Polynomial Rational Root Calculator

Calculation Result

Polynomial

-

Possible Rational Roots

0

Actual Rational Roots Found

0

Degree

0

Leading coefficient factors:

Constant term factors:

Candidate roots from the theorem:

Actual rational roots:

Reduced quotient after dividing actual roots:

Calculator

Polynomial coefficients

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

Graph minimum x

Used for the plotted polynomial view.

Graph maximum x

Choose a range that reveals turning behavior.

Graph points

More points make the graph smoother.

Displayed decimals

Controls numeric display in the table and summary.

Zero tolerance

Values within tolerance count as numerical zero.

Polynomial Plot

The graph helps compare algebraic candidates with visible x-axis crossings. A plotted crossing suggests a real root, but theorem candidates must still be tested exactly.

Candidate Evaluation Table

#	Candidate Root	Decimal Value	P(candidate)	Root Status
Run the calculator to populate results.

Example Data Table

Example Polynomial	Coefficients	Leading Coefficient	Constant Term	Possible Candidates	Actual Rational Roots
2x³ - 3x² - 11x + 6	2, -3, -11, 6	2	6	±1, ±2, ±3, ±6, ±1/2, ±3/2	3, 1/2, -2
x³ - 6x² + 11x - 6	1, -6, 11, -6	1	-6	±1, ±2, ±3, ±6	1, 2, 3
3x² + x - 2	3, 1, -2	3	-2	±1, ±2, ±1/3, ±2/3	2/3, -1

Formula Used

Polynomial form: \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \)

Rational Root Theorem: If p/q is a rational root in lowest terms, then:

p divides the constant term a₀
q divides the leading coefficient aₙ

Candidate list: All unique values of ± p/q after simplification.

Evaluation test: A candidate r is an actual rational root when P(r) = 0.

Synthetic division check: When a root is confirmed, the polynomial can be divided by (x - r) to reduce the equation and inspect remaining factors.

How to Use This Calculator

Enter polynomial coefficients in descending power order.
Set the graph range and plotting density.
Choose display precision and zero tolerance.
Press Calculate Rational Roots to analyze candidates.
Review factor lists, theorem candidates, and actual roots.
Inspect the evaluation table for exact candidate testing.
Use the graph to compare algebraic results with x-intercepts.
Export the table as CSV or summary as PDF.

Frequently Asked Questions

1. What does this calculator actually find?

It lists all rational candidates allowed by the Rational Root Theorem, tests each candidate, and highlights which ones are true rational roots of your polynomial.

2. Do all listed candidates become real roots?

No. The theorem only gives possible rational roots. Each candidate must still be substituted into the polynomial or checked by synthetic division.

3. Can this calculator find irrational or complex roots?

It focuses on rational-root testing. Irrational or nonreal roots may still exist, but they are outside the theorem’s candidate list.

4. Why are fractions simplified before display?

Different factor pairs can generate equivalent fractions. Simplifying them removes duplicates, keeps the result list shorter, and improves interpretation.

5. What if the constant term is zero?

Then zero is automatically a root, and the polynomial has a factor of x. The calculator handles this by including 0 among tested values.

6. Why use a zero tolerance?

Decimal evaluation can create tiny rounding errors. A tolerance treats very small values as numerical zero when determining root status.

7. How does the graph help?

The graph shows where the polynomial crosses or touches the x-axis. That visual cue helps confirm whether tested rational roots make sense.

8. Can I use decimals in the coefficient input?

Yes, but the Rational Root Theorem is naturally designed for integer coefficients. Decimal inputs are accepted, though theorem-style factor logic works best with integers.