Rational Zeros Calculator
Enter a polynomial. The calculator applies the Rational Root Theorem: every rational zero p/q (lowest terms) has p dividing the constant term and q dividing the leading coefficient — then tests every candidate with exact synthetic division.
Accepts comma/space separated coefficients, fractions like 1/2, or expressions like 2x^3-3x^2-11x+6. Max degree 12. All computation is exact — no floating point.
Results
Possible rational zeros (±p/q)
Candidate tests (synthetic division)
| Candidate | Remainder | Root? |
|---|
Rational zeros found
Partial factorization
FAQ
What is the Rational Root Theorem?
For a polynomial with integer coefficients a_n x^n + ... + a_1 x + a_0, every rational root p/q (in lowest terms) satisfies: p divides the constant term a_0, and q divides the leading coefficient a_n. So all possible rational zeros are the combinations +/-p/q.
How do you find all rational zeros of a polynomial?
List all factors p of the constant term and all factors q of the leading coefficient, form every +/-p/q candidate, then test each one with synthetic division or direct substitution. Any candidate with remainder 0 is a rational zero; deflate the polynomial and repeat to find multiplicities.
What are the rational zeros of 2x^3 - 3x^2 - 11x + 6?
The candidates are +/-1, +/-2, +/-3, +/-6, +/-1/2, +/-3/2. Testing them gives the rational zeros x = 3, x = 1/2, and x = -2, so 2x^3 - 3x^2 - 11x + 6 = 2(x - 3)(x - 1/2)(x + 2).
Can a polynomial have no rational zeros?
Yes. For example x^2 - 2 has candidates +/-1 and +/-2, but none of them is a root; its zeros are the irrational numbers sqrt(2) and -sqrt(2). A degree-n polynomial can also have fewer than n real zeros overall.
Does this calculator handle fractions and zero roots?
Yes. Decimal or fractional coefficients are scaled to an equivalent integer-coefficient polynomial first, and a zero constant term is factored as x^m before the candidate test, so a root of 0 is reported with its multiplicity.
Free tool by NRTechStudio