Problem 2 - Precalculus with Calculus Previews

Question

Write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{9 x-8}{x^{2}-1} $$

Step-by-Step Solution

Answer

\( \frac{9x - 8}{x^2 - 1} = \frac{A}{x - 1} + \frac{B}{x + 1} \).

Step 1: Factor the Denominator

Start by factoring the denominator of the rational expression \( x^2 - 1 \). This can be expressed as a difference of squares: \( x^2 - 1 = (x - 1)(x + 1) \).

Step 2: Setup the Form for Partial Fraction Decomposition

Write the partial fraction decomposition for the expression \( \frac{9x - 8}{x^2 - 1} \). Since the denominator factors into \( (x - 1)(x + 1) \), the decomposition will take the form: \[ \frac{A}{x - 1} + \frac{B}{x + 1} \] where \( A \) and \( B \) are constants to be determined.

Key Concepts

Understanding Rational Expressions

A rational expression is like a fraction, but instead of just simple numbers in the numerator and denominator, you can have polynomials instead. Think of it this way: the expression resembles a normal fraction with some added complexity due to the nature of algebra.
In the given problem, the rational expression is \( \frac{9x - 8}{x^2 - 1} \). Here:

• The numerator is \( 9x - 8 \)
• The denominator is \( x^2 - 1 \)

Rational expressions are fundamental in algebra, providing a foundation for understanding more complex equations and systems.
They are especially important in calculus and higher mathematics, where they are used to analyze the behavior of functions.

The Process of Factoring

Factoring is like breaking down a complex entity into simpler, more manageable pieces. In mathematics, when we factor, we are looking for expressions that multiply together to give us the original expression.
In our example, we need to factor the denominator \( x^2 - 1 \). This can be rewritten using factoring techniques.
Notice that it fits the pattern of a difference of squares, which we will explore further:

• \( x^2 - 1 \) can be rewritten as \( (x - 1)(x + 1) \)

Factoring makes it easier to simplify fractions and solve equations. Understanding how to factor efficiently is crucial for successfully manipulating rational expressions.

The Role of the Denominator

In a rational expression, the denominator is crucial because it determines the form and structure of the expression. It's the polynomial at the bottom of the fraction, and its roots significantly influence the expression's behavior.
For the expression \( \frac{9x - 8}{x^2 - 1} \), the denominator is \( x^2 - 1 \), which, after factoring, becomes \( (x - 1)(x + 1) \).
Key facts about the denominator:

• Determines the degrees of complexity in solving or manipulating the expression.
• In partial fraction decomposition, we need to factor the denominator first to set up the decomposition correctly.
• It must not be zero, as division by zero is undefined in mathematics.

Hence, understanding the denominator aids in assessing the scope for splitting the expression into simpler fractions.

Exploring the Difference of Squares

The difference of squares is a special factoring technique used when dealing with expressions in the form \( a^2 - b^2 \). This technique is particularly efficient and useful for simplifying polynomials.
In the exercise solution, the expression \( x^2 - 1 \) perfectly fits this pattern as it can be rewritten as:\( a^2 - b^2 \), where:

• \( a = x \)
• \( b = 1 \)

This results in the factorization:\( x^2 - 1 = (x - 1)(x + 1) \).
Using the difference of squares makes the partial fraction decomposition process straightforward.
This method allows polynomial expressions to be quickly simplified, offering a clear path towards understanding and solving rational expressions.