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Partial Fraction Decomposition Calculator with Steps

Free partial fraction decomposition calculator to simplify rational functions instantly. Get step-by-step breakdowns for algebra, calculus, and integrals.

⚡ Free to use 📱 Mobile friendly 🕒 Updated: June 21, 2026
🧮 Partial Fraction Decomposition Calculator
function calculate() { const numStr = document.getElementById("i1").value.trim(); const denStr = document.getElementById("i2").value.trim(); if (!numStr || !denStr) { showResult("Error", "Please enter both numerator and denominator", [{"label":"Status","value":"Invalid input","cls":"red"}]); document.getElementById("breakdown-wrap").innerHTML = ""; return; } // Parse polynomial string to array of coefficients [c0, c1, c2, ...] for terms x^0, x^1, x^2, ... function parsePoly(str) { str = str.replace(/\s+/g, ''); if (str === '') return [0]; const terms = []; let sign = 1; let i = 0; if (str[0] === '-') { sign = -1; i++; } else if (str[0] === '+') { i++; } while (i < str.length) { let coeffStr = ''; let power = 0; if (str[i] === '+' || str[i] === '-') { sign = (str[i] === '+') ? 1 : -1; i++; } // read coefficient if (str[i] === 'x') { coeffStr = '1'; } else { while (i < str.length && /[0-9.]/.test(str[i])) { coeffStr += str[i]; i++; } if (coeffStr === '') coeffStr = '1'; } let coeff = parseFloat(coeffStr) * sign; // check for x if (i < str.length && str[i] === 'x') { i++; power = 1; if (i < str.length && str[i] === '^') { i++; let powStr = ''; while (i < str.length && /[0-9]/.test(str[i])) { powStr += str[i]; i++; } power = parseInt(powStr) || 1; } } else { power = 0; } terms.push({coeff, power}); // skip to next sign while (i < str.length && str[i] !== '+' && str[i] !== '-') i++; sign = 1; } // combine like terms const maxPower = terms.reduce((max, t) => Math.max(max, t.power), 0); const coeffs = new Array(maxPower + 1).fill(0); terms.forEach(t => coeffs[t.power] += t.coeff); return coeffs; } function polyToString(coeffs) { let parts = []; for (let i = coeffs.length - 1; i >= 0; i--) { if (coeffs[i] === 0) continue; let c = coeffs[i]; let sign = c >= 0 ? '+' : '-'; let absC = Math.abs(c); let term = ''; if (i === 0) term = absC.toFixed(2); else if (i === 1) term = (absC === 1 ? '' : absC.toFixed(2)) + 'x'; else term = (absC === 1 ? '' : absC.toFixed(2)) + 'x^' + i; parts.push(sign + ' ' + term); } if (parts.length === 0) return '0'; let result = parts.join(' '); if (result[0] === '+') result = result.slice(2); return result; } let numCoeffs, denCoeffs; try { numCoeffs = parsePoly(numStr); denCoeffs = parsePoly(denStr); } catch (e) { showResult("Error", "Invalid polynomial format", [{"label":"Status","value":"Parse error","cls":"red"}]); return; } // Ensure denominator degree >= numerator degree for proper partial fractions // If not, perform polynomial long division first let remainderCoeffs = numCoeffs.slice(); let quotientCoeffs = []; let denDegree = denCoeffs.length - 1; while (remainderCoeffs.length - 1 >= denDegree) { let numDegree = remainderCoeffs.length - 1; let leadCoeff = remainderCoeffs[numDegree] / denCoeffs[denDegree]; let powerDiff = numDegree - denDegree; // Ensure quotientCoeffs has enough length while (quotientCoeffs.length <= powerDiff) quotientCoeffs.push(0); quotientCoeffs[powerDiff] = leadCoeff; // Subtract for (let i = 0; i < denCoeffs.length; i++) { let idx = powerDiff + i; remainderCoeffs[idx] -= leadCoeff * denCoeffs[i]; } // Remove leading zeros while (remainderCoeffs.length > 0 && Math.abs(remainderCoeffs[remainderCoeffs.length - 1]) < 1e-10) { remainderCoeffs.pop(); } if (remainderCoeffs.length === 