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Chapter 5: Problem 13

Write the partial fraction decomposition of each rational expression. $$\frac{7 x-4}{x^{2}-x-12}$$

Short Answer

Expert verified
The partial fraction decomposition of the given expression is: \(\frac{1}{x-4} + \frac{2}{x+3}\).

Step by step solution

01

Factor the denominator

Firstly, factor the denominator of the fraction, which is \(x^{2}-x-12\). Upon factoring, we get \((x-4)(x+3)\).So, we can write the given fraction as: \(\frac{7x-4}{(x-4)(x+3)}\).\n
02

Set up the partial fractions

A general pattern that can be used to break down the fraction is based on the factored denominator. We would assume the fraction can be broken down as: \(\frac{7x-4}{(x-4)(x+3)} = \frac{A}{x-4} + \frac{B}{x+3}\), where \(A\) and \(B\) are constants that we need to solve for.\n
03

Equate coefficients

To determine the values of \(A\) and \(B\), multiply both sides of the equation by \((x - 4)(x + 3)\) to clear the fractions: \(7x - 4 = A(x+3)+ B(x-4)\).\n
04

Solve for A and B

This is a linear equation system. By choosing convenient values of \(x\), we can solve for \(A\) and \(B\). \nChoose \(x = -3\) to solve for \(A\): \(7(-3) - 4 = A(-3+3) + 3B \Longrightarrow A = 1 \n Now choose \(x = 4\) to solve for \(B\), \(7(4) - 4 = 4A + B(4+3) \Longrightarrow B = 2\). \n

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Most popular questions from this chapter

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. \(2 x+5 y=-4\) \(3 x-y=11\)

In Exercises \(19-30,\) solve each system by the addition method. \(x+y=6\) \(x-y=-2\)

A person with no more than 15,000 dollars to invest plans to place the money in two investments. One investment is high risk, high yield; the other is low risk, low yield. At least 2000 dollars is to be placed in the high-risk investment. Furthermore, the amount invested at low risk should be at least three times the amount invested at high risk. Find and graph a system of inequalities that describes all possibilities for placing the money in the high- and low-risk investments.

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 small canoes has a fixed cost of \(\$ 18,000 .\) It costs \(\$ 20\) to produce each canoe. The selling price is \(\$ 80\) per canoe. (In solving this exercise, let \(x\) represent the number of canoes produced and sold.)

In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &y=\frac{1}{3} x+\frac{2}{3}\\\ &y=\frac{5}{7} x-2 \end{aligned} $$
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