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Rational Root Theorem Calculator

Work out rational root theorem instantly with clear inputs, formula shown and shareable results.

About Rational Root Theorem Calculator

The rational root theorem says any rational root p/q in lowest terms must have p dividing the constant term and q dividing the leading coefficient. That reduces an infinite search to a short finite list.

Formula

Rational root theorem

possible roots = ±(divisor of the constant) / (divisor of the leading coefficient)

Worked example

With the default values below, Rational Root Theorem Calculator returns Rational roots that actually work of -3, 0.5, 2. Change any input above and every figure here is recalculated in your browser.

Inputs used

Leading coefficientaccepts -1,000,000–1,000,000
2
Constant termaccepts -1,000,000–1,000,000
-6
Full coefficient list to test against
2, 1, -13, 6

Results

Rational roots that actually work
-3, 0.5, 2
Candidate roots ±p/q
-6, -3, -2, -1.5, -1, -0.5, 0.5, 1, 1.5, 2, 3, 6
Number of candidates
12
Rational roots found
3

Step by step

  1. 1Values used
  2. 2Rational root theorem
  3. 3Rational roots that actually work= -3, 0.5, 2
  4. 4Candidate roots ±p/q= -6, -3, -2, -1.5, -1, -0.5, 0.5, 1, 1.5, 2, 3, 6
  5. 5Number of candidates= 12
  6. 6Rational roots found= 3

Assumptions and limits

  • The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For 2x³ + x² − 13x + 6?

Candidates are ±1, ±2, ±3, ±6, ±½, ±3/2, and the actual rational roots are 2, −3 and ½.

What if no candidate works?

The polynomial has no rational roots; its real roots are irrational and need numerical methods or surd forms.

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