Last updated: June 25, 2026
Partial Fraction Decomposition Calculator
| Pole | Order | Residue | Type |
|---|
| F(s) Term | f(t) Inverse | Constant |
|---|
| Method | Constants Found | Operations | Speed |
|---|
For distinct linear factors in Q(x), each factor (x-a) gives one partial fraction term with constant numerator A.
For a repeated linear factor (x-a)^n, include n partial fraction terms with powers 1 through n.
For an irreducible quadratic (discriminant less than 0), the numerator is a linear polynomial Ax+B.
- 1Check degree: if deg(N) >= deg(D), perform polynomial long division first.
- 2Fully factor D(x) into linear and irreducible quadratic factors over the reals.
- 3Write the partial fraction template matching each factor type and multiplicity.
- 4Multiply both sides by D(x) to clear all denominators.
- 5Solve for unknown constants by substituting roots or matching coefficients.
- 6Verify by recombining partial fractions and comparing with the original.
| PFD Form | Integral Result |
|---|---|
| A/(x-a) | A ln|x-a| + C |
| A/(x-a)^n | A/(1-n)(x-a)^(n-1) + C |
| (Ax+B)/(x^2+k^2) | (A/2)ln(x^2+k^2)+(B/k)arctan(x/k)+C |
| A/(x^2+k^2) | (A/k)arctan(x/k) + C |
| Ax/(x^2+k^2) | (A/2)ln(x^2+k^2) + C |
- AAlways verify your answer by combining partial fractions back over a common denominator.
- BComplex conjugate roots always appear in pairs; their residues are also conjugates.
- CIn Laplace transforms, all poles must have negative real parts for a stable (causal) system.
- DIntegration of partial fractions is only valid on intervals containing no poles.
In the algebraic fraction analysis framework, partial fraction decomposition is the integration and transform bridge that makes otherwise unsolvable integrals tractable and Laplace transform inversions straightforward. The technique was formalized by the Swiss mathematician Euler in the eighteenth century and later extended by Laplace in his development of the transform that now carries his name.
A calculus student integrating 1/(x² − 1) and a control engineer inverting a Laplace transform are performing the same decomposition — and both need the step-by-step breakdown to identify each constant correctly. Understanding partial fraction decomposition tells you exactly how to split any rational function into its simplest components.
Use this free Partial Fraction Decomposition Calculator to instantly decompose any rational function, generate the complete step-by-step solution, evaluate residues, analyze pole locations, and prepare results for integration or Laplace inversion. No sign-up required.
What Is Partial Fraction Decomposition?
Partial Fraction Decomposition Definition
Partial Fraction Decomposition is an algebraic technique that expresses a rational function — the ratio of two polynomials — as a finite sum of simpler fractions, each with a lower-degree denominator. It is classified as a polynomial decomposition operation within the broader family of algebraic manipulation and calculus preparation techniques used in mathematics, engineering, and physics.
Partial Fraction Decomposition — Definition Partial fraction decomposition rewrites a rational function P(x)/Q(x) as a sum of partial fractions, each having a factor of the denominator Q(x) as its own denominator and a constant or lower-degree polynomial as its numerator. The decomposition is unique once the degree of P(x) is less than the degree of Q(x).
The Standard Decomposition Form
The general decomposition for a rational function with distinct linear factors in the denominator is:
| P(x) / [(x − r₁)(x − r₂)···(x − rₙ)] = A₁/(x − r₁) + A₂/(x − r₂) + ··· + Aₙ/(x − rₙ) |
Where each Aᵢ is a constant numerator uniquely determined by the original numerator polynomial P(x), and each denominator factor (x − rᵢ) is one linear factor from the full denominator Q(x).
| Proper Condition: degree of P(x) must be strictly less than degree of Q(x) |
What Does a Decomposition of 2/(x+1) + 1/(x+2) Actually Mean?
