Problem 3 - Algebra and Trigonometry

Question

\(3-12\) . Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$ \frac{1}{(x-1)(x+2)} $$

Step-by-Step Solution

Answer

Only set up decomposition: \( \frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2} \).

Step 1: Identify the Denominator

The first step is to carefully examine the denominator of the given function: \[(x-1)(x+2)\].This denominator consists of two linear factors, \((x-1)\) and \((x+2)\).

Step 2: Set Up the Partial Fraction

For partial fraction decomposition, express the given fraction as the sum of two separate fractions. Each fraction will have one of the linear factors in the denominator and a constant in the numerator:\[\frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}\].Here, \(A\) and \(B\) are constants that need to be determined.

Key Concepts

Linear Factors

A linear factor is a polynomial of the first degree, which means it has the form \((ax + b)\). In the context of partial fraction decomposition, linear factors are the building blocks of the denominator of a rational expression. When dealing with linear factors:

• They will always have an exponent of one.
• Each can be represented by distinct constants and variables.

In the given exercise, the denominator \((x-1)(x+2)\) is composed of two linear factors: \((x-1)\) and \((x+2)\).
Identifying these factors is the first step in setting up a partial fraction decomposition. Recognizing linear factors quickly will make the decomposition process much simpler.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. An example of a rational expression is \(\frac{1}{(x-1)(x+2)}\), as seen in the exercise. Rational expressions can be simplified, added, or decomposed into partial fractions, as needed. Key characteristics include:

• The denominator should never be zero
• Any common factors between the numerator and the denominator can be canceled out to simplify the expression.

Simplifying rational expressions is essential for easier computation or integration. It is crucial to correctly identify components of the expression, such as linear factors in the denominator, before proceeding with operations like decomposition.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a complex rational expression as a sum of simpler fractions. This technique is especially useful in calculus for integrating rational functions. The process involves:

• Breaking down the denominator into its factors, usually linear factors or irreducible quadratic factors.
• Expressing the original rational function as a sum of fractions, each having its denominator as one of these factors.
• Finding unknown constants for the numerators of these fractions.

In the exercise provided, we decompose \(\frac{1}{(x-1)(x+2)}\) into the form \(\frac{A}{x-1} + \frac{B}{x+2}\). Here, \(A\) and \(B\) are constants that need to be determined through further calculations. Setting up this equation is crucial for solving the entire expression.