In Exercises 9–42, write the partial fraction decomposition of each rational expression. x^2/(x − 1)² (x + 1)
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Step 1: Identify the form of the partial fraction decomposition.
Step 2: Set up the partial fraction decomposition.
Step 3: Multiply through by the common denominator to clear the fractions.
Step 4: Expand and collect like terms on both sides of the equation.
Step 5: Equate the coefficients of corresponding powers of x to form a system of equations.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for performing operations such as addition, subtraction, multiplication, and division, as well as for decomposing them into simpler components, which is the focus of this question.
Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful for integrating rational functions and solving equations, as it breaks down complex expressions into more manageable parts, allowing for easier manipulation and analysis.
Polynomial factorization involves expressing a polynomial as a product of its factors. In the context of partial fraction decomposition, recognizing the factors of the denominator is essential, as it determines the form of the simpler fractions into which the original expression will be decomposed.