Calculator Input
Enter polynomial coefficients for f(x) and g(x). The tool evaluates h(x) = f(x) / g(x), applies the quotient rule, flags zero-denominator points, and builds a graph table.
Example Data Table
This example uses f(x) = x² + 3x + 2 and g(x) = x - 1. The quotient becomes h(x) = (x² + 3x + 2) / (x - 1).
| x | f(x) | g(x) | h(x) |
|---|---|---|---|
| 0 | 2 | -1 | -2 |
| 2 | 12 | 1 | 12 |
| 3 | 20 | 2 | 10 |
Formula Used
Quotient definition: h(x) = f(x) / g(x)
Domain rule: g(x) ≠ 0
Quotient rule for derivatives: h′(x) = [f′(x)g(x) - f(x)g′(x)] / [g(x)]²
Polynomial model used here: f(x) = a₃x³ + a₂x² + a₁x + a₀, g(x) = b₃x³ + b₂x² + b₁x + b₀
The calculator first evaluates numerator and denominator separately. It then divides them only when the denominator is nonzero. The derivative uses the quotient rule, which measures how the ratio changes at the chosen x value.
How to Use This Calculator
- Enter the numerator polynomial coefficients a₃ through a₀.
- Enter the denominator polynomial coefficients b₃ through b₀.
- Choose the x value where you want the quotient evaluated.
- Set the graph range and step size for the result table.
- Click Calculate Quotient to see the result above the form.
- Review the graph, generated table, intercept estimates, and domain exclusions.
- Use the CSV or PDF buttons to export your current results.
FAQs
1. What does this calculator compute?
It computes the quotient h(x) = f(x) / g(x) for two polynomial functions. It also evaluates the derivative using the quotient rule, estimates intercepts, detects denominator zeros, builds a results table, and draws a graph.
2. Why can the result become undefined?
The quotient is undefined wherever the denominator equals zero. Division by zero is not allowed. The calculator marks those points and skips them in the plotted quotient data.
3. What kind of functions can I enter here?
This version accepts polynomial coefficients up to degree three for both numerator and denominator. That allows many useful rational functions while keeping the page simple and fast.
4. How is the derivative of the quotient found?
The derivative uses the quotient rule. First find f′(x) and g′(x). Then apply h′(x) = [f′(x)g(x) - f(x)g′(x)] / [g(x)]², as long as g(x) is not zero.
5. What are denominator zeros used for?
Denominator zeros identify excluded domain values. They often indicate breaks or vertical asymptote behavior in a rational graph, depending on whether factors cancel.
6. What does the graph show?
The graph compares numerator, denominator, and quotient on the same plotting range. This helps you see where each part changes sign, grows quickly, or creates undefined quotient regions.
7. Why might the table step size change automatically?
Very small steps can create extremely large tables. The page increases the step size when needed, keeping performance steady and the export files easier to manage.
8. Can I use the exported files for reports?
Yes. The CSV file is useful for spreadsheet analysis. The PDF file provides a neat summary with the main formula strings, selected x value, quotient, derivative, and generated table values.