Rational Zeros Calculator
List all possible rational zeros of a polynomial using the Rational Root Theorem: p/q where p|constant and q|leading.
This free online rational zeros calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
Coefficient of the highest-degree term
The constant (degree-zero) term
Results
|Leading coefficient|
1
|Constant term|
1
Max possible rational zeros
4
How to Use This Calculator
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Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
When to Use This Calculator
- •Use the Rational Zeros Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
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About Rational Zeros Calculator
The Rational Zeros Calculator applies the Rational Root Theorem to list all possible rational roots of a polynomial with integer coefficients. The theorem states that if p/q is a rational root of aₙxⁿ + ... + a₁x + a₀ = 0 (in lowest terms), then p divides a₀ and q divides aₙ. This dramatically narrows the search for roots from infinitely many rationals to a finite list. Once candidates are generated, they can be tested using synthetic division or direct substitution. The Rational Root Theorem is a cornerstone technique for factoring polynomials of degree 3 and higher, and it appears prominently in algebra, precalculus, and polynomial theory courses. Combined with synthetic division, it provides a systematic method for completely factoring polynomials over the rationals.
The Math Behind It
Formula Reference
Rational Root Theorem
Possible rational roots = ±(factors of a₀) / (factors of aₙ)
Variables: a₀ = constant term, aₙ = leading coefficient
Worked Examples
Example 1: List and test candidates
Find rational zeros of x³ - 2x² - 5x + 6
Rational zeros: x = 1, 3, -2
Common Mistakes & Tips
- !Forgetting to include both positive and negative candidates (±)
- !Not reducing p/q to lowest terms, leading to duplicate candidates
- !Assuming all candidates are actual roots (they are only possibilities to test)
- !Forgetting fractional candidates when the leading coefficient is not 1
Related Concepts
Used in These Calculators
Calculators that build on or apply the concepts from this page:
Frequently Asked Questions
Does the Rational Root Theorem find all roots?
No. It only finds rational roots. Polynomials can have irrational roots (like √2) or complex roots (like 2+3i) that the theorem does not detect. For those, other methods like the quadratic formula or numerical algorithms are needed.
What if no candidate works?
If none of the rational candidates are roots, the polynomial has no rational roots. It may still have real irrational or complex roots. For cubics and quartics, Cardano's and Ferrari's formulas can find exact roots.
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