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Chapter 7: Problem 13

Writing the Form of the Decomposition. Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{4 x^{2}+3}{(x-5)^{3}}$$

Short Answer

Expert verified
The partial fraction decomposition of the rational expression \(\frac{4 x^{2}+3}{(x-5)^{3}}\) is \(\frac{A}{(x-5)^{3}} + \frac{B}{(x-5)^{2}} + \frac{C}{(x-5)}\). Where \(A\), \(B\), and \(C\) are constants.

Step by step solution

01

Identify the Form of the Denominator

The denominator in the expression is \((x-5)^{3}\). As the power of \((x-5)\) is 3, there will be three fractions in the decomposition. As the power is more than 1, we need to write down decreasing powers of \((x-5)\) starting from 3.
02

Formulate Partial Fractions

There should be as many fractions as the highest power of the denominator. Hence, start with the highest power of 3 and decreasing until 1, each respective fraction should look like these: \(\frac{A}{(x-5)^{3}}\), where \(A\) is the constant for the first fraction. \(\frac{B}{(x-5)^{2}}\), where \(B\) is the constant for the second fraction. \(\frac{C}{(x-5)}\), where \(C\) is the constant for the third fraction.
03

Write the Completed Partial Fraction Decomposition

Now, combine all the fractions from Step 2 to show the complete partial fraction decomposition. Therefore, it becomes: \(\frac{4 x^{2}+3}{(x-5)^{3}} =\frac{A}{(x-5)^{3}} + \frac{B}{(x-5)^{2}} + \frac{C}{(x-5)}\)

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Key Concepts

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

Rational Expressions
Rational expressions are expressions that involve a ratio or fraction of two polynomials. These are in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).
The numerator and the denominator are polynomials. In rational expressions, it's crucial the denominator is not zero since division by zero is undefined.
These expressions can be simplified or decomposed into simpler partial fractions, which help in integration and other operations in calculus.

When dealing with rational expressions, especially in the context of partial fraction decomposition, the focus is often on the polynomial in the denominator.
  • It is necessary to assess the degree of the polynomial.
  • This degree determines the method used for decomposition.
Recognizing the form and degree of these expressions forms the basis of solving many algebraic problems.
Powers of Polynomials
Polynomials, such as \( (x-5)^3 \), involve variables raised to powers. The power or exponent signifies how many times the base, in this case \( x-5 \), is multiplied by itself.
For example, \( (x-5)^3 \) is equivalent to \((x-5) \cdot (x-5) \cdot (x-5)\).

Higher powers indicate more complex behavior in graphs and equations, leading to a more layered approach in decomposition. The process involves:
  • Breaking down the powers sequentially in decreasing order.
  • Starting with the highest power in the denominator.
In the given problem, the polynomial \((x-5)^3\) suggests a specific structure for partial fraction decomposition. Each fraction in the decomposition corresponds to one of these decreasing powers, ensuring that each component is accounted for.
Constants in Partial Fractions
When performing partial fraction decomposition, the constants \( A \), \( B \), and \( C \) are essential parts of the formula. They represent unknown values that are effectively placeholders in the expression.

These constants ensure each term of the original rational expression is represented correctly.
  • Each constant corresponds to a fraction whose denominator is a power of the original polynomial.
  • They are crucial for reconstruction of the original expression when combined.
The form \( \frac{A}{(x-5)^3} + \frac{B}{(x-5)^2} + \frac{C}{(x-5)} \) reflects a strategic placement of these constants to cater for various powers of \( x-5 \). Solving for these constants is the next step after setting up the decomposition, allowing us to equate and solve through substitution or comparison.

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Most popular questions from this chapter

Pulley System A system of pulleys is loaded with 128 -pound and 32 -pound weights (see figure). The tensions \(t _ { 1 }\) and \(t _ { 2 }\) in the ropes and the acceleration \(a\) of the 32 -pound weight are found by solving the system of equations $$\left\\{ \begin{aligned} t _ { 1 } - 2 t _ { 2 } & = 0 \\ t _ { 1 } & \- 2 a = 128 \\ t _ { 2 } + a & = 32 \end{aligned} \right.$$ where \(t _ { 1 }\) and \(t _ { 2 }\) are in pounds and \(a\) is in feet per second squared. Solve this system.

Optimal Profit A manufacturer produces two models of elliptical cross-training exercise machines. The times for assembling, finishing, and packaging model \(\mathrm{X}\) are 3 hours, 3 hours, and 0.8 hour, respectively. The times for model \(Y\) are 4 hours, 2.5 hours, and 0.4 hour. The total times available for assembling, finishing, and packaging are 6000 hours, 4200 hours, and 950 hours, respectively. The profits per unit are \(\$ 300\) for model \(X\) and \(\$ 375\) for model \(Y .\) What is the optimal production level for each model? What is the optimal profit?

Data Analysis A store manager wants to know the demand for a product as a function of the price. The table shows the daily sales \(y\) for different prices \(x\) of the product. $$ \begin{array}{|c|c|}\hline \text { Price, } & {\text { Demand }, y} \\ \hline \$ 1.00 & {45} \\ \hline \$ 1.20 & {37} \\ \hline \$ 1.50 & {23} \\\ \hline\end{array} $$ (a) Find the least squares regression line \(y=a x+b\) for the data by solving the system for \(a\) and \(b .\) $$ \left\\{\begin{array}{l}{3.00 b+3.70 a=105.00} \\ {3.70 b+4.69 a=123.90}\end{array}\right. $$ (b) Use a graphing utility to confirm the result of part (a). (c) Use the linear model from part (a) to predict the demand when the price is \(\$ 1.75 .\)

If a linear programming problem has a solution, then it must occur at a _____ of the set of feasible solutions.

Optimal Revenue An accounting firm has 780 hours of staff time and 272 hours of reviewing time available each week. The firm charges \(\$ 1600\) for an audit and \(\$ 250\) for a tax return. Each audit requires 60 hours of staff time and 16 hours of review time. Each tax return requires 10 hours of staff time and 4 hours of review time. What numbers of audits and tax returns will yield an optimal revenue? What is the optimal revenue?
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