How to use
Enter every polynomial coefficient from the highest power down to the constant term, including zeros for missing powers. For 2x³ − 3x² − 8x + 12, enter 2, -3, -8, 12. The gray coefficients are a working example. Focusing the field clears that example before you type, so sample data is not confused with your own input.
Select Calculate or press Enter to test the current polynomial. Editing the coefficients immediately removes the confirmed output and disables Copy. Clear leaves a genuinely empty field. The input accepts degree 1 through 6 and integer coefficients with absolute value at most 1,000,000.
How the Rational Root Theorem is used
For a polynomial with leading coefficient a and nonzero constant c, every rational root in lowest terms p/q must have p dividing c and q dividing a. The calculator enumerates positive and negative reduced combinations, removes duplicates, and substitutes each candidate using exact integer fraction arithmetic. A displayed root is therefore tested, not merely suggested by the theorem.
When the constant term is zero, x is factored first. The output records how many times zero is a root, then applies the divisor rule to the remaining polynomial. After each exact root is found, synthetic division is repeated, which preserves multiplicity: a double root appears twice rather than being silently collapsed.
Reading the result
Rational roots is the list of candidates that evaluated to zero, including repeats. Candidate list contains every reduced p/q value tested; it may be longer than the roots list because the theorem supplies possibilities, not guarantees. Zero-root multiplicity identifies the initial x factors. Exact checks shows P(r) = 0 for each reported value.
If Rational roots says none, the polynomial may still have irrational or complex roots. This focused page does not approximate those roots and does not claim the polynomial has no roots. The remaining factor is retained internally after exact synthetic division, but the page does not present a complete symbolic factorization.
Worked examples
For 2x³ − 3x² − 8x + 12, the possible values come from divisors of 12 over divisors of 2. Exact substitution finds −2, 3/2, and 2. Multiplying 2(x + 2)(x − 3/2)(x − 2) recovers the original polynomial.
For x² − 2x + 1, the only actual rational root is 1, but synthetic division finds it twice, so the result lists 1, 1. For 2x³ − x², zero is removed twice and the remaining linear factor gives 1/2. For x² + 1, the rational list is empty even though the polynomial has two complex roots.
Limits and troubleshooting
The degree limit and coefficient bound keep divisor enumeration and exact substitution responsive. A polynomial with a zero leading coefficient is rejected instead of silently changing the degree; remove the unintended leading entry yourself. Include internal zero coefficients so powers remain aligned. Decimal coefficients, fraction strings, variable names, and expanded expressions are outside this structured input.
Some permitted coefficients have many divisors. If the unique reduced candidate set exceeds 500, the calculator stops and asks for a smaller polynomial instead of freezing the browser. Copy result contains the tested candidates, verified rational roots, zero multiplicity, and checks as plain text. Use a numerical or symbolic solver when you need every real or complex root.
Output and study use
The candidate list is useful for showing the theorem’s search space, while the root list is the verified conclusion. In homework, record both steps: explain why p/q is eligible, then show the exact zero substitution or synthetic division. Do not report the candidate list as the solution set. The copied output contains no hidden precision or decimal approximation.
FAQ
Are all candidates roots?
No. The theorem creates a finite test list; only exact zeros appear as roots.
Why include a zero coefficient?
It preserves the power position, such as x³ + 0x² − x.
Are repeated roots repeated in the output?
Yes. Exact synthetic division records multiplicity.
Does none mean no real roots?
No. It means no rational roots were found.
Can I enter decimals?
No. This theorem workflow is restricted to integer coefficients.