Polynomial Calculator
Polynomial Root & Factor Calculator works out Degree and Leading coefficient from Input style, Coefficients and Coefficient order. The default figures on this page give all roots of 1, 2, 3; change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
6 steps
- Polynomial: x^3 - 6x^2 + 11x - 6 (degree 3).
- Rational Root Theorem: testing p/q with p dividing the constant term and q dividing the leading coefficient finds 1, 2, 3.
- Dividing those factors out leaves 1.
- Roots: 1, 2, 3.
- Factorisation over the reals: (x - 1)(x - 2)(x - 3).
- Vieta check: sum of roots = -a(n-1)/a(n) = 6, product = (-1)^3 a0/a(n) = 6.
Real rootstable
Root tabletable
| index | real | imaginary | display | type | residual |
|---|---|---|---|---|---|
| 1 | 1 | 0 | 1 | real | 0 |
| 2 | 2 | 0 | 2 | real | 0 |
| 3 | 3 | 0 | 3 | real | 0 |
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
Sources and references
3 sources
What the Polynomial Root & Factor Calculator calculates
Finds every real and complex root of a polynomial of any degree, lists the exact rational roots, and writes the factorisation over the real numbers.
How to use this Polynomial Root & Factor Calculator
Enter Input style, Coefficients, Coefficient order and Polynomial. Degree updates as you type, and “Use a worked example” loads checked figures you can adjust. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind Degree, Leading coefficient and All roots.
The method follows OpenStax, College Algebra 2e, Section 5.5: Zeros of Polynomial Functions (Rational Zero Theorem, Factor Theorem, Descartes’ Rule of Signs), Weierstrass (Durand-Kerner) method for simultaneous polynomial root finding; see Petkovic, Iterative Methods for Simultaneous Inclusion of Polynomial Zeros and Viète’s formulas relating coefficients to sums and products of roots.
x^3 - 6x^2 + 11x - 6 example
With Coefficients: 1,-6,11,-6, the calculator returns Degree: 3, Real roots (with multiplicity): 3, Smallest real root: 1. This example is checked against an independent source in the test suite.
What you can compare
Change one input at a time to see how it affects Degree, Leading coefficient and All roots. The inputs you can vary are Input style, Coefficients, Coefficient order, Polynomial, Variable and Digits shown.
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
See something wrong? Report a problem with this calculator
Frequently asked questions
Can you do polynomials on a calculator?
Finds every real and complex root of a polynomial of any degree, lists the exact rational roots, and writes the factorisation over the real numbers. Put your own figures in above and the working updates beside the result. For scale, the default figures on this page give 1, 2, 3.
How to calculate polynomial in calculator?
Finds every real and complex root of a polynomial of any degree, lists the exact rational roots, and writes the factorisation over the real numbers. Open “How this is worked out” beside the result to see each step for your own figures.
Can a TI-84 solve polynomials?
It needs Input style, Coefficients, Coefficient order, Polynomial, Variable and Digits shown. Nothing else is required, and the figures stay in your browser.
How to calculate a polynomial?
The result gives Degree, Leading coefficient, All roots, Real roots, Real roots (with multiplicity) and Distinct real roots, each labelled with its unit so the figure can be read on its own.
What information do I need for the Polynomial Root & Factor Calculator?
You need Input style, Coefficients, Coefficient order, Polynomial, Variable and Digits shown. The page opens with default figures that give all roots of 1, 2, 3, so you can see the scale before entering your own.
What does this Polynomial Root & Factor Calculator show?
It shows Degree, Leading coefficient, All roots, Real roots, Real roots (with multiplicity), Distinct real roots, Non-real roots and Smallest real root, with the working and any warnings for your entries.