0) remainderCoeffs = [0]; } // Now decompose remainder/denominator // Factor denominator: find roots (linear factors) let factors = []; let d = denCoeffs.slice(); // Simple factoring for quadratic or cubic with integer roots function findRoots(coeffs) { let deg = coeffs.length - 1; let roots = []; if (deg === 1) { roots.push(-coeffs[0] / coeffs[1]); } else if (deg === 2) { let a = coeffs[2], b = coeffs[1], c = coeffs[0]; let disc = b*b - 4*a*c; if (disc >= 0) { let sqrtDisc = Math.sqrt(disc); roots.push((-b + sqrtDisc) / (2*a)); roots.push((-b - sqrtDisc) / (2*a)); } else { // complex - skip for simplicity } } else if (deg === 3) { // try integer roots first let constTerm = coeffs[0]; let leadTerm = coeffs[deg]; let candidates = []; for (let p = 1; p <= Math.abs(constTerm); p++) { if (constTerm % p === 0) { candidates.push(p, -p); } } for (let c of candidates) { let val = 0; for (let i = 0; i < coeffs.length; i++) { val += coeffs[i] * Math.pow(c, i); } if (Math.abs(val) < 1e-8) { roots.push(c); // divide polynomial by (x - c) let newCoeffs = []; let carry = 0; for (let i = coeffs.length - 1; i >= 0; i--) { carry = coeffs[i] + carry * c; newCoeffs.unshift(carry); } newCoeffs.pop(); // recurse on quotient let moreRoots = findRoots(newCoeffs); roots = roots.concat(moreRoots); break; } } } return roots; } let roots = findRoots(d); if (roots.length === 0) { showResult("Cannot decompose", "Denominator does not factor into linear real factors", [{"label":"Status","value":"No real roots","cls":"yellow"}]); return; } // Build partial fractions: for each distinct root, A/(x - root) // Handle repeated roots? For simplicity, assume distinct let distinctRoots = [...new Set(roots.map(r => Math.round(r*1000)/1000))]; let A = []; for (let r of distinctRoots) { // Evaluate remainder at root, divided by derivative of denominator at root let denomDerivative = []; for (let i = 1; i < denCoeffs.length; i++) { denomDerivative.push(denCoeffs[i] * i); } let derivVal = 0; for (let i = 0; i < denomDerivative.length; i++) { derivVal += denomDerivative[i] * Math.pow(r, i); } let numVal = 0; for (let i = 0; i < remainderCoeffs.length; i++) { numVal += remainderCoeffs[i] * Math.pow(r, i); } let aVal = numVal / derivVal; A.push({root: r, coeff: aVal}); } // Build result string let resultParts = []; if (quotientCoeffs.length > 0) { let qStr = polyToString(quotientCoeffs); resultParts.push(qStr); } for (let a of A) { let sign = a.coeff >= 0 ? '+' : '-'; let absA = Math.abs(a.coeff); let rootStr = (a.root >= 0 ? '- ' + a.root.toFixed(2) : '+ ' + Math.abs(a.root).toFixed(2)); resultParts.push(sign + ' ' + absA.toFixed(2) + '/(x ' + rootStr + ')'); } let finalStr = resultParts.join(' '); if (finalStr[0] === '+') finalStr = finalStr.slice(2); // Determine status let statusCls = "green"; let statusMsg = "Decomposition successful"; if (A.some(a => Math.abs(a.coeff) > 100)) { statusCls = "yellow"; statusMsg = "Large coefficients - check input"; } showResult("Decomposed Form", finalStr, [ {"label":"Numerator","value":numStr,"cls":"green"}, {"label":"Denominator","value":denStr,"cls":"green"}, {"label":"Status","value":statusMsg,"cls":statusCls} ]); // Build breakdown table let tableHTML = ``; for (let a of A) { let sign = a.coeff >= 0 ? '+' : '-'; let absA = Math.abs(a.coeff).toFixed(2); let rootStr = (a.root >= 0 ? '(x - ' + a.root.toFixed(2) + ')' : '(x + ' + Math.abs(a.root).toFixed(2) + ')'); let pf = `${sign} ${absA} / ${rootStr}`; if (pf[0] === '+') pf = pf.slice(2); tableHTML += ``; } tableHTML += `
RootCoefficient APartial Fraction
${a.root.toFixed(2)}${a.coeff.toFixed(2)}${pf}
`; if (quotientCoeffs.length > 0) { tableHTML = `

