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Chapter 9: Problem 14

Decompose into partial fractions. Check your answers using a graphing calculator. $$\frac{26 x^{2}+208 x}{\left(x^{2}+1\right)(x+5)}$$

Short Answer

Expert verified
The partial fractions are \(\frac{19}{4(x+5)} + \frac{(-5x+104)/4}{x^2 + 1}\).

Step by step solution

01

Identify the Partial Fraction Form

Express the given fraction as a sum of simpler fractions. For a fraction of the form \(\frac{26x^{2} + 208x}{(x^{2}+1)(x+5)}\), the partial fractions will take the form:\[\frac{A x + B}{x^{2}+1} + \frac{C}{x+5}\].
02

Set the Equation Equal to the Original

Write the equation:\[\frac{26x^2 + 208x}{(x^2 + 1)(x + 5)} = \frac{A x + B}{x^2 + 1} + \frac{C}{x + 5}\].Clear the denominators by multiplying both sides by \((x^2 + 1)(x + 5)\).
03

Expand and Combine Like Terms

After multiplying by \((x^2 + 1)(x + 5)\), we get:\[26x^2 + 208x = (Ax + B)(x + 5) + C(x^{2} + 1)\].Expand and combine like terms on the right-hand side.
04

Match Coefficients

Equate the coefficients of like terms from both sides:\(x^2\text{ terms: } 26 = A + C\),\(x\text{ terms: } 208 = 5A + B\),\(\text{Constant terms: } 0 = 5B + C\).
05

Solve the System of Equations

Solve the system of equations:1. \(26 = A + C\)2. \(208 = 5A + B\)3. \(0 = B + 5C\).First solve equation 1 for \(C\), then substitute into equations 2 and 3 to find \(A\) and \(B\).
06

Check Solution with Graphing Calculator

Verify the solutions by comparing the original fraction and the sum of the partial fractions using a graphing calculator.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebra
In algebra, decomposing complex expressions into their simpler parts can make them more manageable. In this problem, we use partial fractions to break down a rational function into simpler fractions. This involves expressing the given fraction as a sum of simpler fractions, each with a polynomial numerator and a simpler polynomial denominator. The initial fraction is \(\frac{26x^2 + 208x}{(x^2 + 1)(x + 5)}\). By expressing it in partial fractions, we can simplify and solve various algebraic equations.
Rational Functions
Rational functions are ratios of two polynomials. They appear frequently in calculus and algebra. In this exercise, the given function is a rational function because it represents the quotient of a polynomial in the numerator and a product of polynomials in the denominator. One key step is to express \(\frac{26x^2 + 208x}{(x^2 + 1)(x + 5)}\) as a sum of partial fractions: \(\frac{A x + B}{x^2 + 1} + \frac{C}{x + 5}\). This decomposition makes solving for the coefficients \(A\), \(B\), and \(C\) possible. It further allows us to perform operations like integration and differentiation more easily.
System of Equations
Solving for the coefficients \(A\), \(B\), and \(C\) involves forming a system of linear equations. Once the rational function has been expressed in partial fractions, we can match the coefficients of like terms in the polynomials. We get a system of equations:
  • Equation 1: \(26 = A + C\)
  • Equation 2: \(208 = 5A + B\)
  • Equation 3: \(0 = B + 5C\)
Solving this system involves using substitution or elimination. First, solve equation 1 for \(C\), then substitute into equations 2 and 3 to find the specific values of \(A\), \(B\), and \(C\). This process requires a basic understanding of algebraic techniques for solving systems.
Graphing Calculator Verification
After finding the coefficients, it's crucial to verify the solution with a graphing calculator. This step ensures the decomposition is correct. To do this, input both the original fraction and the sum of the partial fractions into the calculator. By graphing both expressions, you can check if their graphs overlap completely. If they do, the partial fraction decomposition is verified. This method provides a visual confirmation, making it easier for students to understand the accuracy of their algebraic solutions.

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Most popular questions from this chapter

Decompose into partial fractions. Check your answers using a graphing calculator. $$\frac{11 x^{2}-39 x+16}{\left(x^{2}+4\right)(x-8)}$$

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