Rational Zeros Calculator

Enter a polynomial with integer coefficients and find all possible rational zeros using the Rational Root Theorem. Input your polynomial in the polynomial field (e.g. 2x^4 + x^3 - 15x^2 - 7x + 7) and get back a complete list of candidate rational roots in ±p/q form, plus the actual rational zeros confirmed by substitution.

Enter a polynomial with integer coefficients. Use ^ for exponents (e.g. x^3). Supports up to degree 6.

Enter coefficients from highest to lowest degree, separated by commas.

Results

Possible Rational Zeros (±p/q)

--

Actual Rational Zeros

--

Number of Candidates

--

Factors of Constant Term (p)

--

Factors of Leading Coefficient (q)

--

Results Table

Frequently Asked Questions

What is a rational zero?

A rational zero (or rational root) of a polynomial is a value x = p/q — where p and q are integers and q ≠ 0 — that makes the polynomial equal to zero. Not every polynomial has rational zeros; some roots may be irrational or complex.

What is the Rational Zeros Theorem (Rational Root Theorem)?

The Rational Root Theorem states that if a polynomial with integer coefficients has a rational zero p/q (in lowest terms), then p must be a divisor of the constant term and q must be a divisor of the leading coefficient. This gives a finite list of candidates to test, rather than searching infinitely many values.

How do you find all possible rational zeros of a polynomial?

First, list all integer factors of the constant term (these are your p values). Then list all integer factors of the leading coefficient (these are your q values). Form all fractions ±p/q and reduce them to lowest terms. Every rational zero of the polynomial must appear in this list.

How do you determine which possible rational zeros are actual zeros?

Substitute each candidate ±p/q into the polynomial. If f(p/q) = 0, that value is an actual zero. You can also use synthetic division to test candidates efficiently — if the remainder is zero after dividing, the candidate is a root.

Can a polynomial have no rational zeros?

Yes. The Rational Root Theorem tells you which rational numbers could be zeros, but testing them may reveal none actually satisfies the equation. In that case, all roots are irrational or complex numbers.

Does the Rational Zeros Theorem work for all polynomials?

It applies specifically to polynomials with integer (whole number) coefficients. If the coefficients are fractions or irrational numbers, you would first need to multiply through to obtain integer coefficients before applying the theorem.

What is synthetic division and how does it relate to rational zeros?

Synthetic division is a shorthand method for dividing a polynomial by a linear factor (x − c). When testing a candidate rational zero c, if synthetic division yields a remainder of 0, then c is a root. It also produces the reduced (depressed) polynomial, which you can continue factoring.

Why might the list of possible rational zeros be very long?

The number of candidates grows with the number of divisors of both the constant term and the leading coefficient. A constant term like 12 has 6 positive divisors and a leading coefficient like 6 also has 4, producing up to 48 candidate fractions (±p/q). Reducing to lowest terms removes duplicates, but the list can still be large for highly composite coefficients.

More Math Tools