In Exercises 9–42, write the partial fraction decomposition of each rational expression. x/(x^2 +2x -3)
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<Step 1: Factor the denominator.> The denominator can be factored into .
<Step 2: Set up the partial fraction decomposition.> Since the denominator factors into two distinct linear factors, express the rational expression as .
<Step 3: Clear the fractions by multiplying through by the common denominator.> Multiply both sides by to get .
<Step 4: Expand and collect like terms.> Expand the right side to get , which simplifies to .
<Step 5: Equate coefficients and solve for A and B.> Compare coefficients: and . Solve this system of equations to find the values of and .
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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 often necessary for integration or solving equations.
Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful when integrating rational expressions, as it allows for easier manipulation and integration of each term. The process involves factoring the denominator and expressing the original fraction in terms of its simpler components.
Factoring polynomials is the process of breaking down a polynomial into simpler polynomial factors that, when multiplied together, yield the original polynomial. This is essential in partial fraction decomposition, as the first step involves factoring the denominator of the rational expression to identify the appropriate form for the decomposition.