P.4 Rational Expressions

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1 7_0P04.qp /7/06 9:4 AM Page 7 Section P.4 Rational Epressions 7 P.4 Rational Epressions Domain of an Algebraic Epression Te set of real numbers for wic an algebraic epression is defined is te domain of te epression. Two algebraic epressions are equivalent if tey ave te same domain and yield te same values for all numbers in teir domain. For instance, te epressions and are equivalent because. Eample Finding te Domain of an Algebraic Epression a. Te domain of te polynomial 4 is te set of all real numbers. In fact, te domain of any polynomial is te set of all real numbers, unless te domain is specifically restricted. b. Te domain of te radical epression Wat you sould learn Find domains of algebraic epressions. Simplify rational epressions. Add, subtract, multiply, and divide rational epressions. Simplify comple fractions. Wy you sould learn it Rational epressions are useful in estimating te temperature of food as it cools. For instance, a rational epression is used in Eercise 96 on page 46 to model te temperature of food as it cools in a refrigerator set at 40F. is te set of real numbers greater tan or equal to, because te square root of a negative number is not a real number. c. Te domain of te epression is te set of all real numbers ecept, wic would result in division by zero, wic is undefined. Now try Eercise 5. Dwayne Newton/PotoEdit Te quotient of two algebraic epressions is a fractional epression. Moreover, te quotient of two polynomials suc as,, is a rational epression. or Alert students to te importance of te domain of an epression in graping functions later in tis course and in calculus. Simplifying Rational Epressions Recall tat a fraction is in simplest form if its numerator and denominator ave no factors in common aside from ±. To write a fraction in simplest form, divide out common factors. a c b c a b, c 0.

2 7_0P04.qp //07 :7 AM Page 8 8 Capter P Prerequisites Te key to success in simplifying rational epressions lies in your ability to factor polynomials. Wen simplifying rational epressions, be sure to factor eac polynomial completely before concluding tat te numerator and denominator ave no factors in common. Eample Write Simplifying a Rational Epression 4 6 in simplest form , Factor completely. Divide out common factors. Note tat te original epression is undefined wen (because division by zero is undefined). To make sure tat te simplified epression is equivalent to te original epression, you must restrict te domain of te simplified epression by ecluding te value. Now try Eercise 7. It may sometimes be necessary to cange te sign of a factor by factoring out to simplify a rational epression, as sown in Eample. Eample Write Simplifying a Rational Epression 9 4 in simplest form , 4 Now try Eercise 5. Factor completely. 4 4 Divide out common factors. STUDY TIP In tis tet, wen a rational epression is written, te domain is usually not listed wit te epression. It is implied tat te real numbers tat make te denominator zero are ecluded from te epression. Also, wen performing operations wit rational epressions, tis tet follows te convention of listing beside te simplified epression all values of tat must be specifically ecluded from te domain in order to make te domains of te simplified and original epressions agree. In Eample, for instance, te restriction 4 is listed beside te simplified epression to make te two domains agree. Note tat te value is ecluded from bot domains, so it is not necessary to list tis value. Operations wit Rational Epressions To multiply or divide rational epressions, you can use te properties of fractions discussed in Section P.. Recall tat to divide fractions you invert te divisor and multiply. Te factoring tecnique used in equating 4 and 4 in Eample is not always obvious to students. Identify te tecnique in one or two eamples.

3 7_0P04.qp /7/06 9:4 AM Page 9 Section P.4 Rational Epressions 9 Eample 4 Multiplying Rational Epressions Now try Eercise 5., 0,, 5 Eample 5 Dividing Rational Epressions Divide 8 4 by Invert and multiply , ± Divide out common factors. Now try Eercise 5. To add or subtract rational epressions, you can use te LCD (least common denominator) metod or te basic definition a b ± c d ad ± bc, bd b 0 and d 0. Basic definition Tis definition provides an efficient way of adding or subtracting two fractions tat ave no common factors in teir denominators. Eample 6 Subtracting Rational Epressions Subtract 4 from Now try Eercise 57. Basic definition Distributive Property Combine like terms. STUDY TIP Wen subtracting rational epressions, remember to distribute te negative sign to all te terms in te quantity tat is being subtracted.

