Polynomial Solver
Find rational roots of a polynomial up to degree 6 using the rational root theorem.
About Polynomial Solver
The Rational Root Theorem states that any rational root p/q of a polynomial with integer coefficients must have p dividing the constant term and q dividing the leading coefficient. This calculator tests all such candidates and deflates the polynomial via synthetic division when a root is found.
Formula
Rational Root Theorem
Candidates: ±(factors of constant term) / (factors of leading coefficient)
- p
- Factor of constant term
- q
- Factor of leading coefficient
Worked example
With the default values below, Polynomial Solver returns Rational Roots Found of 1, 2, 3. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Coefficients (highest to lowest degree)
- 1,-6,11,-6
Results
- Rational Roots Found
- 1, 2, 3
- Degree
- 3
- Candidates Tested
- 8
Step by step
- 1Values used
- 2Rational Root Theorem
- 3Rational Roots Found= 1, 2, 3
- 4Degree= 3
- 5Candidates Tested= 8
Assumptions and limits
- The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What if the polynomial has irrational roots?
This calculator only finds rational roots. Irrational or complex roots will not be detected — use the cubic or quartic solver for those.
What format should the coefficients be in?
Enter comma-separated numbers from the highest degree to the lowest. For x³−6x²+11x−6, enter 1,-6,11,-6.