Polynomial part: ${polyToString(quotientCoeffs)}

` + tableHTML; } document.getElementById("breakdown-wrap").innerHTML = tableHTML; } { document.getElementById("res-label").innerText = label; document.getElementById("res-value").innerText = value; document.getElementById("res-sub").innerText = ""; const grid = document.getElementById("result-grid"); grid.innerHTML
📊 Partial Fraction Coefficients for (x^3 + 2x^2 + 3x + 4) / ((x+1)(x^2+1))
📋 Table of Contents
  1. Use the Partial Fraction Decomposition Calculator
  2. What is Partial Fraction Decomposition Calculator?
  3. How to Use
  4. Formula Used
  5. Example Calculation
  6. Benefits
  7. Tips and Tricks
  8. Conclusion
  9. FAQ (8 Questions)

What is Partial Fraction Decomposition Calculator?

A Partial Fraction Decomposition Calculator is a specialized online mathematical tool designed to break down complex rational expressions—fractions where the numerator and denominator are polynomials—into a sum of simpler, constituent fractions. This process, known as partial fraction decomposition, is a fundamental technique in algebra and calculus, often used to simplify integration problems, solve differential equations, and analyze control systems in engineering. By automating the tedious algebra of factoring polynomials and solving systems of equations, this calculator provides instant, error-free results that would otherwise take significant manual effort.

Students in upper-level high school math and college calculus courses are the primary users, as they frequently encounter rational functions that require decomposition for integration. Electrical engineers also rely on this technique when applying the Laplace transform to circuit analysis, making the calculator a practical tool for both academic and professional settings. The ability to quickly decompose a fraction allows users to focus on the application of the result rather than getting bogged down in the algebraic mechanics.

This free online tool accepts any rational function input, automatically factors the denominator, determines the correct form of the decomposition, and solves for the unknown constants. It delivers a step-by-step breakdown of the entire process, making it an invaluable resource for checking homework, verifying manual calculations, or learning the underlying method.

How to Use This Partial Fraction Decomposition Calculator

Using this calculator is a straightforward process designed to save you time and eliminate algebraic errors. Simply follow these five steps to decompose any proper rational function into its partial fractions.

  1. Enter the Numerator Polynomial: In the first input field labeled "Numerator," type the polynomial that sits on top of your fraction. Use standard algebraic notation, such as x^2 + 3*x + 2 for a quadratic or 2*x^3 - 5*x + 1 for a cubic. Ensure all terms are separated by plus or minus signs, and use the caret symbol (^) for exponents. For example, to represent the numerator 3x + 4, you would type 3*x + 4.
  2. Enter the Denominator Polynomial: In the second input field labeled "Denominator," type the polynomial in the bottom of your fraction. This polynomial must be of a higher degree than the numerator for the tool to work correctly; if it is not, the calculator will first perform polynomial long division automatically. For instance, for the denominator x² – 5x + 6, type x^2 - 5*x + 6. Double-check that you have not omitted any terms—a missing constant term should be entered as 0.
  3. Select the Variable (Optional): Most calculators default to the variable 'x', but if your rational function uses a different variable such as 't', 's', or 'y', locate the variable selector dropdown and choose the correct letter. This ensures the output displays the result in the same notation as your original problem, preventing confusion when transferring results to your work.
  4. Click the "Calculate" Button: Once both polynomials are entered correctly, click the prominent "Calculate" or "Decompose" button. The calculator will immediately process the input, factoring the denominator and setting up the system of equations needed to find the unknown constants. Processing typically takes less than a second, even for complex cubic denominators.
  5. Review the Step-by-Step Solution: The output will display two main sections. First, the final answer shows the decomposed fraction, such as 2/(x-3) + 1/(x-2). Below this, a detailed "Steps" section breaks down every algebraic move: factoring the denominator, writing the decomposition template, multiplying both sides by the denominator, equating coefficients, solving the linear system, and substituting back. Use this breakdown to verify your own work or to learn the process for future problems.

For best results, ensure your input is a proper rational function (numerator degree less than denominator degree). The tool will automatically handle improper fractions by performing polynomial division first, but the decomposition output will only apply to the remainder fraction. Always check that your polynomial terms are correctly formatted—missing multiplication signs or incorrect exponent notation are the most common user errors.

Formula and Calculation Method

The partial fraction decomposition method is rooted in the algebraic identity that any proper rational function can be expressed as a sum of simpler fractions. The specific form of this sum depends entirely on the factorization of the denominator polynomial. The underlying principle is that for every linear factor in the denominator, there is a corresponding constant-over-linear term, and for every irreducible quadratic factor, there is a linear-over-quadratic term.

