Turn quartic coefficients into a clean resolvent cubic. Inspect intermediate p, q, r values instantly. Download tables, share proofs, and reduce mistakes today quickly.
| Example | a | b | c | d | e | p | q | r | Resolvent |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 0 | -5 | 0 | 4 | -5 | 0 | 4 | z³ + 5 z² − 16 z − 80 |
| 2 | 2 | -3 | 0 | 7 | -1 | (computed) | (computed) | (computed) | (computed) |
Resolvent polynomials convert a hard quartic into a structured cubic whose solutions guide factorization. After the x = y − A/4 shift, the depressed parameters p, q, r summarize the quartic’s geometry. The resolvent’s roots indicate a value that balances y⁴ against a perfect-square decomposition. In practice, that balancing step is the heart of Ferrari’s method and is also useful for symbolic checking in computer algebra workflows.
Parameter p controls the curvature of the even part, q controls asymmetry, and r sets the baseline level. When q is near zero, the equation behaves like a biquadratic and often yields cleaner radicals. Large |p| with small |q| can still produce sharp sensitivity, so rounding may change inferred roots. Reporting p, q, r alongside the original coefficients improves auditability in homework, reports, and engineering notes.
In this tool, the resolvent z³ − p z² − 4 r z + (4 r p − q²) is assembled directly from p, q, r. The z² term tracks −p, the linear term tracks −4r, and the constant term captures the coupling between r and p, penalized by q². This coupling is the signal that odd terms prevent a simple biquadratic split. Comparing these coefficients across scenarios helps detect transcription errors.
The cubic discriminant is a quick diagnostic of root structure. A positive value typically indicates three distinct real roots, while a negative value suggests one real root and a complex pair. The displayed real-root estimates are numerical helpers for selecting a resolvent root that leads to stable square roots when continuing Ferrari’s method. For documentation, store the chosen z, then record the derived quadratic factors before final root extraction.
CSV export is ideal for spreadsheets, notebooks, and batch comparisons of multiple quartics. The PDF export provides a clean, shareable snapshot for assignments, peer review, or internal validation. For reproducibility, keep the input coefficients, the normalized A, B, C, D values, and the depressed p, q, r values together. When results disagree with another source, these intermediate parameters usually reveal where conventions or rounding diverge. If you test multiple inputs, keep units consistent and label each run. Small changes in b can shift q strongly, so track revisions carefully. over time.
What polynomial degree does this tool target?
It targets quartic inputs a x⁴ + b x³ + c x² + d x + e. The tool computes the depressed-form parameters and one standard cubic resolvent for that quartic.
Does the tool solve the quartic completely?
No. It builds the resolvent cubic and gives helpful real-root estimates. You can continue with Ferrari’s method to form quadratic factors and then extract x-roots.
Why do I see different resolvent formulas elsewhere?
Multiple resolvent forms exist and are related by substitutions. If you compute p, q, r correctly, each valid resolvent encodes the same factorization structure.
How should I interpret the cubic discriminant?
It indicates the likely nature of the cubic’s roots. Positive values often mean three distinct real roots; negative values often mean one real root and a complex pair.
Why can results change slightly with similar inputs?
Floating-point arithmetic rounds decimals. Near-degenerate cases can amplify rounding, especially when q is small or when subtraction causes cancellation in p, q, or r.
What is the best way to document my work?
Export the table, then record your chosen resolvent root z and any subsequent quadratic factors. Keeping A, B, C, D and p, q, r together makes comparisons easy.
Note: Floating-point rounding can affect very large inputs.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.