Find The Partial Fraction Decomposition Calculator

What is Partial Fraction Decomposition?

Partial Fraction Decomposition is a method used in algebra to break down a complex rational expression (a fraction of two polynomials) into a sum of simpler fractions. This technique is particularly useful in calculus for integrating rational functions, and in other areas of mathematics like finding the inverse Laplace transform or solving certain types of differential equations.

The core idea is to express a fraction where the denominator is a product of factors into a sum of fractions whose denominators are those factors (or powers of those factors).

Who should use it?

Calculus students learning integration techniques.
Engineers and scientists dealing with differential equations or control systems (via Laplace transforms).
Anyone needing to simplify complex rational expressions for further analysis.

Common Misconceptions

It works for any fraction: Partial Fraction Decomposition is primarily for *proper* rational functions (where the degree of the numerator is less than the degree of the denominator). If it's improper, you first perform polynomial long division.
It's always easy: While the concept is straightforward, the algebra to find the coefficients of the simpler fractions can become complex, especially with repeated or irreducible quadratic factors in the denominator. This calculator handles distinct and repeated linear factors.

Partial Fraction Decomposition Formula and Mathematical Explanation

The form of the Partial Fraction Decomposition depends on the factors of the denominator of the original rational function P(x)/Q(x), where P(x) and Q(x) are polynomials and the degree of P(x) is less than the degree of Q(x).

1. Distinct Linear Factors

If the denominator Q(x) can be factored into distinct linear factors like Q(x) = (x – r₁)(x – r₂)…(x – rₙ), then the decomposition is:

P(x) / Q(x) = A₁/(x – r₁) + A₂/(x – r₂) + … + Aₙ/(x – rₙ)

Where A₁, A₂, …, Aₙ are constants to be determined.

Example: (x+7) / ((x-1)(x+2)) = A/(x-1) + B/(x+2)

2. Repeated Linear Factors

If Q(x) has a repeated linear factor (x – r)ᵐ, then the decomposition includes terms for each power from 1 to m:

… + A₁/(x – r) + A₂/(x – r)² + … + Aₘ/(x – r)ᵐ + …

Example: (x+7) / (x-1)² = A/(x-1) + B/(x-1)²

3. Irreducible Quadratic Factors

If Q(x) has an irreducible quadratic factor (ax² + bx + c), the corresponding term in the decomposition is (Ax + B)/(ax² + bx + c). This calculator focuses on linear factors.

To find the constants (A, B, C, etc.), we multiply both sides of the equation by the original denominator Q(x), simplify, and then either substitute convenient values for x (the roots r₁, r₂, etc.) or equate coefficients of like powers of x to form a system of linear equations.

Variables Table

Variables in Partial Fraction Decomposition
Variable	Meaning	Unit	Typical Range
P(x)	Numerator polynomial	N/A	Polynomial expression
Q(x)	Denominator polynomial	N/A	Polynomial expression
r₁, r₂, r, …	Roots of the denominator factors	N/A	Real numbers
m	Multiplicity of a repeated root	N/A	Integer ≥ 2
A, B, C, …	Constants in the decomposed fractions	N/A	Real numbers

Practical Examples (Real-World Use Cases)

Example 1: Integration in Calculus

Suppose we want to integrate ∫ (x+7) / (x² + x – 2) dx. First, factor the denominator: x² + x – 2 = (x-1)(x+2).

The rational function is (x+7) / ((x-1)(x+2)). We perform Partial Fraction Decomposition:

(x+7) / ((x-1)(x+2)) = A/(x-1) + B/(x+2)

Multiplying by (x-1)(x+2): x+7 = A(x+2) + B(x-1)

If x=1: 1+7 = A(1+2) + B(0) => 8 = 3A => A = 8/3

If x=-2: -2+7 = A(0) + B(-2-1) => 5 = -3B => B = -5/3

So, (x+7) / ((x-1)(x+2)) = (8/3)/(x-1) + (-5/3)/(x+2)

The integral becomes ∫ (8/3)/(x-1) dx + ∫ (-5/3)/(x+2) dx = (8/3)ln|x-1| – (5/3)ln|x+2| + C, which is much easier to solve.

Example 2: Inverse Laplace Transforms

In control systems or circuit analysis, we might encounter a Laplace transform F(s) = (s+1) / (s(s+2)). To find the inverse Laplace transform, we first use Partial Fraction Decomposition on F(s):

(s+1) / (s(s+2)) = A/s + B/(s+2)

s+1 = A(s+2) + Bs

If s=0: 1 = 2A => A = 1/2

If s=-2: -1 = -2B => B = 1/2

So, F(s) = (1/2)/s + (1/2)/(s+2). The inverse Laplace transform is then easily found term by term: f(t) = (1/2) * 1 + (1/2) * e^-2t = 1/2 + 1/2 e^-2t for t ≥ 0.

