Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x³ + 4)/(9x³ - 4x)

Verified step by step guidance

1. First, factor the denominator completely. The denominator is \$9x^3 - 4x\(. Factor out the common factor \)x\( to get \)x(9x^2 - 4)\(. Recognize that \)9x^2 - 4\( is a difference of squares, so factor it as \)(3x - 2)(3x + 2)\(. Thus, the full factorization of the denominator is \)x(3x - 2)(3x + 2)$.
2. Set up the partial fraction decomposition using the factors from the denominator. Since all factors are linear and distinct, write the decomposition as: \(\frac{A}{x} + \frac{B}{3x - 2} + \frac{C}{3x + 2}\), where \(A\), \(B\), and \(C\) are constants to be determined.
3. Multiply both sides of the equation by the original denominator \(x(3x - 2)(3x + 2)\) to clear the fractions. This gives: \(x^3 + 4 = A(3x - 2)(3x + 2) + B x (3x + 2) + C x (3x - 2)\).
4. Expand the right-hand side by multiplying out each term carefully. For example, expand \((3x - 2)(3x + 2)\) using the difference of squares formula, then distribute \(A\), \(B x\), and \(C x\) accordingly. After expansion, collect like terms of powers of \(x\) (i.e., \(x^3\), \(x^2\), \(x\), and constants).
5. Equate the coefficients of corresponding powers of \(x\) from both sides of the equation to form a system of linear equations in \(A\), \(B\), and \(C\). Solve this system to find the values of \(A\), \(B\), and \(C\), which completes 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 writing the original fraction as a sum of fractions with those factors as denominators.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors, which can be linear or quadratic expressions. For partial fraction decomposition, factoring the denominator completely is essential to identify the terms for the decomposition. Common factoring techniques include factoring out the greatest common factor, difference of squares, and factoring by grouping.

Setting Up and Solving Systems of Equations

After expressing the rational expression as a sum of partial fractions with unknown coefficients, you multiply both sides by the common denominator to clear fractions. This results in a polynomial equation where coefficients of corresponding powers of x must be equal. Equating these coefficients leads to a system of linear equations, which you solve to find the unknown constants.