Rational Expressions


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1 C H A P T E R Rtionl Epressions nformtion is everywhere in the newsppers nd mgzines we red, the televisions we wtch, nd the computers we use. And I now people re tlking bout the Informtion Superhighwy, which will deliver vst mounts of informtion directly to consumers homes. In the future the combintion of telephone, television, nd computer will give us onthespot helth cre recommendtions, video conferences, home shopping, nd perhps even electronic voting nd driver s license renewl, to nme just few. There is even tlk of 500 television chnnels! Some eperts re concerned tht the consumer will give up privcy for this technology. Others worry bout regultion, ccess, nd content of the enormous interntionl computer network. Whtever the future of this technology, few people understnd how ll their electronic devices work. However, this vst rry of electronics rests on physicl principles, which re described by mthemticl formuls. In Eercises 49 nd 50 of Section. we will see tht the formul governing resistnce for receivers connected in prllel involves rtionl epressions, which re the subject of this chpter.
2 4 () Chpter Rtionl Epressions In this section Definition of Rtionl Epressions Domin Reducing to Lowest Terms Building Up the Denomintor Rtionl Functions Applictions If the domin consists of ll rel numbers ecept 5, some people write R 5 for the domin. Even though there re severl wys to indicte the domin, you should keep prcticing intervl nottion becuse it is used in lgebr, trigonometry, nd clculus. E X A M P L E. PROPERTIES OF RATIONAL EXPRESSIONS A rtio of two integers is clled rtionl number; rtio of two polynomils is clled rtionl epression. Rtionl epressions re s fundmentl to lgebr s rtionl numbers re to rithmetic. In this section we look crefully t some of the properties of rtionl numbers nd see how they etend to rtionl epressions. Definition of Rtionl Epressions A rtionl epression is the rtio of two polynomils with the denomintor not equl to zero. For emple,, 5,, y 5, y nd re rtionl epressions. The rtionl number is rtionl epression becuse nd re monomils nd is rtio of two monomils. If the denomintor of rtionl epression is, it is usully omitted, s in the epression 5. Domin The domin of rtionl epression is the set of ll rel numbers tht cn be used in plce of the vrible. Becuse the denomintor of rtionl epression cnnot be zero, the domin of rtionl epression consists of the set of rel numbers ecept those tht cuse the denomintor to be zero. The domin of 5 is the set of ll rel numbers ecluding 5. In setbuilder nottion this set is written s 5, nd in intervl nottion it is written s (, 5) (5, ). Domin Find the domin of ech rtionl epression. ) b) y 9 5y c) ) The denomintor is zero if 9 0or 9. The domin is 9 or (, 9) (9, ). b) The denomintor is zero if 5y 0 or y 0. The domin is y y 0 or (, 0) (0, ).
3 . Properties of Rtionl Epressions () 5 c) The denomintor is zero if 0. Solve this eqution. 0 ( ) 0 Fctor out. ( )( ) 0 Fctor completely. 0 or 0 Zero fctor property or The domin is the set of ll rel numbers ecept nd. This set is written s nd, or in intervl nottion s (, ) (, ) (, ). CAUTION The numbers tht you find when you set the denomintor equl to zero nd solve for re not in the domin of the rtionl epression. The solutions to tht eqution re ecluded from the domin. Most students lern to convert into 4 by dividing into to get nd then multiply by to get 4. In lgebr it is better to do this conversion by multiplying the numertor nd denomintor of by s shown here. Reducing to Lowest Terms Ech rtionl number cn be written in infinitely mny equivlent forms. For emple, Ech equivlent form of is obtined from by multiplying both numertor nd denomintor by the sme nonzero number. For emple, 4 nd 9. Note tht we re ctully multiplying by equivlent forms of, the multiplictive identity. If we strt with 4 nd convert it into, we re simplifying by reducing 4 to its lowest terms. We cn reduce s follows: 4 A rtionl number is epressed in its lowest terms when the numertor nd denomintor hve no common fctors other thn. In reducing 4, we divide the numertor nd denomintor by the common fctor, or divide out the common fctor. We cn multiply or divide both numertor nd denomintor of rtionl number by the sme nonzero number without chnging the vlue of the rtionl number. This fct is clled the bsic principle of rtionl numbers. If b Bsic Principle of Rtionl Numbers is rtionl number nd c is nonzero rel number, then b c. bc
4 (4) Chpter Rtionl Epressions CAUTION Although it is true tht 5, 4 we cnnot divide out the s in this epression becuse the s re not fctors. We cn divide out only common fctors when reducing frctions. Just s rtionl number hs infinitely mny equivlent forms, rtionl epression lso hs infinitely mny equivlent forms. To reduce rtionl epressions to its lowest terms, we follow ectly the sme procedure s we do for rtionl numbers: Fctor the numertor nd denomintor completely, then divide out ll common fctors. E X A M P L E A negtive sign in frction cn be plced in three loctions: The sme goes for rtionl epressions: 5y 5y 5y Reducing Reduce ech rtionl epression to its lowest terms. ) 8 b) 7 b 4 b ) Fctor 8 s nd 4 s 7: Fctor. Divide out the common fctors. b) Becuse this epression is lredy fctored, we use the quotient rule for eponents to reduce: 7 b 5 b b In the net emple we use the techniques for fctoring polynomils tht we lerned in Chpter 5. E X A M P L E Reducing Reduce ech rtionl epression to its lowest terms. 8 ) b) w c) w 4 8 ( 9) ) ( ) ( ) Fctor. ( )( ) ( )( ) Fctor completely. Divide out the common fctors.
