Introduction. x 7. Partial fraction decomposition x 7 of. x 7. x 3 1. x 2. Decomposition of N x /D x into Partial Fractions. N x. D x polynomial N 1 x
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1 333202_0704.qd 2/5/05 9:43 M Page 533 Section 7.4 Partial Fractions Partial Fractions What you should learn Recognize partial fraction decompositions of rational epressions. Find partial fraction decompositions of rational epressions. Why you should learn it Partial fractions can help you analyze the behavior of a rational function. For instance, in Eercise 57 on page 540, you can analyze the ehaust temperatures of a diesel engine using partial fractions. Michael Rosenfeld/Getty Images Section.4, shows you how to combine epressions such as 2 3 The method of partial fractions shows you how to reverse this process ? 2? 3 Introduction In this section, you will learn to write a rational epression as the sum of two or more simpler rational epressions. For eample, the rational epression can be written as the sum of two fractions with first-degree denominators. That is, Partial fraction decomposition 7 of Partial fraction Partial fraction Each fraction on the right side of the equation is a partial fraction, and together they make up the partial fraction decomposition of the left side. Decomposition of N/D into Partial Fractions. Divide if improper: If ND is an improper fraction degree of N degree of D, divide the denominator into the numerator to obtain N D polynomial N D and apply Steps 2, 3, and 4 below to the proper rational epression N D. Note that N is the remainder from the division of N by D. 2. Factor the denominator: Completely factor the denominator into factors of the form p q m and a 2 b c n where a 2 b c is irreducible. 3. Linear factors: For each factor of the form p q m, the partial fraction decomposition must include the following sum of m fractions. p q 2 p q... m 2 p q m 4. Quadratic factors: For each factor of the form a 2 b c n, the partial fraction decomposition must include the following sum of n fractions. B C a 2 b c B 2 C 2 a 2 b c... B n C n 2 a 2 b c n
2 333202_0704.qd 2/5/05 9:43 M Page Chapter 7 Systems of Equations and Inequalities Partial Fraction Decomposition lgebraic techniques for determining the constants in the numerators of partial fractions are demonstrated in the eamples that follow. Note that the techniques vary slightly, depending on the type of factors of the denominator: linear or quadratic, distinct or repeated. Eample Distinct Linear Factors You can use a graphing utility to check graphically the decomposition found in Eample. To do this, graph and in the same viewing window. The graphs should be identical, as shown below. 9 Technology y y Write the partial fraction decomposition of 2 6. Solution The epression is proper, so be sure to factor the denominator. Because , you should include one partial fraction with a constant numerator for each linear factor of the denominator. Write the form of the decomposition as follows Write form of decomposition. Multiplying each side of this equation by the least common denominator, 3 2, leads to the basic equation 7 2 B 3. Because this equation is true for all, you can substitute any convenient values of that will help determine the constants and B. Values of that are especially convenient are ones that make the factors 2 and 3 equal to zero. For instance, let 2. Then B B5 5 5B B. To solve for, let 3 and obtain B B B 2 So, the partial fraction decomposition is Substitute 2 for. Substitute 3 for. Check this result by combining the two partial fractions on the right side of the equation, or by using your graphing utility. Now try Eercise 5.
3 333202_0704.qd 2/5/05 9:43 M Page 535 Section 7.4 Partial Fractions 535 The net eample shows how to find the partial fraction decomposition of a rational epression whose denominator has a repeated linear factor. Eample 2 Repeated Linear Factors Have students check some of their answers to Eercises 5 52 or the answers to some of the eamples by combining the partial fractions. Write the partial fraction decomposition of Solution This rational epression is improper, so you should begin by dividing the numerator by the denominator to obtain Because the denominator of the remainder factors as you should include one partial fraction with a constant numerator for each power of and and write the form of the decomposition as follows B C 2 Multiplying by the LCD, 2, leads to the basic equation B C. Letting eliminates the - and B-terms and yields C Letting 0 eliminates the B- and C-terms and yields B00 C Write form of decomposition B C t this point, you have ehausted the most convenient choices for, so to find the value of B, use any other value for along with the known values of and C. So, using, 6, and C 9, B B 9 2 2B B. C 9. So, the partial fraction decomposition is Now try Eercise
4 333202_0704.qd 2/5/05 9:43 M Page Chapter 7 Systems of Equations and Inequalities The procedure used to solve for the constants in Eamples and 2 works well when the factors of the denominator are linear. However, when the denominator contains irreducible quadratic factors, you should use a different procedure, which involves writing the right side of the basic equation in polynomial form and equating the coefficients of like terms. Then you can use a system of equations to solve for the coefficients. Eample 3 Distinct Linear and Quadratic Factors The Granger Collection Historical Note John Bernoulli ( ), a Swiss mathematician, introduced the method of partial fractions and was instrumental in the early development of calculus. Bernoulli was a professor at the University of Basel and taught many outstanding students, the most famous of whom was Leonhard Euler. Write the partial fraction decomposition of Solution This epression is proper, so factor the denominator. Because the denominator factors as you should include one partial fraction with a constant numerator and one partial fraction with a linear numerator and write the form of the decomposition as follows B C 2 4 Multiplying by the LCD, 2 4, yields the basic equation B C. Write form of decomposition. Epanding this basic equation and collecting like terms produces B 2 C B 2 C 4. Polynomial form Finally, because two polynomials are equal if and only if the coefficients of like terms are equal, you can equate the coefficients of like terms on opposite sides of the equation B 2 C 4 Equate coefficients of like terms. You can now write the following system of linear equations. Equation Equation 2 B 3 C Equation 3 From this system you can see that and C 4. Moreover, substituting into Equation yields B 3 B 2. So, the partial fraction decomposition is Now try Eercise 29.
