In Exercises 16–24, write the partial fraction decomposition of each rational expression. (7x^2 - 7x + 23)/(x - 3)(x^2 + 4)
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Identify the form of the partial fraction decomposition. Since the denominator is (x - 3)(x^2 + 4), the decomposition will be of the form .
Multiply both sides of the equation by the common denominator to eliminate the fractions.
Set up the equation: .
Expand the right side of the equation: and .
Combine like terms and equate the coefficients of corresponding powers of from both sides 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 context, the expression (7x^2 - 7x + 23)/(x - 3)(x^2 + 4) is a rational expression that needs to be decomposed into simpler fractions.
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 or simplifying complex expressions. The goal is to break down the given rational expression into fractions whose denominators are the factors of the original denominator, making it easier to work with.
Polynomial factorization involves breaking down a polynomial into its constituent factors, which can be linear or irreducible quadratic expressions. In the given rational expression, the denominator (x - 3)(x^2 + 4) consists of a linear factor and an irreducible quadratic factor. Understanding how to factor polynomials is essential for correctly applying partial fraction decomposition.