5 . Properties of Rtionl Epressions (5) 7 study tip Studying in quiet plce is better thn studying in noisy plce. There re very few people who cn listen to music or converstion nd study t the sme time. b) Fctor out from the numertor to get common fctor: c) ( 8) 4 4( 4) ( )( 4) ( )( ) 4 4 w ( w) w ( w) The rtionl epressions in Emple () re equivlent becuse they hve the sme vlue for ny replcement of the vribles, provided tht the replcement is in the domin of both epressions. In other words, the eqution 8 Fctoring out 4 will give the common fctor. Difference of two cubes, difference of two squres Divide out common fctors. is n identity. It is true for ny vlue of ecept nd. The min points to remember for reducing rtionl epressions re summrized s follows. Since ( b) b, plcement of negtive sign in rtionl epression chnges the ppernce of the epression: ( ) ( ) Strtegy for Reducing Rtionl Epressions. All reducing is done by dividing out common fctors.. Fctor the numertor nd denomintor completely to see the common fctors.. Use the quotient rule to reduce rtio of two monomils involving eponents. 4. We my hve to fctor out common fctor with negtive sign to get identicl fctors in the numertor nd denomintor. Building Up the Denomintor In Section. we will see tht only rtionl epressions with identicl denomintors cn be dded or subtrcted. Frctions without identicl denomintors cn be converted to equivlent frctions with common denomintor by reversing the procedure for reducing frctions to its lowest terms. This procedure is clled building up the denomintor. Consider converting the frction into n equivlent frction with denomintor of 5. Any frction tht is equivlent to cn be obtined by multiplying the numertor nd denomintor of by the sme nonzero number. Becuse 5 7, we multiply the numertor nd denomintor of by 7 to get n equivlent frction with denomintor of 5:
6 8 () Chpter Rtionl Epressions E X A M P L E 4 Notice tht reducing nd building up re ectly the opposite of ech other. In reducing you remove fctor tht is common to the numertor nd denomintor, nd in buildingup you put common fctor into the numertor nd denomintor. E X A M P L E 5 Multiplying the numertor nd denomintor of rtionl epression by chnges the ppernce of the epression: ( ) 7 ( 7) 7 y 5 ( y 5) 4 y (4 y) 5 y 4 y Building up the denomintor Convert ech rtionl epression into n equivlent rtionl epression tht hs the indicted denomintor. ) 7,? 5? b), 4 b 9 b 4 ) Fctor 4 s 4 7, then multiply the numertor nd denomintor of 7 by the missing fctors, nd : b) Becuse 9 b 4 b b, we multiply the numertor nd denomintor by b : 5 5 b b b b 5b 9 4 b When building up denomintor to mtch more complicted denomintor, we fctor both denomintors completely to see which fctors re missing from the simpler denomintor. Then we multiply the numertor nd denomintor of the simpler epression by the missing fctors. Building up the denomintor Convert ech rtionl epression into n equivlent rtionl epression tht hs the indicted denomintor. 5? ), b)?, b b 7 ) Fctor both b nd b to see which fctor is missing in b. Note tht we fctor out from b to get the fctor b: b ( b) b ( b) ( b) Now multiply the numertor nd denomintor by the missing fctor, : 5 5() 5 b ( b) () b b) Becuse 7 ( )( 4), multiply the numertor nd denomintor by 4: ( ) ( 4) ( ) ( 4) 8 7 Rtionl Functions A rtionl epression cn be used to determine the vlue of vrible. For emple, if y, 4
