Rational Root Theorem Calculator
Enter the constant term and leading coefficient of a polynomial to list all possible rational roots given by the Rational Root Theorem.
Total candidate ±p/q values to test
- 1
Divisors of |constant term a₀|
|6| = 4p ∈ {1, 2, 3, 6} - 2
Divisors of |leading coefficient aₙ|
|2| = 2q ∈ {1, 2} - 3
Maximum ±p/q candidates
2 × 4 × 2 = 16Upper bound before reducing fractions and removing duplicates. - 4
Unique candidates after reduction
12
How does this calculator work?
By the Rational Root Theorem, every possible rational root of an integer-coefficient polynomial is ±p/q, where p divides the constant term and q divides the leading coefficient. This tool lists all such reduced, de-duplicated candidates and counts them, so you know exactly which values to test for actual roots.
Formula
How this is calculated
The Rational Root Theorem states that any rational root p/q (in lowest terms) of a polynomial with integer coefficients must have a numerator p that divides the constant term a0 and a denominator q that divides the leading coefficient an. Enter a0 (the constant term) and an (the leading coefficient); both must be nonzero integers, since dividing by zero is undefined and a zero constant term shifts the polynomial.
The calculator finds every positive divisor p of |a0| and every positive divisor q of |an|, then forms each fraction ±p/q. Each fraction is reduced to lowest terms using the greatest common divisor and duplicate values are removed, so for example 2/2 and 1/1 collapse to a single candidate. The remaining values are sorted from least to greatest.
These candidates are only the possible rational roots, not guaranteed roots: you still substitute each one into the full polynomial (or use synthetic division) to check which actually evaluate to zero. Irrational and complex roots are never captured by this list, and the count grows with the number of divisors of the two coefficients.
Frequently asked questions
No. It gives every value that could be a rational root. You must test each candidate in the polynomial; many will not be roots, and irrational or complex roots never appear.
The theorem requires integer coefficients. A zero leading coefficient means it is not actually the leading term, and a zero constant term makes 0 a root and changes the divisor structure.
Candidates are reduced to lowest terms and de-duplicated, so equivalent fractions like 4/2 and 2/1 are shown only once as a single value.
Also known as
TG we-Calculate Editorial Team. (2026). Rational Root Theorem Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/rational-root-theorem-calculator
TG we-Calculate Editorial Team. "Rational Root Theorem Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/rational-root-theorem-calculator.
TG we-Calculate Editorial Team, "Rational Root Theorem Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/rational-root-theorem-calculator
@misc{wecalculate_rational_root_theorem_calculator, title = {Rational Root Theorem Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/rational-root-theorem-calculator}}, year = {2026}, note = {TG we-Calculate} }
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