Calculator Input
Enter coefficients from highest power to constant. Example: x³ + 1 becomes 1, 0, 0, 1.
Example Data Table
| Example | Numerator coefficients | Denominator coefficients | Purpose |
|---|---|---|---|
| Example A | 2, 3, 1 | 1, 0, 0, 1 | Shows one real pole and a complex conjugate pair. |
| Example B | 1, 0, 5 | 1, 0, 0, 0, 16 | Shows four complex poles from x⁴ + 16. |
| Example C | 3, 2, 7, 4 | 1, 1, 1, 1 | Shows decomposition after polynomial long division first. |
Formula Used
Start with a rational function F(x) = P(x) / Q(x).
If deg(P) is at least deg(Q), long division extracts the polynomial part.
The remaining fraction is written over complex factors of Q(x).
When Q(x) = Π(x - rj)^mj, the decomposition becomes ΣΣ Aj,k / (x - rj)^k.
For simple poles, the residue is Aj,1 = P(rj) / Q′(rj).
This page computes poles numerically, then solves the coefficients from sampled equations.
That approach handles complex poles directly and checks reconstruction error numerically.
How to Use This Calculator
Enter numerator coefficients in descending power order.
Enter denominator coefficients the same way.
Set the real axis plot range and point count.
Choose a display precision that fits your coursework.
Press Compute Decomposition to generate the result above the form.
Review the decomposition string, pole map, and verification table.
Use CSV for tabular export and PDF for a clean report copy.
The example button loads a function with complex poles instantly.
Frequently Asked Questions
1. What does this calculator decompose?
It decomposes rational functions with real coefficients into complex linear factors. It also performs polynomial long division first when the fraction is improper.
2. Why are complex poles useful?
Complex poles reveal the full factor structure of the denominator. They also make residue calculations cleaner in advanced algebra, transforms, and control work.
3. Does the tool handle repeated poles?
Yes, repeated poles are estimated when close roots are grouped numerically. The output then includes higher powers like 1/(x − r)^2 when needed.
4. Why can tiny imaginary drift appear?
The solver is numerical, so very small rounding artifacts can appear. The verification table reports that drift, which should usually remain close to zero.
5. What if my numerator degree is larger?
The page first performs long division. It then decomposes only the remaining proper fraction and adds the polynomial quotient back into the final answer.
6. What coefficient order should I enter?
Always enter coefficients from highest power to constant term. For example, x^4 − 2x + 3 becomes 1, 0, 0, -2, 3.
7. Can I use fractions or decimals?
Decimals work directly. Fractions should be converted to decimals before entry because the input parser accepts real numeric tokens only.
8. What does the Plotly graph show?
The pole map shows each complex pole in the complex plane. The real axis graph below it shows how the original function behaves across your chosen interval.