Problem 62 - Precalculus

Question

Apply a graphing utility to graph \(y_{1}=\frac{x^{3}+2 x+6}{(x+3)\left(x^{2}-4\right)^{3}}\) and \(y_{2}=\frac{2}{x+3}+\frac{x+3}{\left(x^{2}-4\right)^{3}}\) in the same viewing rectangle. Is \(y_{2}\) the partial-fraction decomposition of \(y_{1} ?\)

Step-by-Step Solution

Answer

Graph \(y_1\) and \(y_2\) to verify if they match perfectly. If they do, \(y_2\) is the partial fraction decomposition of \(y_1\).

Step 1: Factor the Denominator

Let's begin by factoring the denominators of both functions. For function \(y_1\):The denominator is \((x+3)(x^2-4)^3\). The term \((x^2-4)\) can be factored further as \((x-2)(x+2)\), so our denominator becomes \((x+3)(x-2)^3(x+2)^3\).For function \(y_2\), the denominator is already presented in partial fraction form.

Step 2: Apply a Graphing Utility

Using a graphing calculator or software, enter the two functions:\(y_1=\frac{x^3+2x+6}{(x+3)(x^2-4)^3}\)\(y_2=\frac{2}{x+3} + \frac{x+3}{(x^2-4)^3}\)Graph these functions in the same viewing rectangle to visually compare them.

Step 3: Compare Graphical Output

Observe the graphs of \(y_1\) and \(y_2\). Check if the lines overlap perfectly in the viewing rectangle. If they do, it suggests that \(y_2\) is indeed a partial fraction decomposition of \(y_1\).

Step 4: Conclusion Based on Graphs

If the graph of \(y_2\) matches exactly with \(y_1\), then \(y_2\) represents the partial fraction decomposition of \(y_1\). If there are any deviations in their graphs, then \(y_2\) is not an exact partial fraction decomposition of \(y_1\).

Key Concepts

Rational Functions

Rational functions are a special category of expressions in mathematics, where both the numerator and the denominator are polynomials. They are expressed in the form \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0 \). These functions are fascinating because they can take complex shapes and exhibit unique behaviors, such as asymptotes and discontinuities. Some important features to remember about rational functions include:

• The degree of the polynomials in the numerator and denominator can define the function's end behavior.
• Rational functions can have vertical asymptotes where the denominator equals zero.
• Horizontal or slant asymptotes describe the behavior of the function at large values of \(x\).

Understanding these behaviors is essential when analyzing or graphing rational functions, as they provide insight into the function's characteristics and how it interacts with the graphing axes.

Graphing Utilities

Graphing utilities are powerful tools in mathematics that help visualize complex functions. They allow you to enter functions, such as rational ones, and quickly see how they behave on a graph. This is incredibly useful for verifying partial fraction decompositions and seeing any overlaps or deviations visually. When using a graphing utility:

• Ensure the entire function is entered correctly, with correct parentheses and syntax.
• Choose an appropriate viewing rectangle that includes all important features of the graph, like asymptotes and intercepts.
• Compare different graphs to verify equivalency or identify differences.

Graphing utilities are not only useful for checking solutions in exercises like partial fraction decomposition but also for gaining a deeper understanding of the function's behavior across different values of \(x\).

Factorization

Factorization involves breaking down a polynomial into a product of simpler polynomials. It's a crucial step in solving rational functions and verifying conjectures like partial fraction decomposition. When you factor a polynomial completely, you can often reveal the structure of the rational function, simplifying it into a form that is easier to manipulate or integrate.For instance, in the example provided, we factor the denominator \((x^2 - 4)^3\) to \((x-2)^3(x+2)^3\). Such factorization helps identify singularities or zeros of the function, aiding in sketching the graph and understanding its discontinuities. Tips for effective factorization:

• Look for common factors across polynomial terms first.
• Use algebraic identities, such as difference of squares: \(a^2 - b^2 = (a-b)(a+b)\).
• Practice on diverse polynomial expressions to build confidence in recognizing patterns.

Denominator Analysis

Denominator analysis is a critical part of studying rational functions, as the behavior of the function is significantly influenced by the zeros of the denominator. Zeros indicate where the function might have asymptotes or undefined points. Understanding the structure of the denominator through analysis is vital for successful partial fraction decomposition.Key points in denominator analysis include:

• Identify the zeros of the denominator by setting it equal to zero and solving for \(x\).
• Recognize that these zeros are the potential points of vertical asymptotes on the graph.
• Analyze repeated factors in the denominator that might lead to differences in the behavior of asymptotes.

By thoroughly analyzing the denominator, you gain insight into the function’s potential indeterminate points and how the function might behave near these points. This analysis can inform how you simplify and interpret rational functions and their decompositions.