Rational Zeros Calculator
Enter the degree and integer coefficients of your polynomial. The calculator applies the Rational Root Theorem to list every possible rational zero (the p/q candidates), then tests each one and identifies which are actual roots. A show-your-work panel walks through the synthetic division for each confirmed root.
Formula
Worked example
For P(x) = x^3 - 6x^2 + 11x - 6: constant = -6 (divisors: ±1, ±2, ±3, ±6), leading = 1 (divisors: ±1). Candidates: ±1, ±2, ±3, ±6. Testing: P(1) = 1-6+11-6 = 0, P(2) = 8-24+22-6 = 0, P(3) = 27-54+33-6 = 0. Actual zeros: x = 1, 2, 3.
What the Rational Root Theorem says
If a polynomial P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 has integer coefficients, then every rational zero of P(x) must be expressible as p/q in lowest terms, where p is a factor of the constant term a_0 and q is a factor of the leading coefficient a_n. This gives a finite, testable list of candidates. The theorem does not guarantee any rational roots exist - it only narrows the search to a small set of fractions. Once you have the list, you verify each candidate by evaluating P(p/q) or by performing synthetic division. If the remainder is zero, p/q is a root; otherwise it is not.
How to find rational zeros step by step
Step 1 - Write down the constant term and the leading coefficient. Step 2 - List all positive integer divisors of each. Step 3 - Form every unique fraction p/q where p comes from the divisors of the constant and q from the divisors of the leading coefficient, including both positive and negative versions. Reduce each fraction to lowest terms and remove duplicates. Step 4 - Test each candidate using synthetic division or direct substitution. Any candidate that gives a remainder of zero is an actual rational root. Step 5 - After finding a root r, divide the polynomial by (x - r) using synthetic division to reduce its degree, then repeat the process on the quotient until you reach a quadratic or linear factor.
Synthetic division: verifying and reducing
Synthetic division is the fastest way to both test a candidate and reduce the polynomial at the same time. Write the coefficients of P(x) from highest to lowest power (use 0 for any missing term). Bring down the leading coefficient. Multiply it by the candidate value r and add to the next coefficient. Repeat across all coefficients. The last number in the bottom row is the remainder - if it is zero, r is a root and the remaining numbers are the coefficients of the reduced quotient polynomial of degree n-1. Dividing out each confirmed root and repeating on the quotient is efficient because the candidate list for the smaller polynomial is usually shorter.
When the theorem cannot find all roots
The Rational Root Theorem only covers rational roots. Many polynomials have irrational roots (like square roots) or complex roots (involving i) that do not appear in the p/q candidate list. For a degree-2 polynomial, any roots not found rationally can be found exactly with the quadratic formula. For degree 3 or 4, Cardano's and Ferrari's formulas apply but are complex. For degree 5 and above, the Abel-Ruffini theorem shows there is no general radical formula, so numerical methods (Newton's method, bisection) or computer algebra systems are needed for the non-rational roots. The rational zeros this calculator finds can still be factored out first, which reduces the problem to a lower-degree polynomial.
Rational Root Theorem - p/q candidate structure
| Polynomial type | Leading coefficient | Constant term | Candidate form |
|---|---|---|---|
| Monic (a_n = 1) | 1 | a_0 | integers: ±1, ±2, ... (divisors of a_0) |
| General integer | a_n | a_0 | ±(divisors of a_0) / ±(divisors of a_n) |
| Leading coeff = 2 | 2 | a_0 | integers and halves: p/1 and p/2 |
| Leading coeff = 6 | 6 | a_0 | p/1, p/2, p/3, p/6 (four sets) |
Every rational zero of a polynomial with integer coefficients must have the form p/q in lowest terms, where p divides the constant term and q divides the leading coefficient.
Frequently asked questions
What is the Rational Root Theorem?
The Rational Root Theorem (also called the Rational Zero Theorem) states that if a polynomial with integer coefficients has a rational root p/q in lowest terms, then p must divide the constant term and q must divide the leading coefficient. This limits the search for rational roots to a finite list of fractions that can be systematically tested.
How do I use this calculator?
Select the degree of your polynomial, then enter the integer coefficients from highest power to lowest (including zeros for any missing terms). The calculator lists all p/q candidates, tests each one, and shows you which are actual rational zeros. The show-your-work panel walks through the synthetic division for each confirmed root.
What if my polynomial has non-integer coefficients?
The Rational Root Theorem requires integer coefficients. If your polynomial has fractional coefficients, multiply every term by the least common denominator (LCD) to convert them to integers first. For example, (1/2)x^2 + 3x - 2 becomes x^2 + 6x - 4 after multiplying by 2. The roots of the scaled polynomial are identical to those of the original.
Why might the calculator show no rational zeros?
A polynomial can have no rational roots even though candidates exist. All its roots may be irrational (such as square roots or cube roots) or complex (involving i). For example, x^2 + 1 has candidates x = ±1 from the theorem, but P(1) = 2 and P(-1) = 2, so neither is a root - the actual roots are the imaginary numbers i and -i.
What does the candidate count tell me?
The candidate count is the number of unique p/q fractions the theorem produces. For a monic polynomial (leading coefficient 1), candidates are just the integer divisors of the constant term, so the count is small. For polynomials with a large leading coefficient and large constant term, the list can be long. A larger list means more testing but does not mean more roots - a degree-3 polynomial has at most 3 roots regardless of how many candidates the theorem generates.
Can a rational zero appear more than once (multiplicity)?
Yes. If a root r appears k times - meaning (x - r)^k divides the polynomial - it is called a root of multiplicity k. The Rational Root Theorem still finds r as a candidate. After dividing out (x - r) once and confirming the quotient also has r as a root, it appears again in the actual zeros list. Graphically, a root of even multiplicity just touches the x-axis without crossing it.
How is rational zeros different from finding all roots?
Rational zeros are only the subset of roots that are rational numbers (integers and fractions). A polynomial of degree n has exactly n roots counting multiplicity, but some or all of those roots may be irrational or complex. This calculator finds only the rational ones. To find all roots, pair this result with the quadratic formula (for leftover degree-2 quotients) or numerical root-finding tools.