Analyze polynomial structure, roots, and factors fast instantly. Enter coefficients, compare forms, and verify solutions. Clear results, graphs, downloads, and algebra guidance always together.
Use integer coefficients for the most reliable exact factorization over rational numbers.
This sample shows how one common polynomial is entered and factored.
| Degree | Coefficients | Polynomial | Factored Form | Real Roots |
|---|---|---|---|---|
| 3 | 1, -6, 11, -6 | x3 - 6x2 + 11x - 6 | (x - 1)(x - 2)(x - 3) | 1, 2, 3 |
| 2 | 2, 5, 2 | 2x2 + 5x + 2 | (2x + 1)(x + 1) | -1/2, -1 |
| 4 | 1, 0, -5, 0, 4 | x4 - 5x2 + 4 | (x - 2)(x - 1)(x + 1)(x + 2) | -2, -1, 1, 2 |
For a polynomial in standard form P(x) = anxn + an-1xn-1 + ... + a1x + a0, this calculator follows a structured factoring workflow.
1. Extract the greatest common factor from all coefficients whenever possible.
2. Apply the Rational Root Theorem. Any rational root has the form p/q, where p divides the constant term and q divides the leading coefficient.
3. Each valid root creates a linear factor (qx - p).
4. Divide the polynomial by each discovered factor to reduce the degree.
5. For a remaining quadratic ax2 + bx + c, inspect the discriminant Δ = b2 - 4ac.
6. If the discriminant is negative, no real quadratic roots exist. If it is a perfect square, the quadratic factors exactly over the rationals.
Choose the polynomial degree first. Then enter the coefficients from the highest power down to the constant term.
Use integer coefficients whenever you want exact factorization over rational numbers. Change the variable symbol if needed.
Set a graph interval to inspect how the polynomial behaves across a chosen x-range.
Keep the GCF option enabled to pull out a common factor automatically. Keep the sign option enabled if you prefer a positive leading coefficient.
Click Factor Polynomial to show the result above the form. Review the standard form, factorization, candidate roots, step list, notes, and Plotly graph.
Use the CSV and PDF buttons to save the current result summary for later reference.
This version handles linear, quadratic, cubic, and quartic polynomials entered with integer coefficients. It is best for factorization over rational numbers, while still reporting useful remainder and discriminant information when complete exact factorization is not available.
The Rational Root Theorem depends on integer divisors of the constant term and leading coefficient. That makes exact testing reliable and step-based. Decimal inputs may hide exact rational structure and reduce the usefulness of divisor-based candidate generation.
It pulls out any shared integer factor from all coefficients before further factoring begins. This simplifies the remaining polynomial and often reveals cleaner linear or quadratic factors afterward.
Some expressions are irreducible over the rationals, even though they may have irrational or complex roots. In those cases, the calculator still shows the simplified remainder, real-root notes for quadratics, and a graph for visual analysis.
The calculator builds candidates using ± factors of the constant term over factors of the leading coefficient. Each candidate is substituted exactly, and a successful test creates a linear factor and triggers degree reduction.
The graph helps you verify turning behavior, intercepts, and likely real roots. It is especially useful when a polynomial does not fully factor into rational linear terms, because the curve still reveals sign changes and shape.
They save a compact summary of the polynomial, factored form, real roots, and graph interval. This is helpful for study notes, assignments, tutoring sessions, or quick verification records.
Yes. The displayed steps show the standard form, extracted factors, rational-root discoveries, and any leftover factor. This makes the result easier to inspect and compare against your original input.
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.