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Chapter 9: Problem 1

Find the partial fraction decomposition for each rational expression. \(=\) $$\frac{5}{3 x(2 x+1)}$$

Short Answer

Expert verified
\(\frac{5}{3x} - \frac{10}{3(2x+1)}\)

Step by step solution

01

Identify the Denominator

First, write the given rational expression and identify the factors in the denominator. The expression given is: \(\frac{5}{3x(2x+1)}\). The denominator is factored into \(3x\) and \(2x + 1\).
02

Set Up Partial Fractions

Set up the partial fractions based on the factors of the denominator. Each factor will have a corresponding term in the decomposition: \(\frac{A}{3x} + \frac{B}{2x+1}\). We assume constants \(A\) and \(B\).
03

Combine Fractions

Combine the fractions on the right-hand side so they have a common denominator: \(\frac{A}{3x} + \frac{B}{2x+1} = \frac{A(2x+1) + B(3x)}{3x(2x+1)}\).
04

Equate Numerators

Equate the numerator of the combined fractions to the numerator of the original expression: \(5 = A(2x+1) + B(3x)\).
05

Solve for Constants

Create a system of equations by equating the coefficients from both sides of the equation. Solve this system to find \(A\) and \(B\). For the constant term: \(A = 5\). Coefficient of \(x\) term: \(2A + 3B = 0\) leads to \(2(5) + 3B = 0\), solving for \(B\): \(B = -\frac{10}{3}\).
06

Write the Decomposition

Substitute the values of \(A\) and \(B\) back into the partial fractions: \(\frac{5}{3x(2x+1)} = \frac{5}{3x} - \frac{10/3}{2x+1}\) or \(\frac{5}{3x} - \frac{10}{3(2x+1)}\).

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Key Concepts

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

Rational Expressions
A rational expression is simply a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is essential since they form the basis of partial fraction decomposition.
Partial fraction decomposition allows us to break down complex rational expressions into simpler ones, making them easier to integrate or differentiate.
For example, in the rational expression \(\frac{5}{3x(2x+1)}\), the numerator is \(5\) and the denominator is a product of two polynomials: \(3x\) and \(2x+1\).
Decomposing this expression helps us understand its structure.
Common Denominator
To combine multiple rational expressions into a single fraction, we need a common denominator. This is crucial when working with partial fraction decomposition.
In our example, we initially set up the partial fractions as \(\frac{A}{3x} + \frac{B}{2x+1}\).
The common denominator of these fractions is \(3x(2x+1)\).
We then combine these fractions under this common denominator:
\(\frac{A(2x+1) + B(3x)}{3x(2x+1)}\).
This step simplifies the process of solving for the constants.
System of Equations
A system of equations is a set of equations with multiple variables. Solving a system of equations simultaneously determines the values of these variables.
In partial fraction decomposition, once the numerators are equated, we encounter a system of equations.
For our example, by equating the numerators, we get: \(5 = A(2x+1) + B(3x)\).
This creates the system:
  • For the constants: \(A = 5\)
  • Coefficient of \(x\) term: \(2A + 3B = 0\)
Solving this system gives us the values for \(A\) and \(B\), which are essential for completing the decomposition.
Constants
Constants are fixed values that appear in mathematical expressions. In partial fraction decomposition, these constants \(A\) and \(B\) represent the numerators of the simpler fractions.
The goal of decomposition is to find these constants.
From our system of equations:
  • We first find \(A = 5\)
  • Then solve for \(B\) using \(2(5) + 3B = 0\), which simplifies to \(B = -\frac{10}{3}\)
Substituting these values back into the partial fractions, we get: \(\frac{5}{3x} - \frac{10/3}{2x+1}\) or \(\frac{5}{3x} - \frac{10}{3(2x+1)}\).
These constants make the fractions easier to handle individually.

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Most popular questions from this chapter

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} x+y+z &=4 \\ 2 x-y+3 z &=4 \\ 4 x+2 y-z &=-15 \end{aligned}$$

$$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right], \quad B=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right], \quad \text { and } \quad C=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] $$ where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A+(B+C)=(A+B)+C\) (associative property)

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product when possible. $$A^{2}$$

Use the determinant theorems to find the value of each determinant. $$\left|\begin{array}{rrrr} 3 & -6 & 5 & -1 \\ 0 & 2 & -1 & 3 \\ -6 & 4 & 2 & 0 \\ -7 & 3 & 1 & 1 \end{array}\right|$$

Solve each problem. In certain parts of the Rocky Mountains, deer provide the main food source for mountain lions. When the deer population is large, the mountain lions thrive. However, a large mountain lion population reduces the size of the deer population. Suppose the fluctuations of the two populations from year to year can be modeled with the matrix equation $$ \left[\begin{array}{c} m_{n+1} \\ d_{n+1} \end{array}\right]=\left[\begin{array}{rr} 0.51 & 0.4 \\ -0.05 & 1.05 \end{array}\right]\left[\begin{array}{l} m_{n} \\ d_{n} \end{array}\right] $$ The numbers in the column matrices give the numbers of animals in the two populations after \(n\) years and \(n+1\) years, where the number of deer is measured in hundreds. (a) Give the equation for \(d_{n+1}\) obtained from the second row of the square matrix. Use this equation to determine the rate at which the deer population will grow from year to year if there are no mountain lions. (b) Suppose we start with a mountain lion population of 2000 and a deer population of \(500,000\) (that is, 5000 hundred deer). How large would each population be after 1 yr? 2 yr? (c) Consider part (b) but change the initial mountain lion population to \(4000 .\) Show that the populations would both grow at a steady annual rate of 1.01
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