Problem 56
Question
Explain how to find the partial fraction decomposition of a rational expression with a prime quadratic factor in the denominator.
Step-by-Step Solution
Verified Answer
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.
1Step 1: 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.
2Step 2: 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.
3Step 3: 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.
4Step 4: Check your answer
Add the partial fractions together and simplify to verify that the result matches the original rational expression.
Other exercises in this chapter
Problem 55
A planet's orbit follows a path described by \(16 x^{2}+4 y^{2}-64\) A comet follows the parabolic path \(y-x^{2}-4 .\) Where might the comet intersect the orbi
View solution Problem 56
Exercises \(55-57\) will help you prepare for the material covered in the next section. $$ \text { Add }: \frac{5 x-3}{x^{2}+1}+\frac{2 x}{\left(x^{2}+1\right)^
View solution Problem 56
In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x^{2}+y^{2} \l
View solution Problem 56
A system for tracking ships indicates that a ship lies on a path described by \(2 y^{2}-x^{2}-1 .\) The process is repeated and the ship is found to lie on a pa
View solution