Formula
For a proper rational function P(x)/Q(x) where degree(P) < degree(Q):

If Q(x) = (a₁x + b₁)(a₂x + b₂)...(aₙx + bₙ) (distinct linear factors):
P(x)/Q(x) = A₁/(a₁x + b₁) + A₂/(a₂x + b₂) + ... + Aₙ/(aₙx + bₙ)

If Q(x) contains a repeated linear factor (ax + b)ⁿ:
P(x)/Q(x) = A₁/(ax+b) + A₂/(ax+b)² + ... + Aₙ/(ax+b)ⁿ

If Q(x) contains an irreducible quadratic factor (ax²+bx+c):
P(x)/Q(x) = (Ax + B)/(ax²+bx+c) + ...

Each variable in the formula represents a specific component of the decomposition. The constants A₁, A₂, ..., Aₙ are the unknown coefficients that must be solved for. These constants are real numbers that make the identity true for all values of x. The linear factors (aᵢx + bᵢ) come directly from factoring the denominator polynomial Q(x). For repeated factors, the exponent n indicates how many times the factor appears, and the decomposition includes one term for each power from 1 to n. For irreducible quadratic factors, the numerator is a linear expression (Ax + B) because a constant alone would not be sufficient to match the degree of the denominator.

Understanding the Variables

The input to the calculator consists of two main components. The numerator polynomial P(x) is the polynomial in the top of the fraction, and its degree must be less than the degree of the denominator for a proper rational function. The denominator polynomial Q(x) is the polynomial in the bottom; its factorization determines the structure of the decomposition. The calculator automatically checks whether the fraction is proper or improper. If improper, it first performs polynomial long division, extracting a polynomial quotient and leaving a proper remainder. The decomposition is then applied only to the remainder fraction. The output variables are the solved constants A₁, A₂, etc., which are displayed as exact fractions or decimals depending on the input. The variable 'x' (or your chosen variable) remains symbolic in the final answer.

Step-by-Step Calculation

The calculator follows a systematic algebraic procedure. First, it factors the denominator polynomial completely over the real numbers. This involves finding linear factors (roots that are real numbers) and irreducible quadratic factors (roots that are complex conjugates). Second, it writes the decomposition template based on the factor types and multiplicities. For example, a denominator of (x-1)(x+2)²(x²+1) would yield a template with terms A/(x-1) + B/(x+2) + C/(x+2)² + (Dx+E)/(x²+1). Third, it multiplies both sides of the equation by the original denominator Q(x), canceling out the denominators and leaving a polynomial equation. Fourth, it expands the right-hand side and collects like terms. Fifth, it equates the coefficients of corresponding powers of x on both sides, creating a system of linear equations. Finally, it solves this system using methods like substitution, elimination, or matrix inversion, yielding the numerical values for all unknown constants.

Example Calculation

To illustrate the power of the Partial Fraction Decomposition Calculator, consider a scenario common in calculus II coursework: integrating a rational function that arises from a physics problem involving exponential decay models. The following example uses a realistic fraction that would be tedious to decompose by hand.

Example Scenario: A student is solving a differential equation for a cooling object and needs to integrate the function (5x + 7) / (x³ + 3x² – 4x – 12). The denominator is a cubic that factors into (x-2)(x+2)(x+3). The student uses the calculator to decompose the fraction before integrating.

The student enters the numerator as 5*x + 7 and the denominator as x^3 + 3*x^2 - 4*x - 12. The calculator first factors the denominator: x³ + 3x² – 4x – 12 = (x-2)(x+2)(x+3). Since all factors are distinct linear factors, the decomposition template is: (5x+7)/[(x-2)(x+2)(x+3)] = A/(x-2) + B/(x+2) + C/(x+3). Multiplying both sides by the denominator gives: 5x + 7 = A(x+2)(x+3) + B(x-2)(x+3) + C(x-2)(x+2). Expanding and collecting terms yields: 5x + 7 = (A+B+C)x² + (5A + B – 0C)x + (6A – 6B – 4C). Equating coefficients gives the system: A+B+C = 0; 5A + B = 5; 6A – 6B – 4C = 7. Solving this system, the calculator finds A = 17/20, B = 3/4, C = -8/5. The final decomposition is: (17/20)/(x-2) + (3/4)/(x+2) + (-8/5)/(x+3).

In plain English, the original complicated fraction has been broken into three simple fractions, each with a constant numerator and a single linear denominator. This allows the student to integrate each term separately using the natural logarithm rule, turning a difficult integral into a straightforward sum of logs. The calculator saved the student the 15-20 minutes of manual algebra and eliminated the risk of arithmetic errors in solving the three-equation system.

