Rational Zeros Calculator (rational root theorem: candidates ±p/q & actual zeros)

Understanding the rational zeros calculator — concept and principle

This calculator applies the rational zero (rational root) theorem to a polynomial with whole-number coefficients. It lists every possible rational zero ±p/q, tests each one exactly, and reports the actual rational zeros and the factor that remains. Use it for algebra and precalculus homework, for factoring cubics and quartics, and as a first step before using the quadratic formula on what is left.

How it is calculated

For P(x) = aₙxⁿ + … + a₁x + a₀:

1. If a₀ = 0, factor out x first; 0 is then a zero.
2. List the factors p of the constant term a₀ and q of the leading coefficient aₙ.
3. Form every candidate ±p/q in lowest terms.
4. Test each: p/q is a zero exactly when P(p/q) = 0, which is the same as synthetic division leaving remainder 0.
5. Divide out each zero found (as often as it repeats) to get the remaining factor.

Tests use exact whole-number arithmetic, so rounding cannot hide or invent a zero.

The principle

Suppose p/q in lowest terms is a zero. Multiplying P(p/q) = 0 by qⁿ gives aₙpⁿ + aₙ₋₁pⁿ⁻¹q + … + a₀qⁿ = 0. Every term except a₀qⁿ contains p, so p must divide a₀qⁿ; since p and q share no factors, p divides a₀. In the same way every term except aₙpⁿ contains q, so q divides aₙ. That leaves only finitely many candidates, which turns a hard search into a short checklist.

Worked example

Default input: P(x) = 2x³ − 3x² − 11x + 6.

1. Factors of the constant 6: p = 1, 2, 3, 6. Factors of the leading 2: q = 1, 2.
2. Candidates: ±1, ±2, ±3, ±6, ±1/2, ±3/2, which is 12 in total.
3. P(3) = 54 − 27 − 33 + 6 = 0; P(−2) = −16 − 12 + 22 + 6 = 0; P(1/2) = 0.25 − 0.75 − 5.5 + 6 = 0.
4. Rational zeros: −2, 1/2, 3.
5. P(x) = (x + 2)(2x − 1)(x − 3), so nothing remains to factor.

How to read the result

The rational zeros give linear factors: a zero p/q gives the factor (qx − p). If a remaining factor is shown, its zeros are irrational or complex; for a quadratic remainder use the quadratic formula. If no candidate works, the polynomial has no rational zeros, though it may still have real ones.

Common mistakes

• Forgetting the negative candidates.
• Swapping p and q: p comes from the constant term, q from the leading coefficient.
• Applying the theorem to polynomials with fractional or decimal coefficients without first clearing denominators.
• Stopping after one zero instead of dividing it out and continuing.

Frequently asked questions

Do I need to test every candidate?

You can stop when the remaining factor is a quadratic and solve it directly; a graph also helps you test the likely candidates first.

What if the leading coefficient is 1?

Then q = 1, and every rational zero must be a whole number that divides the constant term.

Can a rational zero repeat?

Yes. The calculator divides it out as many times as it occurs.

Sources: OpenStax, College Algebra 2e (2021), §5.5 Zeros of Polynomial Functions