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Chapter 6: Problem 10

Write the partial fraction decomposition for the expression. $$ \frac{3 x-4}{(x-5)^{2}} $$

Short Answer

Expert verified
The partial fraction decomposition of the given expression is \(\frac{3 x-4}{(x-5)^{2}} = \frac{3}{x-5} + \frac{11}{(x - 5)^2}\).

Step by step solution

01

Identify the Coefficients

Since the denominator is \((x-5)^{2}\), it can be rewritten as two separate terms: (x-5) and (x-5). So, we propose that \(\frac{3 x-4}{(x-5)^{2}} = \frac{A}{x-5} + \frac{B}{(x - 5)^2}\) where A and B are coefficients that need to be found, and can be any real number.
02

Clear the Fractions

Multiply each side by the denominator of the right-hand side, leading to the equation: \(3x - 4 = A(x - 5) + B\). This clears away the fraction, leaving an equation with variables A and B instead.
03

Solve for the Coefficients

This step can be accomplished by solving the system of equations provided by substituting convenient values for x. Choose x = 5, since it nullifies A. Doing so, we arrive with \(B = 3*5 - 4 = 11\). Substitute x = 0 (or any other value that seem fit) to find A using the equation: \(3*x - 4 = A*(x - 5) + B\). So, \(A = \frac{3x - 4 - B}{x - 5} = \frac{3*0 - 4 - 11}{0 - 5} = 3\).
04

Write the final decomposition

Now place the found coefficients back into the proposed partial fractions: \(\frac{3 x-4}{(x-5)^{2}} = \frac{3}{x-5} + \frac{11}{(x - 5)^2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficients
In partial fraction decomposition, coefficients are the values you place in front of the fractions in the decomposition process. Think of them as the pieces you're trying to solve for. They represent the unknowns in your equation when you transform a complex fraction into simpler parts. For example, when breaking down the fraction \( \frac{3x - 4}{(x-5)^2} \), we use placeholders \( A \) and \( B \) as coefficients in the expression \( \frac{A}{x-5} + \frac{B}{(x-5)^2} \). These coefficients are not just random numbers; they need to be calculated to satisfy the equation across all values.
Denominator
The denominator in partial fraction decomposition is crucial because it dictates the structure of the decomposition. In this exercise, the original denominator is \((x-5)^2\), a repeated linear factor. This repetition means we must account for it differently. Instead of just one term, we write it as \( \frac{A}{x-5} + \frac{B}{(x-5)^2} \). Each unique or repeated factor in the denominator guides how we set up the equation with different terms. This structure helps isolate different powers of \( x \) for further simplification and solving.
Solve for the Coefficients
Solving for the coefficients involves finding the values of \( A \) and \( B \) that make the equation true for all \( x \). First, you clear the fraction by multiplying both sides by the original denominator, resulting in the equation \( 3x - 4 = A(x - 5) + B \). A smart trick is to choose values of \( x \) that simplify the computation, like substituting \( x = 5 \) to directly find \( B \). This gives \( B = 11 \). Choosing another value, such as \( x = 0 \), allows you to solve for \( A \), which equals 3. By strategically picking values for \( x \), you easily pinpoint the coefficients, leading to the final decomposition of \( \frac{3}{x-5} + \frac{11}{(x-5)^2} \).

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Most popular questions from this chapter

Women's Height The mean height of American women between the ages of 30 and 39 is 64.5 inches, and the standard deviation is 2.7 inches. Find the probability that a 30 - to 39 -year-old woman chosen at random is (a) between 5 and 6 feet tall. (b) 5 feet 8 inches or taller. (c) 6 feet or taller.

Use the error formulas to find \(n\) such that the error in the approximation of the definite integral is less than 0.0001 using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{1}^{3} \frac{1}{x} d x $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{2} \frac{1}{(x-1)^{2 / 3}} d x $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n .\) (Round your answers to three significant digits.) $$ \int_{0}^{2} \frac{1}{\sqrt{1+x^{3}}} d x, n=4 $$

Present Value Use a program similar to the Simpson's Rule program on page 454 with \(n=8\) to approximate the present value 454 the income \(c(t)\) over \(t_{1}\) years at the given annual interest rate \(r .\) Then use the integration capabilities of a graphing utility to approximate the present value. Compare the results. (Present value is defined in Section \(6.1 .)\) $$ c(t)=6000+200 \sqrt{t}, r=7 \%, t_{1}=4 $$
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