Find the Zeros of the Function Calculator

Calculator Inputs

Coefficient of x⁶

Coefficient of x⁵

Coefficient of x⁴

Coefficient of x³

Coefficient of x²

Coefficient of x

Constant Term

Minimum x

Maximum x

Step Size

Tolerance

Maximum Iterations

Example Data Table

This example uses f(x) = x³ - 6x² + 11x - 6 over the interval [0, 4].

x	f(x)	Observation
0	-6	Below the axis
1	0	Zero at x = 1
2	0	Zero at x = 2
3	0	Zero at x = 3
4	6	Above the axis

Formula Used

The calculator evaluates a polynomial function in the form:

f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀

A zero of the function is any value r that makes f(r) = 0.

To detect zeros, the calculator samples the selected interval, checks for sign changes between nearby points, refines each candidate with bisection, and also tries numerical iteration with the Newton update:

x_next = x - f(x) / f'(x)

This mixed method helps capture both visible crossings and roots that need refinement.

How to Use This Calculator

Enter the polynomial coefficients from x⁶ down to the constant term.
Set the minimum and maximum x values for the search interval.
Choose a step size. Smaller steps usually improve detection.
Set the tolerance and maximum iterations for refinement.
Click Find Zeros to generate the roots, graph, and sample table.
Review the listed zeros and confirm them on the plot.
Use the CSV or PDF button to export the results.

FAQs

1. What is a zero of a function?

A zero is an x-value that makes the function equal to zero. On a graph, it is where the curve touches or crosses the horizontal axis.

2. Does this calculator work for every function type?

This version is designed for polynomial functions entered through coefficients. It does not directly accept trigonometric, logarithmic, or exponential expressions.

3. Why do I need a search interval?

The interval tells the calculator where to look. Roots outside that range will not appear, so a wider interval can reveal additional zeros.

4. Why does step size matter?

A smaller step checks more sample points and improves root detection. A larger step is faster, but it can miss narrow crossings or closely spaced zeros.

5. What if no zero is found?

The function may have no real zero in that interval, or the settings may be too coarse. Try expanding the interval or reducing the step size.

6. Can the calculator detect repeated roots?

It can detect some repeated roots through numerical refinement, but repeated roots are harder because the graph may only touch the axis without changing sign.

7. Why is the graph useful?

The graph gives a visual check of where the curve approaches or crosses zero. It helps confirm whether the listed roots match the function behavior.

8. What is shown in the exports?

The export includes the function summary, detected roots, and sampled value table. That makes it useful for notes, homework checks, and reporting.