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Chapter 5: Problem 47

Explain how to find the partial fraction decomposition of a rational expression with a repeated linear factor in the denominator.

Short Answer

Expert verified
The main steps to finding the partial fraction decomposition of a rational expression with a repeated linear factor in the denominator are: Define the general form of the decomposition, solve for the coefficients, and express the final decomposition. The complexity will depend on the degree of the repeated factor and the nature of the polynomial in the numerator.

Step by step solution

01

Identify The Form of The Rational Expression

First, observe the form of the rational function given. If the denominator has a repeated linear factor, then it should appear as \( \frac{P(x)}{(ax+b)^n} \), where \( P(x) \) is a polynomial of degree less than \( n \), \( ax+b \) is the repeated linear factor, and \( n \) is the number of times the factor is repeated.
02

Formulate the General Form of Decomposition

Once the form has been identified, formulate the general form of the decomposition. This general form should include several fractions in addition, each with a constant in the numerator and the factor in the denominator raised to a power. The powers on the factor in the denominators should descend from \( n \) down to 1. It should look like: \( \frac{A}{ax+b} + \frac{B}{(ax+b)^2} + \frac{C}{(ax+b)^3} + ... + \frac{Z}{(ax+b)^n} \) where \( A, B, C...Z \) are constants to be determined.
03

Solve for Coefficients

Next, equate the original rational expression with the general form derived. Cross multiply to get rid of the denominators and isolate terms to form a system of equations. These equations can then be solved to find the values of \( A, B, C...Z \).
04

Express Final Decomposition

After obtaining the values of \( A, B, C...Z \), substitute these values back into the general form formulated in Step 2. This obtains the partial fraction decomposition of the original rational function. Simplify if necessary.

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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 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 \(43-46,\) let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers. Three times a first number decreased by a second number is 1. The first number increased by twice the second number is \(12 .\) Find the numbers.

In Exercises \(19-30,\) solve each system by the addition method. \(3 x+2 y=14\) \(3 x-2 y=10\)

In Exercises \(1-4,\) determine whether the given ordered pair is a solution of the system. $$ \begin{aligned} &(2,3)\\\ &x+3 y=11\\\ &x-5 y=-13 \end{aligned} $$

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