What Is Partial Fraction Decomposition?
Partial fraction decomposition is a way to rewrite a rational function into a sum of simpler fractions. A rational function is any expression of the form N(x)/D(x), where N(x) and D(x) are polynomials and D(x) is not zero. The key idea is that complicated denominators become much easier to work with when you split them into pieces tied to the denominator’s factors.
Why does this matter? Because many algebra and calculus tasks become simpler when the denominator is “separated.” Integration is the classic example: an integral that looks intimidating as one fraction often becomes a short list of easy integrals after decomposition. The same is true for Laplace transforms, differential equations, and simplifying expressions before substitution.
Why Would You Use Partial Fractions Instead of Factoring Alone?
Factoring shows you the structure of the denominator, but it doesn’t automatically tell you how to break the whole rational function into usable pieces. Partial fractions goes one step further: it produces an equivalent expression that is often more practical for doing math. For example, factoring tells you where the denominator is zero; partial fractions tells you how the numerator “distributes” across those factors.
What if you are studying for an exam or checking homework? Partial fractions is also a verification tool. If you can decompose and recombine correctly, you’ve demonstrated a strong understanding of polynomial algebra and rational expressions.
How Do You Know If a Rational Function Is Proper or Improper?
A rational function is proper if the degree of the numerator is strictly less than the degree of the denominator. It is improper if the numerator degree is greater than or equal to the denominator degree. This distinction matters because a proper fraction can be decomposed directly, while an improper fraction should be converted into:
polynomial part + proper rational part
That conversion is done using polynomial long division. What if you skip that step? You can still write a partial fraction template, but the coefficients become messy and you risk building the wrong structure. This calculator performs the division automatically so you get the polynomial part first, then the clean decomposition of the remainder.
What Factor Types Create Which Partial Fraction Terms?
Partial fractions follows a predictable template that depends on the factorization of the denominator:
Distinct Linear Factors
If the denominator has distinct linear factors like (x − r₁)(x − r₂)…(x − rₙ), the decomposition uses constants on top:
A₁/(x − r₁) + A₂/(x − r₂) + … + Aₙ/(x − rₙ)
Repeated Linear Factors
What if a factor repeats, like (x − r)³? Then you must include one term for each power:
A₁/(x − r) + A₂/(x − r)² + A₃/(x − r)³
This looks redundant at first, but it is required for the identity to hold for all x.
Irreducible Quadratic Factors
If the denominator has a quadratic factor that does not factor over the reals, such as x² + px + q, the numerator must be linear:
(Bx + C)/(x² + px + q)
Repeated Quadratic Factors
What if the quadratic factor repeats, like (x² + px + q)²? Then you include a linear numerator for each power:
(B₁x + C₁)/(Q) + (B₂x + C₂)/(Q²)
How This Calculator Solves the Coefficients
Once the denominator factors are known, the remaining problem is to find the coefficient values that make the decomposition exactly equal to the original rational function. One common classroom method is to multiply through by the denominator and then compare coefficients of like powers of x. Another approach is to plug in convenient x-values and solve a system of equations.
This calculator uses a stable numeric strategy: it builds the correct partial-fraction template for your factor list, generates enough valid x-values (avoiding denominator zeros), and solves the resulting linear system for the unknown coefficients. If your rational function is improper, it first performs polynomial division so the system is built only for the proper remainder.
Why does that work? Because if two polynomials are equal for enough distinct x-values, they must be the same polynomial. Partial fractions leverages that identity logic in a structured way.
How to Enter Your Problem So the Output Matches Your Expectations
Step 1: Choose the variable
The variable is cosmetic (x is standard), but it helps you read results naturally, especially when you copy the decomposition into notes or an assignment.
Step 2: Enter numerator coefficients
Coefficients are entered from highest degree down to constant. If the highest-degree coefficient is zero, your true degree is lower, so either remove the leading zero or reduce the numerator degree selection.
Step 3: Enter denominator factors and multiplicities
This calculator expects you to provide the denominator in factored form. That’s intentional because partial fractions is defined in terms of denominator factors. If your denominator is expanded, factor it first.
For linear factors, enter r in (x − r). What if you have (x + 5)? Then r = −5.
For quadratic factors, enter p and q in x² + px + q. If your quadratic is written differently, rewrite it into that standard shape before entering values.
What If the Denominator Factorization Is Wrong?
A partial-fraction template is only correct if the factorization is correct. If you accidentally enter the wrong root, miss a multiplicity, or input a quadratic that actually factors into linear terms, the system can become inconsistent or the coefficients may not match what you expect.
A quick sanity check is to compute the total denominator degree from your factors. Each linear factor contributes degree 1 per multiplicity, and each quadratic factor contributes degree 2 per multiplicity. If that degree does not match the original denominator degree, something is missing or duplicated.
