Find likely rational roots from entered coefficients fast. Review factors, candidates, and exact substituted values. Understand root testing before deeper factoring or division methods.
| Polynomial | Coefficients | Possible Rational Zeros | Actual Rational Roots |
|---|---|---|---|
| x3 - 6x2 + 11x - 6 | 1, -6, 11, -6 | ±1, ±2, ±3, ±6 | 1, 2, 3 |
| 2x3 - 3x2 - 11x + 6 | 2, -3, -11, 6 | ±1, ±2, ±3, ±6, ±1/2, ±3/2 | 3, -2, 1/2 |
| x4 - 5x2 + 4 | 1, 0, -5, 0, 4 | ±1, ±2, ±4 | -2, -1, 1, 2 |
For a polynomial written as:
anxn + an-1xn-1 + ... + a1x + a0
Any rational root must be written in lowest terms as p/q.
If the constant term is zero, then zero is a root. The polynomial is reduced first, then tested again.
The Rational Zero Root Theorem helps you test possible rational roots of a polynomial. It is a standard tool in algebra. Students use it before factoring, synthetic division, or graphing. Teachers also use it to show structured polynomial analysis.
This calculator accepts polynomial coefficients in descending order. It reads the leading coefficient and constant term. Then it builds the factor sets needed by the theorem. From those sets, it creates every simplified rational candidate. It also checks each candidate by substitution.
The possible rational zeros are only candidates. They are not guaranteed roots. A number becomes a confirmed rational root only when the polynomial evaluates to zero. That is why the evaluation table matters. It separates possible answers from actual answers.
Some fractions repeat in different forms. For example, 2/4 and 1/2 represent the same value. The calculator simplifies fractions first. That keeps the candidate list short and useful. It also makes classroom checking easier.
If the constant term equals zero, then zero is already a root. In that case, the polynomial contains a factor of the variable. The calculator removes that zero root first. Then it applies the theorem to the reduced polynomial. This gives a cleaner set of remaining candidates.
This page works well for homework, exam review, and lesson planning. It is also helpful when you want to compare several polynomials quickly. Since the output can be exported, you can keep a record of tested values and confirmed roots.
The theorem only lists rational possibilities. Some polynomials have irrational or complex roots instead. When no candidate works, the polynomial may still have real roots. They are just not rational values. In that case, use factoring methods, graphing, or numerical root tools next.
It finds all possible rational zeros from the theorem, then checks each one by substitution. It also shows confirmed rational roots when any candidate makes the polynomial equal zero.
Yes. Enter coefficients from the highest degree term down to the constant term. Missing terms should use zero coefficients, such as 1,0,-5,0,4.
No. The Rational Root Theorem is normally applied to integer coefficients. This calculator validates integers only so the factor logic stays correct.
The theorem produces candidates, not guaranteed answers. Each candidate must still be tested. Only the values that make the polynomial equal zero are actual rational roots.
Zero is automatically a rational root. The calculator removes that factor first and then applies the theorem again to the reduced polynomial.
No. It is designed for rational root candidates and rational root checking. Irrational or complex roots need other methods after this step.
Simplification removes duplicate fractions such as 2/4 and 1/2. That makes the result easier to read and avoids repeated testing.
The CSV export saves the evaluation table for records or classwork. The PDF option lets you print or save the page for offline review.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.