A Partial Fraction Decomposition Calculator is a tool that takes a rational function (a polynomial numerator divided by a polynomial denominator) and expresses it as a sum of simpler rational terms (partial fractions) — making integration, inverse Laplace transforms and analysis much easier.

What this calculator does (and why it helps)

This interactive online calculator accepts polynomial numerator and denominator coefficients and computes a numerical partial-fraction decomposition. It performs polynomial division (if necessary), finds approximate roots of the denominator, computes residues for simple roots using the formula A=P(r)Q′(r)A=Q′(r)P(r)​, and plots the original rational function together with its partial-fraction components — all inside your WordPress post or page. The visual breakdown helps students, engineers and teachers verify algebraic decompositions and see where vertical asymptotes or dominant components occur.

Where to place the calculator on your WordPress site

The delivered HTML/JS file is built to fit a typical WordPress content column between sidebars (max-width 760px) and is intended for the Custom HTML block or a theme’s custom-code area. It uses a white background and modern, accessible UI. To add it:

1. Copy the entire HTML file contents created in the provided canvas file.
2. In the WordPress editor, add a Custom HTML block and paste the code.
3. Update and preview the page. The widget is responsive and will adapt to smaller screens.

How to use the calculator — step by step

1. Enter numerator & denominator

Provide polynomial coefficients as comma-separated lists in highest-to-lowest degree order. Example:

• Numerator 1,0,-2 represents x2−2x2−2.
• Denominator 1,-3,2 represents x2−3x+2x2−3x+2.

This coefficient format is compact, unambiguous, and easy to type on mobile or desktop.

2. Plot range and compute

Set the plot range (min and max x) and click Compute & Plot. The tool:

• Performs polynomial long division if the numerator degree is greater than or equal to the denominator degree and shows the polynomial part.
• Finds denominator roots numerically (using a companion-matrix eigenvalue method).
• Computes residues for simple roots using A=P(r)Q′(r)A=Q′(r)P(r)​.
• Displays a readable decomposition, the approximate roots and multiplicities, and plots the original rational function. When roots are distinct and real, individual partial fraction terms are shown on the plot as dotted traces.

3. Interpreting the output

• Polynomial part: If there was a polynomial obtained from division (an improper fraction), this will be shown separately.
• Partial fraction expression: The calculator prints the numerical partial-fraction representation. For simple real roots the form is ∑Akx−rk∑x−rk​Ak​​.
• Roots list: Approximate root values and multiplicities help you spot asymptotes and repeated factors.
• Plots: The main rational function is plotted. When possible, individual simple-term contributions are plotted as dotted lines so you can see which terms dominate in different regions.

Limitations & best practices

This tool is designed for robust numerical decomposition and visualization, but it has important constraints:

• Best results: When the denominator factors into distinct real linear factors (e.g., (x−1)(x−2)(x−1)(x−2)), the output looks clean and residues are real and exact-looking.
• Complex roots: If denominator roots are complex, residues may be complex. The tool will still compute numerical residues but will present complex values. Complex conjugate factors can combine into real quadratic terms — if you need symbolic quadratic denominators, factor and handle them by hand or use a symbolic CAS.
• Repeated roots: For repeated factors (multiplicity > 1) the calculator reports multiplicity but the simplified symbolic form for repeated power denominators can be more involved; the tool attempts numeric evaluation and will warn if results look non-simple.
• Symbolic exactness: This is primarily a numeric / educational tool. For exact symbolic decomposition (fractions with algebraic radicals or symbolic parameters) use a CAS that does symbolic factoring (for example, SymPy, Maxima, or a symbolic plugin).

Why this approach is useful (E-E-A-T and technical details)

The calculator uses well-known numerical linear algebra (companion matrix eigenvalues) to find polynomial roots and the standard residue formula A=P(r)Q′(r)A=Q′(r)P(r)​ for simple poles. This is a standard, reliable approach used in numeric math libraries and engineering tools for decomposing rational functions. The visual interpretation provided by Plotly.js helps users validate algebraic manipulations and teaches the connection between algebraic decomposition and graph behavior (asymptotes, dominant poles). The implementation emphasizes clarity, accessibility (ARIA), and performance for interactive web use.

Troubleshooting common issues

1. Strange complex coefficients: Check whether the denominator has complex roots; factor it manually if you expect only real linear factors.
2. Division not shown: If the numerator degree is lower than denominator’s, there’s no polynomial (quotient) term.
3. Vertical asymptote spikes on the plot: These arise near real roots of the denominator; zoom or narrow the plotting window to inspect behavior away from vertical asymptotes.

Summary

This calculator gives a practical, visual approach to partial fraction decomposition directly inside WordPress. It is well suited for education, verification, and quick numeric decomposition tasks. For exact symbolic proofs and manipulations involving symbolic parameters, combine this tool with a symbolic algebra system.

FAQ

Q1: What input format should I use?
A1: Use comma-separated coefficients in highest-to-lowest order. Example 2,0,-3 = 2x2−32x2−3.

Q2: Does the calculator handle complex roots?
A2: Yes — it will compute numeric residues for complex roots, but complex residues display as complex numbers. If you need a real-form with irreducible quadratics, factor the denominator into real quadratics and handle symbolically.

Q3: What about repeated roots (multiplicity)?
A3: The tool detects multiplicities and reports them. For repeated poles you may see higher-power denominators in the printed form; for fully symbolic coefficient extraction with repeated roots use a CAS.

Q4: Is this symbolic or numeric?
A4: Primarily numeric and visual. It uses numeric linear algebra to find roots and then applies the residue formula for decomposition. For symbolic output use a dedicated symbolic algebra package.

Q5: Can I copy the code into a WordPress Custom HTML block?
A5: Yes. Copy the single-file HTML+JS content that was created and paste into a Custom HTML block or a custom-code area of your theme. The component is responsive and sized to fit standard content columns.