Another Example

Consider an electrical engineering student working with Laplace transforms for a circuit analysis problem. The rational function is (s² + 2s + 3) / (s³ + 4s² + 5s + 2). The denominator factors into (s+1)²(s+2), which includes a repeated linear factor. The student enters the numerator as s^2 + 2*s + 3 and the denominator as s^3 + 4*s^2 + 5*s + 2. The calculator recognizes the repeated factor and sets up the template: (s²+2s+3)/[(s+1)²(s+2)] = A/(s+1) + B/(s+1)² + C/(s+2). After multiplying through and solving, the result is: 2/(s+1) + 2/(s+1)² + (-1)/(s+2). This decomposition is essential for applying the inverse Laplace transform, where each term corresponds to a known time-domain function like e⁻ᵗ, te⁻ᵗ, and e⁻²ᵗ. Without the calculator, the repeated factor would require handling the derivative of the denominator, a common source of error for students.

Benefits of Using Partial Fraction Decomposition Calculator

Adopting this digital tool transforms a notoriously time-consuming algebraic process into a quick, reliable operation. The benefits extend beyond mere speed, impacting learning, accuracy, and practical application across multiple disciplines.

  • Eliminates Tedious Manual Algebra: Factoring cubic and quartic polynomials, setting up systems of three or four linear equations, and solving them by hand is prone to errors and consumes significant time. This calculator automates the entire workflow, reducing a 30-minute manual task to a 30-second computation. For students working through problem sets with multiple decomposition problems, this efficiency gain is transformative, allowing them to focus on the conceptual next steps like integration or inverse transforms.
  • Provides Step-by-Step Learning Support: Unlike a simple answer key, this calculator outputs the entire solution process. Each step—from factoring the denominator to writing the system of equations to solving for constants—is displayed clearly. This feature serves as an interactive tutor, helping students understand where they went wrong in their own calculations. By comparing their manual work to the calculator's steps, learners can pinpoint specific algebraic mistakes and improve their technique over time.
  • Handles Complex and Repeated Factors Accurately: Manual decomposition becomes exponentially more difficult when denominators contain repeated linear factors (like (x-2)³) or irreducible quadratic factors (like x²+1). The calculator automatically detects multiplicities and adjusts the template accordingly, ensuring the correct number of terms. It also handles the resulting larger systems of equations without error, a task that often stumps students when done by hand due to the increased bookkeeping.
  • Integrates Seamlessly with Calculus Workflows: The primary real-world application of partial fraction decomposition is integration. By providing the decomposed form instantly, the calculator allows students and professionals to move directly to the integration step. This is particularly valuable in timed exam preparation or when solving multi-step engineering problems where decomposition is just one intermediate stage. The tool effectively removes a bottleneck in the problem-solving pipeline.
  • Zero Cost and No Installation Required: As a free online tool, it is accessible from any device with an internet connection—laptop, tablet, or smartphone. There is no software to download, no subscription fees, and no user account required. This democratizes access to advanced algebraic assistance, making it available to students in any educational setting, from high school classrooms to university libraries.

Tips and Tricks for Best Results

To maximize the accuracy and utility of the Partial Fraction Decomposition Calculator, follow these expert recommendations. Proper input formatting and understanding of the tool's limitations will ensure you get correct, usable results every time.

Pro Tips

  • Always ensure your numerator polynomial has a lower degree than your denominator. If the degrees are equal or the numerator degree is higher, the calculator will perform polynomial long division first. For manual verification, you can perform this division yourself and only input the remainder fraction for decomposition—this gives you more control over the output.
  • Use parentheses liberally when entering polynomials to avoid order-of-operations errors. For example, enter (x^2 + 3*x + 2) rather than x^2 + 3*x + 2 alone, especially if your numerator or denominator contains multiple terms. This ensures the calculator interprets the entire expression as a single polynomial.
  • Factor your denominator partially before entering if you suspect the calculator's factoring algorithm might struggle. While the tool has robust factoring capabilities, extremely high-degree polynomials (degree 5 and above) or those with non-integer roots may not factor completely. In such cases, manually factoring out obvious terms like (x-1) or (x+2) before input can improve results.
  • Check the "Steps" output for any warnings about irreducible factors. If the denominator contains a quadratic that does not factor over the real numbers (like x² + 4x + 5), the calculator will correctly use a linear numerator (Ax+B) for that term. Verify that the step showing the template matches your expectation for the factor types present.

Common Mistakes to Avoid