Enter coefficients and inspect rational candidates with confidence. Track quotient terms, remainders, and zero checks. Export clean results for homework, revision, and classroom practice.
| Polynomial | Leading coefficient | Constant term | Candidate roots | Chosen divisor | Quotient | Remainder |
|---|---|---|---|---|---|---|
| x3 - 6x2 + 11x - 6 | 1 | -6 | ±1, ±2, ±3, ±6 | 2 | x2 - 4x + 3 | 0 |
Rational Zero Theorem: If a polynomial has integer coefficients, every rational zero can be written as p/q. The value p must divide the constant term. The value q must divide the leading coefficient.
Synthetic division rule: Bring down the first coefficient. Multiply it by c. Add that product to the next coefficient. Repeat across the row. The last value is the remainder. The earlier values form the quotient coefficients.
Exact zero check: A tested candidate p/q is a true rational zero when P(p/q) = 0. This calculator checks that exactly by evaluating the scaled numerator.
Polynomials often look harder than they are. A structured method makes them easier. This calculator lists possible rational zeros first. It then applies synthetic division quickly. You can test, compare, and confirm roots in one place. That saves class time and reduces copying mistakes. It also helps students understand how algebraic patterns connect.
The Rational Zero Theorem gives possible rational roots. It uses factors from the constant term. It also uses factors from the leading coefficient. Each possible root has the form p over q. Here, p divides the constant term. Here, q divides the leading coefficient. The theorem does not guarantee every candidate works. It creates a complete shortlist to test.
Synthetic division is a fast division shortcut. It works when dividing by x minus c. Bring down the leading coefficient first. Multiply by the test value. Add the next coefficient. Repeat until the last column appears. The final number is the remainder. A remainder of zero confirms a root. The earlier values become the quotient coefficients.
Many students guess roots randomly. That approach wastes time. Candidate lists narrow the search immediately. Reduced fractions also remove duplicates. Negative and positive possibilities are both important. Zero can be a root too. When the constant term is zero, x is a factor. Then the reduced polynomial can be tested again.
Use this tool during homework checks. Use it during lesson revision. Use it before graphing a polynomial function. It is also useful for factorization practice. Teachers can demonstrate every step live. Students can compare manual work with computed results. That makes error spotting much easier.
Start with the candidate root list. Next, review any confirmed rational zeros. Then inspect the synthetic division table. Check the quotient polynomial and remainder carefully. A zero remainder means exact divisibility. A nonzero remainder means the test value failed. Export options help save worked examples. The example table also shows a complete solved case. This supports exam preparation and independent algebra practice.
It first builds the complete rational candidate list from the leading coefficient and constant term. After that, it tests those candidates and performs synthetic division with your chosen value.
No. The theorem gives possible rational zeros only. A candidate becomes a real rational root only when evaluation or synthetic division gives a remainder of zero.
Yes. Enter negative whole numbers where needed. The calculator handles positive, negative, and zero coefficients while keeping the candidate search and synthetic division steps consistent.
The Rational Zero Theorem is stated for polynomials with integer coefficients. Whole-number input keeps the candidate list valid and ensures the factor logic works correctly.
If the constant term is zero, then zero is already a rational root. The calculator counts that factor and still generates more candidates from the reduced polynomial.
The remainder tells you whether your test value works. A remainder of zero confirms exact divisibility. Any other remainder shows the tested value is not a root.
Yes. You can enter values like 3/2 or -5/3 in the synthetic divisor field. The table is calculated with exact reduced fractions.
The export tools save the main result summary, candidate list, confirmed zeros, synthetic division rows, and evaluation data. That is useful for study notes and worked examples.
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.