Question

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

Accepted Answer

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 $$9x^2 - 4$ is a difference of squares, so factor it as (3x - 2)(3x + 2$$)$$. Thus, the full factorization of the denominator is x(3x - 2)(3x + 2$$)$$. Set up the partial fraction decomposition using the factors from the denominator. Since all factors are linear and distinct, write the decomposition as: $$\frac{A}{x} + \frac{B}{3x - 2} + \frac{C}{3x + 2}$$, where A$, B$, and C$ are constants to be determined. Multiply both sides of the equation by the original denominator x(3x - 2)(3x + 2$$)$$ to clear the fractions. This gives: x^3$ + 4 = A(3x - 2)(3x + 2) + B x (3x + 2) + C x (3x - 2). $$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). 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.

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

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Key Concepts

Partial Fraction Decomposition

Factoring Polynomials

Setting Up and Solving Systems of Equations