Enter coefficients and inspect possible rational roots. See exact evaluations root checks and exports instantly. Solve equations with organized steps and practical examples today.
| Equation | Coefficients | Candidate set | One rational root |
|---|---|---|---|
| x^3 - 6x^2 + 11x - 6 = 0 | 1, -6, 11, -6 | ±1, ±2, ±3, ±6 | 1 |
| 2x^3 - x^2 - 8x + 4 = 0 | 2, -1, -8, 4 | ±1, ±2, ±4, ±1/2 | 2 |
| 3x^2 - 10x + 3 = 0 | 3, -10, 3 | ±1, ±3, ±1/3 | 3 |
The calculator applies the Rational Root Theorem. For a polynomial anxn + an-1xn-1 + ... + a0 = 0, every rational root has the form p/q.
Here, p is a factor of the constant term a0. The value q is a factor of the leading coefficient an. Each unique candidate ±p/q is tested by exact substitution.
The exact numerator check is anpn + an-1pn-1q + ... + a0qn. If that value equals zero, the candidate is a rational root.
This one rational root of the given equation calculator helps students, teachers, tutors, and exam learners test polynomial equations with speed and structure. Many algebra questions ask for one rational root before factorization continues. Manual checking works, but it often becomes slow when an equation has many valid candidates. This page reduces that effort and keeps every step readable. The layout is simple, so attention stays on the equation, the theorem, and the final verified root.
The calculator uses the Rational Root Theorem. It starts with the leading coefficient and the constant term. Then it collects their factors and builds valid candidates in the form plus or minus p over q. Every reduced candidate is tested with exact arithmetic, not rough rounding. That matters because exact checks avoid decimal error and clearly show whether the substituted value makes the polynomial equal zero. When a root is found, the page also shows a quotient after division.
The result section is useful for both learning and checking. You can see the entered equation, the factor lists for p and q, the valid candidate rule, and the first rational root discovered by the calculator. The candidate table adds more detail. It displays each possible value, a decimal view, the exact test expression, and the final status. This makes the tool helpful for homework review, classroom demonstrations, self study, revision sessions, and worked examples.
Use this calculator for quadratic, cubic, quartic, and higher degree expressions with integer coefficients. It is especially helpful when you want a fast algebra checker, a theorem practice helper, or a clean report for notes. The CSV and PDF export buttons support lesson planning, assignment records, and printable study material. Because the page also includes formula notes, usage guidance, and sample data, it can serve as both a calculator and a compact learning reference.
It also supports step based verification in a way that matches common algebra instruction. Instead of only showing an answer, it shows why a candidate works. That transparency builds confidence and helps users prepare for quizzes, tests, interviews, and daily practice where accurate root checking matters. It saves time while reinforcing correct theorem based polynomial analysis.
It finds one rational root from a polynomial with integer coefficients. It also lists valid theorem candidates, tests each one exactly, and shows whether a rational root exists.
A rational root is a solution that can be written as a fraction of two integers. Whole numbers are also rational roots because they can be written over one.
Yes. It creates fraction candidates using factors of the constant term and factors of the leading coefficient. Then it tests each reduced fraction exactly.
They come from the Rational Root Theorem. The value p divides the constant term, and q divides the leading coefficient. Any rational root must match that pattern.
Yes. In that case, it reports that no rational root was found among valid candidates. The polynomial may still have irrational or complex roots.
Enter coefficients from the highest power down to the constant term. For x^3 - 6x^2 + 11x - 6, enter 1, -6, 11, and -6.
If a rational root is found, the calculator divides the polynomial by x minus that root. The quotient helps you continue solving the remaining equation.
Export when you need a printable summary, a worksheet record, or a file for class notes. The download keeps the equation, detected root, and tested candidates.
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.