A decomposition of 2/(x + 1) + 1/(x + 2) for the rational function (3x + 5)/[(x + 1)(x + 2)] means:
- Adding the two fractions back together: 2(x + 2)/[(x+1)(x+2)] + 1(x + 1)/[(x+1)(x+2)] = (2x + 4 + x + 1)/[(x+1)(x+2)] = (3x + 5)/[(x+1)(x+2)] ✓
- Each simple fraction is directly integrable: ∫ 2/(x+1) dx = 2 ln|x+1| + C
- The decomposition immediately reveals the poles of the rational function at x = −1 and x = −2
- Each constant in the numerator is the residue of the function at that pole
Partial Fraction Decomposition vs. Polynomial Long Division — Key Difference
| Method | Partial Fraction Decomposition | Polynomial Long Division |
| Applies to | Proper rational functions | Improper rational functions (deg P ≥ deg Q) |
| Output | Sum of simple fractions | Polynomial quotient plus remainder fraction |
| Purpose | Integration and Laplace inversion | Simplification of improper fractions |
| Required first step when | Denominator degree exceeds numerator | Numerator degree ≥ denominator degree |
| Result form | A/(x−r) + B/(x−s) + ··· | Polynomial + proper fraction |
Why Partial Fraction Decomposition Is Important
For Calculus Students Computing Integrals
The primary application of partial fraction decomposition in calculus is integration. Rational functions of the form P(x)/Q(x) are generally not directly integrable in standard form, but each partial fraction of the form A/(x − r) integrates immediately to A·ln|x − r| + C. A single decomposition step converts an otherwise difficult or impossible integral into a sum of elementary logarithmic integrals.
- Converts integrals of rational functions into sums of immediately integrable terms
- Provides the standard method for the partial fraction decomposition integral evaluated in definite and indefinite form
- Appears directly in calculus textbooks, university examinations, and engineering mathematics courses worldwide
For Engineers Using the Laplace Transform
In control engineering, signal processing, and differential equation solving, the Laplace Transform converts differential equations into algebraic equations in the frequency domain. The inverse Laplace transform requires recognizing standard forms — and partial fraction decomposition converts any rational Laplace-domain expression into terms whose inverse transforms are immediately readable from transform tables.
- Enables direct inversion of any rational Laplace transform expression to its time-domain function
- Identifies system poles that determine stability, oscillation frequency, and decay rate
- Used daily in the design of filters, controllers, and dynamic system models by engineers worldwide
For Algebra Students Mastering Rational Expressions
Even before reaching calculus, students encounter partial fraction decomposition as a pure algebra technique for simplifying and analyzing complex rational expressions. Understanding how any algebraic fraction can be decomposed into simpler components builds the structural intuition for polynomial factoring, root finding, and rational function behavior.
- Reinforces polynomial factoring skills through denominator analysis
- Connects root locations of the denominator polynomial to the structure of the decomposition
- Bridges the algebra and calculus curriculum by showing exactly what makes a rational function integrable
How to Use the Partial Fraction Decomposition Calculator (Step-by-Step)
Step 1 — Write Your Rational Function in Standard Form
Express your function as P(x)/Q(x) where P(x) is the numerator polynomial and Q(x) is the denominator polynomial. Verify that the degree of P(x) is strictly less than the degree of Q(x). If not, perform polynomial long division first to extract the polynomial part, then apply decomposition to the remaining proper fraction.
Step 2 — Factor the Denominator
Factor Q(x) completely into linear factors (x − r) and irreducible quadratic factors (x² + bx + c). This step determines the form that the decomposition will take. Each distinct linear factor contributes one partial fraction of the form A/(x − r). Each irreducible quadratic factor contributes one fraction of the form (Bx + C)/(x² + bx + c).
Step 3 — Enter the Numerator and Denominator
Enter P(x) in the numerator field and the factored form of Q(x) in the denominator field. The calculator accepts polynomial expressions in standard notation. The equation preview confirms that the entry reflects the correct rational function before calculating.
Step 4 — Click Calculate
Press the Calculate button. The solver applies the cover-up method for simple linear factors and solves the system of equations for quadratic or repeated factors. The result displays the complete decomposed form, the individual constants A, B, C for each partial fraction, and the full step-by-step solution showing every algebraic operation.
