Partial Fraction Decomposition Calculator

What this partial fraction decomposition calculator does

This partial fraction decomposition calculator rewrites rational functions of the form (ax + b) / ((x − r₁)(x − r₂)) as a sum of simpler fractions. It uses a denominator already factored into linear terms and a first-degree numerator. For distinct roots, the output is A / (x − r₁) + B / (x − r₂); when r₁ = r₂, it uses the required repeated-root form.

Splitting this particular rational function makes its individual linear-factor contributions visible. That is useful before integrating with respect to x, checking a hand-derived decomposition, or matching simple terms in a Laplace-transform calculation.

Partial fraction decomposition formulas for two linear factors

The partial fraction calculation starts with the following rational function, where a, b, r₁, and r₂ are real input values:

f(x) = (ax + b) / ((x − r₁)(x − r₂))

For distinct roots r₁ ≠ r₂, this calculator seeks the decomposition

(ax + b) / ((x − r₁)(x − r₂)) = A / (x − r₁) + B / (x − r₂).

Clearing the two denominator factors produces an identity whose coefficients determine A and B:

ax + b = A(x − r₂) + B(x − r₁).

The coefficients A and B make this equality true for every permitted value of x, not merely at one selected point.

The same two-linear-factor decomposition in MathML is

For a repeated root, r₁ = r₂ = r, partial fraction decomposition requires both powers of the repeated linear factor:

(ax + b) / (x − r)² = A / (x − r) + B / (x − r)².

In this case the calculator still determines two coefficients, but it uses the repeated-root identity rather than the distinct-root coefficient formula.

Partial fraction decomposition: how the calculator works step by step

For the entered numerator and roots, the calculator follows the same algebra used to decompose this rational function by hand:

1. Model the rational function. The values a, b, r₁, and r₂ specify (ax + b) / ((x − r₁)(x − r₂)).
2. Choose the appropriate identity. With distinct roots, it uses ax + b = A(x − r₂) + B(x − r₁); equal roots use the repeated-root identity instead.
3. Match the numerator terms. The x coefficient and constant term on the cleared-denominator identity determine the two unknown coefficients.
4. Solve for A and B. For distinct roots it evaluates the corresponding coefficient solution; for a repeated root it uses A = a and B = b + ar.
5. Display the decomposition. The result identifies whether the roots are distinct or repeated and writes the appropriate partial fractions.

Entering coefficients and roots for partial fraction decomposition

To describe the rational function used by this partial fraction calculator, enter values for (ax + b) / ((x − r₁)(x − r₂)) as follows:

• Numerator coefficient a is the coefficient multiplying x in ax + b.
• Numerator coefficient b is the constant term in that numerator.
• Root r₁ defines the first denominator factor, (x − r₁).
• Root r₂ defines the second denominator factor, (x − r₂). Enter r₂ = r₁ when the denominator is a repeated factor.

After you submit the coefficients and roots, the calculator reports A and B and writes the partial fraction expression corresponding to those inputs.

Interpreting partial fraction decomposition results

A partial fraction result from this calculator has one of two forms, determined solely by whether the two entered roots are equal:

• Distinct roots: (ax + b) / ((x − r₁)(x − r₂)) = A / (x − r₁) + B / (x − r₂).
• Repeated root (r₁ = r₂): (ax + b) / (x − r)² = A / (x − r) + B / (x − r)².

For integration, the displayed terms can be integrated separately: a term with 1 / (x − r) produces a logarithm, while a term with 1 / (x − r)² has a reciprocal antiderivative. For transform work, A and B are the coefficients attached to the separate linear-factor terms. As an algebra check, combine the displayed fractions over the original denominator and confirm that the numerator becomes ax + b.

Worked partial fraction example with distinct roots

Consider the rational function (3x + 5) / ((x − 1)(x − 2)). Its calculator inputs are

• a = 3,
• b = 5,
• r₁ = 1,
• r₂ = 2.

The distinct-root partial fraction identity is

3x + 5 = A(x − 2) + B(x − 1).

Expanding its right-hand side gives

A(x − 2) + B(x − 1) = (A + B)x + (−2A − B).

Matching the numerator coefficients gives the system

• A + B = 3,
• −2A − B = 5.

Solving produces A = −8 and B = 11, so the decomposition is

(3x + 5) / ((x − 1)(x − 2)) = −8 / (x − 1) + 11 / (x − 2).

Entering a = 3, b = 5, r₁ = 1, and r₂ = 2 produces these same distinct-root coefficients in the calculator.

Worked partial fraction example with a repeated root

For a repeated linear factor, take (2x + 3) / (x − 4)². The root 4 has multiplicity two, so use

• a = 2,
• b = 3,
• r₁ = 4,
• r₂ = 4.

The required repeated-root decomposition is

(2x + 3) / (x − 4)² = A / (x − 4) + B / (x − 4)².

Clearing the squared denominator yields

2x + 3 = A(x − 4) + B.

Expanding the right-hand side gives

A(x − 4) + B = Ax − 4A + B.

Matching the x coefficient and constant term gives

• A = 2,
• −4A + B = 3.

With A = 2, the second equation is −8 + B = 3, so B = 11. The repeated-root partial fractions are therefore

(2x + 3) / (x − 4)² = 2 / (x − 4) + 11 / (x − 4)².

Setting both root fields to 4 makes the calculator select this repeated-root form automatically.

Comparing hand calculations with this partial fraction calculator

For rational functions with two supplied linear roots, manual algebra and this focused calculator serve different parts of the partial-fraction workflow:

When this partial fraction calculator is most useful

This two-linear-factor partial fraction calculator is particularly useful in the following settings:

• Calculus: separating a rational integrand into terms that can be integrated one at a time.
• Differential equations: simplifying rational expressions that occur during solution methods.
• Control theory and circuits: separating simple transfer-function terms before applying transform relationships.
• Homework checks: verifying a hand-computed pair of partial fraction coefficients against the entered roots and numerator.

Assumptions and limitations of this partial fraction decomposition calculator

This partial fraction calculator intentionally handles one specific rational form, so its inputs and output rely on these algebraic assumptions:

• Linear numerator only. The numerator must be ax + b. A numerator of higher degree must first be divided by the denominator when appropriate; that preliminary step is not performed here.
• Two factored linear denominator terms. The denominator must be (x − r₁)(x − r₂). Enter the roots, rather than coefficients of an unfactored quadratic.
• Real-number inputs. The fields and displayed decomposition are designed for real a, b, r₁, and r₂ values. Complex-root cases are outside this calculator's stated form.
• Repeated roots through r₁ = r₂. Equal root entries represent (x − r)² and cause the calculator to use both 1 / (x − r) and 1 / (x − r)² terms.
• Valid rational-function domain. As with the original expression, the decomposition is not defined at denominator zeros. The identity is understood where the denominator is nonzero.

If the denominator has more factors, is not already factored, or uses complex coefficients, use additional algebra before applying this tool or choose a more general symbolic partial-fractions method.

Typical follow-up steps after partial fraction decomposition

Once this calculator has separated the rational function into partial fractions, common next steps include:

• Integrating each displayed term. Apply the standard antiderivatives for 1 / (x − r) and 1 / (x − r)² separately.
• Using transform pairs. When x represents a transform variable, match each separated denominator factor to the relevant transform relationship.
• Checking the decomposition. Put the result over the common denominator and verify that its numerator is the original ax + b.