Partial Fraction Decomposition Calculator
Enter numerator coefficients and explicit denominator factors to obtain polynomial division, repeated-factor terms, exact fractions and a residual check.
Partial Fraction Decomposition Calculator inputs
Change any assumption and this calculator recomputes its own complete report. Invalid relationships are stopped before a result is shown.
Partial Fraction Decomposition Calculator results
How the partial fraction decomposition calculator works
The partial fraction decomposition calculator is built around this task’s actual formulas, choices and edge cases. Review the assumptions and interpretation before acting on the result.
Coefficient and factor notation
The numerator is entered as coefficients from highest degree to constant. For example, 2,3 means 2x + 3, while 1,0,-4 means x squared minus 4. Denominator lines describe already-factored linear or quadratic expressions. A linear line uses L,a,b,power for ax+b. A quadratic line uses Q,a,b,c,power for ax squared plus bx plus c. This explicit format prevents the calculator from pretending that every symbolic denominator can be factored reliably.
Polynomial division first
When the numerator degree is at least the denominator degree, the partial fraction decomposition calculator performs polynomial long division before solving the proper fraction. The polynomial quotient stays outside the decomposed terms and the remainder becomes the new numerator. Skipping this step is a common source of incomplete answers. The result reports the quotient, expanded denominator and residual so the full rational identity can be checked rather than accepted on appearance.
Repeated factors
A repeated linear factor requires one term for every power. For example, a cubic power needs terms over the first, second and third powers. A repeated irreducible quadratic requires a linear numerator over every power. The partial fraction decomposition calculator builds this full basis automatically. This is more complete than a cover-up-only method, which works cleanly for distinct linear factors but does not by itself handle repeated or quadratic factors.
Coefficient system
The engine multiplies every unknown partial-fraction term by the complementary denominator polynomial. Matching equal powers of x creates a linear system. Gaussian elimination solves that system, and rational reduction presents simple fractions where possible. A residual of zero means that reconstructing the numerator from the returned terms matches the entered function. That verification is especially valuable before using the result in integration, inverse Laplace transforms or differential-equation work.
Validation and limitations
Factors with zero leading coefficients, missing powers or redundant descriptions can make the system invalid or singular. The partial fraction decomposition calculator stops with an error instead of fabricating a coefficient. It works over the real-number factorization supplied by the visitor. Denominators should be factored first, with irreducible real quadratics left intact. Complex linear factors, symbolic parameters and automatic factoring of arbitrary expressions are outside this focused engine.
Using the decomposition
After verifying the identity, integrate or transform each simpler term using the appropriate rule. Linear denominators commonly produce logarithms or negative powers. Quadratic numerators can often be separated into a derivative-of-denominator part and a constant remainder that leads to an inverse-tangent term. The partial fraction decomposition calculator completes the algebraic split; it does not claim that every later calculus step has the same form.
Partial Fraction Decomposition Calculator questions
Can it handle an improper rational function?
Yes. Polynomial division is performed before the remainder is decomposed.
How is a repeated factor entered?
Increase the final power value, such as L,1,-1,3 for a third power.
Can it use quadratic factors?
Yes. Q,a,b,c,power creates a linear numerator over each quadratic power.
What does zero residual mean?
The returned decomposition reconstructs the entered numerator over the expanded denominator.