Step 5 — Use the Result for Integration or Laplace Inversion
Read the decomposed form directly and apply it to your intended calculation. For integration, integrate each term individually. For Laplace inversion, match each term against standard transform table entries. Use the pole-zero chart to visualize where each denominator factor places its root on the complex plane.
Partial Fraction Decomposition Formula and Cases
Case 1 — Distinct Linear Factors
When the denominator Q(x) factors into n distinct linear factors with no repetition:
| P(x) / [(x − r₁)(x − r₂)···(x − rₙ)] = A₁/(x − r₁) + A₂/(x − r₂) + ··· + Aₙ/(x − rₙ) |
Each constant Aᵢ is found by the cover-up method: multiply both sides by (x − rᵢ), then substitute x = rᵢ:
| Aᵢ = P(rᵢ) / ∏ⱼ≠ᵢ (rᵢ − rⱼ) |
Case 2 — Repeated Factor
When the denominator contains a repeated factor (x − r)ⁿ, the decomposition includes n separate terms for that factor, one for each power from 1 to n:
| A/(x − r) + B/(x − r)² + C/(x − r)³ + ··· + N/(x − r)ⁿ |
The constants A, B, C, …, N for a repeated factor cannot be found by the cover-up method alone and require expanding and matching coefficients or differentiating.
Case 3 — Irreducible Quadratic Factor
When the denominator contains an irreducible quadratic factor x² + bx + c (where b² − 4c < 0, meaning no real roots), the corresponding partial fraction has a linear numerator:
| (Bx + C) / (x² + bx + c) |
This form is required because a constant numerator over a quadratic denominator cannot represent the full range of rational functions with quadratic denominators.
Case 4 — Mixed: Linear and Quadratic Factors
When the denominator contains both linear and quadratic factors, all three term types appear simultaneously in the decomposition. Each factor type contributes its own term structure, and the constants for all terms are solved simultaneously by equating coefficients of each power of x.
| P(x) / [(x − r)(x² + bx + c)] = A/(x − r) + (Bx + C)/(x² + bx + c) |
Partial Fraction Decomposition Example Calculations
Example 1 — Two Distinct Linear Factors
Decompose (3x + 5) / [(x + 1)(x + 2)].
| Step | Operation | Result |
| Set up decomposition | (3x+5)/[(x+1)(x+2)] = A/(x+1) + B/(x+2) | Template established |
| Multiply through | 3x + 5 = A(x+2) + B(x+1) | Numerator equation |
| Cover-up for A (x = −1) | 3(−1)+5 = A(−1+2) → 2 = A | A = 2 |
| Cover-up for B (x = −2) | 3(−2)+5 = B(−2+1) → −1 = −B | B = 1 |
| Write result | 2/(x+1) + 1/(x+2) | Decomposition complete |
| Verify | 2(x+2)+1(x+1) = 2x+4+x+1 = 3x+5 ✓ | Correct |
Integration result: ∫ (3x+5)/[(x+1)(x+2)] dx = 2 ln|x+1| + ln|x+2| + C
Example 2 — Three Distinct Linear Factors
Decompose 1 / [x(x+1)(x−1)].
| Cover-up at x=0: A = 1/[(0+1)(0−1)] = 1/(−1) = −1 |
| Cover-up at x=−1: B = 1/[(−1)(−1−1)] = 1/2 = 1/2 |
| Cover-up at x=1: C = 1/[(1)(1+1)] = 1/2 = 1/2 |
Result: −1/x + (1/2)/(x+1) + (1/2)/(x−1)
Example 3 — Repeated Factor
Decompose 1 / [(x+1)²(x−2)].
| Set up: A/(x+1) + B/(x+1)² + C/(x−2) |
Cover-up at x = 2: C = 1/(2+1)² = 1/9. Cover-up at x = −1: B = 1/(−1−2) = −1/3. Expanding and matching x² coefficients: A + C = 0, so A = −1/9.
