Calculator Input
2, -3, -8, 12 represents 2x³ - 3x² - 8x + 12.
Formula Used
The Rational Root Theorem states that every rational zero of a polynomial with integer coefficients must have the form ±p/q.
Here, p is any factor of the constant term, and q is any factor of the leading coefficient.
The calculator builds all unique reduced candidates, then substitutes each candidate into the polynomial. A candidate is a confirmed rational zero when f(x) = 0.
If the constant term is zero, then x = 0 is a root. The calculator removes repeated zero factors first, then tests the remaining polynomial.
How to Use This Calculator
- Enter polynomial coefficients from highest power to constant.
- Use commas or spaces between the integer values.
- Set the minimum and maximum x values for plotting.
- Choose how many points the graph should evaluate.
- Click Calculate Rational Zeros to generate candidates.
- Review the candidate table and identify confirmed zeros.
- Inspect the graph to see x-axis crossings visually.
- Use the export buttons to save the result table.
Example Data Table
| Polynomial | Leading coefficient | Constant term | Possible candidates | Confirmed rational zeros |
|---|---|---|---|---|
| 2x³ - 3x² - 8x + 12 | 2 | 12 | ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2 | -2, 3/2, 2 |
| x³ - 6x² + 11x - 6 | 1 | -6 | ±1, ±2, ±3, ±6 | 1, 2, 3 |
| x⁴ - 5x³ + 6x² | 1 | 0 | 0, ±1, ±2, ±3, ±6 | 0, 2, 3 |
Frequently Asked Questions
1. What does this calculator find?
It lists all possible rational zeros from the Rational Root Theorem. It then tests each candidate and identifies which values truly make the polynomial equal zero.
2. Why must coefficients be integers?
The theorem depends on factors of the constant term and leading coefficient. Factor-based candidate generation only works directly when polynomial coefficients are integers.
3. Does every candidate become an actual root?
No. The theorem gives only possible rational zeros. Each candidate must be substituted into the polynomial to confirm whether its output is exactly zero.
4. What happens when the constant term is zero?
Then x = 0 is automatically a root. The calculator removes repeated zero factors first, then applies the rational root method to the remaining polynomial.
5. Can the graph prove the root?
The graph helps you visualize crossings and turning points. However, the exact confirmation still comes from substitution and checking whether f(x) equals zero.
6. Why are some roots written as fractions?
Rational zeros can be whole numbers or fractions. Fractions appear when a factor of the constant term is divided by a factor of the leading coefficient.
7. Does this calculator find irrational roots too?
No. It focuses on rational candidates only. A polynomial can still have irrational or complex roots that do not appear in the rational candidate list.
8. What should I enter for a missing term?
Enter zero for every missing power. For example, x³ - 5x + 6 becomes
1, 0, -5, 6.