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

Verified step by step guidance

1. Identify the form of the partial fraction decomposition. Since the denominator is \((x + 4)(3x^2 + 1)\), where \(x + 4\) is a linear factor and \(3x^2 + 1\) is an irreducible quadratic factor, the decomposition will be of the form: \[ \frac{3x - 2}{(x + 4)(3x^2 + 1)} = \frac{A}{x + 4} + \frac{Bx + C}{3x^2 + 1} \] where \(A\), \(B\), and \(C\) are constants to be determined.
2. Multiply both sides of the equation by the common denominator \((x + 4)(3x^2 + 1)\) to clear the fractions: \[ 3x - 2 = A(3x^2 + 1) + (Bx + C)(x + 4) \] This step eliminates the denominators and allows us to work with polynomials.
3. Expand the right-hand side by distributing: \[ A(3x^2 + 1) = 3A x^2 + A \] and \[ (Bx + C)(x + 4) = Bx^2 + 4Bx + Cx + 4C \]. Combine like terms to write the right side as a polynomial in standard form: \[ (3A + B) x^2 + (4B + C) x + (A + 4C) \].
4. Set up a system of equations by equating the coefficients of corresponding powers of \(x\) from both sides. On the left, the polynomial is \(3x - 2\), which can be written as \(0x^2 + 3x - 2\). So, equate coefficients: \[ \text{Coefficient of } x^2: 0 = 3A + B \] \[ \text{Coefficient of } x: 3 = 4B + C \] \[ \text{Constant term}: -2 = A + 4C \].
5. Solve the system of equations for \(A\), \(B\), and \(C\). Once these constants are found, substitute them back into the partial fraction form \( \frac{A}{x + 4} + \frac{Bx + C}{3x^2 + 1} \) to complete the 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 equations. It involves breaking down the denominator into factors and assigning unknown constants to each fraction.

Factoring and Types of Denominator Factors

Understanding the factorization of the denominator is crucial. Denominators can have linear factors (like x + 4) or irreducible quadratic factors (like 3x^2 + 1). Each type requires a different form in the decomposition: linear factors correspond to constants in the numerator, while quadratic factors require linear expressions.

Setting Up and Solving Systems of Equations

After expressing the rational function as a sum of partial fractions with unknown coefficients, you multiply both sides by the denominator to clear fractions. Then, equate coefficients of corresponding powers of x to form a system of linear equations. Solving this system yields the values of the unknown constants.