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Chapter 5: Problem 50

How can you verify your result for the partial fraction decomposition for a given rational expression without using a graphing utility?

Short Answer

Expert verified
To verify your result for the partial fraction decomposition, perform the decomposition first, then add the resulting fractions back together. If the original rational expression is regained, it confirms that the decomposition was done correctly.

Step by step solution

01

Understanding Partial Fraction Decomposition

Partial fraction decomposition involves expressing a given rational expression as a sum of simpler fractions. It is typically done when we are trying to integrate a complicated rational function or when we are solving certain types of differential equations.
02

Perform the Partial Fraction Decomposition

Carry out the partial fraction decomposition. This could involve factoring the denominator, separating the rational function into a sum of simpler fractions, and then equating coefficients and solving for constants.
03

Recombine the Fractions

After finding the constants, recombine the simpler fractions to form a rational expression.
04

Compare with Original Rational Expression

If the recombined rational expression is identical to the original expression, then it verifies the partial fraction decomposition is correct. This method does not require a graphing utility and purely relies on algebraic operations.

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Most popular questions from this chapter

If \(a, b,\) and \(c\) are constants, find the partial fraction decomposition of $$\frac{a x+b}{(x-c)^{2}}$$

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. You invest in a new play. The cost includes an overhead of \(\$ 30,000,\) plus production costs of \(\$ 2500\) per performance. A sold-out performance brings in \(\$ 3125\). (In solving this exercise, let \(x\) represent the number of sold-out performances.)

Use a system of linear equations to solve Exercises \(57-67\) The graph shows the calories in some favorite fast foods. Use the information in Exercises \(57-58\) to find the exact caloric content of the specified foods. (GRAPH CAN'T COPY) A rectangular lot whose perimeter is 360 feet is fenced along three sides. An expensive fencing along the lot's length costs \(\$ 20\) per foot, and an inexpensive fencing along the two side widths costs only \(\$ 8\) per foot. The total cost of the fencing along the three sides comes to \(\$ 3280-\) What are the lot's dimensions?

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.)

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