7 . Properties of Rtionl Epressions (7) 9 then we sy tht y is rtionl function of. We cn lso use function nottion s shown in the net emple. E X A M P L E clcultor closeup To check, use Y= to enter y ( )( 4). Then use the vribles feture (VARS) to find y () nd y (). E X A M P L E 7 Evluting rtionl function Find R(), R(), nd R() for the rtionl function R(). 4 To find R(), replce by in the formul: R() To find R(), replce by in the formul: R() () () 4 We cnnot find R() becuse is not in the domin of the rtionl epression. Applictions A rtionl epression cn occur in finding n verge cost. The verge cost of mking product is the totl cost divided by the number of products mde. Averge cost function Mercedes Benz spent $700 million to develop its new 999 M clss SUV, which will sell for round $40,000 (Motor Trend, July 998, If the cost of mnufcturing the SUV is $0,000 ech, then wht rtionl function gives the verge cost of developing nd mnufcturing vehicles? Compre the verge cost per vehicle for mnufcturing levels of 0,000 vehicles nd 00,000 vehicles. The polynomil 0, ,000,000 gives the cost in dollrs of developing nd mnufcturing vehicles. The verge cost per vehicle is given by the rtionl function 0, ,000,000 AC(). If 0,000, then 0,000(0,000) 700,000,000 AC(0,000) 00,000. 0,000 If 00,000, then 0,000(00,000) 700,000,000 AC(00,000) 7, ,000 The verge cost per vehicle when 0,000 vehicles re mde is $00,000, wheres the verge cost per vehicle when 00,000 vehicles re mde is $7,000. 4
8 40 (8) Chpter Rtionl Epressions WARMUPS True or flse? Eplin.. A rtionl number is rtionl epression. True. The epression is rtionl epression. True. The domin of the rtionl epression is. Flse 5 4. The domin of is ( 9)( ) 9 nd. True 5. The domin of is (, ) (, ) (, ). Flse. The rtionl epression 5 5. Flse 7. Multiplying the numertor nd denomintor of by yields. Flse 8. The epression is equivlent to. True 9. The eqution 4 is n identity. True 0. The epression y reduced to its lowest terms is y. Flse y. EXERCISES Reding nd Writing After reding this section, write out the nswers to these questions. Use complete sentences.. Wht is rtionl epression? A rtionl epression is rtio of two polynomils with the denomintor not equl to zero.. Wht is the domin of rtionl epression? The domin of rtionl epression is ll rel numbers ecept those tht cuse the denomintor to be zero.. Wht is the bsic principle of rtionl numbers? The bsic principle of rtionl numbers sys tht (b)(c) bc, provided nd c re not zero. 4. How do we reduce rtionl epression to lowest terms? To reduce rtionl epression, fctor the numertor nd denomintor completely nd then divide out the common fctors. 5. How do you build up the denomintor of rtionl epression? We build up the denomintor by multiplying the numertor nd denomintor by the sme epression.. Wht is verge cost? Averge cost is totl cost divided by the number of items. Find the domin of ech rtionl epression. See Emple z 5 z z 0 7z 0. z z z 0 4z. 5 y y y nd y y 4. y y y nd y y 9. 5 nd b 4. b 7b 4 b b nd b nd 0 4. nd nd 0 nd 8. 4 nd 0 nd 5 Reduce ech rtionl epression to its lowest terms. See Emples nd
9 . Properties of Rtionl Epressions (9) 4 y 5.. 5b 0 7. b 0y 5 b b 5 5 b 8. 8 y 54y z 9 z 9. w y b c 0. 5 w y 4 8b c y w b 8 z y 4c 4 b.. b8 b 5 4 b 5 b b b. 4. m n b 4n 4m b b 8. 7 y y b b 9 y y b 4 4 b b b ( ) ( 4) b by y y 5 b y Convert ech rtionl epression into n equivlent rtionl epression tht hs the indicted denomintor. See Emples 4 nd ,? ,? ,?? 50., b b 5 b 5 b 5?? 5., 5., ?? 5., 54., ?? 55. 5, 5., 5 57.??, 58., ?? 59., 0., b b 7 b??.,., ??., 4., Find the indicted vlue for ech given rtionl epression. See Emple. 5. R() 5, R() T(), T(9) 5 y 5 7. H(y), H() y G() 5, G(5) 7 7 4b 9. W(b), W() Undefined b b 70. N(), N() Undefined In plce of ech question mrk in Eercises 7 90, put n epression tht will mke the rtionl epressions equivlent. 7.??
