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Chapter 5: Problem 53

Write an informal explanation of partial fraction decomposition.

Short Answer

Expert verified
Partial fraction decomposition breaks down a fraction into simpler parts, solving for unknown constants, and reconstructing the fraction.

Step by step solution

01

Definition

Partial fraction decomposition is a method used to break down a complex rational function into simpler fractions, which are easier to integrate or evaluate.
02

Identify the Denominator

First, write down the denominator of the rational function. Factorize it completely. For example, if the denominator is \(x^2 - x - 6\), factor it into \((x-3)(x+2)\).
03

Set Up Partial Fractions

Write the given rational function as a sum of fractions with unknown constants. For instance, for \(\frac{5x + 3}{(x-3)(x+2)}\), set it up as \(\frac{A}{x-3} + \frac{B}{x+2}\).
04

Determine Constants

Multiply both sides by the common denominator to eliminate the fractions. Then, equate the coefficients of corresponding powers of \(x\) on both sides of the equation to form a system of linear equations. Solve this system to find the values of the constants \(A\) and \(B\).
05

Construct the Decomposition

Substitute the values of constants back into the partial fraction form. So if \(A=2\) and \(B=-1\), the decomposition of \(\frac{5x + 3}{(x-3)(x+2)}\) would be \(\frac{2}{x-3} - \frac{1}{x+2}\).

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Key Concepts

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

Rational Functions
Rational functions are expressions formed by dividing two polynomials. They often appear in the form \(\frac{P(x)}{Q(x)}\). Here, \(P(x)\) and \(Q(x)\) are polynomials. One major characteristic of rational functions is that they can have asymptotes and discontinuities where the denominator is zero. Learning to work with these functions involves understanding how to simplify them for easier manipulation, which is crucial when applying techniques like partial fraction decomposition.
Factoring Polynomials
Factoring polynomials is key in simplifying rational functions. By breaking down a polynomial like \(x^2 - x - 6\) into factors such as \((x-3)(x+2)\), we can better manage the terms and apply partial fraction decomposition. Factoring involves finding the roots of the polynomial and expressing it as a product of its factors. Recognizing patterns like the difference of squares ((a^2 - b^2 = (a - b)(a + b))), trinomial patterns, or using methods like synthetic division helps in factorizing polynomials efficiently.
Algebraic Techniques
Algebraic techniques are fundamental when working with partial fractions. After factoring the denominator in a rational function, our next step is to set up an equation with unknown constants. For instance, if you have \(\frac{5x + 3}{(x-3)(x+2)}\), you set it as \(\frac{A}{x-3} + \frac{B}{x+2}\). Multiplying through by the common denominator helps clear fractions and enables solving for these unknowns. It often leads to a system of linear equations, where you need to equate the coefficients of like terms on both sides to find values of $$A$$ and $$B$$.
Integrating Rational Functions
Integrating rational functions becomes much easier after partial fraction decomposition. Once the rational function is expressed as a sum of simpler fractions, integration can be performed term-by-term. For example, integrating \(\frac{5x + 3}{(x-3)(x+2)}\) by breaking it into \(\frac{2}{x-3} - \frac{1}{x+2}\) simplifies the process. Each partial fraction can be integrated using standard integral rules, such as \(\frac{1}{x-a}\)'s integral being \(\text{ln}|x-a| + C\). This step-by-step approach is practical and makes complex integrals manageable.

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Most popular questions from this chapter

A patient undergoing a heart scan is given a sample of fluorine- \(18\left({ }^{18} \mathrm{~F}\right)\). After \(4 \mathrm{hr}\), the radioactivity level in the patient is \(44.1 \mathrm{MBq}\) (megabecquerel). After \(5 \mathrm{hr}\), the radioactivity level drops to \(30.2 \mathrm{MBq}\). The radioactivity level \(Q(t)\) can be approximated by \(Q(t)=Q_{0} e^{-k t},\) where \(t\) is the time in hours after the initial dose \(Q_{0}\) is administered. a. Determine the value of \(k\). Round to 4 decimal places. b. Determine the initial dose, \(Q_{0}\). Round to the nearest whole unit. c. Determine the radioactivity level after \(12 \mathrm{hr}\). Round to 1 decimal place.

Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples \(5-6\) ) $$ \begin{array}{l} 2 x+4=4-5 y \\ 2+4(x+y)=7 y+2 \end{array} $$

A farmer has 1200 acres of land and plans to plant corn and soybeans. The input cost (cost of seed, fertilizer, herbicide, and insecticide) for 1 acre for each crop is given in the table along with the cost of machinery and labor. The profit for 1 acre of each crop is given in the last column. $$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Input Cost } \\ \text { per Acre } \end{array} & \begin{array}{c} \text { Labor/Machinery } \\ \text { Cost per Acre } \end{array} & \begin{array}{c} \text { Profit } \\ \text { per Acre } \end{array} \\ \hline \text { Corn } & \$ 180 & \$ 80 & \$ 120 \\ \hline \text { Soybeans } & \$ 120 & \$ 100 & \$ 100 \\ \hline \end{array} $$ Suppose the farmer has budgeted a maximum of $$\$ 198,000$$ for input costs and a maximum of $$\$ 110,000$$ for labor and machinery. a. Determine the number of acres of each crop that the farmer should plant to maximize profit. (Assume that all crops will be sold.) b. What is the maximum profit? c. If the profit per acre were reversed between the two crops (that is, $$\$ 100$$ per acre for corn and $$\$ 120$$ per acre for soybeans), how many acres of each crop should be planted to maximize profit?

Solve the system using any method. $$ \begin{array}{l} 4(x-2)=6 y+3 \\ \frac{1}{4} x-\frac{3}{8} y=-\frac{1}{2} \end{array} $$

a. Sketch the lines defined by \(y=x+2\) and \(y=-\frac{1}{2} x+2\) b. Find the area of the triangle bounded by the lines in part (a) and the \(x\) -axis.
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