Partial Fraction Decomposition Calculator

What is Partial Fraction Decomposition Calculator?

A Partial Fraction Decomposition Calculator is a tool used in algebra and calculus to break a rational function (a fraction where numerator and denominator are polynomials) into a sum of simpler fractions. This process, called partial fraction decomposition, simplifies complex rational expressions, making them easier to integrate, differentiate, or analyze.

For example, a rational function \(\frac{P(x)}{Q(x)}\) where \(\deg(P)<\deg(Q)\) can be decomposed as:

\[ \frac{P(x)}{Q(x)} = \frac{A_1}{x-r_1} + \frac{A_2}{x-r_2} + \dots + \frac{B_1 x + C_1}{x^2 + px + q} + \dots \]

About the Partial Fraction Decomposition Calculator

This Partial Fraction Decomposition Calculator automates the process of splitting rational functions into simpler fractions. It applies rules for distinct linear factors, repeated factors, and irreducible quadratic factors. By inputting the numerator and denominator polynomials, the calculator returns a step-by-step decomposition, showing the constants or numerators of each partial fraction. It is extremely helpful for students and professionals dealing with integration, Laplace transforms, and algebraic simplification.

Key formulas for decomposition include:

Distinct Linear Factors: \(\frac{P(x)}{(x-r_1)(x-r_2)\dots(x-r_n)} = \frac{A_1}{x-r_1} + \frac{A_2}{x-r_2} + \dots + \frac{A_n}{x-r_n}\)

Repeated Linear Factors: \(\frac{P(x)}{(x-r)^m} = \frac{A_1}{x-r} + \frac{A_2}{(x-r)^2} + \dots + \frac{A_m}{(x-r)^m}\)

Irreducible Quadratic Factors: \(\frac{P(x)}{(x^2+px+q)^n} = \frac{B_1x+C_1}{x^2+px+q} + \dots + \frac{B_nx+C_n}{(x^2+px+q)^n}\)

How to Use this Partial Fraction Decomposition Calculator

1. Enter the numerator polynomial \(P(x)\) and the denominator polynomial \(Q(x)\) of your rational function.
2. Ensure that the degree of \(P(x)\) is less than \(Q(x)\). If not, perform long division first.
3. Click the Calculate button to see the partial fraction decomposition.
4. The calculator will provide step-by-step results, showing constants or coefficients for each partial fraction.
5. Use these simplified fractions for easier integration, solving differential equations, or further algebraic manipulation.

Example: Decompose \(\frac{2x+3}{(x-1)(x+2)}\)

\[ \frac{2x+3}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2} \quad \Rightarrow \quad A = 1, B = 1 \]

Result: \(\frac{2x+3}{(x-1)(x+2)} = \frac{1}{x-1} + \frac{1}{x+2}\)