Problem 31 - A Graphical Approach to College Algebra

Question

Determine whether each partial fraction decomposition is correct by graphing the left side and the right side of the equation on the same coordinate axes and observing whether the graphs coincide. $$\frac{4 x^{2}-3 x-4}{x^{3}+x^{2}-2 x}=\frac{2}{x}+\frac{-1}{x-1}+\frac{3}{x+2}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is correct; both expressions are equivalent.

Step 1: Set Up the Equation

We start with the equation \( \frac{4x^{2}-3x-4}{x^{3}+x^{2}-2x} = \frac{2}{x} + \frac{-1}{x-1} + \frac{3}{x+2} \). Our goal is to check whether these two expressions represent the same function by graphing.

Step 2: Simplify the Denominator

Factor the denominator \( x^3 + x^2 - 2x \). This can be simplified to \( x(x + 2)(x - 1) \).

Step 3: Confirm Common Denominator

The common denominator for the terms on the right-hand side is \( x(x+2)(x-1) \), which matches the denominator on the left. This confirms that the decomposition has the correct structure.

Step 4: Evaluate Numerators

Combine the fractions on the right-hand side: \[ \frac{2(x+2)(x-1) - (x)(x+2) + 3(x)(x-1)}{x(x+2)(x-1)} \]Simplify the numerator step by step.

Step 5: Simplify Right-Hand Side Numerator

1. \( 2(x+2)(x-1) = 2(x^2 + x - 2) = 2x^2 + 2x - 4 \)2. \( - (x)(x+2) = -x^2 - 2x \)3. \( 3(x)(x-1) = 3x^2 - 3x \)Add these to get the right-hand numerator: \[ (2x^2 + 2x - 4) + (-x^2 - 2x) + (3x^2 - 3x) = 4x^2 - 3x - 4 \]

Step 6: Compare Numerators

The numerator from the right-hand side \( 4x^2 - 3x - 4 \) matches the left-hand side numerator. Thus, both sides are equivalent algebraically.

Step 7: Graph Verification

Graph \( \frac{4x^2 - 3x - 4}{x^3 + x^2 - 2x} \) and \( \frac{2}{x} + \frac{-1}{x-1} + \frac{3}{x+2} \) on the same axes. Check if the graphs overlap completely, indicating they are equivalent.

Key Concepts

Graphing Functions

Graphing functions is a method to visually analyze the properties of equations by plotting points on a coordinate system. This technique helps in comprehending the behavior of functions and validating mathematical equivalences, such as partial fraction decomposition.

When graphing, each function's equation is represented as a line or curve on a set of axes. For the exercise given, we graph two expressions: the original fraction and its partial fraction decomposition. The idea is to ensure that both graphs overlap entirely. Overlapping graphs mean both expressions are equivalent.

• Always begin by identifying the range of values you want to graph.
• Determine any asymptotes or intercepts.
• Choose suitable scales for the x and y axes, keeping in mind the function's behavior.
• Plot sufficient points to capture the overall shape of the function.
• Connect these points smoothly to form the graph.

Graphing confirms that the analytical work is correct, providing a clear visual check of function equivalence.

Algebraic Expressions

An algebraic expression involves numbers, variables, and arithmetic operations. Understanding these expressions is crucial when working with equations and performing operations such as partial fraction decomposition.

In algebraic terms, equations are manipulated to simplify, evaluate, or solve them. For instance, converting a complex rational expression into a sum of simpler fractions makes it easier to analyze or integrate.

• Recognize parts of the expression: coefficients, constants, and variables.
• Simplification might involve factoring polynomials or finding a common denominator, as shown in the exercise.
• The goal is to express complex terms in a more digestible form.

These simplifications often help in checking equivalence or solving expressions, reinforcing mathematical understanding.

Factoring Polynomials

Factoring is a key tool when dealing with polynomial expressions. It involves breaking down a polynomial into simpler, multiply-able components. This approach is indispensable in simplifying expressions and performing partial fraction decomposition.

In the exercise, the denominator is factored to reveal its essential components, which becomes crucial in confirming the correctness of the partial fraction breakdown.

• Identify polynomials that can be factored, such as quadratics or higher orders.
• Seek common factors, difference of squares, or use techniques like grouping or solving for zeros.
• Factoring is not just about breaking down expressions but understanding inherent structures within them.

Recognizing these structures simplifies not only the original problem but also aids in future algebraic operations, such as integration or solving equations.