4 7_0P04.qp /7/06 9:4 AM Page Capter P Prerequisites For tree or more fractions, or for fractions wit a repeated factor in te denominators, te LCD metod works well. Recall tat te least common denominator of several fractions consists of te product of all prime factors in te denominators, wit eac factor given te igest power of its occurrence in any denominator. Here is a numerical eample Te LCD is. 4 Sometimes te numerator of te answer as a factor in common wit te denominator. In suc cases te answer sould be simplified. For instance, in te eample above, was simplified to 4. Eample 7 Combining Rational Epressions: Te LCD Metod Perform te operations and simplify. Using te factored denominators,, and, you can see tat te LCD is. 6 Now try Eercise 6. Distributive Property Group like terms. Combine like terms. Factor.

5 7_0P04.qp /7/06 9:4 AM Page 4 Comple Fractions Fractional epressions wit separate fractions in te numerator, denominator, or bot are called comple fractions. Here are two eamples. and A comple fraction can be simplified by combining te fractions in its numerator into a single fraction and ten combining te fractions in its denominator into a single fraction. Ten invert te denominator and multiply. Section P.4 Rational Epressions 4 Eample 8 Simplifying a Comple Fraction, Now try Eercise 69. Combine fractions. Simplify. Invert and multiply. Additional Eample , 0 In Eample 8, te restriction is added to te final epression to make its domain agree wit te domain of te original epression. Anoter way to simplify a comple fraction is to multiply eac term in its numerator and denominator by te LCD of all fractions in its numerator and denominator. Tis metod is applied to te fraction in Eample 8 as follows., LCD is. Combine fractions. Simplify. Point out to your students tat two different metods are sown ere for simplifying te same comple fraction. Empasize tat bot yield te same result.

6 7_0P04.qp /7/06 9:4 AM Page 4 4 Capter P Prerequisites Te net four eamples illustrate some metods for simplifying rational epressions involving negative eponents and radicals. Tese types of epressions occur frequently in calculus. To simplify an epression wit negative eponents, one metod is to begin by factoring out te common factor wit te smaller eponent. Remember tat wen factoring, you subtract eponents. For instance, in 5 te smaller eponent is and te common factor is Eample 9 Simplifying an Epression wit Negative Eponents Simplify. Begin by factoring out te common factor wit te smaller eponent. Now try Eercise 75. Activities. Wat is te domain of? Answer: Te set of all real numbers less tan or equal to.. Te implied domain ecludes wat values of from te domain of 6 9? 9 Answer:,. Simplify:. Answer: A second metod for simplifying tis type of epression involves multiplying te numerator and denominator by a term to eliminate te negative eponent. Eample 0 Simplify Simplifying an Epression wit Negative Eponents Now try Eercise 79.

7 7_0P04.qp /7/06 9:4 AM Page 4 Eample Rewriting a Difference Quotient Te following epression from calculus is an eample of a difference quotient. Rewrite tis epression by rationalizing its numerator., 0 Notice tat te original epression is undefined wen 0. So, you must eclude 0 from te domain of te simplified epression so tat te epressions are equivalent. Now try Eercise 85. Section P.4 Rational Epressions 4 Difference quotients, like tat in Eample, occur frequently in calculus. Often, tey need to be rewritten in an equivalent form tat can be evaluated wen 0. Note tat te equivalent form is not simpler tan te original form, but it as te advantage tat it is defined wen 0. Eample Rewriting a Difference Quotient Rewrite te epression by rationalizing its numerator Now try Eercise

8 7_0P04.qp /7/06 9:5 AM Page Capter P Prerequisites P.4 Eercises See for worked-out solutions to odd-numbered eercises. Vocabulary Ceck Fill in te blanks.. Te set of real numbers for wic an algebraic epression is defined is te of te epression.. Te quotient of two algebraic epressions is a fractional epression and te quotient of two polynomials is a.. Fractional epressions wit separate fractions in te numerator, denominator, or bot are called. 4. To simplify an epression wit negative eponents, it is possible to begin by factoring out te common factor wit te eponent. 5. Two algebraic epressions tat ave te same domain and yield te same values for all numbers in teir domains are called. In Eercises 6, find te domain of te epression , , > In Eercises 7, find te missing factor in te numerator suc tat te two fractions are equivalent In Eercises 40, write te rational epression in simplest form y y y 60y 5 y y y 4y 8y y y. 6. y y 7y y y z z z 4 In Eercises 4 and 4, complete te table. Wat can you conclude? y y y y

