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Chapter 5: Problem 56

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

Short Answer

Expert verified
The partial fraction decomposition of a rational expression with a prime quadratic factor involves identifying the quadratic factor, expressing the rational expression as a sum of simpler fractions, solving a system of equations to find the numerators of the fractions, and verifying that the result matches the original expression.

Step by step solution

01

Identify the quadratic factor in the denominator

Look at the denominator of the rational expression and find the prime quadratic factor. A prime quadratic factor is a quadratic expression that cannot be factored further.
02

Express the rational expression as an equivalent sum of fractions

The goal of partial fraction decomposition is to express the given rational expression as an equivalent sum of fractions, where each fraction has a simpler denominator. If the prime quadratic factor in the denominator is \(ax^2+bx+c\), the corresponding term in the sum of fractions will have that factor in the denominator and a linear term \(mx+n\) in the numerator.
03

Solve system of equations

To find the linear numerator term of the partial fraction, set the given rational expression equal to the sum of fractions and solve the resulting system of equations.
04

Check your answer

Add the partial fractions together and simplify to verify that the result matches the original rational expression.

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Most popular questions from this chapter

Involve supply and demand. Although Social Security is a problem, some projections indicate that there's a much bigger time bomb ticking in the federal budget, and that's Medicare. In \(2000,\) the cost of Social Security was \(5.48 \%\) of the gross domestic product, increasing by \(0.04 \%\) of the GDP per year. In \(2000,\) the cost of Medicare was \(1.84 \%\) of the gross domestic product, increasing by \(0.17 \%\) of the GDP per year. a. Write a function that models the cost of Social Security as a percentage of the GDP \(x\) years after 2000 . b. Write a function that models the cost of Medicare as a percentage of the GDP \(x\) years after 2000 . c. In which year will the cost of Medicare and Social Security be the same? For that year, what will be the cost of each program as a percentage of the GDP? Which program will have the greater cost after that year?

Exercises 116-118 will help you prepare for the material covered in the next section. a. Graph the solution set of the system: $$\left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x-2 x & \leq 6 \\ y & \leq-x+7 .\end{aligned}\right.$$ b. List the points that form the corners of the graphed region in part (a). c. Evaluate \(2 x+5 y\) at each of the points obtained in part (b).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Partial fraction decomposition involves finding a single rational expression for a given sum or difference of rational expressions.

What kinds of problems are solved using the linear programming method?

The group should write four different word problems that can be solved using a system of linear equations in two variables. All of the problems should be on different topics. The group should turn in the four problems and their algebraic solutions.
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