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
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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.
Formula
Candidates: ±(factor of a₀) / (factor of aₙ)
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.
Frequently asked questions
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.