Mathematics, Statistics & Geometry

Partial Fractions Calculator

Perform partial fraction decomposition on rational expressions with two distinct linear denominators to simplify complex integrals.

Decomposition
1.3333 / (x - 1) + 3.6667 / (x + 2)
Numerator C1.333
Numerator D3.667
Calculation StepsOriginal: (5x + -1) / ((x - 1)(x - -2)) Set up decomposition: (5x + -1) / ((x - 1)(x - -2)) = C/(x - 1) + D/(x - -2) Multiply by denominator: 5x + -1 = C(x - -2) + D(x - 1) Solve for C (let x = 1): 5(1) + -1 = C(1 - -2) 4 = C(3) => C = 1.3333 Solve for D (let x = -2): 5(-2) + -1 = D(-2 - 1) -11 = D(-3) => D = 3.6667

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Reversing the Common Denominator

The Partial Fractions Calculator is an essential algebraic tool for simplifying complex rational expressions. By utilizing the Heaviside method to reverse-engineer common denominators, it splits difficult fractions into easily integrable components.

Px+Q(x−a)(x−b)=Cx−a+Dx−b\begin{aligned} \frac{Px + Q}{(x-a)(x-b)} = \frac{C}{x-a} + \frac{D}{x-b} \end{aligned}

Where:
Px + Q=
The original linear polynomial top half of the fraction
a, b=
The zeroes of the original denominator
C, D=
The new numerators that perfectly balance the split fractions

The Integration Cheat Code

In Calculus II, students are hit with a wall when asked to integrate ∫5x−1(x−1)(x+2)dx\int \frac{5x - 1}{(x-1)(x+2)} dx. Standard U-substitution fails completely.

Partial fraction decomposition acts as a mathematical cheat code. By algebraically proving that the fraction is exactly equal to Cx−1+Dx+2\frac{C}{x-1} + \frac{D}{x+2}, the impossible integral transforms into two basic natural logarithms.

Real-World Applications

  • Control Systems Engineering: Used constantly when dealing with Laplace Transforms to convert complex 's-domain' transfer functions back into understandable 'time-domain' outputs.
  • Electrical Circuits: Analyzing the voltage response of RLC circuits over time requires splitting complex frequency impedance fractions into partial sums.
  • Chemical Kinetics: Solving differential equations that model complex, multi-stage chemical reactions over time.

Frequently Asked Questions

It is the exact opposite of finding a common denominator. It takes a complex, combined fraction and splits it apart into two or more simpler, separate fractions that add up to the original.

It is almost exclusively used in calculus to make integration possible. Integrating a massive combined polynomial fraction is incredibly difficult. Integrating C/(x-a) is trivially easy (it's just C * ln|x-a|).

You set up a generic equation with unknown variables (C and D) on top of the split denominators. By multiplying both sides by the original denominator, you can algebraically solve for C and D.

Yes, but the setup changes. If the denominator is (x-a)², you must split it into C/(x-a) + D/(x-a)². This calculator specifically handles distinct linear roots.

It's a fast algebraic shortcut for finding C and D. You simply cover up the (x-a) part of the original fraction and plug in x=a to instantly find the value of C.

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