Partial Fraction Decomposition Calculator

Decompose rational functions into partial fractions. Simplified version for two linear factors.

Enter coefficients separated by commas (highest degree first)

Coefficients of first linear factor: a,b for (ax+b)

Coefficients of second linear factor: a,b for (ax+b)

For distinct linear factors:\nN(x)/((a₁x+b₁)(a₂x+b₂)) = A/(a₁x+b₁) + B/(a₂x+b₂)\n\nCover-up method:\nA = N(-b₁/a₁) / (a₂(-b₁/a₁)+b₂)\nB = N(-b₂/a₂) / (a₁(-b₂/a₂)+b₁)\n\nWhere N(x) is the numerator polynomial
Example:\n(3x+5)/((x-1)(x+1))\n\nNumerator: 3,5 (for 3x+5)\nFactor 1: 1,-1 (for x-1)\nFactor 2: 1,1 (for x+1)\n\nAt x=1: A = (3·1+5)/(1+1) = 8/2 = 4\nAt x=-1: B = (3·(-1)+5)/(-1-1) = 2/(-2) = -1\n\nResult: 4/(x-1) + (-1)/(x+1)\nSimplified: 4/(x-1) - 1/(x+1)

What is partial fraction decomposition?

Partial fraction decomposition breaks a complex rational function into simpler fractions. Example: (3x+5)/(x^2-1) = A/(x-1) + B/(x+1). Used in calculus for integration, inverse Laplace transforms, and solving differential equations. Makes complex fractions easier to work with.

When can I use partial fractions?

Requirements: 1) Rational function (polynomial/polynomial), 2) Degree of numerator < degree of denominator (proper fraction), 3) Denominator can be factored. If improper (numerator degree >= denominator), first do polynomial long division, then decompose the remainder.

What are the types of partial fraction decomposition?

Type 1: Distinct linear factors (x-a)(x-b) → A/(x-a) + B/(x-b). Type 2: Repeated linear factors (x-a)^2 → A/(x-a) + B/(x-a)^2. Type 3: Irreducible quadratic (ax^2+bx+c) → (Ax+B)/(ax^2+bx+c). Type 4: Combinations of above.

How do I find the constants A, B, C in partial fractions?

Method 1: Multiply both sides by denominator, then equate coefficients. Method 2: Substitute convenient x values. Method 3: Cover-up method (for simple cases). Example: 1/((x-1)(x+2)) = A/(x-1) + B/(x+2). Multiply by denominator: 1 = A(x+2) + B(x-1). Set x=1: A=1/3. Set x=-2: B=-1/3.

Why is partial fraction decomposition useful?

Applications: 1) Integration - breaks complex rational functions into simple logarithms and arctangents, 2) Laplace transforms - inverse transforms easier, 3) Differential equations - solution methods, 4) Control theory - transfer functions, 5) Circuit analysis - impedance calculations.