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The Rational Root Theorem narrows down an otherwise infinite search for a polynomial's roots into a short, specific list of candidates — every possible rational root must be a factor of the constant term divided by a factor of the leading coefficient. This tool builds that list and tests each candidate directly.
How it works
Every factor of the constant term (p) is paired over every factor of the leading coefficient (q) to form every possible ±p/q candidate. Each candidate is then substituted directly into the polynomial — if it evaluates to zero, it's confirmed as an actual root.
- Enter x⁵ coefficient (optional).
- Enter x⁴ coefficient (optional).
- Enter x³ coefficient.
- Enter x² coefficient.
- Enter x coefficient.
- Enter constant.
- Click Calculate to see your results.
Examples
A polynomial with rational roots
x³-2x²-x+2 has candidates ±1, ±2 — testing each confirms 1, -1, and 2 are all actual roots.
A polynomial with no rational roots
x²+x+1 has candidates ±1, but neither evaluates to zero, so it has no rational roots (its roots are complex).
Who should use it
- Algebra 2 and precalculus coursework on the Rational Root Theorem.
- Finding a starting root to enable synthetic division on a higher-degree polynomial.
Industry applications
- Mathematics education
Advantages
- Automatically generates and tests every candidate rather than requiring manual trial and error.
- Clearly reports which candidates are confirmed roots.
Limitations
- Requires a non-zero constant term (a constant of 0 makes the theorem's candidate list undefined) and integer coefficients.
Common mistakes to avoid
- Forgetting to test the negative candidates as well as the positive ones — every candidate should be checked with both signs.
Best practices
- Combine this with Descartes' Rule of Signs first — it narrows down how many positive and negative roots to expect, cutting down which candidates are worth testing.
Tips
- Once you find one rational root, use Synthetic Division to reduce the polynomial's degree by one — repeating this process is often the fastest way to fully factor it.