Problem 48 - College Algebra Essentials

Question

write the partial fraction decomposition of each rational expression. $$ \frac{a x+b}{x^{2}-c^{2}} \quad(c \neq 0) $$

Step-by-Step Solution

Answer

The partial fraction decomposition of the rational expression \(\frac{a x+b}{x^{2}-c^{2}}\) is \(\frac{A}{x+c} + \frac{B}{x-c}\), where A and B are constants that can be determined by setting \(x = c\) and \(x = -c\) in the equation \(ax+b = A(x-c) + B(x+c)\) and solving.

Step 1: Understand the Difference of Squares Rule

Examine the denominator of the given fraction and note that it follows the difference of squares rule: \(a^2 - b^2 = (a+b)(a-b)\). In this case \(x^{2}-c^{2}=(x+c)(x-c)\), we express the denominator in this factorized form.

Step 2: Setting Up the Partial Fraction Decomposition

Start with the rational expression \(\frac{a x+b}{x^{2}-c^{2}}\). Using the difference of squares, rewrite the denominator to get \(\frac{a x+b}{(x+c)(x-c)}\). Set this equal to the sum of two simpler fractions with undetermined coefficients: \(\frac{A}{x+c} + \frac{B}{x-c}\), where A and B are constants that need to be determined. This will look like: \(\frac{a x+b}{(x+c)(x-c)} = \frac{A}{x+c} + \frac{B}{x-c}\).

Step 3: Solve for A and B

Multiply both sides by the common denominator \((x+c)(x-c)\) to obtain \(ax+b = A(x-c) + B(x+c)\), then set \(x = c\) and \(x = -c\) to find equations for A and B and solve for the constants.

Key Concepts

Rational Expressions

A rational expression is a fraction where the numerator and the denominator are both polynomials. These expressions can often be complex, but breaking them down into smaller parts can make them easier to understand and work with.

• The term "rational" denotes that the expression is in a ratio or fraction form.
• Polynomials in the numerator and denominator can vary in degree, but the denominator cannot be zero as this would make the expression undefined.
• Rational expressions are often encountered in algebra and calculus, requiring manipulation such as factoring or simplification to solve equations or to integrate further.

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler fractions. This is particularly useful in calculus when integrating rational expressions.

Difference of Squares

The difference of squares is a unique pattern in algebra that makes factoring certain expressions straightforward. It is based on the identity \[a^2 - b^2 = (a+b)(a-b)\]In simpler terms, any expression that fits the template of a squared term minus another squared term can be factored into the product of a sum and a difference.

• For example, \(x^2 - c^2\) can be rewritten as \((x+c)(x-c)\).
• This pattern is essential when working with rational expressions because it allows for the denominator to be simplified through factoring.
• Recognizing the difference of squares quickly allows you to apply it to partial fraction decomposition by breaking down the denominator into its components.

Denominator Factoring

Factoring the denominator is a crucial step in simplifying rational expressions, especially when setting up partial fraction decompositions. By breaking the denominator into its factors, you can express the fraction as a sum of simpler terms.

• Start by identifying any common patterns such as the difference of squares or common monomials that can be factored out.
• Once the denominator is factored, it becomes easier to set up equations that allow you to solve for unknown variables or to integrate if necessary.
• In partial fraction decomposition, successfully factoring the denominator allows you to set up an equation where the original expression equals a sum of fractions, each with one of these factors as the new denominators.

Understanding and applying these factoring techniques are essential in algebra and calculus, allowing for the simplification and resolution of complex rational expressions.