Math
Rational Zeros Calculator
Type any polynomial and the rational zeros calculator lists the candidates ±p/q from the rational root theorem (p divides the constant term, q divides the leading coefficient), tests every candidate with exact arithmetic, reports which ones are real zeros and their multiplicities, and divides them out to show the remaining factor and the factored form.
Any degree up to 12. Fractions and decimals are cleared automatically. An = 0 is optional.
Try an example
Result
Rational zeros
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- Candidates ±p/q
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- Candidates tested
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- How many zeros are rational
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- Zeros as decimals
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- Factored form
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- Remaining (deflated) factor
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More details (1 more)
- About the remaining factor
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Student quick launch
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Study path
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What the rational zeros calculator solves
The rational root theorem (rational zeros theorem) narrows the search for zeros of a polynomial with integer The number multiplying a variable, like the 3 in 3x. to a finite list: every rational zero p/q has p dividing the constant term and q dividing the leading coefficient. Testing that list finds every rational zero; whatever remains after dividing them out has only A number that cannot be written as a fraction of two whole numbers, like π or √2. or complex zeros.
Worked example: 2x^3 - 3x^2 - 3x + 2
| Step | Result |
|---|---|
| Factors of the constant 2 | 1, 2 |
| Factors of the leading coefficient 2 | 1, 2 |
| Candidates ±p/q | ±1, ±2, ±1/2 |
| Test | f(-1) = 0, f(1/2) = 0, f(2) = 0 |
| Factored form | (x + 1)(2x - 1)(x - 2) |
Common mistakes
- Forgetting the negative candidates or the fractions when the leading coefficient is not 1.
- Assuming every candidate is a zero. The theorem only says where rational zeros can be; most candidates fail.
- Not dividing out a zero you found. The deflated polynomial is smaller and may reveal repeated zeros.
- Applying the theorem to a polynomial with fraction coefficients without clearing them first.
How to use it
- Type the polynomial, for example 2x^3 - 3x^2 - 3x + 2. An = 0 on the end is optional.
- Read the rational zeros at the top, then the candidate list and the test of each candidate in the steps.
- Use the factored form and the remaining factor to finish solving: a leftover quadratic can be solved with the quadratic formula.
How to read the answer
Each rational zero p/q corresponds to a linear factor (qx - p). Multiplicity counts how many times that factor appears. The remaining factor collects the zeros that are not rational; if it is quadratic, the quadratic formula finishes the job.
Common mistakes and edge cases
- Listing p/q with p from the leading coefficient and q from the constant (backwards).
- Stopping the test after the first zero without checking for multiplicity.
- Reporting candidates as answers without testing them.
Worked examples
Three rational zeros
2x^3 - 3x^2 - 3x + 2 → -1, 1/2, 2
Rational zeros
x = -1, x = 1/2, x = 2
Quartic
x^4 - 5x^2 + 4 → ±1, ±2
Rational zeros
x = -2, x = -1, x = 1, x = 2
Repeated zero
x^3 - 3x + 2 = (x - 1)^2 (x + 2)
Rational zeros
x = -2, x = 1 (multiplicity 2)
One rational zero, quadratic left
x^3 - 2x^2 - 2x - 3 → 3 and a complex pair
Rational zeros
x = 3
No rational zeros
x^2 - 2 has only irrational zeros
Rational zeros
No rational zeros
Zero constant term
x^3 - x = x(x - 1)(x + 1)
Rational zeros
x = 0, x = -1, x = 1
Fraction coefficients
(1/2)x^2 - (3/2)x + 1 → 1 and 2
Rational zeros
x = 1, x = 2
Not a polynomial
1/x is rejected
Rational zeros
Error
Frequently asked questions
What is the rational root theorem?+
If a polynomial has integer coefficients and a rational zero p/q in lowest terms, then p must divide the constant term and q must divide the leading coefficient. It gives a finite list of candidates to test.
Does every candidate have to be a zero?+
No. The theorem only limits where rational zeros can be. Many polynomials have candidates but no rational zeros at all, such as x^2 - 2.
What does multiplicity mean?+
How many times a zero's factor appears. x^3 - 3x + 2 = (x - 1)^2 (x + 2) has the zero 1 with multiplicity 2. The graph touches the x-axis there without crossing.
What is the deflated polynomial?+
What is left after dividing out every rational zero with synthetic division. Its zeros are the irrational and complex zeros of the original polynomial.
What if the constant term is 0?+
Then x = 0 is a zero. Factor out the highest power of x first, then apply the theorem to what remains. The calculator does this automatically.
Can I use decimals or fractions as coefficients?+
Yes. The calculator multiplies through by a common denominator so the coefficients are integers; the zeros do not change.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026