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Chapter 5: Problem 20

Write the partial fraction decomposition of each rational expression. $$\frac{2 x^{2}-18 x-12}{x^{3}-4 x}$$

Short Answer

Expert verified
The partial fraction decomposition of \(\frac{2x^{2} - 18x - 12}{x(x - 2)(x + 2)}\) will be the sum of fractions of the form \(\frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 2}\), where A, B, and C are the coefficients obtained after solving the system of equations from step 3.

Step by step solution

01

Factorisation of the denominator

The denominator of the given fraction can be factored as follows:\n\(x^{3} - 4x = x(x^{2} - 4) = x(x - 2)(x + 2)\). Hence, the given fraction can be written as: \(\frac{2x^{2} - 18x - 12}{x(x - 2)(x + 2)}\).
02

Setup of the partial fraction decomposition

Set up the partial fraction decomposition as the sum of fractions. Each fraction must have a numerator of one degree lower than the degree of its denominator and the denominators are the factors from the previous step: \(\frac{2x^{2} - 18x - 12}{x(x - 2)(x + 2)} = \frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 2}\)
03

Determination of coefficients

Multiply through by the denominator of the left-hand side to clear the fractions and equate coefficients: \(2x^{2} - 18x - 12 = Ax(x - 2)(x + 2) + Bx(x + 2) + Cx(x - 2)\). Expand this equation and group like terms. By comparing the coefficients of similar terms on both sides, a system of linear equations is obtained. Solving this system yields values for A, B, and C.
04

Writing the final decomposition

The final partial fraction decomposition is then written by substituting the coefficients obtained from step 3 into the setup from step 2.

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Most popular questions from this chapter

In Exercises \(19-30,\) solve each system by the addition method. \(4 x+3 y=15\) \(2 x-5 y=1\)

Solve the system for \(x\) and \(y\) in terms of \(a_{1}, b_{1}, c_{1}, a_{2}, b_{2}\) and \(c_{2}\) $$ \begin{array}{l} a_{1} x+b_{1} y=c_{1} \\ a_{2} x+b_{2} y=c_{2} \end{array} $$

Find the partial fraction decomposition for \(\frac{1}{x(x+1)}\) and use the result to find the following sum: $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\dots+\frac{1}{99 \cdot 100}$$

What is a system of linear equations? Provide an example with your description.

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(x=9-2 y\) \(x+2 y=13\)
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