Result: −(1/9)/(x+1) − (1/3)/(x+1)² + (1/9)/(x−2)
Example 4 — Irreducible Quadratic Factor
Decompose (2x + 1) / [(x − 1)(x² + x + 1)].
| Set up: A/(x−1) + (Bx + C)/(x²+x+1) |
Cover-up at x = 1: A = (2+1)/(1+1+1) = 1. Expanding and matching x² and constant terms: B = −1, C = 0.
Result: 1/(x−1) + (−x)/(x²+x+1)
Decompose into Partial Fractions — Step-by-Step Process Summary
When you use this calculator to find partial fraction decomposition results with full steps, the process follows these stages automatically:
| Stage | What the Calculator Does |
| Degree check | Confirms numerator degree < denominator degree |
| Denominator parsing | Identifies all linear factors, quadratic factors, and repeated factors |
| Template construction | Builds the partial fraction template with the correct number of unknown constants |
| System setup | Expands the right-hand side and equates to the numerator |
| Constant solving | Applies cover-up method and/or solves the linear system |
| Verification | Multiplies all partial fractions back together and confirms equality with original |
The partial fraction decomposition calculator step by step output shows every algebraic detail at each stage, making it a complete learning and verification tool for both students and practitioners.
What Is a Good Decomposition Result? — Interpretation Guide
When All Constants Are Simple Integers or Fractions
When the decomposition produces clean integer or simple fractional constants like A = 2, B = −1, or C = 1/3, the original rational function had a well-structured denominator with rational roots. These results integrate to simple logarithmic expressions and invert under the Laplace transform to clean exponential or polynomial terms.
When a Quadratic Factor Produces a Linear Numerator
When the decomposition includes a term like (Bx + C)/(x² + bx + c), the irreducible quadratic factor has no real roots. The integration of this term requires completing the square in the denominator and produces arctangent or logarithmic-plus-arctangent results. In Laplace inversion, it corresponds to oscillatory (sinusoidal) time-domain components.
When the Decomposition Fails — Improper Rational Functions
If the degree of the numerator polynomial equals or exceeds the degree of the denominator polynomial, the rational function is improper and must be reduced by polynomial long division before decomposition. The calculator automatically detects this condition and prompts for the division step. Attempting decomposition on an improper fraction without prior division produces incorrect constants.
When Poles Are in the Left Half-Plane — Stability Indication
In Laplace transform applications, the location of poles (roots of the denominator) on the complex plane indicates system stability. Poles with negative real parts (left half-plane) correspond to stable, decaying exponential terms. Poles on the imaginary axis correspond to sustained oscillations. Poles in the right half-plane indicate unstable, growing responses.
Benefits of Using This Partial Fraction Decomposition Calculator
- Instant decomposition — enter numerator and denominator for an immediate result across all factor types
- Step-by-step solution — the partial fraction decomposition calculator with steps output shows every algebraic detail from template setup to constant verification
- Linear, quadratic, and repeated factor support — handles all standard decomposition cases including mixed denominators
- Integration-ready output — each partial fraction is displayed with its immediate integral form for partial fraction decomposition integral applications
- Laplace transform mode — formats results for direct use in inverse Laplace transform calculations
- Pole-zero visualization — displays all denominator roots on a complex plane chart with stability classification
- Residue computation — calculates the residue at each pole, which equals the partial fraction constant for simple poles
- Method comparison — compares cover-up method, system-of-equations method, and residue theorem for the same input
- Degree and complexity analysis — reports numerator degree, denominator degree, factor types, and decomposition complexity
- No registration required — completely free to use immediately
Common Mistakes to Avoid
Mistake 1 — Not Checking the Degree Condition First
The most critical prerequisite for partial fraction decomposition is that the rational function must be proper — the degree of the numerator must be strictly less than the degree of the denominator. Applying decomposition to an improper fraction like (x³ + 2x) / (x² − 1) without first performing polynomial long division produces wrong constants. Always verify the degree condition before beginning.
Mistake 2 — Using a Constant Numerator for a Quadratic Denominator Factor
When the denominator contains an irreducible quadratic factor like x² + 4, the corresponding partial fraction must have a linear numerator Bx + C, not just a constant B. Writing B/(x² + 4) instead of (Bx + C)/(x² + 4) leaves one degree of freedom unaccounted for and produces an incorrect system of equations. The rule is: the numerator of each partial fraction must have degree exactly one less than its denominator.
