Problem 48 - College Algebra

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 is given by \[\frac{a x+b}{x^{2}-c^{2}}=\frac{A}{x-c}+\frac{B}{x+c}\] where A and B can be found by solving the system of equations: \(a = A + B\) , \(b = Ac - Bc\).

Step 1: Identify the denominator's factors

The denominator of given rational expression is a difference of two squares \(x^{2}-c^{2}\) which can be factored into \((x-c)(x+c)\).

Step 2: Set up the expression for partial fractions

Using the principle of partial fractions, we decompose the given fraction into the sum of simpler fractions. We represent each factor of the denominator with a new fraction, where its numerator is an undetermined constant since the original numerator is a linear expression. This gives the following: \[\frac{a x+b}{x^{2}-c^{2}}=\frac{A}{x-c}+\frac{B}{x+c}\]. Where A and B are constants that we need to find.

Step 3: Cross multiply to get rid of the denominators

Cross multiply to get an equation that is easier to solve for the constants A and B. \[a x+b = A(x+c) + B(x-c)\]

Step 4: Solve for A and B

To solve for A and B, we can equate coefficients from the left and right sides of the equation. This gives two equations: 1. Coefficient of x: \(a = A + B\) 2. Independent term: \(b = Ac - Bc\) From these, we can solve for A and B.

Key Concepts

Rational Expression

A rational expression is similar to a rational number—a fraction where the numerator and the denominator are both integers. In algebra, a rational expression is a fraction where the numerator and the denominator are polynomials. For example, in the expression \( \frac{ax+b}{x^2-c^2} \), \( ax+b \) is the numerator, which is a polynomial of degree 1 (linear), and \( x^2-c^2 \) is the denominator, a polynomial of degree 2 (quadratic).

Understanding the structure of rational expressions is crucial for performing operations such as partial fraction decomposition, where the goal is to rewrite a complex fraction as a sum of simpler fractions. This makes it easier to integrate or differentiate mathematical functions when dealing with calculus problems.

Factorization

Factorization is breaking down a composite number or a polynomial into a product of its nontrivial factors. This process makes it easier to simplify equations and solve algebraic problems.

Take the denominator of our rational expression \( x^2-c^2 \). It's not immediately obvious how this should be broken down, but recognizing it as a difference of two squares—a special factorization case—allows us to write it as \( (x-c)(x+c) \).

Recognizing Patterns

When factorizing polynomials, look for patterns such as:

• Common factors in all terms.
• Special products like the difference of two squares.
• Trinomial patterns that allow factoring into binomials.

Mastering these patterns can greatly simplify the process of partial fraction decomposition.

Difference of Two Squares

The difference of two squares is a specific polynomial form and a shortcut in factorization. It refers to an expression like \( x^2 - y^2 \) that can be factored into \( (x+y)(x-y) \). Here, \( x \) and \( y \) are any algebraic expressions.

In our exercise, the denominator \( x^2 - c^2 \) fits this pattern exactly, allowing it to be factored as described in the factorization section. Recognizing when a polynomial is a difference of two squares is a key skill that simplifies many algebraic processes, including the simplification of rational expressions and solving of equations.

Equating Coefficients

Equating coefficients is a technique used to find the values of unknown constants within algebraic expressions. It's based on the principle that if two polynomials are equal, then their corresponding coefficients must be equal.

In the context of the partial fraction decomposition from the exercise, after cross multiplying to combine the partial fractions, we obtain a polynomial expression with undetermined constants (A and B). By equating the coefficients of like terms on both sides of the resulting equation, we can create a system of equations to solve for these constants.

This is a critical step in partial fraction decomposition because it allows us to determine the exact values for the constants that will make our decomposed expression equivalent to the original rational expression.