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Chapter 5: Problem 55

Find the partial fraction decomposition of $$\frac{4 x^{2}+5 x-9}{x^{3}-6 x-9}$$

Short Answer

Expert verified
The partial fraction decomposition of \( \frac{4 x^{2}+5 x-9}{x^{3}-6 x-9} \) can be written in the form \( \frac{A}{x} + \frac{B}{x-3} + \frac{C}{x+3} \). The coefficients A, B, and C can be found by solving equations obtained from equating coefficients of similar terms.

Step by step solution

01

Factor the Denominator

The first step in performing a partial fraction decomposition is to factor the denominator if possible. However, the denominator of the given function, \(x^{3}-6 x-9\), cannot be factorised in simple manner. So, we will leave it as it is.
02

Set Up Partial Fractions

Express the given fraction as sum of simpler fractions, where each fraction has denominator equal to a factor of the original denominator. In this case, we will consider \(x^{3}-6 x-9\) as three linear factors, even though they cannot be explicitly written, and write the fraction as: \[ \frac{4 x^{2}+5 x-9}{x^{3}-6 x-9} = \frac{A}{x} + \frac{B}{x-3} + \frac{C}{x+3}\]
03

Cross-Multiply And Expand

Cross multiply to remove the denominator and then expand the expression, this gives: \[4 x^{2}+5 x-9 = A(x-3)(x+3) + B*x*(x+3) + C*x*(x-3)\]
04

Group Similar Terms

Group similar terms together. This step will give an equation formed by collecting coefficients of similar terms.
05

Solve for Coefficients A, B, and C

Solve the coefficients A, B, and C from the equation obtained above. In this case, coefficients can be found by substitution method by putting suitable values of x. For example, \(x = 0, x = 3, x = -3\).

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Most popular questions from this chapter

Exercises \(47-50\) describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\) c. Determine the break-even point. Describe what this means. A company that manufactures bicycles has a fixed cost of \(\$ 100,000 .\) It costs \(\$ 100\) to produce each bicycle. The selling price is \(\$ 300\) per bike. (In solving this exercise, let \(x\) represent the number of bicycles produced and sold.)

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