Rational Root Theorem Calculator
Find all possible rational roots of a polynomial using the Rational Root Theorem.
p(x) = 2x^3 − 3x^2 − 2x + 3
All calculations run live in your browser. Roots verified by polynomial evaluation with tolerance 10⁻⁹.
What Is the Rational Root Theorem Calculator?
The Rational Root 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 converts an infinite search into a finite candidate list.
- ›Works only for polynomials with integer coefficients
- ›Lists candidates, not all candidates are actual roots
- ›Green-highlighted candidates in the tool are confirmed actual roots
- ›Synthetic division verifies each root and reduces polynomial degree
Formula
Rational Root Theorem
Possible roots = ±p/q
p = factors of constant term · q = factors of leading coefficient
If p(x) = aₙxⁿ+…+a₀
Integer coefficients required
Candidate: ±p/q
p | a₀, q | aₙ (in lowest terms)
Test via substitution
p(c) = 0 → c is a root
Verify via synthetic div
Remainder = 0 confirms root
How to Use
- 1Enter coefficients from highest degree to lowest, space-separated (e.g. "2 -3 -2 3" for 2x³−3x²−2x+3)
- 2Click "Find Rational Roots", the tool extracts leading and constant terms, finds all factors
- 3Review the ±p/q candidate list, green badges are confirmed actual roots
- 4Click any green badge to run synthetic division and see step-by-step verification
- 5Use the quotient polynomial to continue factoring for remaining roots
Example Calculation
Find rational roots of 2x³ − 3x² − 2x + 3:
Constant term = 3 → factors of |3|: 1, 3
Candidates ±p/q: ±1, ±3, ±1/2, ±3/2
Test x=1: 2−3−2+3 = 0 ✓
Test x=3: 54−27−6+3 = 24 ✗
Test x=3/2: 2(27/8)−3(9/4)−3+3 = 0 ✓
Test x=−1/2: 2(−1/8)−3(1/4)+1+3 = 0 ✓
Actual roots: x = 1, 3/2, −1/2
After finding x = 1
Synthetic division by (x−1) gives 2x²−x−3, which factors as (2x−3)(x+1).
Full factorization: (x−1)(2x−3)(x+1)
Understanding Rational Root Theorem
Candidate Count vs. Polynomial Degree
| Polynomial | Leading | Constant | Max candidates |
|---|---|---|---|
| x³ − 6x² + 11x − 6 | 1 | 6 | 8 (±1,±2,±3,±6) |
| 2x³ − 3x² − 2x + 3 | 2 | 3 | 8 (±1,±3,±1/2,±3/2) |
| x⁴ − 5x² + 4 | 1 | 4 | 6 (±1,±2,±4) |
| 6x³ − 11x² − 3x + 2 | 6 | 2 | 16 |
Frequently Asked Questions
What is the Rational Root Theorem?
A foundational theorem in algebra that reduces polynomial root-finding from an infinite problem to a finite one. It works by exploiting the structure of integer-coefficient polynomials.
- ›Applies only to polynomials with integer coefficients
- ›p divides the constant term a₀; q divides the leading coefficient aₙ
- ›The root must be in lowest terms, GCD(p, q) = 1
- ›Negative candidates (−p/q) must also be tested
Does the theorem guarantee rational roots exist?
The Rational Root Theorem is a necessary condition, not a sufficient one. It guarantees: IF a rational root exists, it must be in the candidate list. But the candidate list may be entirely non-roots.
- ›x² − 2 = 0 has candidates ±1, ±2, none are roots; actual roots are ±√2
- ›x² + 1 = 0 has candidates ±1, neither is a root; actual roots are ±i
- ›All candidates failing means the polynomial has no rational roots
- ›In that case, use numerical methods (Newton's method) to find irrational roots
How do I test a candidate root?
Two equivalent methods work for testing candidates:
- ›Direct substitution: compute p(c), if p(c) = 0, c is a root
- ›Synthetic division: divide p(x) by (x − c); if remainder = 0, c is a root
- ›Synthetic division is preferred as it also gives the quotient polynomial
- ›The tool uses numerical tolerance 10⁻⁹ to handle floating-point fractions
What is synthetic division and why use it?
Synthetic division is a compact algorithm: write the coefficients, bring down the first, then repeatedly multiply by c and add to the next coefficient. The final value is the remainder.
- ›Works only for linear divisors (x − c), not for higher-degree factors
- ›If remainder = 0: c is a root, quotient is p(x) / (x−c)
- ›Reduces degree by 1 each time, making further factoring easier
- ›The tool shows every step of the synthetic division process
Can I find all roots of a polynomial this way?
The complete strategy for finding all roots depends on the degree and nature of the polynomial:
- ›Degree 1: root is directly x = −b/a
- ›Degree 2: use quadratic formula for all roots including irrational/complex
- ›Degree 3–4: rational root theorem + deflation + quadratic/cubic formula
- ›Degree 5+: rational roots + numerical methods (Newton, Durand-Kerner)
What does "deflation" mean in polynomial root-finding?
After confirming a root r, the polynomial factors as p(x) = (x − r) × q(x), where q has degree one less. Apply the rational root theorem again to q(x) for remaining roots.
- ›Deflate once per confirmed root, the quotient becomes the new polynomial
- ›Repeated roots appear multiple times: p(x) = (x−r)² × q(x)
- ›The tool shows the quotient polynomial after dividing out the first root
- ›Continue deflating until a quadratic remains, then use the quadratic formula
Why do I need integer coefficients?
The theorem uses number-theoretic properties (divisibility of integers) to limit candidates. Non-integer coefficients break the divisibility argument.
- ›Rational coefficients: multiply by LCM of denominators (e.g., ½x²+x = x²+2x when multiplied by 2)
- ›The roots of the transformed polynomial are identical to the original
- ›Decimal coefficients: round to nearest integer only if the polynomial is exact
- ›Floating-point polynomials are better handled with numerical root-finders