Free Calculator Tool

Rational Zeros Calculator

Find all possible and actual rational zeros of a polynomial using the Rational Zeros Theorem. Step-by-step solutions.

Want to solve ANY math problem? Just take a screenshot.

Math.Photos handles algebra, calculus, geometry, statistics, and more — all from a photo or your handwritten notes.

20 free solves · no credit card · no account needed to start

Install

Find Rational Zeros

f(x) =

How It Works

Rational Zeros Theorem

If a polynomial has integer coefficients, any rational zero p/q must have p as a factor of the constant term and q as a factor of the leading coefficient.

Finding candidates

List all factors of the constant term (p) and all factors of the leading coefficient (q). Form all possible fractions p/q with both positive and negative signs.

Testing candidates

Substitute each candidate into the polynomial. If f(p/q) = 0, then p/q is a rational zero. Use synthetic division for efficiency.

After finding zeros

Once you find a rational zero, factor it out using synthetic division to reduce the polynomial degree. Repeat until all zeros are found.

Related Tools

Equation SolverInverse Function CalculatorExtrema CalculatorQuadratic Formula Calculator

Already did the work? Get it checked instead.

A calculator gives you an answer. It doesn't tell you which line of your working went wrong. Photograph your own steps — handwritten notebook, whiteboard, or a screenshot of an online assignment — and Math.Photos checks them line by line, then points at the exact step where the sign flipped.

Reads handwriting

Photo of your notebook, not a retyped expression.

Step-level verdict

"Line 3 dropped a negative", not just "wrong".

Teach Me mode

Questions that lead you to the fix, not the answer.

Install How the math checker works → First 20 checks free

Pricing

Start free — 20 solves, no card. Upgrade only when you run out.

Lite

$1.99 /month

50 solves/month

Enough for a weekly problem set.

Most popular

Student

$4.99 /month

500 solves/month

Every assignment, all semester, plus Teach Me mode.

Pro

$9.99 /month

3,000 solves/month

Multi-device sync and LaTeX/PDF export.

Annual billing saves 37%. Cancel anytime — see full plan comparison.

Need more than a rational zeros calculator?

Math.Photos solves any math problem from a screenshot — algebra, calculus, geometry, statistics, and more. With step-by-step explanations.

Install

20 free solves · no credit card

Frequently Asked Questions

How do you find all possible rational zeros?

Use the Rational Zeros Theorem: list all factors p of the constant term and all factors q of the leading coefficient. The possible rational zeros are all values of the form p/q (with both + and - signs). Then test each candidate by plugging it into the polynomial.

What is the Rational Zeros Theorem?

The Rational Zeros Theorem states that if a polynomial f(x) = a_n*x^n + ... + a_1*x + a_0 has integer coefficients, then every rational zero has the form p/q, where p divides a_0 (constant term) and q divides a_n (leading coefficient).

Can a polynomial have no rational zeros?

Yes. For example, x^2 + 1 = 0 has no real zeros at all (only complex: x = i and x = -i). And x^2 - 2 = 0 has zeros x = +/-sqrt(2), which are irrational. The Rational Zeros Theorem only finds rational zeros.

How do you use synthetic division after finding a zero?

Once you find that x = r is a zero, divide the polynomial by (x - r) using synthetic division. Write r on the left, the coefficients across the top, bring down the first, multiply-and-add across. The result is a polynomial of one lower degree. Repeat to find remaining zeros.