Calculator Inputs
Formula Used
The Rational Root Theorem lists every possible rational zero. Write each candidate as p divided by q. Use constant factors for p and leading factors for q.
Possible rational zeros = ± factors(constant term) / factors(leading coefficient)
Each candidate must make the polynomial equal zero. Synthetic division confirms candidates and reduces the polynomial. Repeated successful divisions reveal root multiplicity.
How to Use
Enter a polynomial expression or its ordered coefficients. Select the variable, analysis mode, format, and precision. Press calculate to generate the complete solution.
Review candidate roots before checking confirmed zeros. Inspect synthetic tables for every successful division. Export the summary using CSV or PDF.
Example Data
| Polynomial | Possible rational zeros | Confirmed rational zeros |
|---|---|---|
| 2x^3 - 3x^2 - 8x + 12 | ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2 | -2, 3/2, 2 |
| x^3 - 6x^2 + 11x - 6 | ±1, ±2, ±3, ±6 | 1, 2, 3 |
| 3x^3 + x + 1 | ±1, ±1/3 | None |
Calculation History
Frequently Asked Questions
What is a rational zero?
A rational zero is an integer or fraction. It makes the polynomial value equal zero. Its numerator and denominator are whole numbers.
Does every polynomial have rational zeros?
No, many polynomials have no rational zeros. Their roots may be irrational or complex. The theorem only creates possible rational candidates.
What are p and q?
The value p divides the constant term. The value q divides the leading coefficient. Reduced fractions form the possible rational roots.
Why are candidates tested?
The theorem creates possibilities, not guaranteed zeros. Substitution or division tests each candidate. A zero remainder confirms the root.
How are repeated zeros detected?
The calculator divides repeatedly by the same factor. Every successful division increases that root's multiplicity. The remaining quotient is tested again.
Can coefficients be fractions?
Yes, fractions and decimals are supported. Denominators are cleared before candidate generation. Exact rational arithmetic reduces rounding errors.
What happens when the constant is zero?
Zero becomes an immediate rational root. The calculator factors out the variable automatically. It then continues with the reduced polynomial.
How are complex roots approximated?
The calculator uses a simultaneous numerical root method. It processes the remaining polynomial after rational division. Results depend on selected decimal precision.
Why must missing terms use zero?
Coefficient order represents every descending polynomial power. A missing power still needs its position. Enter zero to preserve correct alignment.