Decomposing fraction...
Enter numerator and denominator coefficients to build the partial fraction decomposition.
What Is Partial Fraction Decomposition?
A partial fraction decomposition rewrites one rational function as a sum of simpler rational functions. The process reverses fraction addition: instead of combining terms over a common denominator, it starts with the combined fraction and recovers terms whose denominators are linear or irreducible quadratic factors.
The method is especially useful in calculus because the simpler pieces often match logarithm, inverse tangent, or substitution-based integration formulas. This calculator handles proper and improper rational functions, repeated factors, and real quadratic factors while showing the denominator structure and coefficient-matching steps.
Choose the Correct Partial Fraction Form
The denominator factor type determines both the number of terms and the degree allowed in each numerator.
Denominator powers
A repeated factor needs a separate term at every power through its multiplicity.
Numerator degree
Linear factors use constants; irreducible quadratic factors use linear numerators.
Improper fractions
Divide first whenever the numerator degree is at least the denominator degree.
How to Decompose Partial Fractions
First compare polynomial degrees. If the numerator degree is greater than or equal to the denominator degree, polynomial long division produces a polynomial part and a proper remainder. Only the proper rational remainder is decomposed.
Next factor the denominator over the real numbers and write the complete decomposition template. Multiply by the common denominator to clear every fraction. The resulting equation is a polynomial identity, so coefficients attached to equal powers of x must match.
Solving that coefficient system gives the unknown constants. The calculator then rebuilds the numerator from the solved terms and compares every coefficient before displaying the answer.
Repeated and Irreducible Factors
A repeated linear factor such as (x−a)³ requires terms with denominators (x−a), (x−a)², and (x−a)³. Omitting an intermediate power makes the template incomplete and may leave the coefficient system without a solution.
An irreducible quadratic has no real linear factors. Its partial-fraction numerator must be one degree lower, so a quadratic denominator receives Ax+B. When the quadratic repeats, every power receives its own independent linear numerator.
Complex conjugate denominator roots combine into real quadratic factors. The displayed decomposition therefore retains real coefficients even when factor detection uses complex roots internally.
Partial Fraction Forms Reference
| Denominator type | Required numerator | Decomposition pattern |
|---|---|---|
| Distinct linear | Constant | A/(x-a) + B/(x-b) |
| Repeated linear | Constant at every power | A/(x-a) + B/(x-a)^2 |
| Irreducible quadratic | Linear | (Ax+B)/(x^2+px+q) |
| Repeated quadratic | Linear at every power | (Ax+B)/q(x) + (Cx+D)/q(x)^2 |
Partial Fractions for Integration
Integration motivates many partial fraction problems. Terms such as A/(x−a) integrate to logarithms, while linear numerators over quadratic factors can often be split into a derivative-matching logarithm term and a remaining inverse-tangent term.
Decomposition does not perform the integration itself; it transforms a difficult rational integrand into recognizable building blocks. Long division remains essential because a rational function must be proper before the standard decomposition templates apply.
Verification matters in calculus work because one incorrect sign changes the antiderivative. Recombining the displayed terms over the original denominator is the most direct algebraic check.
Coefficient Matching and Verification
After the correct template is written, multiplying by the common denominator removes every fraction. What remains is an identity between two polynomials. Coefficients of x³, x², x, and the constant term must agree independently, producing a linear system whose unknowns are the partial-fraction numerators.
This calculator solves that system with pivoted elimination rather than relying on a few substituted x-values. Coefficient matching works uniformly with linear factors, quadratic factors, and mixtures of repeated powers. It also avoids accidentally substituting a denominator zero.
The final check reconstructs the proper numerator from every solved term. Additional safe sample values compare the original and decomposed rational functions numerically, while points where the denominator vanishes are deliberately skipped.
Partial Fraction Decomposition FAQ
What is partial fraction decomposition?
It rewrites a rational function as a sum of simpler rational terms. The sum has the same value as the original function everywhere both expressions are defined.
When can you use partial fractions?
The expression must be a rational function whose denominator can be factored into real linear and irreducible quadratic factors. An improper function must be divided first.
What do you do with an improper rational function?
Use polynomial long division to separate a polynomial quotient and a proper rational remainder. Decompose only the remainder over the original denominator.
How do repeated factors change partial fractions?
Include one term for every positive power of the repeated factor. A factor raised to the third power therefore produces three separate denominator powers.
Why does an irreducible quadratic need Ax + B?
The numerator must have lower degree than its denominator factor. A quadratic factor therefore requires the most general linear numerator, Ax+B.
How do you check a partial fraction decomposition?
Combine the terms over their common denominator and simplify the numerator. It must reproduce the proper remainder coefficient by coefficient.
How are partial fractions used in integration?
They turn one complicated rational integrand into simpler fractions with standard antiderivatives. Common results involve logarithms and inverse tangent functions.
Related Calculators
Polynomial Roots Calculator
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Quadratic Formula Calculator
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Long Division Calculator
Work through quotient and remainder calculations.
Fraction Calculator
Calculate and simplify ordinary fraction operations.
System of Equations Calculator
Solve coefficient systems with clear steps.
External Resources
OpenStax Calculus Volume 2 — partial fractions and integration