Problem 19 - Precalculus: Functions and Graphs

Question

Find the partial fraction decomposition. \(\frac{9 x^{2}-3 x+8}{x^{3}+2 x}\)

Step-by-Step Solution

Answer

\( \frac{4}{x} + \frac{5x - 3}{x^2 + 2} \)

Step 1: Factor the Denominator

First, factor the denominator of the function \(x^3 + 2x\). Note that the common factor is \(x\). Thus, the expression becomes \(x(x^2 + 2)\).

Step 2: Set Up the Partial Fraction Decomposition

Write the fraction as a sum of simpler fractions based on the factors obtained. With the denominator factored as \(x(x^2 + 2)\), the partial fraction decomposition will be: \[ \frac{9x^2 - 3x + 8}{x(x^2 + 2)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 2} \], where \(A\), \(B\), and \(C\) are constants to be determined.

Step 3: Combine Fractions on the Right Side

Combine the fractions on the right: \[ \frac{A(x^2 + 2) + (Bx + C)x}{x(x^2 + 2)} \]. This should equal the left-hand side \( \frac{9x^2 - 3x + 8}{x(x^2 + 2)} \).

Step 4: Simplify and Match Coefficients

Expand the numerator on the right-hand side: \( A(x^2 + 2) + (Bx + C)x = Ax^2 + 2A + Bx^2 + Cx \). Arrange by powers of \(x\): \((A + B)x^2 + Cx + 2A\). Now, match these coefficients with the left side: 1. \(A + B = 9\)2. \(C = -3\)3. \(2A = 8\).

Step 5: Solve for Constants

From \(2A = 8\), find \(A = 4\). Substitute \(A = 4\) into \(A + B = 9\) to get \(4 + B = 9\), which gives \(B = 5\). Already, from \(C = -3\).

Step 6: Write the Final Decomposition

Now that we have \(A = 4\), \(B = 5\), and \(C = -3\), we can write the decomposition: \[ \frac{9x^2 - 3x + 8}{x(x^2 + 2)} = \frac{4}{x} + \frac{5x - 3}{x^2 + 2} \].

Key Concepts

Fraction Decomposition Steps

Partial fraction decomposition is a method used to express a rational expression as a sum of simpler fractions. This process is incredibly helpful in calculus and differential equations.
Here are the basic steps:

• Factor the Denominator: Start by factoring the denominator of the rational expression. This is crucial as it dictates the form of partial fractions you'll use.
• Set Up Partial Fractions: Based on the factors in the denominator, write expressions for each fraction, involving unknown constants.
• Combine and Simplify: Use algebra to combine fractions and simplify them, matching the left-hand side of the equation.
• Matching Coefficients: Solve for unknown constants by setting the numerators equal and solving the resulting system.

Working through these steps with care will lead to the correct partial fraction decomposition, simplifying complex rational expressions effectively.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into a product of simpler polynomials. These simpler polynomials are often crucial for analyses and solving.
Here's how you can tackle factoring:

• Identify Common Factors: Check and extract common factors in all terms first. For example, in the polynomial \(x^3 + 2x\), the common factor is \(x\).
• Factor Further: After removing common factors, try factoring the remaining expression. Some expressions might not factor further, as seen in \(x^2 + 2\).
• Check Completeness: Ensure that the polynomial is fully factored. Polynomials might require special techniques like grouping or using identities like a difference of squares.

Factoring is a fundamental skill in mathematics that unlocks the simplicity hidden in complex expressions, especially useful in further algebraic manipulation.

Matching Coefficients

Matching coefficients is a technique used to find the unknown constants in partial fraction decomposition. It involves equating the two sides of an equation after expanding and combining like terms. Here is a simple way to understand it:

• Expand the Numerator: Once you have set up your partial fractions, expand the numerators on your terms similarly to how you would expand a polynomial.
• Align the Powers: Make sure that terms from both sides of the equation are arranged according to their powers of \(x\).
• Match the Coefficients: By comparing the coefficients of each power of \(x\), set up equations to solve for unknowns. In the exercise, this process yielded equations like \(A + B = 9\) and \(2A = 8\).

By carefully comparing coefficients of similar powers, you can effectively solve for any unknown constants, providing a clear path to the solution.

Rational Expressions

Rational expressions are quotients of two polynomials, often appearing in algebra and calculus. Understanding how to work with them is important:

• Recognize the Structure: These expressions generally take the form \(\frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials, and \(q(x)\) \(eq 0\).
• Simplification: Before working on operations like addition or decomposition, simplify the rational expression by factoring both the numerator and the denominator.
• Partial Fraction Decomposition: It's often easier to understand and integrate rational expressions when broken down into partial fractions.

Handling rational expressions requires precision in factoring and performing algebraic manipulations, which ultimately simplify their evaluation or integration.