Mistake 3 — Missing Terms for Repeated Factors
When a repeated factor like (x + 2)³ appears in the denominator, the decomposition requires three separate terms: A/(x+2), B/(x+2)², and C/(x+2)³. Writing only one or two terms for a repeated factor produces an incomplete partial fraction form. The number of terms for a repeated factor always equals the multiplicity of that factor.
Mistake 4 — Applying Cover-Up Method to Non-Simple Poles
The cover-up method — substituting x = r to find constant A for the factor (x − r) — works only when (x − r) is a simple (non-repeated) linear factor. For repeated factors (x − r)², (x − r)³, and so on, the cover-up method finds only the highest-power constant directly. The lower-power constants require expanding and matching coefficients or repeated differentiation.
Real-World Applications of Partial Fraction Decomposition
Integral Calculus — Evaluating Definite and Indefinite Integrals
The integral partial fraction decomposition calculator approach is the standard technique for integrating rational functions in calculus courses and applied mathematics. Any integral of the form ∫ P(x)/Q(x) dx where Q(x) factors over the reals can be evaluated by first decomposing and then integrating each simple term. This is how integrals like ∫ 1/(x² − 1) dx, ∫ (x + 3)/(x² + 3x + 2) dx, and all similar forms are evaluated in closed form.
Laplace Transform Inversion in Control Engineering
Every control system transfer function is a rational function in the Laplace variable s. Inverting the Laplace transform to find the time-domain step response, impulse response, or ramp response requires decomposing into partial fractions and reading the inverse transform of each simple term from standard tables. This application is used millions of times daily across the global engineering profession.
Signal Processing — Partial Fraction Expansion of Z-Transforms
In digital signal processing, the Z-transform converts discrete-time systems into rational functions of z. Decomposing the Z-transform into partial fractions identifies the individual poles that determine the frequency response and stability of digital filters. Every smartphone’s audio processing, noise cancellation algorithm, and image stabilization system relies on Z-transform partial fraction analysis.
Differential Equations — Solving with Laplace Methods
When solving linear ordinary differential equations using Laplace transforms, the solution in the frequency domain is always a rational function of s. The decomposition into partial fractions step converts this rational expression into terms whose inverse Laplace transforms are immediately recognizable, giving the time-domain solution of the differential equation directly.
How Intelligent Calculator Compares to Other Tools
Several well-known mathematical tools offer partial fraction decomposition capabilities, each with different strengths. Wolfram Alpha provides decomposition results with symbolic computation power and supports complex symbolic inputs, but the step-by-step detail requires a paid subscription. Symbolab offers step-by-step solutions with a strong educational interface focused on student guidance through each algebraic stage. Mathway provides clean results with a mobile-friendly interface suited for quick answers. Desmos, while primarily a graphing tool, can visualize the behavior of rational functions but does not perform symbolic decomposition directly.
| Tool | Step-by-Step | Integration Output | Laplace Mode | Free Access |
| Intelligent Calculator | Full steps shown | Yes, each term | Yes | Fully free |
| Wolfram Alpha | Paid subscription | Limited | Limited | Paid for steps |
| Symbolab | With account | Basic | No | Limited free |
| Mathway | With account | No | No | Limited free |
| Desmos | No | No | No | Free (graphing only) |
The IntelCalculator partial fraction decomposition calculator with steps is designed to provide the complete algebraic derivation, integration output, and Laplace transform support in a single free tool without requiring registration.
Final Thoughts
Partial fraction decomposition is the algebraic technique that makes rational functions workable in calculus and engineering — converting a single complex fraction into a sum of simple fractions, each directly integrable and each directly invertible under the Laplace transform. The degree condition must be satisfied first, the denominator must be fully factored, and the correct template must reflect every linear factor, repeated factor, and irreducible quadratic factor in the denominator.
