Question

Explain how to find the partial fraction decomposition of a rational expression. Include in your explanation a discussion of each of the four cases that arise.

Answer 1

To find the partial fraction decomposition of a rational expression, one must follow these steps:

1. Factor the denominator into irreducible factors.

2. Express the fraction as a sum of partial fractions, where each partial fraction has a denominator corresponding to one of the irreducible factors.

3. Determine the unknown constants in the numerators by equating coefficients.

Step-by-step explanation:

Partial fraction decomposition is a technique used to break down a complex rational expression into simpler fractions, making it easier to integrate or manipulate. The process involves factoring the denominator into irreducible factors, which fall into four main cases:

1. Distinct Linear Factors:

If the denominator has distinct linear factors, express the rational function as the sum of fractions with numerators that are constants. For example, for the expression
`\( (A)/(x-a) + (B)/(x-b) + \ldots \).`

2. Repeated Linear Factors:

When the denominator has repeated linear factors, include a fraction for each power of the repeated factor. For instance,
`\( (A)/((x-a)^2) + (B)/((x-a)^3) + \ldots \).`

3. Distinct Quadratic Factors:

If the denominator has distinct quadratic factors, express the partial fractions with numerators as linear polynomials. For example,
`\( (Ax + B)/(x^2 + px + q) + (Cx + D)/(x^2 + rx + s) + \ldots \).`

4. Irreducible Quadratic Factors:

For irreducible quadratic factors, use numerators with linear terms. For instance,
`\( (Ax + B)/(x^2 + px + q) + \ldots \).`

After expressing the original rational function in partial fraction form, solve for the unknown constants by equating coefficients. This process simplifies the integration or manipulation of the rational expression in further mathematical operations.