5 333202_0704.qd 2/5/05 9:43 M Page 537 Section 7.4 Partial Fractions 537 The net eample shows how to find the partial fraction decomposition of a rational epression whose denominator has a repeated quadratic factor. dditional Eamples Partial fraction decompositions can be confusing to students. You may want to go over several additional eamples. 2 a b c Eample 4 Repeated Quadratic Factors Write the partial fraction decomposition of Solution You need to include one partial fraction with a linear numerator for each power of B C D Write form of decomposition. Multiplying by the LCD, 2 2 2, yields the basic equation B 2 2 C D B 2 2B C D 3 B 2 2 C 2B D. Equating coefficients of like terms on opposite sides of the equation Polynomial form B 2 2 C 2B D produces the following system of linear equations. B 2 C 2B Finally, use the values 8 and B 0 to obtain the following. 28 C 3 C 3 D Equation Equation 2 Equation 3 Equation 4 Substitute 8 for in Equation 3. Check students understanding. Why do you use two different procedures? When is each procedure more effective? Is it mathematically correct (although not necessarily as efficient) to use either procedure regardless of whether the factors of the denominator are linear or quadratic? 20 D 0 D 0 Substitute 0 for B in Equation 4. So, using 8, B 0, C 3, and D 0, the partial fraction decomposition is Check this result by combining the two partial fractions on the right side of the equation, or by using your graphing utility. Now try Eercise 49.
6 333202_0704.qd 2/5/05 9:43 M Page Chapter 7 Systems of Equations and Inequalities Guidelines for Solving the Basic Equation Linear Factors. Substitute the zeros of the distinct linear factors into the basic equation. 2. For repeated linear factors, use the coefficients determined in Step to rewrite the basic equation. Then substitute other convenient values of and solve for the remaining coefficients. Quadratic Factors. Epand the basic equation. 2. Collect terms according to powers of. 3. Equate the coefficients of like terms to obtain equations involving, B, C, and so on. 4. Use a system of linear equations to solve for, B, C,.... ctivities. Write the partial fraction decomposition of nswer: Write the partial fraction decomposition of nswer: Keep in mind that for improper rational epressions such as N D you must first divide before applying partial fraction decomposition. W RITING BOUT MTHEMTICS Error nalysis You are tutoring a student in algebra. In trying to find a partial fraction decomposition, the student writes the following. 2 B 2 B 2 B By substituting 0 and into the basic equation, the student concludes that and B 2. However, in checking this solution, the student obtains the following. 2 2 What has gone wrong? 2
7 333202_0704.qd 2/5/05 9:43 M Page 539 Section 7.4 Partial Fractions Eercises VOCBULRY CHECK: Fill in the blanks.. The process of writing a rational epression as the sum or difference of two or more simpler rational epressions is called. 2. If the degree of the numerator of a rational epression is greater than or equal to the degree of the denominator, then the fraction is called. 3. In order to find the partial fraction decomposition of a rational epression, the denominator must be completely factored into factors of the form p q m and factors of the form a 2 b c n, which are over the rationals. 4. The is derived after multiplying each side of the partial fraction decomposition form by the least common denominator. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Eercises 4, match the rational epression with the form of its decomposition. [The decompositions are labeled (a), (b), (c), and (d).] (a) (c) B 2 C 2 B C 2 4 (b) (d) In Eercises 5 4, write the form of the partial fraction decomposition of the rational epression. Do not solve for the constants In Eercises 5 38, write the partial fraction decomposition of the rational epression. Check your result algebraically B 4 B C In Eercises 39 44, write the partial fraction decomposition of the improper rational epression
8 333202_0704.qd 2/5/05 9:43 M Page Chapter 7 Systems of Equations and Inequalities In Eercises 45 52, write the partial fraction decomposition of the rational epression. Use a graphing utility to check your result graphically Graphical nalysis In Eercises 53 56, (a) write the partial fraction decomposition of the rational function, (b) identify the graph of the rational function and the graph of each term of its decomposition, and (c) state any relationship between the vertical asymptotes of the graph of the rational function and the vertical asymptotes of the graphs of the terms of the decomposition. To print an enlarged copy of the graph, go to the website 2 2( )2 53. y 54. y 4 2 y y y y y y Thermodynamics The magnitude of the range R of ehaust temperatures (in degrees Fahrenheit) in an eperimental diesel engine is approimated by the model R , Model It 0 < Synthesis 58. Writing Describe two ways of solving for the constants in a partial fraction decomposition. True or False? In Eercises 59 and 60, determine whether the statement is true or false. Justify your answer. 59. For the rational epression the partial fraction decomposition is of the form 0 B When writing the partial fraction decomposition of the epression 3 2 the first step is to factor the denominator. In Eercises 6 64, write the partial fraction decomposition of the rational epression. Check your result algebraically. Then assign a value to the constant a to check the result graphically. 6. a ya y 64. Skills Review Model It (continued) where is the relative load (in foot-pounds). (a) Write the partial fraction decomposition of the equation. (b) The decomposition in part (a) is the difference of two fractions. The absolute values of the terms give the epected maimum and minimum temperatures of the ehaust gases for different loads. Yma st term Ymin 2nd term Write the equations for Yma and Ymin. (c) Use a graphing utility to graph each equation from part (b) in the same viewing window. (d) Determine the epected maimum and minimum temperatures for a relative load of 0.5. a a In Eercises 65 70, sketch the graph of the function. 65. f f f f f f
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