Partial Fraction Decomposition Calculator

Calculate partial fraction coefficients, denominator factors, and the decomposed rational expression based on numerator coefficients and distinct linear denominator roots. Partial Fraction Decomposition Calculator helps students, educators, and professionals simplify rational expressions and prepare integrals for easier evaluation.

Highlights

  • Decomposes rational expressions into three simpler partial fractions.
  • Calculates partial fraction coefficients from numerator and denominator values.
  • Supports customizable quadratic numerators and three distinct linear denominator roots.
  • Displays the expanded cubic denominator and original rational expression.
  • Helps simplify algebraic expressions and prepare them for integration.

Table of Contents

  1. How Partial Fraction Decomposition Works?
  2. Practical Use Cases
  3. Accuracy and Limitations
  4. Quick Examples
  5. Frequently Asked Questions (FAQs)
  6. Cite This Calculator

How Partial Fraction Decomposition Works?

Partial fraction decomposition rewrites a rational expression as a sum of simpler fractions. This calculator assumes a quadratic numerator and a denominator formed from three distinct linear factors:

R(x)=n2x2+n1x+n0a(x−r1)(x−r2)(x−r3)

Here, n2, n1, and n0 are the numerator coefficients, a is the denominator leading coefficient, and r1, r2, and r3 are the distinct denominator roots. The decomposition has the form:

R(x)=A1x−r1+A2x−r2+A3x−r3

Each coefficient is found by substituting its corresponding root into the numerator and dividing by the remaining denominator factors:

Ai=n2ri2+n1ri+n0a∏j=1j≠i(ri−rj)

The expanded denominator is obtained by multiplying the three factors. If the roots are r1, r2, and r3, it is:

a[x3−(r1+r2+r3)x2+(r1r2+r1r3+r2r3)x−r1r2r3]

Calculator valueMathematical role
Numerator x² coefficientn2
Numerator x coefficientn1
Numerator constantn0
Denominator leading coefficienta
Denominator rootsr1, r2, r3

The calculator reports the three partial fraction coefficients, the expanded cubic denominator, the original rational expression, and the decomposed expression.

Practical Use Cases

Students can use the calculator to verify hand calculations, compare the original rational expression with its three simpler fractions, and identify how denominator roots determine each coefficient.

Educators can use it to create worked examples involving different quadratic numerators, leading coefficients, and distinct linear denominator roots for algebra or calculus lessons.

Calculus learners can use the decomposed expression as a starting point for integration, because each term has a simpler linear denominator and can often be handled with logarithmic antiderivatives.

Engineers and applied mathematicians can use partial fraction coefficients when simplifying transfer functions, rational models, or algebraic expressions before further symbolic analysis.

Software developers can use the displayed expanded denominator and coefficient results to test symbolic algebra routines for rational expressions with three distinct linear factors.

Accuracy and Limitations

The calculator evaluates the supplied coefficient formulas using numerical input values and is suitable for ordinary decimal calculations within the stated input range of −1,000,000 to 1,000,000. It specifically assumes a quadratic numerator and exactly three distinct linear denominator roots. It does not directly handle repeated roots, irreducible quadratic factors, complex-root factorization, or higher-degree numerators and denominators. When two roots are very close, the factors ri − rj become small, so floating-point rounding can make the reported partial fraction coefficients highly sensitive to small input changes.

Quick Examples

Example #1: Basic Positive Roots

Decompose a rational expression with a quadratic numerator and three positive linear denominator roots.

Example #2: Mixed-Sign Roots

Analyze a rational expression whose denominator has distinct roots on both sides of zero.

Example #3: Fractional Coefficients

Decompose a rational expression using fractional numerator coefficients and noninteger denominator roots.

Frequently Asked Questions

Partial fraction decomposition rewrites a rational expression as a sum of simpler fractions. Each fraction corresponds to a factor of the denominator.

The denominator must be factored, and the numerator’s degree should be less than the denominator’s degree. If not, polynomial division is performed first.

Each distinct linear factor such as x − a contributes a term with a constant numerator, written as A/(x − a).

Repeated factors require a separate term for every power. For example, (x − a)² produces terms such as A/(x − a) and B/(x − a)².

It breaks a complicated rational function into simpler terms that are often easier to integrate using logarithms, inverse trigonometric functions, or basic power rules.

Cite This Calculator