and leading coefficient.
4. What does the discriminant tell me?
It classifies the root structure. Positive means three distinct real roots. Zero means repeated real roots. Negative means one real root with two non-real complex roots.
5. Why is a common factor taken out first?
Extracting a common factor simplifies the polynomial before deeper factoring. It makes later root testing cleaner and often reveals one factor immediately.
6. What are turning points in the result?
Turning points are the local maximum and minimum positions of the cubic. They come from solving the derivative and help explain the graph’s shape.
7. Can I use decimal coefficients?
Yes. The calculator accepts decimal inputs. Rational candidate generation is only meaningful when the leading and constant coefficients are integer-like values.
8. Why do repeated roots appear more than once?
Repeated roots indicate repeated factors. For example, a root of 2 appearing three times means the cubic contains the factor (x - 2)^3.