10 4 (0) Chpter Rtionl Epressions ? 4? 0 75.? 7. 5 y 0? 0 y?? b b 4 4 b 4? 5? w? 5? w ? 84. 4? 4 85.? 8.? 4 87.? 88. 9?? ? Reduce ech rtionl epression to its lowest terms. Vribles used in eponents represent integers m w wm m w m b b b b b 9. b b 8 b b b Solve ech problem. See Emple Driving speed. If Jeremy drives 500 miles in hours, then wht rtionl epression represents his speed in miles per hour (mph)? 50 mph 98. Filing suit. If Mrsh files 48 suits in work dys, then wht rtionl epression represents the rte (in suits per dy) t which she is filing suits? 4 suits per dy 99. Wedding bells. Wheeler Printing Co. chrges $45 plus $0.50 per invittion to print wedding invittions. ) Write rtionl function tht gives the verge cost in dollrs per invittion for printing n invittions. b) How much less does it cost per invittion to print 00 invittions rther thn 00 invittions? c) As the number of invittions increses, does the verge cost per invittion increse or decrese? d) As the number of invittions increses, does the totl cost of the invittions increse or decrese? ) A(n) 0.50n 45 dollrs n b) 7.5 cents c) decreses d) increses Cost per invittion (in dollrs) Number of invittions FIGURE FOR EXERCISE Rose Bowl bound. A trvel gent offers Rose Bowl pckge including hotel, tickets, nd trnsporttion. It costs the trvel gent $50,000 plus $00 per person to chrter the irplne. Find rtionl function tht gives the verge cost in dollrs per person for the chrter flight. How much lower is the verge cost per person when 00 people go compred to 00 people? A(n) 50,000 00n dollrs, $50 per person n 0. Solid wste recovery. The mount of municipl solid wste generted in the United Sttes in the yer 90 n is given by the polynomil.4n 87.4, wheres the mount recycled is given by the polynomil 0.05n 0.4n.7,
11 . Multipliction nd Division () 4 where the mounts re in millions of tons (U.S. Environmentl Protection Agency, ) Write rtionl function p(n) tht gives the frction of solid wste tht is recovered in the yer 90 n. b) Find p(0), p(0), nd p(50). 0.05n 0.4n.7 ) p(n).4n 87.4 b) 7.7%, 8.5%, 4.4% 0. Higher eduction. The totl number of degrees wrded in U.S. higher eduction in the yer 990 n is given in thousnds by the polynomil 4.7n 49, wheres the number of bchelor s degrees wrded is given in thousnds by the polynomil 5.n 09 (Ntionl Center for Eduction Sttistics, ) Write rtionl function p(n) tht gives the percentge of bchelor s degrees mong the totl number of degrees conferred for the yer 990 n. b) Wht percentge of the degrees wrded in 00 will be bchelor s degrees? ) p(n) 5. n 09 b) 9.5% 4. 7n 49 GETTING MORE INVOLVED 0. Eplortion. Use clcultor to find R(), R(0), R(500), R(9,000), nd R(80,000) for the rtionl epression R(). Round nswers to four deciml plces. Wht cn you conclude bout the vlue of R() s gets lrger nd lrger without bound? The vlue of R() gets closer nd closer to. 04. Eplortion. Use clcultor to find H(,000), H(00,000), H(,000,000), nd H(0,000,000) for the rtionl epression H() Round nswers to four deciml plces. Wht cn you conclude bout the vlue of H() s gets lrger nd lrger without bound? The vlue of H() gets closer nd closer to 7.. MULTIPLICATION AND DIVISION In this section In Chpter 5 you lerned to dd, subtrct, multiply, nd divide polynomils. In this chpter you will lern to perform the sme opertions with rtionl epressions. We begin in this section with multipliction nd division. Multiplying Rtionl Epressions Dividing b by b Dividing Rtionl Epressions Multiplying Rtionl Epressions We multiply two rtionl numbers by multiplying their numertors nd multiplying their denomintors. For emple, Insted of reducing the rtionl number fter multiplying, it is often esier to reduce before multiplying. We first fctor ll terms, then divide out the common fctors, then multiply: When we multiply rtionl numbers, we use the following definition. If b Multipliction of Rtionl Numbers nd c re rtionl numbers, then d b c c d b. d We multiply rtionl epressions in the sme wy tht we multiply rtionl numbers: Fctor ll polynomils, divide out the common fctors, then multiply the remining fctors.
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