Question

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (2x2 -18x -12)/x³- 4x

Accepted Answer

First, factor the denominator x3-4x. Notice that x is a common factor, so factor it out: x(x2-4). Then recognize that x2-4 is a difference of squares, which factors as (x-2)(x+2). So the fully factored denominator is x(x-2)(x+2). Next, set up the partial fraction decomposition. Since the denominator factors into three distinct linear factors, the decomposition will be of the form: \frac{$$2x^{2} - 18x - 12$$}{x(x-2)(x+2)} = \frac{A}{x} + \frac{B}{x-2} + \frac{C}{x+2}, where A, B, and C are constants to be determined. Multiply both sides of the equation by the common denominator x(x-2)(x+2) to clear the fractions. This gives: $$2x^{2} - 18x - 12 = A(x-2)(x+2) + B x (x+2) + C x (x-2). $$Expand the right-hand side by multiplying out each term: $$A(x^{2} - 4) + B x (x+2) + C x (x-2). $$Then further expand to get: A $$x^{2} - 4A + B$$ $$x^{2} + 2 B x + C$$ $$x^{2} - 2 C x. $$Group like terms on the right-hand side: (A + B + C) $$x^{2} + (2 B - 2 C) x - 4 A. $$Now, equate the coefficients of corresponding powers of x from both sides to form a system of equations: 2 = A + B + C (coefficient of $$x^{2}$$), -18 = 2 B - 2 C (coefficient of x), and -12 = -4 A (constant term). These equations can be solved to find the values of A, B, and C.

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (2x² -18x -12)/x³- 4x

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