Calculator Input
Enter whole-number coefficients from highest degree to constant term.
Example Data Table
| Polynomial | Leading Coefficient | Constant Term | Possible Candidates | Actual Rational Roots |
|---|---|---|---|---|
| x3 - 6x2 + 11x - 6 | 1 | -6 | ±1, ±2, ±3, ±6 | 1, 2, 3 |
| 2x3 - 3x2 - 8x + 12 | 2 | 12 | ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2 | 2, -2, 3/2 |
| x4 - 5x3 + 6x2 | 1 | 0 | 0 first, then theorem on x2 - 5x + 6 | 0, 0, 2, 3 |
Formula Used
Rational Root Theorem Rule
If a polynomial has integer coefficients, every rational root must have the form:
x = ± p / q
Here, p divides the constant term and q divides the leading coefficient.
Exact Candidate Test
The calculator checks each candidate using the scaled value:
qnP(p/q)
A result of zero confirms the candidate is a true rational root.
Zero Root Handling
If the constant term is zero, the calculator removes x factors first. Then it applies the theorem to the reduced polynomial.
How to Use This Calculator
- Enter coefficients from highest power to constant term.
- Use commas or spaces between whole-number coefficients.
- Set the graph minimum and maximum x-values.
- Choose graph samples and decimal places.
- Press the calculate button.
- Review factor lists and possible candidates.
- Check the test table for confirmed rational roots.
- Use the graph and exports for review or reports.
FAQs
1. What does this calculator actually find?
It lists possible rational roots from the theorem and tests each one. Confirmed candidates are shown as actual rational roots in the result table.
2. Does the theorem find every root?
No. It only identifies possible rational roots. Irrational and complex roots require other methods, such as factoring, graphing, or numerical solving.
3. Can I enter decimal coefficients?
This tool expects integer coefficients. Decimal coefficients should be cleared first by multiplying through by a common denominator before using the theorem.
4. Why is zero treated separately?
When the constant term is zero, x is automatically a factor. The calculator removes zero roots first, then applies the theorem to what remains.
5. Why are duplicate fractions removed?
Equivalent fractions represent the same candidate. The calculator simplifies them, so 2/4 and 1/2 appear only once in the final list.
6. What does qⁿP(p/q) mean?
It is a scaled exact test value. If qⁿP(p/q) equals zero, the candidate p/q is a genuine rational root.
7. Can the graph prove a root by itself?
The graph helps you see intercepts and overall behavior. The exact confirmation still comes from the theorem test table, not the picture alone.
8. What if no rational roots are confirmed?
Then the polynomial may only have irrational or complex roots, or it may need another algebraic method to continue factoring.