Calculator Form
Enter integer coefficients from highest degree to constant. Example: 2, -3, -8, 12
Example Data Table
| Example Polynomial | Candidate Set | Confirmed Rational Roots | Notes |
|---|---|---|---|
| 2x3 - 3x2 - 8x + 12 | ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2 | -2, 3/2, 2 | Enter coefficients as: 2, -3, -8, 12 |
| x3 - 6x2 + 11x - 6 | ±1, ±2, ±3, ±6 | 1, 2, 3 | Enter coefficients as: 1, -6, 11, -6 |
| x3 - 2x + 4 | ±1, ±2, ±4 | -2 | One rational root appears in the theorem list. |
Formula Used
The calculator uses the Rational Root Theorem. For a polynomial with integer coefficients, every rational root can be written as ±p/q, where p divides the constant term and q divides the leading coefficient.
Each candidate is reduced to lowest terms and tested exactly. For r = p/q, the tool evaluates a common-denominator numerator. If that numerator equals zero, the candidate is a confirmed rational root.
This method finds rational roots only. A polynomial may still have irrational or complex roots even when no rational root is confirmed.
How to Use This Calculator
- Type coefficients from highest degree to constant.
- Use commas between all integers.
- Set the x-range for the graph.
- Choose plot points and decimal places.
- Press Find Rational Roots.
- Read the confirmed roots in the result summary.
- Review the candidate test table for exact checks.
- Download CSV or PDF when needed.
FAQs
1. What is a rational root?
A rational root is a solution that can be written as a fraction of integers. This calculator lists theorem-based candidates and tests each one exactly.
2. Does this work for any polynomial?
It works best for nonzero polynomials with integer coefficients. The theorem relies on integer leading and constant terms, though the graph still plots the entered polynomial.
3. Why can the graph cross near zero without a listed root?
The graph can suggest irrational or rounded intercepts. This calculator confirms only rational roots from the theorem-based candidate set.
4. Why does zero sometimes appear as a root?
If the constant term is zero, x is a factor. The calculator detects that immediately and reports 0 as a rational root candidate.
5. How are the candidates generated?
Candidates come from ±p/q, where p divides the constant term and q divides the leading coefficient. Equivalent fractions are reduced and duplicates removed.
6. What does the exact test numerator mean?
It is the integer numerator produced after substitution with a common denominator. A value of zero proves the candidate is an exact rational root.
7. Can I export the results?
Yes. Use CSV for spreadsheet-style review or PDF for a compact printable summary of the current polynomial and tested candidates.
8. What input format should I use?
Enter coefficients from highest degree to constant, separated by commas. For x2 - 5x + 6, type 1, -5, 6.