Find all possible rational zeros of a polynomial using the Rational Root Theorem. Test ±p/q candidates, identify actual zeros, and see step-by-step analysis.
The Rational Zeros Calculator applies the Rational Root Theorem to any polynomial with integer coefficients, giving you every possible rational zero before you even start testing. The theorem states that if a polynomial f(x) = aₙxⁿ + … + a₁x + a₀ has a rational zero p/q in lowest terms, then p must be a factor of the constant term a₀ and q must be a factor of the leading coefficient aₙ. This dramatically narrows the search space compared to guessing random values.
Enter your polynomial coefficients from highest to lowest degree and the calculator instantly generates all ±p/q candidates. It then evaluates f(x) at every candidate to determine which are actual zeros, highlighting them in a color-coded results table and visual bar chart. You can see the Rational Root Theorem applied step by step — identifying factors of a₀ and aₙ, forming the candidate list, and testing each one.
This calculator is essential for precalculus and college algebra courses where you need to factor higher-degree polynomials. Once you find the rational zeros, you can perform synthetic division to reduce the polynomial's degree and find remaining irrational or complex roots. Eight built-in presets let you explore classic textbook polynomials instantly, and sorting options help you organize candidates by value or magnitude for efficient analysis.
Rational Zeros Calculator helps you solve rational zeros problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Polynomial Coefficients (highest to lowest degree, comma-separated), Max Degree Limit, Zero Tolerance (ε) once and immediately inspect Polynomial Degree, Leading Coefficient (aₙ), Constant Term (a₀) to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
If f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ has a rational zero p/q (in lowest terms), then p divides a₀ and q divides aₙ. Possible rational zeros = {±p/q : p ∈ factors(|a₀|), q ∈ factors(|aₙ|)}.
Result: Polynomial Degree shown by the calculator
Using the preset "x³−6x²+11x−6", the calculator evaluates the rational zeros setup, applies the selected algebra rules, and reports Polynomial Degree with supporting checks so you can verify each transformation.
This calculator takes Polynomial Coefficients (highest to lowest degree, comma-separated), Max Degree Limit, Zero Tolerance (ε), Show Steps and applies the relevant rational zeros relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Polynomial Degree, Leading Coefficient (aₙ), Constant Term (a₀), Possible Rational Zeros to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
It states that any rational zero p/q of a polynomial with integer coefficients must have p dividing the constant term and q dividing the leading coefficient. Use this as a practical reminder before finalizing the result.
No — it only finds rational roots. Irrational roots (like √2) and complex roots require other methods such as the quadratic formula or numerical approximation.
The calculator handles any integer leading coefficient. More factors in aₙ simply produce more ±p/q candidates to test.
Include a 0 for every missing power. For x⁴ − 10x² + 9, enter "1,0,-10,0,9".
Because of floating-point arithmetic, f(x) at a true zero may not be exactly 0. The tolerance sets how close to zero counts as a root.
It identifies rational linear factors. Combine with synthetic division and the quadratic formula to factor completely.