Euler formalized the technique in the eighteenth century, Laplace extended it to transform theory, and today it appears in every calculus course and every control engineering textbook worldwide. Use the partial fraction decomposition calculator step by step above to decompose any rational function instantly, verify the constants, prepare the result for integration, and understand every algebraic step in the process.
Use our free Integral Calculator to evaluate the integral of your decomposed partial fractions — with automatic logarithm, arctangent, and combination form results for all standard partial fraction types.
Frequently Asked Questions
What is partial fraction decomposition?
Partial fraction decomposition is an algebraic technique that rewrites a rational function — the ratio of two polynomials — as a sum of simpler algebraic fractions. Each simple fraction has one factor of the original denominator as its own denominator and a constant or lower-degree polynomial as its numerator. The technique is used primarily to enable integration of rational functions and to invert Laplace transforms in engineering and applied mathematics.
How do I find partial fraction decomposition with a calculator?
To find partial fraction decomposition using this calculator, enter the numerator polynomial and the factored denominator in the input fields, then press Calculate. The calculator returns the complete decomposition with all constants, the full step-by-step solution showing how each constant was determined, and the integration-ready form of each partial fraction. For best results, factor the denominator before entering it.
What are the steps in partial fraction decomposition?
The partial fraction decomposition steps are: first, verify the rational function is proper (numerator degree less than denominator degree); second, factor the denominator completely into linear and irreducible quadratic factors; third, write the decomposition template with one unknown constant per factor (two unknowns for each quadratic factor); fourth, multiply through by the full denominator; fifth, solve for each constant using the cover-up method or by equating coefficients; and sixth, verify by adding the partial fractions back together.
How is partial fraction decomposition used in integration?
In integration, partial fraction decomposition converts a difficult rational function integral into a sum of simple integrals. Each term A/(x − r) integrates to A·ln
What is the difference between a linear factor and a quadratic factor in decomposition?
A linear factor is a first-degree polynomial of the form (x − r) that contributes one partial fraction A/(x − r) with a constant numerator. A quadratic factor is a second-degree polynomial of the form (x² + bx + c) that is irreducible — it has no real roots because its discriminant is negative — and contributes one partial fraction (Bx + C)/(x² + bx + c) with a linear numerator. The quadratic factor requires a linear numerator because a constant alone would be insufficient to represent all rational functions with that quadratic denominator.
What happens when the denominator has a repeated factor?
When the denominator contains a repeated factor (x − r)ⁿ with multiplicity n greater than one, the decomposition must include n separate partial fraction terms: A/(x−r), B/(x−r)², C/(x−r)³, up through N/(x−r)ⁿ. Each power from one to n requires its own term. Omitting any of these terms produces an incorrect decomposition. The constants for a repeated factor cannot all be found by the simple cover-up method and require either expanding and matching coefficients or successive differentiation.
Can partial fraction decomposition be applied to improper fractions?
Partial fraction decomposition in its standard form applies only to proper rational functions where the numerator degree is strictly less than the denominator degree. For improper rational functions where the numerator degree is greater than or equal to the denominator degree, polynomial long division must be performed first. The long division produces an integer polynomial quotient plus a proper remainder fraction, and partial fraction decomposition is then applied to that proper remainder fraction only.
How does partial fraction decomposition relate to the Laplace Transform?
In Laplace transform analysis, the frequency-domain solution of a linear differential equation is always a rational function in the variable s. Applying partial fraction decomposition to that rational function separates it into terms whose inverse Laplace transforms are immediately recognizable from standard tables: each term A/(s − a) inverts to Ae^(at), each term (Bs + C)/(s² + ω²) inverts to sinusoidal functions, and repeated poles invert to polynomial-times-exponential terms. Partial fraction decomposition is therefore the essential bridge between the Laplace domain and the time domain for any rational transfer function.
About This Calculator: This partial fraction decomposition calculator is part of Intelligent Calculator’s Mathematics suite — built on classical algebra theory developed by Euler, extended by Laplace, and formalized in modern calculus and engineering mathematics curricula. The calculator implements the cover-up method, coefficient-matching, and residue computation for all standard partial fraction cases. Free. No sign-up required.