Math July 13, 2026 · 8 Min Read

Partial Fraction Decomposition Calculator – Guide & Formulas

Decompose rational expressions into partial fractions instantly. Enter a fraction like (3x + 5) / ((x + 1)(x + 2)) and see the decomposition with step-by-step solving.

Decompose any rational expression into partial fractions instantly with our free calculator. Enter numerator and denominator to see the decomposition with full step-by-step solving.

Key Takeaway

Use the free Partial Fraction Decomposition Calculator to decompose rational expressions into partial fractions instantly. enter a fraction like (3x + 5) / ((x + 1)(x + 2)) and see the decomposition with step-by-step solving. Get instant results with step-by-step explanations.

How to Use the Partial Fraction Decomposition Calculator

  1. Enter the numerator polynomial (e.g., 3x + 5).
  2. Enter the denominator polynomial (e.g., x^2 + 3x + 2).
  3. Review the factored denominator and partial fraction decomposition.
  4. Check the step-by-step solving process.

The Formula

A rational expression P(x)/Q(x) with distinct linear factors in Q(x) decomposes into simpler fractions. Each factor (x - rᵢ) gets a constant numerator Aᵢ found by the cover-up method.

Variable Definitions

  • P(x): The numerator polynomial
  • Q(x): The denominator polynomial
  • rᵢ: Roots (zeros) of the denominator
  • Aᵢ: Constants found by evaluating P(rᵢ)/Q'(rᵢ) or cover-up method
  • Cover-up method: Set x = rᵢ in P(x)/[(x - rᵢ) · remaining factors] to find Aᵢ

Decomposing (3x + 5) / ((x + 1)(x + 2))

Find partial fractions for a rational expression with distinct linear factors.

  1. Step 1: Write the form: (3x + 5) / ((x + 1)(x + 2)) = A/(x + 1) + B/(x + 2).
  2. Step 2: Find A by covering (x + 1) and setting x = -1: A = (3(-1) + 5) / (-1 + 2) = 2/1 = 2.
  3. Step 3: Find B by covering (x + 2) and setting x = -2: B = (3(-2) + 5) / (-2 + 1) = (-1)/(-1) = 1.
  4. Step 4: Result: (3x + 5) / ((x + 1)(x + 2)) = 2/(x + 1) + 1/(x + 2).

Frequently Asked Questions

What is partial fraction decomposition?

Partial fraction decomposition breaks a complex rational expression into a sum of simpler fractions. It is essential for integrating rational functions in calculus and simplifying Laplace transforms.

When can I use partial fraction decomposition?

When the degree of the numerator is less than the degree of the denominator, and the denominator can be factored. If the numerator degree is higher, perform polynomial long division first.

What if the denominator has repeated factors?

For a repeated factor like (x - r)², include terms A/(x - r) + B/(x - r)². Each power up to the multiplicity gets its own term.

What if the denominator has irreducible quadratic factors?

For factors like (x² + 1) that cannot be factored over reals, use the form (Ax + B)/(x² + 1) in the decomposition.

How do I verify my decomposition?

Combine the partial fractions over a common denominator. If the result equals the original rational expression, the decomposition is correct.

What is the cover-up method?

To find the constant A for factor (x - r), cover up (x - r) in the denominator, then evaluate the remaining expression at x = r. This gives A directly for distinct linear factors.

Can I use this for improper fractions?

If the numerator degree ≥ denominator degree, first perform polynomial long division to get a polynomial plus a proper fraction, then decompose the proper fraction part.

What are the applications of partial fractions?

Key applications include: integration of rational functions (calculus), Laplace transforms (differential equations), signal processing (z-transforms), and solving systems of linear equations.