Enter coefficients to list possible rational zeros. See factor pairs, reduced fractions, and root tests. Use this calculator for faster polynomial analysis and practice.
Possible Rational Zeros = ± p / q
Here, p is any factor of the constant term, and q is any factor of the leading coefficient.
Every fraction must be reduced to lowest terms before listing unique candidates.
If the constant term is 0, then 0 is a rational root. Factor out the variable first. Then apply the theorem to the reduced polynomial.
1. Enter polynomial coefficients in descending order.
2. Use commas or spaces between values.
3. Keep all coefficients as integers.
4. Click Calculate.
5. Review factors of the constant term and leading coefficient.
6. Study the possible rational zeros list.
7. Check the evaluation table to see which candidates are true roots.
8. Download the result table as CSV or PDF when needed.
| Polynomial | Leading Coefficient | Constant Term | p Factors | q Factors | Possible Rational Zeros | Confirmed Rational Roots |
|---|---|---|---|---|---|---|
| 2x3 - 3x2 - 8x + 12 | 2 | 12 | 1, 2, 3, 4, 6, 12 | 1, 2 | ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2 | -2, 3/2, 2 |
This calculator helps you analyze polynomial equations with integer coefficients. It applies the Rational Zero Theorem to build a clean list of possible rational roots. You enter the coefficients once. The tool returns candidate zeros, exact polynomial values, and confirmed rational roots. This makes polynomial testing faster and more organized.
The Rational Zero Theorem is useful in algebra, precalculus, and polynomial factorization. It narrows the search space. Instead of guessing many values, you only test fractions built from factors of the constant term and the leading coefficient. This saves time and improves accuracy during homework, exam preparation, and classroom demonstrations.
The results section lists the polynomial, degree, leading coefficient, and constant term. It also shows factor sets for p and q. From there, the calculator builds all unique reduced fractions. Each candidate is checked by direct evaluation. The final table tells you which values are actual rational roots and which values are only possibilities.
Many students can state the theorem but still struggle with organized testing. This page solves that problem. It keeps the setup clear. It reduces duplicate fractions. It highlights true roots quickly. If the constant term is zero, the calculator also explains that zero is a rational root before continuing with the reduced polynomial.
You can use this rational zero theorem calculator for algebra practice, factorization review, and root checking. The CSV export supports record keeping. The PDF export helps with printing or sharing. The example table and formula section also make this page a simple learning reference for polynomial root analysis.
It lists possible rational zeros for a polynomial with integer coefficients. It also evaluates each candidate and marks true rational roots.
Yes. The Rational Zero Theorem is based on integer coefficients. Decimal inputs do not fit the standard theorem setup used here.
The theorem gives candidates only. A candidate becomes a true root only if substituting it into the polynomial makes the result equal zero.
Then zero is a rational root. The calculator notes that first and then applies the theorem to the reduced polynomial after factoring out the variable.
It confirms rational roots from the candidate list. Full factorization may need extra algebra steps, especially when irrational or complex roots remain.
Yes. A rational root can have multiplicity greater than one. This calculator confirms root values, while multiplicity usually needs further division or factoring work.
Enter them from the highest power down to the constant term. For example, x² - 5x + 6 becomes 1, -5, 6.
Reduced fractions remove duplicates. For example, 2/2 and 1/1 represent the same candidate. Simplifying keeps the candidate list accurate and shorter.