Function: p(x) = x^3 - 6x^2 + 11x - 6
Interval: -1 to 5
Summary: Estimated real zeros: 1, 2, 3
| # | Estimated Zero | p(x) at Zero | Note |
|---|---|---|---|
| 1 | 1 | 0 | Simple root estimate |
| 2 | 2 | 0 | Simple root estimate |
| 3 | 3 | 0 | Simple root estimate |
Example polynomial: p(x) = x^3 - 6x^2 + 11x - 6
| x | p(x) |
|---|---|
| 0 | -6 |
| 1 | 0 |
| 2 | 0 |
| 3 | 0 |
| 4 | 6 |
| 5 | 24 |
Polynomial model: p(x) = anx^n + an-1x^(n-1) + ... + a1x + a0
Zero condition: A zero occurs when p(x) = 0.
Numerical approach: The calculator scans the chosen interval, checks sign changes, applies bisection to bracket roots, then refines estimates with Newton's method.
Important note: This tool reports real zeros inside the selected interval. Complex zeros are not listed here.
- Choose the polynomial degree.
- Enter coefficients from highest power to constant term.
- Set the interval where you want to search.
- Adjust samples for finer scanning accuracy.
- Set tolerance and iteration controls if needed.
- Press Find Zeros to calculate results.
- Review the root table and graph below.
- Use the export buttons for CSV or PDF output.
1. What does a zero of a function mean?
A zero is an x-value where the function output becomes zero. On the graph, it is where the curve touches or crosses the x-axis.
2. Does this calculator find complex zeros?
No. This version estimates real zeros only within your chosen interval. Complex roots are not displayed in the result table.
3. Why should I widen the interval sometimes?
If a real zero lies outside your selected range, the scan cannot detect it. A wider interval helps reveal missing roots.
4. What does scanning samples control?
It controls how many checkpoints the calculator uses across the interval. More samples usually improve detection of narrow sign changes or near-axis behavior.
5. Can repeated roots be missed?
Yes, repeated roots can be harder to detect because the graph may touch the axis without changing sign. Increasing samples helps.
6. Why is p(x) at the reported zero not exactly zero?
Numerical methods produce approximations. Small remaining values are normal and reflect tolerance settings, rounding, and finite iteration limits.
7. Which coefficients should I enter first?
Enter coefficients from the highest degree down to the constant term. For x^3 - 6x^2 + 11x - 6, enter 1, -6, 11, -6.
8. What does the graph help me verify?
The graph lets you visually confirm where the curve crosses or touches the x-axis. It also helps you choose better intervals.