Calculator Inputs
Example Data Table
| Degree | Polynomial | Discriminant | Meaning |
|---|---|---|---|
| 2 | x² - 5x + 6 | 1 | Two distinct real roots. |
| 3 | x³ - 6x² + 11x - 6 | 4 | Three distinct real roots. |
| 4 | x⁴ - 5x² + 4 | 5184 | All roots are distinct. |
| 4 | x⁴ - 4x³ + 6x² - 4x + 1 | 0 | Repeated root occurs at x = 1. |
Formula Used
General discriminant formula:
Δ(P) = aₙ^(2n-2) × ∏(rᵢ - rⱼ)² for all i < j.
Here, aₙ is the leading coefficient,
n is the degree,
and rᵢ are the roots.
Resultant relation used internally:
Δ(P) = (-1)^[n(n-1)/2] × Res(P, P′) / aₙ
The calculator builds the Sylvester matrix of
P(x) and
P′(x),
computes its determinant, and converts that determinant into the discriminant.
Special low-degree forms:
Quadratic: Δ = b² - 4ac
Cubic: Δ = b²c² - 4ac³ - 4b³d - 27a²d² + 18abcd
Quartic discriminants are longer, so the calculator evaluates them through the general resultant method for better consistency.
How to Use This Calculator
- Select a polynomial degree from 2 to 8.
- Enter coefficients from the highest power down to the constant term.
- Set the graph range and sample count if needed.
- Choose a display precision for reported values.
- Press Calculate Discriminant to generate the result section above the form.
- Review the discriminant, approximate roots, sign meaning, and graph.
- Use the CSV or PDF buttons to export the summary.
Frequently Asked Questions
1) What does a polynomial discriminant measure?
It measures root separation. A zero discriminant means the polynomial has at least one repeated root. A nonzero value means all roots are distinct.
2) Why is the discriminant useful?
It helps detect repeated roots, compare root behavior, and study how sensitive a polynomial is to coefficient changes. It is widely used in algebra, control, and numerical analysis.
3) Does a positive discriminant always mean all roots are real?
No. That shortcut works only for certain low-degree cases, especially quadratics and cubics with real coefficients. For higher degrees, sign alone is not enough.
4) Why can the graph look flat or steep?
Large coefficients or wide x-ranges can make polynomial values grow quickly. Narrow the plotting interval to inspect root neighborhoods more clearly.
5) Are the listed roots exact?
They are numerical approximations. The discriminant itself is computed from the resultant, while the root list is estimated for interpretation and plotting support.
6) What happens when the discriminant is nearly zero?
That usually signals repeated roots or extremely close roots. In practical computation, small rounding errors can produce tiny values instead of exact zeros.
7) Why must the leading coefficient be nonzero?
A zero leading coefficient lowers the polynomial degree. The discriminant depends on the actual degree, so the highest-order coefficient must stay nonzero.
8) Can I use decimal coefficients?
Yes. The calculator accepts integers and decimals. Very large or tiny values may display in scientific notation to keep the output readable.