How to Verify Your Decomposition Without Recombining Everything
Recombining is the most direct proof, but you can often verify correctness faster by checking values. Pick several x-values that do not make the denominator zero, evaluate the original rational function, then evaluate the decomposition. The values should match (up to rounding from displayed precision).
What if your results are extremely close but not identical? That is typically rounding. Increase precision and check again. If the mismatch grows or changes unpredictably, revisit your factorization and inputs.
Where Partial Fractions Shows Up in Calculus and Engineering
Integration
Partial fractions turns a complicated rational integral into a sum of simpler integrals, often involving ln terms for linear factors and arctan terms for irreducible quadratics. This is one of the main reasons the technique is taught early in calculus.
Laplace transforms and differential equations
In systems and controls, you often invert Laplace transforms by decomposing rational expressions in s. Each partial fraction corresponds to a time-domain component such as exponentials or damped oscillations.
Series and approximation work
Partial fractions can also help when expanding rational functions into series, because the simpler terms have known expansion patterns.
Common Mistakes and How to Avoid Them
Mixing up signs in linear factors
The factor (x + 3) corresponds to (x − (−3)). If you enter r=3 instead of r=−3, you move the pole to the wrong location and the entire decomposition changes.
Forgetting repeated terms
If (x − 2)² is in the denominator, you need A/(x − 2) and B/(x − 2)². Leaving one out makes the template incomplete.
Using a constant numerator for a quadratic factor
For irreducible quadratics over the reals, you need (Bx + C), not just a constant. That linear numerator is required to match degrees properly.
Ignoring improper fractions
If degree(N) ≥ degree(D), divide first. This tool does it automatically, but it’s still important conceptually.
What If You Want “Exact” Coefficients Instead of Decimals?
Many partial fraction results can be expressed as exact fractions, but deriving exact values requires symbolic arithmetic and rational simplification rules. This calculator focuses on accurate numeric coefficients with adjustable precision and clear steps. If you need exact rational outputs, you can often recognize simple fractions from the decimal results (for example, 0.5 as 1/2) or use a symbolic algebra tool afterward.
Practical Examples to Try
Example 1: Distinct linear factors
Try N(x)=2x+3 and denominator factors (x−1)(x+2). Enter numerator degree 1 with coefficients 2, 3, then add linear factors with r=1 and r=−2. The decomposition will be a sum of two simple terms.
Example 2: Repeated linear factor
Try N(x)=1 and denominator (x−1)². Add one linear factor with r=1 and multiplicity 2. You should see two terms: A/(x−1) + B/(x−1)².
Example 3: Quadratic factor
Try a denominator like x² + 4x + 5 (which is irreducible over the reals) and a small numerator like 3x+1. Add one quadratic factor with p=4 and q=5. The numerator for the partial fraction will be linear.
Limitations and Safe Use Notes
This calculator solves coefficients numerically and formats output to your chosen precision. For most study and planning uses, that’s exactly what you want. If your coefficients are very large, the factorization is ill-conditioned, or the function has poles extremely close together, numeric methods can become sensitive. In those cases, increase precision, double-check factor inputs, and verify by plugging values back into the original expression.
FAQ
Partial Fractions Calculator – Frequently Asked Questions
Answers to common questions about factoring, repeated terms, quadratics, improper fractions, and how to verify results.
Partial fraction decomposition rewrites a rational function N(x)/D(x) as a sum of simpler fractions whose denominators are factors of D(x). This is commonly used for integration, Laplace transforms, and simplifying algebra.
For standard partial fractions, yes. This calculator expects the denominator in factored form using linear factors (x−r), repeated factors (x−r)^m, and/or quadratic factors (x²+px+q)^m.
Then it has a polynomial part plus a proper fraction. The calculator performs polynomial division first, then decomposes the remaining proper fraction.
A linear factor is entered as (x − r). For (x + 3), use r = −3 because x + 3 = x − (−3).
For a quadratic factor Q(x)=x²+px+q, the partial-fractions terms use linear numerators: (Bx + C)/Q(x), and for repeats, (B₂x + C₂)/Q(x)², etc.
Yes. Coefficients can be real or complex depending on your inputs and factorization. Most real-world uses keep quadratics irreducible over the reals, which typically yields real coefficients.
A quick check is to recombine the partial fractions over a common denominator and verify you get back the original rational function. You can also test by plugging in a few x-values (avoiding denominator zeros).
That usually happens if the factorization is inconsistent, there are too few independent equations (rare), or the system becomes numerically unstable. Try verifying factors, multiplicities, and coefficients.
No. History is kept only in your browser session for convenience and can be cleared at any time.
Students use it for calculus and algebra, engineers use it for Laplace transforms and system analysis, and data/science workflows sometimes use it for symbolic simplification and model derivations.