9 7_0P04.qp /7/06 9:5 AM Page 45 Section P.4 Rational Epressions Error Analysis Describe te error. 44. Error Analysis Describe te error. Geometry In Eercises 45 and 46, find te ratio of te area of te saded portion of te figure to te total area of te figure In Eercises 47 54, perform te multiplication or division and simplify r r r r t t 6 t t 6t 9 t 4 y 5. y In Eercises 55 64, perform te addition or subtraction and simplify r y 8 y y 6 4 y 5y 5 y y y 5y In Eercises 65 7, simplify te comple fraction In Eercises 7 78, simplify te epression by removing te common factor wit te smaller eponent In Eercises 79 84, simplify te epression ( ) t t t t

10 7_0P04.qp /7/06 9:5 AM Page Capter P Prerequisites In Eercises 85 90, rationalize te numerator of te epression Probability In Eercises 9 and 9, consider an eperiment in wic a marble is tossed into a bo wose base is sown in te figure. Te probability tat te marble will come to rest in te saded portion of te bo is equal to te ratio of te saded area to te total area of te figure. Find te probability ( + ) 9. Rate A potocopier copies at a rate of 6 pages per minute. (a) Find te time required to copy page. (b) Find te time required to copy pages. (c) Find te time required to copy 60 pages. 94. Montly Payment Te formula tat approimates te annual interest rate r of a montly installment loan is given by r 4NM P N NM P z z were N is te total number of payments, M is te montly payment, and P is te amount financed. (a) Approimate te annual interest rate for a five-year car loan of $0,000 tat as montly payments of $400. (b) Simplify te epression for te annual interest rate r, and ten rework part (a). 95. Resistance Te formula for te total resistance (in oms) of a parallel circuit is given by R T R R R were R, R, and R are te resistance values of te first, second, and tird resistors, respectively. (a) Simplify te total resistance formula. (b) Find te total resistance in te parallel circuit wen R 6 oms, R 4 oms, and R oms. 96. Refrigeration Wen food (at room temperature) is placed in a refrigerator, te time required for te food to cool depends on te amount of food, te air circulation in te refrigerator, te original temperature of te food, and te temperature of te refrigerator. Consider te model tat gives te temperature of food tat is at 75F and is placed in a 40F refrigerator as T 0 4t 6t 75 t 4t 0 were T is te temperature (in degrees Fareneit) and t is te time (in ours). (a) Complete te table. t T t T (b) Wat value of T does te matematical model appear to be approacing? 97. Plants Te table sows te numbers of endangered and treatened plant species in te United States for te years 000 troug 005. (Source: U.S. Fis and Wildlife Service) R T Year Endangered, E Treatened, T

11 7_0P04.qp /7/06 9:5 AM Page 47 Section P.4 Rational Epressions 47 Matematical models for te data are Endangered plants: E 4.5t 565.9t and Treatened plants: T 4.48t 9.65t were t represents te year, wit t 0 corresponding to 000. (a) Using te models, create a table to estimate te numbers of endangered plant species and te numbers of treatened plant species for te given years. Compare tese estimates wit te actual data. (b) Determine a model for te ratio of te number of treatened plant species to te number of endangered plant species. Use te model to find tis ratio for eac of te given years. 98. Marriages and Divorces Te table sows te rates (per 000 of te total population) of marriages and divorces in te United States for te years 990 troug 004. (Source: U.S. National Center for Healt Statistics) Matematical models for te data are Marriages: and Divorces: Year Marriages, M Divorces, D M t 9,897.8t t 0,47.65t D 0.00t 0.06t 4.8 were t represents te year, wit t 0 corresponding to 990. (a) Using te models, create a table to estimate te number of marriages and te number of divorces for eac of te given years. Compare tese estimates wit te actual data. (b) Determine a model for te ratio of te number of marriages to te number of divorces. Use te model to find tis ratio for eac of te given years. In Eercises 99 04, simplify te epression In Eercises 05 and 06, simplify te given epression , 0, 0, 0, 0, 0, 0 4 nn n n 9 n Syntesis nn n True or False? In Eercises 07 and 08, determine weter te statement is true or false. Justify your answer. 07. n n 08. n n n n 09. Tink About It How do you determine weter a rational epression is in simplest form? 0. Tink About It Is te following statement true for all nonzero real numbers a and b? Eplain. a b b a 6 n 4 n 6 n n n n n n n n. Writing Write a paragrap eplaining to a classmate wy y y.. Writing Write a paragrap eplaining to a classmate wy y y.