Find the partial fraction decomposition for each rational expression. See Examples 1–4. 5/(3x(2x + 1))

Verified step by step guidance

1. Identify the denominator factors of the rational expression. Here, the denominator is \$3x(2x + 1)\(, which consists of the linear factors \)3x\( and \)2x + 1$.
2. Set up the partial fraction decomposition by expressing the given fraction as a sum of fractions with unknown constants in the numerators over each factor in the denominator. Since both factors are linear, write: \(\frac{5}{3x(2x + 1)} = \frac{A}{3x} + \frac{B}{2x + 1}\), where \(A\) and \(B\) are constants to be determined.
3. Multiply both sides of the equation by the common denominator \$3x(2x + 1)\( to clear the denominators. This gives: \)5 = A(2x + 1) + B(3x)$.
4. Expand the right side: \(5 = A \cdot 2x + A \cdot 1 + B \cdot 3x = 2Ax + A + 3Bx\).
5. Group like terms (terms with \(x\) and constant terms) and equate the coefficients on both sides to form a system of equations: For \(x\) terms, \$0 = 2A + 3B\(; for constants, \)5 = A\(. Solve this system to find \)A\( and \)B$.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a complex rational expression as a sum of simpler fractions. This technique is especially useful for integrating rational functions or solving algebraic equations. It involves breaking down a fraction into components with simpler denominators.

Factorization of Denominators

Understanding how to factor the denominator into linear or irreducible quadratic factors is essential. In this problem, the denominator is already factored as 3x(2x + 1), which guides the form of the partial fractions. Each factor corresponds to a term in the decomposition.

Setting Up and Solving Equations for Coefficients

After expressing the rational expression as a sum of unknown coefficients over each factor, you multiply both sides by the common denominator to clear fractions. Then, equate coefficients of like terms or substitute values for x to solve for the unknown constants.