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Chapter 7: Problem 54

Writing the Partial Fraction Decomposition, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result. $$\frac{3 x+1}{2 x^{3}+3 x^{2}}$$

Short Answer

Expert verified
The partial fraction decomposition of \(\frac{3x+1}{2x^3 + 3x^2}\) is \(\frac{1}{x} + \frac{1/3}{x^2} - \frac{2}{2x + 3}\).

Step by step solution

01

Factorize the denominator

Factorize the denominator polynomial \(2x^3 + 3x^2\), which becomes \(x^2 (2x + 3)\).
02

Formulate partial fraction decomposition

Set up the partial fraction decomposition for the given fraction. With the denominator now factorized, it becomes easier. We write \(\frac{3x+1}{x^2 (2x + 3)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{2x + 3}\), where A, B and C are coefficients we need to find.
03

Determine the coefficients

To find the values of A, B and C, we multiply every term by the denominator on the left, set it equal to the numerator on the left, and solve for the coefficients. Doing this yields 3x + 1 = Ax(x + 3) + B(2x + 3) + Cx^2. Setting x = 0 gives B = 1/3. Setting x = -3/2 gives C = -2. Lastly, putting x = 1 gives A = 1.
04

Write the final decomposition

The partial fraction decomposition of the given function is \(\frac{1}{x} + \frac{1/3}{x^2} - \frac{2}{2x + 3}\).
05

Validate the result using a graphing utility

The validity of the decomposition can be checked by comparing the graphs of the original and decomposed functions. Post validation, we can confirm that the decomposition is correct. Using a graphing utility is recommended to accurately compare.

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Key Concepts

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

Rational Expressions
Rational expressions are simply fractions where both the numerator and the denominator are polynomials. They follow rules similar to regular fractions, such as being simplified or decomposed into simpler parts. Understanding rational expressions is fundamental when dealing with partial fraction decomposition, as it involves breaking down complex fractions into sums of simpler fractions.

The key is the denominator of the rational expression, which determines how the expression can be split. In the case of the expression \(\frac{3x+1}{2x^3+3x^2}\), the denominator \(2x^3+3x^2\) hints at the parts we will break it into after factorization.
Polynomial Factorization
Polynomial factorization is crucial in simplifying complex algebraic expressions. It's the process of breaking down a polynomial into a product of simpler polynomials. This is the first step in partial fraction decomposition.

For the expression \(\frac{3x+1}{2x^3+3x^2}\), we factorized the denominator \(2x^3+3x^2\) to find \(x^2(2x+3)\).
  • Factorization helps to identify the possible terms in the partial fraction decomposition.
  • It reveals multiple roots and simplifies further calculations.
This breakdown is essential for setting up the equation to find the unknown coefficients later.
Coefficients
Once the expression is set for decomposition, we need to determine the coefficients of each term, labeled as \(A\), \(B\), and \(C\). These coefficients scale the parts of decomposed fractions, acting as weights adjusting their values.

To find these coefficients, multiply through by the denominator \(x^2(2x+3)\) to eliminate the fractions.
  • Balance the equation by making the new equation equal to the original numerator, \(3x + 1\).
  • Substitute strategic values of \(x\) (e.g., \(x = 0, -\frac{3}{2}, 1\)) to solve for \(A\), \(B\), and \(C\).
  • This process leverages properties of polynomial equations to isolate and calculate each coefficient efficiently.
Graphing Utilities
Graphing utilities are powerful tools for verifying mathematical solutions. By comparing the graph of the original rational expression with its decomposed form, you can visually confirm the accuracy of the partial fraction decomposition.

Using a graphing utility, plot both the original \(\frac{3x+1}{2x^3+3x^2}\) and the decomposition \(\frac{1}{x} + \frac{1/3}{x^2} - \frac{2}{2x+3}\). The curves should overlap perfectly.
  • Graphing utilities help identify discrepancies and ensure the mathematical integrity of the decomposition.
  • They provide an easy-to-understand visual confirmation.
  • These tools are especially useful for complex functions where manual validation might be prone to errors.

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Most popular questions from this chapter

Think About It Are the following two systems of equations equivalent? Give reasons for your answer. $$ \left\\{ \begin{aligned} x + 3 y - z & = 6 \\ 2 x - y + 2 z & = 1 \\ 3 x + 2 y - z & = 2 \end{aligned} \right. $$ $$ \left\\{ \begin{aligned} x + 3 y - z = & 6 \\ - 7 y + 4 z = & 1 \\ - 7 y - 4 z = & \- 16 \end{aligned} \right. $$

Pulley System A system of pulleys is loaded with 128 -pound and 32 -pound weights (see figure). The tensions \(t _ { 1 }\) and \(t _ { 2 }\) in the ropes and the acceleration \(a\) of the 32 -pound weight are found by solving the system of equations $$\left\\{ \begin{aligned} t _ { 1 } - 2 t _ { 2 } & = 0 \\ t _ { 1 } & \- 2 a = 128 \\ t _ { 2 } + a & = 32 \end{aligned} \right.$$ where \(t _ { 1 }\) and \(t _ { 2 }\) are in pounds and \(a\) is in feet per second squared. Solve this system.

Think About It In Exercises 67 and \(68,\) the graphs of the two equations appear to be parallel. Yet, when you solve the system algebraically, you find that the system does have a solution. Find the solution and explain why it does not appear on the portion of the graph shown. $$ \left\\{\begin{aligned} 21 x-20 y &=0 \\ 13 x-12 y &=120 \end{aligned}\right. $$

Media Selection A company has budgeted a maximum of \(\$ 1,000,000\) for national advertising of an allergy medication. Each minute of television time costs \(\$ 100,000\) and each one-page newspaper ad costs \(\$ 20,000 .\) Each television ad is expected to be viewed by 20 million viewers, and each newspaper ad is expected to be seen by 5 million readers. The company's market research department recommends that at most 80\(\%\) of the advertising budget be spent on television ads. What is the optimal amount that should be spent on each type of ad? What is the optimal total audience?

Sports In Super Bowl I, on January \(15,1967\) , the Green Bay Packers defeated the Kansas City Chiefs by a score of 35 to \(10 .\) The total points scored came from a combination of touchdowns, extra-point kicks, and field goals, worth \(6,1 ,\) and 3 points, respectively. The numbers of touchdowns and extra-point kicks were equal. There were six times as many touchdowns as field goals. Find the numbers of touchdowns, extra-point kicks, and field goals scored. (Source: National Football League)
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