Find the partial fraction decomposition for each rational expression. See Examples 1–4. 1/(x(2x + 1)(3x² + 4))

Verified step by step guidance

1. Identify the denominator factors of the rational expression \(\frac{1}{x(2x + 1)(3x^2 + 4)}\). Here, the factors are \(x\), \$2x + 1\(, and \)3x^2 + 4\(. Note that \)x\( and \)2x + 1\( are linear factors, while \)3x^2 + 4$ is an irreducible quadratic factor.
2. Set up the partial fraction decomposition form. For each linear factor, assign a constant numerator, and for the irreducible quadratic factor, assign a linear numerator. So, write the decomposition as: \(\frac{1}{x(2x + 1)(3x^2 + 4)} = \frac{A}{x} + \frac{B}{2x + 1} + \frac{Cx + D}{3x^2 + 4}\), where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.
3. Multiply both sides of the equation by the common denominator \(x(2x + 1)(3x^2 + 4)\) to clear the denominators. This gives: \$1 = A(2x + 1)(3x^2 + 4) + B x (3x^2 + 4) + (Cx + D) x (2x + 1)$.
4. Expand the right-hand side by distributing each term carefully. This involves multiplying polynomials and combining like terms to express the right side as a polynomial in powers of \(x\).
5. Equate the coefficients of corresponding powers of \(x\) on both sides of the equation. This will give a system of linear equations in terms of \(A\), \(B\), \(C\), and \(D\). Solve this system to find the values of these constants, completing the partial fraction decomposition.

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 the denominator into factors and assigning unknown constants to each fraction.

Types of Factors in the Denominator

Understanding the types of factors in the denominator is crucial. Linear factors (like x or 2x + 1) correspond to terms with constants in the numerator, while irreducible quadratic factors (like 3x^2 + 4) require linear expressions in the numerator. Recognizing these helps set up the correct form for the decomposition.

Solving for Unknown Coefficients

After setting up the partial fractions, the next step is to find the unknown coefficients by multiplying both sides by the common denominator and equating coefficients of like terms. This often involves substituting convenient values of x or comparing coefficients to form a system of equations.