How to Use This Partial Fraction Decomposition Calculator

Enter Numerator Coefficients: Input the coefficients 'a' (for x²), 'b' (for x), and 'c' (constant) of the numerator polynomial ax² + bx + c. If your numerator is linear (like bx+c), enter 0 for 'a'.
Choose Denominator Type:
- Distinct Linear Factors: If your denominator is like (x-r1)(x-r2)…, enter the roots r1, r2, and r3 in the corresponding fields. Leave r3 blank if you only have two distinct factors, or r2 and r3 blank for one.
- Repeated Linear Factor: If your denominator is like (x-r)², enter the root 'r' and select multiplicity '2'. If it's (x-r)³, enter 'r' and select '3'. When using this option, make sure the distinct root fields are empty.
Calculate: Click the "Calculate" button. The calculator will automatically perform the Partial Fraction Decomposition.
View Results:
- The "Primary Result" shows the decomposed form.
- "Intermediate Results" display the calculated values of the constants A, B, C, etc.
- "Formula Used" briefly describes the type of decomposition performed.
- A table summarizes the coefficients found.
- A chart compares the plot of the original function and the sum of its partial fractions over a small range to visually confirm the decomposition.
Reset: Click "Reset" to clear inputs and start over with default values.
Copy Results: Click "Copy Results" to copy the main result and intermediate values to your clipboard.

Decision-making guidance: Ensure the degree of the numerator is less than the degree of the denominator. If not, perform polynomial long division first before using this calculator on the remainder term. This calculator is designed for denominators that factor into linear or repeated linear terms.

Key Factors That Affect Partial Fraction Decomposition Results

Degree of Numerator vs. Denominator: The method applies directly only when the degree of the numerator is less than the degree of the denominator (proper rational function). If not, polynomial division is needed first.
Factors of the Denominator: The form of the decomposition is entirely determined by the types of factors in the denominator:
- Distinct linear factors (x-a) lead to terms A/(x-a).
- Repeated linear factors (x-a)^m lead to terms A₁/(x-a) + A₂/(x-a)² + … + Aₘ/(x-a)^m.
- Irreducible quadratic factors (ax²+bx+c) lead to terms (Ax+B)/(ax²+bx+c) (not fully handled by this basic calculator beyond identification).
Roots of the Denominator: The values of the roots (r₁, r₂, r, etc.) directly influence the values of the constants A, B, C. Small changes in roots can lead to significant changes in these constants.
Multiplicity of Roots: The power to which a linear factor is raised determines how many terms are associated with that factor in the decomposition.
Coefficients of the Numerator: The coefficients of the numerator polynomial are used along with the denominator roots to solve for the constants A, B, C.
Algebraic Manipulation Accuracy: When solving manually, the process involves either substituting roots or equating coefficients, both of which require careful algebraic manipulation. Errors here lead to incorrect constants. Our calculator automates this.

Frequently Asked Questions (FAQ)

Q1: What if the degree of the numerator is greater than or equal to the degree of the denominator? A1: The rational function is "improper." You must first perform polynomial long division to get a polynomial plus a proper rational function. Then, apply Partial Fraction Decomposition to the proper rational function remainder.

Q2: What if the denominator has irreducible quadratic factors? A2: If the denominator has a factor like (ax²+bx+c) that cannot be factored into real linear factors (i.e., b²-4ac < 0), the decomposition will include a term of the form (Ax+B)/(ax²+bx+c). This calculator focuses on linear factors but acknowledges quadratics. More advanced tools or manual methods are needed for full decomposition with quadratics.

Q3: How are the constants A, B, C found? A3: After setting up the decomposition form, multiply both sides by the original denominator. Then, you can either substitute the roots of the denominator to solve for some constants directly (Heaviside cover-up method) or expand and equate coefficients of corresponding powers of x to get a system of linear equations for the constants.

Q4: Why is Partial Fraction Decomposition useful in integration? A4: It breaks down a complex fraction into simpler ones (like A/(x-r) or B/(x-r)²) whose integrals are standard and easily found (ln|x-r| or -B/(x-r)). See our integration calculator for examples.

Q5: Can this calculator handle complex roots? A5: This calculator is designed primarily for real roots in linear factors. Irreducible quadratic factors often involve complex roots, leading to terms that are usually kept in quadratic form with real coefficients in the decomposition unless complex numbers are explicitly used.

Q6: What if the denominator is hard to factor? A6: Factoring the denominator is the first crucial step. If it's a high-degree polynomial, finding the roots can be difficult and might require numerical methods or a polynomial calculator or solver. This calculator assumes you have the factored form or roots.

Q7: Is the Partial Fraction Decomposition unique? A7: Yes, for a given proper rational function, the Partial Fraction Decomposition into the sum of fractions with denominators as powers of the factors of the original denominator is unique.

Q8: What are the limitations of this online calculator? A8: This calculator handles distinct and repeated linear factors well. It does not fully decompose rational functions with irreducible quadratic factors in the denominator, though it can be adapted. It also assumes you know the roots or factored form of the denominator.

Related Tools and Internal Resources

Integration Calculator: Once you have the partial fractions, use this to integrate term by term.
Polynomial Calculator: Useful for operations with polynomials, including finding roots which is essential for factoring the denominator.
Equation Solver: Helps in solving the system of linear equations that can arise when finding the coefficients A, B, C.
Laplace Transform Calculator: Apply partial fractions before finding the inverse Laplace transform.
Differential Equation Solver: Partial fractions are sometimes used when solving linear differential equations.
Algebra Basics Guide: Refresh your understanding of fundamental algebraic manipulations.

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