Enter Polynomial Details
Example Data Table
| Polynomial | Input coefficients | Possible candidates | Verified rational zeros |
|---|---|---|---|
| 2x^3 - 3x^2 - 11x + 6 | 2, -3, -11, 6 | ±1, ±2, ±3, ±6, ±1/2, ±3/2 | -2, 1/2, 3 |
| x^3 - 6x^2 + 11x - 6 | 1, -6, 11, -6 | ±1, ±2, ±3, ±6 | 1, 2, 3 |
| x^4 - 5x^2 + 4 | 1, 0, -5, 0, 4 | ±1, ±2, ±4 | -2, -1, 1, 2 |
Formula Used
The calculator uses the Rational Root Theorem. For a polynomial with integer coefficients:
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
If p/q is a rational zero in lowest terms, then p must divide the constant term a₀, and q must divide the leading coefficient aₙ.
Possible rational zeros = ± factors of a₀ / factors of aₙ
Each candidate is tested in P(x). A verified zero is then used in synthetic division. The process repeats to find multiplicity and the remaining quotient.
How to Use This Calculator
- Enter every coefficient of the polynomial.
- Select whether coefficients are listed from highest power or constant term.
- Choose a tolerance and decimal display level.
- Press the calculate button to test rational candidates.
- Review verified zeros, multiplicities, and synthetic division steps.
- Use CSV or PDF export for study notes and reports.
Rational Zeros Guide
What This Tool Finds
A rational zero is a root that can be written as a fraction. It makes a polynomial equal zero. This calculator uses the Rational Root Theorem to build a safe candidate list. Then it tests every candidate with direct substitution and synthetic division.
Why Candidate Testing Matters
The method works best when coefficients are integers. The leading coefficient gives possible denominators. The constant term gives possible numerators. Every reduced fraction is tested with both signs. This avoids random guessing and keeps the work organized.
Advanced Result Details
Advanced options help you check answers with more control. You can choose coefficient order, tolerance, decimals, and displayed candidate limits. The result includes verified zeros, multiplicities, synthetic steps, and the remaining quotient. It also shows sign change estimates, which help compare positive and negative root possibilities.
Export and Study Use
Use the tool when factoring higher degree equations. It is helpful for algebra homework, tutoring, engineering models, and quick lesson checks. You can export the candidate table as a CSV file. You can also save a PDF summary for records or reports.
Multiplicity and Quotients
A zero with multiplicity greater than one appears more than once in factor form. The calculator divides the polynomial again after a zero is found. This shows whether the same rational root repeats. If a quotient remains, its roots may be irrational or complex. The tool reports that part instead of pretending the factorization is complete.
Coefficient Entry Tips
Always enter coefficients for all powers. Use zero for missing terms. For example, x^4 - 5x^2 + 4 should be entered as 1, 0, -5, 0, 4 in descending order. This keeps each term aligned with its power.
Check the Output
The calculator is not a replacement for mathematical judgment. It is a step helper. Review the possible roots, the tested values, and the final quotient. If no rational zeros appear, the polynomial may still have real or complex zeros. They are simply not rational zeros found by this theorem.
Reading the Output
Read the output from top to bottom. Start with the polynomial summary. Then check the candidate list. A value near zero means the candidate passed. The synthetic row confirms the division. The quotient shows what remains. This flow makes mistakes easier to find before you copy the answer correctly.
FAQs
1. What is a rational zero?
A rational zero is a polynomial root that can be written as p/q, where p and q are integers and q is not zero. When substituted into the polynomial, it makes the polynomial value equal zero.
2. What coefficients should I enter?
Enter coefficients for every power of the variable. Include zero where a term is missing. For x^3 + 2x - 5, enter 1, 0, 2, -5 in descending order.
3. Why are possible zeros only candidates?
The theorem gives a list of values that could be rational zeros. Each candidate still needs testing. Many candidates fail because they do not make the polynomial equal zero.
4. Does the tool find irrational zeros?
No. This calculator focuses on rational zeros from the Rational Root Theorem. A remaining quotient may still have irrational or complex roots that require other methods.
5. What does multiplicity mean?
Multiplicity shows how many times the same zero repeats as a factor. If x = 2 has multiplicity 3, then the polynomial contains the factor (x - 2)^3.
6. Why use synthetic division?
Synthetic division quickly confirms a zero and reduces the polynomial degree. It also shows the quotient left after dividing by the matching linear factor.
7. What tolerance should I use?
The default tolerance works for most integer coefficient problems. Increase it slightly if rounding creates tiny decimal remainders. Avoid large tolerance values because they can mark weak candidates as zeros.
8. Can I export my result?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a printable summary that includes key results and candidate tests.