Problem 23 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition. Write the partial fraction decomposition of the rational expression. Check your result algebraically. $$\frac{1}{x^{2}-1}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \( \frac{1}{x^{2}-1} \) is \( \frac{1/2}{x-1} - \frac{1/2}{x+1} \)

Step 1: Decompose the denominator as a difference of squares

Recognize the denominator \( x^{2}-1 \) as the difference of two squares. As such, it can be rewritten as \( x^{2}-1 = (x-1)(x+1) \).

Step 2: Write the expression as partial fractions

Write the given expression, \( \frac{1}{x^{2}-1} \), as the sum of simpler fractions. We can write this as: \( \frac{1}{x^{2}-1} = \frac{A}{(x-1)} + \frac{B}{(x+1)} \), where A and B are constants that we need to find.

Step 3: Find the values of A and B

First, multiply through by the common denominator to eliminate the denominators on the right side. This yields \( 1 = A(x+1) + B(x-1) \). Then set x to values that will allow you to solve for A and B.\nSetting x = 1 gives \( 1 = 2A \) so \( A = \frac{1}{2} \).Setting x = -1 gives \( 1 = -2B \) so \( B = -\frac{1}{2} \)

Step 4: Write the final partial fractions

Substitute the values of A and B back into the partial fractions. This gives: \( \frac{1}{x^{2}-1} = \frac{1/2}{x-1} - \frac{1/2}{x+1} \)

Key Concepts

Rational Expressions

Rational expressions are similar to fractions, where both the numerator and the denominator are polynomials. They represent the division of two polynomials.
For example, \( \frac{1}{x^2 - 1} \) is a rational expression because it is the division of 1 by the polynomial \( x^2 - 1 \).

Key points about rational expressions include:

• They can be simplified by factoring and canceling common factors in the numerator and denominator.
• Understanding them helps in performing operations such as addition, subtraction, multiplication, and division.
• They often require partial fraction decomposition to break down into simpler parts for easier integration or solving.

By decomposing into partial fractions, complex rational expressions are broken into simpler additive components. This is especially useful in calculus and algebra when performing integration or solving equations.

Difference of Squares

Recognizing and using the difference of squares is crucial for simplifying and solving expressions. A difference of squares is a special algebraic identity: \( a^2 - b^2 = (a - b)(a + b) \).

This concept was key in the decomposition process:

• The denominator \( x^2 - 1 \) can be expressed as \( (x - 1)(x + 1) \) because it fits the difference of squares pattern where \( a = x \) and \( b = 1 \).

Understanding the difference of squares helps in:

• Simplifying expressions by factoring.
• Solving quadratic equations.
• Identifying patterns in algebraic equations that appear frequently.

By rewriting expressions using this concept, solving and simplifying become much more manageable.

Algebraic Equations

Algebraic equations are statements of equality that involve variables and constants. Solving these equations involves finding the values of variables that make the equation true.

In the context of partial fraction decomposition:

• You set up an equation like \( 1 = A(x + 1) + B(x - 1) \) to solve for constants A and B.
• By choosing strategic values for \( x \), such as \( x = 1 \) and \( x = -1 \), you can simplify the equation to solve for A and B easily.

Key aspects of solving algebraic equations include:

• Transposing terms to isolate variables or constants as needed.
• Substituting values that simplify calculations, such as eliminating one variable at a time.
• Understanding the properties of equality to manipulate equations effectively.

Mastery of algebraic equations is not only essential in pure algebra but also in real-world problem-solving across various fields.