How To Do Partial Fraction Decomposition

Partial Fraction Decomposition:

\[ \frac{P(x)}{Q(x)} = \sum_{i=1}^{n} \frac{A_i}{(x - r_i)} + \sum_{j=1}^{m} \frac{B_jx + C_j}{(x^2 + b_jx + c_j)} \]

Partial Fraction Decomposition:

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1. What Is Partial Fraction Decomposition?

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler fractions that are easier to work with in calculus and other areas of mathematics.

2. Steps For Partial Fraction Decomposition

The general steps for partial fraction decomposition are:

Factor the denominator Q(x) completely
Determine the form of the decomposition based on the factors
Set up equations for the unknown coefficients
Solve the system of equations for the coefficients
Write the final decomposed form

3. Types Of Partial Fractions

Different forms appear based on the denominator's factors:

Linear factors: \[ \frac{A}{x - r} \]

Repeated linear factors: \[ \frac{A_1}{x - r} + \frac{A_2}{(x - r)^2} + \cdots \]

Irreducible quadratic factors: \[ \frac{Bx + C}{x^2 + bx + c} \]

4. Example Problems

Example 1: Simple linear factors
\[ \frac{3x + 5}{(x + 1)(x - 2)} = \frac{A}{x + 1} + \frac{B}{x - 2} \]

Example 2: Repeated linear factor
\[ \frac{x^2 + 1}{(x - 3)^2(x + 2)} = \frac{A}{x - 3} + \frac{B}{(x - 3)^2} + \frac{C}{x + 2} \]

Example 3: Quadratic factor
\[ \frac{2x^2 - x + 4}{(x - 1)(x^2 + 4)} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + 4} \]

5. Frequently Asked Questions (FAQ)

Q1: When is partial fraction decomposition used?
A: Commonly used in integral calculus, Laplace transforms, and solving differential equations.

Q2: What if the degree of P(x) ≥ Q(x)?
A: You must first perform polynomial long division before decomposing.

Q3: How do you handle repeated roots?
A: Each power of the repeated factor gets its own term in the decomposition.

Q4: What about complex roots?
A: They can be kept as quadratic factors or separated into complex linear terms.

Q5: Is there a general formula for decomposition?
A: The form depends on the denominator's factorization, so each case must be handled individually.