# Rational Zeros Calculator

URL: https://speedycalculator.com/calculators/math/rational-zeros/
Last updated: 2026-10-30

> The rational zeros calculator lists every candidate rational zero ±p/q of a polynomial with integer coefficients and tests which ones are actual zeros.

## Inputs

- Degree of the polynomial
- a₆ (x⁶)
- a₅ (x⁵)
- a₄ (x⁴)
- a₃ (x³)
- a₂ (x²)
- a₁ (x)
- a₀

## Results

- Rational zeros
- Possible rational zeros
- Number of candidates tested

## Formula

```
Candidates: ±(factor of a₀) / (factor of aₙ)
```

## Explanation

The rational root theorem narrows the search for exact roots of a polynomial to a short list of fractions. Testing them, often with synthetic division, is the standard way to begin factoring cubics and quartics by hand.

### How to use the Rational Zeros Calculator

- **Degree of the polynomial**
- **a₆ (x⁶)**
- **a₅ (x⁵)**
- **a₄ (x⁴)**
- **a₃ (x³)**
- **a₂ (x²)**
- **a₁ (x)**
- **a₀**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The calculator lists the positive divisors p of the constant term and q of the leading coefficient, forms every reduced fraction p/q and tests ±p/q with exact fraction arithmetic. A factor of x is taken out first if the constant term is zero.

## Worked example

Inputs — Degree of the polynomial: 3; a₃ (x³): 2; a₂ (x²): -3; a₁ (x): -3; a₀: 2.

Results — **Rational zeros = 1/2, −1, 2**; Possible rational zeros = ±1/2, ±1, ±2; Number of candidates tested = 6.

## Key terms

- **Rational root theorem**: Any rational zero p/q in lowest terms has p dividing a₀ and q dividing aₙ.

## FAQ

**Does every polynomial have rational zeros?**
No. x² − 2 has none, because its zeros are ±√2.

**Why must coefficients be integers?**
The theorem relies on divisibility; multiply through by a common denominator first.

**What next once I find a zero r?**
Divide by (x − r) with synthetic division and continue with the smaller polynomial.

**Can I skip the negative candidates?**
Use Descartes' rule of signs to see whether negative zeros are possible.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [Wolfram MathWorld – Rational zero theorem](https://mathworld.wolfram.com/RationalZeroTheorem.html)