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Magazine Chapter 8: Problem 19
Find the partial fraction decomposition. \(\frac{9 x^{2}-3 x+8}{x^{3}+2 x}\)
Short Answer
Expert verified
\( \frac{4}{x} + \frac{5x - 3}{x^2 + 2} \)
Step by step solution
01
Factor the Denominator
First, factor the denominator of the function \(x^3 + 2x\). Note that the common factor is \(x\). Thus, the expression becomes \(x(x^2 + 2)\).
02
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.
03
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 \( \frac{9x^2 - 3x + 8}{x(x^2 + 2)} \).
04
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. \(2A = 8\).
05
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 \(C = -3\).
06
Write the Final Decomposition
Now that we have \(A = 4\), \(B = 5\), and \(C = -3\), we can write the decomposition: \[ \frac{9x^2 - 3x + 8}{x(x^2 + 2)} = \frac{4}{x} + \frac{5x - 3}{x^2 + 2} \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
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.
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 \(2A = 8\).
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.
