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Rational Zeros Theorem Calculator — result sheet
List rational-root candidates for an integer polynomial and test them exactly to identify the rational zeros of a linear, quadratic, or cubic polynomial.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: For an integer polynomial with leading coefficient a and constant d, every rational zero p/q in lowest terms has p dividing d and q dividing a; each candidate is then checked by exact substitution.
The calculator accepts coefficients through degree three, trims leading zero coefficients to determine the actual degree, generates reduced ±p/q candidates, and evaluates them with integer arithmetic so a decimal rounding artifact cannot be mistaken for a zero.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Coefficient a (x³) — minimum -1000000; maximum 1000000
- Coefficient b (x²) — minimum -1000000; maximum 1000000
- Coefficient c (x) — minimum -1000000; maximum 1000000
- Constant d — minimum -1000000; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Coefficient a (x³) | 1 |
| Coefficient b (x²) | -6 |
| Coefficient c (x) | 11 |
| Constant d | -6 |
The polynomial x³−6x²+11x−6 has rational zeros x = 1, 2, and 3.
Assumptions and limits
- All coefficients are integers; a leading zero is allowed when the polynomial is actually quadratic or linear.
- The zero polynomial is rejected because every number would be a zero and the rational-zero list would not be finite.
- Only rational zeros are reported; irrational or complex zeros are not approximated by this tool.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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