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Chapter 5: Problem 34

Write the partial fraction decomposition of each rational expression. $$\frac{10 x^{2}+2 x}{(x-1)^{2}\left(x^{2}+2\right)}$$

Short Answer

Expert verified
\(\frac{10x^2+2x}{(x-1)^2(x^2+2)} = \frac{1}{x-1} + \frac{6}{(x-1)^2} + \frac{-8x-4}{x^2+2}\)

Step by step solution

01

Setup Partial Fractions

First, let's break the given fraction into the sum of partial fractions. We can write it in the following form: \(\frac{10x^2+2x}{(x-1)^2(x^2+2)} = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{Cx+D}{x^2+2}\). A, B, C, and D are constants which we need to find. The first two terms are due to the linear term \((x-1)^2\) and the third term is due to the quadratic term \(x^2+2\).
02

Clear the Fraction

Now, to find the constants A, B, C, and D, we will clear the fraction and group the like terms together. To do this, we can multiply both sides of the equation by the denominator \((x-1)^2(x^2+2)\) of the left side. We get, \(10x^2+2x = A(x-1)(x^2+2) + B(x^2+2) + (Cx+D)(x-1)^2\).
03

Evaluate constants

Now we will choose specific values for x that will simplify the equations and solve for A, B, C, and D. Let's start with x = 1, then the equation will be only in terms of B, \(12 = 2B\), solving this we get B = 6. Next by differentiating the equation with respect to x and then substituting x=1, will cancel A giving us \(C = -8\). Now to find A and D substitute x=i, we can solve the system of equations giving us A = 1 and D = -4.
04

Write the final answer

Now that we have the values of A, B, C, and D, substitute them back into the partial fractions from step 1. The partial fraction decomposition of the given expression is equal to \(\frac{1}{x-1} + \frac{6}{(x-1)^2} + \frac{-8x-4}{x^2+2}\).

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Most popular questions from this chapter

A person with no more than 15,000 dollars to invest plans to place the money in two investments. One investment is high risk, high yield; the other is low risk, low yield. At least 2000 dollars is to be placed in the high-risk investment. Furthermore, the amount invested at low risk should be at least three times the amount invested at high risk. Find and graph a system of inequalities that describes all possibilities for placing the money in the high- and low-risk investments.

Explain how to find the partial fraction decomposition of a rational expression with a prime quadratic factor in the denominator.

Sketch the graph of the solution set for the following system of inequalities: $$ \begin{array}{r}|x+y| \leq 3 \\\|y| \leq 2\end{array} $$

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &y=\frac{1}{3} x+\frac{2}{3}\\\ &y=\frac{5}{7} x-2 \end{aligned} $$

In \(1985,\) college graduates averaged \(\$ 508\) in weekly earnings. This amount has increased by approximately \(\$ 25\) in weekly earnings per year. By contrast, in 1985 , people with less than four years of high school averaged \(\$ 270\) in weekly earnings. This amount has only increased by approximately \(\$ 4\) in weekly earnings per year. a. Write a function that models weekly earnings, \(E,\) for college graduates \(x\) years after 1985 b. Write a function that models weekly earnings, \(E,\) for people with less than four years of high school \(x\) years after 1985 c. How many years after 1985 will college graduates be earning three times as much per week as people with less than four years of high school? (Round to the nearest whole number.) In which year will this occur? What will be the weekly earnings for each group at that time? (GRAPH CAN'T COPY)
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