Calculator Inputs
Example Data Table
| Example Polynomial | Coefficients (a, b, c, d, e) | Factorization | Roots |
|---|---|---|---|
| x4 - 5x2 + 4 | (1, 0, -5, 0, 4) | (x - 2)(x - 1)(x + 1)(x + 2) | -2, -1, 1, 2 |
| x4 + 2x2 + 1 | (1, 0, 2, 0, 1) | (x2 + 1)2 | i, i, -i, -i |
| 2x4 + 3x3 - 3x2 - 8x - 4 | (2, 3, -3, -8, -4) | (2x2 - x - 2)(x2 + 2x + 2) | 1.2808, -0.7808, -1 ± i |
Formula Used
This calculator first tests exact integer-coefficient quadratic pair matches. When exact matching does not succeed, it computes numerical roots and rebuilds an approximate factorization from real roots and complex conjugate pairs.
How to Use This Calculator
- Enter the five coefficients for ax4 + bx3 + cx2 + dx + e.
- Choose graph minimum x, maximum x, and the number of sample points.
- Set the decimal precision for the displayed roots and factors.
- Press Factor Quartic to show the result section above the form.
- Review the exact factorization result, approximate factors, roots table, and graph.
- Use the CSV button for spreadsheet analysis or the PDF button for printing and sharing.
FAQs
1. What does this calculator factor?
It analyzes quartic polynomials of the form ax⁴ + bx³ + cx² + dx + e. It searches for exact quadratic pairs first, then rebuilds factors from numerical roots when needed.
2. Will it always return an exact factorization?
No. Some quartics do not factor neatly using integers, and others require irrational or complex expressions. In such cases, the calculator returns a stable approximate factorization based on computed roots.
3. Can I use decimal coefficients?
Yes. Decimal coefficients are accepted. Exact integer pair detection works best for whole-number coefficients, while the numerical root engine handles decimal inputs effectively.
4. Why are some factors shown as quadratics?
Complex roots occur in conjugate pairs for real-coefficient polynomials. Grouping each pair into one quadratic keeps the factorization in real terms and makes the result easier to read.
5. What does the graph help me verify?
The graph shows the real-axis behavior of the quartic. You can inspect intercepts, turning regions, and whether each displayed real root matches an x-axis crossing or touchpoint.
6. Does the tool handle repeated roots?
Yes. Repeated roots are included in the numerical solution set. They may appear as repeated linear factors or repeated complex values, depending on the polynomial.
7. What does the residual column mean?
The residual is |P(root)| after substituting each recovered root back into the quartic. Very small residuals confirm that the computed root closely satisfies the polynomial.
8. Why use the CSV and PDF exports?
CSV is useful for records and spreadsheet work. PDF is useful for homework, reports, tutoring notes, and clean printing of the final factoring result.