In Exercises 9–42, write the partial fraction decomposition of each rational expression. (x^2-6x+3)/(x − 2)³
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Identify the form of the partial fraction decomposition. Since the denominator is , the decomposition will have terms of the form .
Set up the equation: .
Multiply through by to clear the denominators: .
Expand the right-hand side: .
Combine like terms and equate coefficients with the left-hand side to solve for , , 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 like addition, subtraction, and decomposition. In this case, the expression (x^2 - 6x + 3)/(x - 2)³ is a rational expression that requires decomposition into simpler fractions.
Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions. This technique is particularly useful for integrating rational functions or simplifying complex expressions. The goal is to break down the given rational expression into components that are easier to work with, especially when the denominator has repeated factors, as seen in (x - 2)³.
Polynomial long division is a process used to divide one polynomial by another, similar to numerical long division. It is essential when the degree of the numerator is greater than or equal to the degree of the denominator. In the context of the given expression, if necessary, polynomial long division can simplify the rational expression before applying partial fraction decomposition.