Find the partial fraction decomposition for each rational expression. See Examples 1–4. 5/(3x(2x + 1))
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1
Identify the form of the partial fraction decomposition. Since the denominator is a product of distinct linear factors, and , the decomposition will be of the form .
Set up the equation: .
Multiply through by the common denominator to clear the fractions: .
Expand and simplify the right side: .
Equate the coefficients of like terms from both sides of the equation to form a system of equations: 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 5/(3x(2x + 1)) is a rational expression that needs to be decomposed 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 expressions or simplifying complex algebraic fractions. The goal is to break down the given expression into components that are easier to work with, based on the factors of the denominator.
Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential in partial fraction decomposition, as the form of the denominator determines how the expression can be split. In the given expression, recognizing that the denominator 3x(2x + 1) can be factored helps in setting up the correct form for the partial fractions.