Mini Lecture 8.1 Rational Expressions and Their Simplification

Transcription

1 Mini Lecture 8. Rtionl Epressions nd Their Simplifiction Lerning Objectives:. Find numbers for which rtionl epression is undefine. Simplif rtionl epressions.. Solve pplied problems involving rtionl epressions. Emples: Find ll the numbers for which the rtionl epression is undefine If the rtionl epression is defined for ll rel numbers, so stt Simpli c c c c 0. The rtionl epression describes the time, in hours, to trvel 0 miles t rte of r 0 (r 0) miles per hour.. Determine the vlue of r tht would cuse the epression to be undefine Find the time in hours if r 0. Teching Notes: Rtionl epressions re quotients of two polnomils. The indicte division nd division b zero is undefine We must lws eclude n vlue(s) of the vrible tht mke denomintor zero. When simplifing rtionl epressions, first, fctor the numertor nd denomintor completel, then divide both the numertor nd the denomintor b n common fctors. A rtionl epression is simplified if its numertor nd denomintor hve no common fctors other thn nd. When reducing rtionl epressions, onl fctors, not common terms, tht re common to the entire numertor nd the entire denomintor cn be divided out. Answers:.. nd defined for ll rel numbers.. or c c r 0 time hours ML-0

2 Lerning Objectives:. Multipl rtionl epressions.. Divide rtionl epressions. Mini Lecture 8. Multipling nd Dividing Rtionl Epressions Emples: Multipl s indicte Divide s indicte ( ) ( ) ( 8) Teching Notes: When in doubt, fctor it out. Fctor first before multipling. Remind students tht in order to divide frctions, (in this section, rtionl epressions) ou must multipl b the reciprocl of the divisor. Answers:.... ( 8) 8 0 ( ) 7 7 ( )( ) ( ) ( )( ).... ( ) ( 6) ( ) ( ) ( ) ( ) ML-

3 Mini Lecture 8. Adding nd Subtrcting Rtionl Epressions with the Sme Denomintor Lerning Objectives:. Add rtionl epressions with the sme denomintor.. Subtrct rtionl epressions with the sme denomintor.. Add nd subtrct rtionl epressions with opposite denomintors. Emples: Ad Simplif if possibl Subtrct, simplif if possibl Add, mking sure to find common denomintor first nd simplif if possibl. Teching Notes: To dd rtionl epressions with the sme denomintor, dd numertors nd plce nd sum over the common denomintor. Simplif the nswer if possibl To subtrct rtionl epressions with the sme denomintor, subtrct numertors nd plce the difference over the common denomintor. Simplif the nswer if possibl When subtrcting numertors with common denomintor, mke sure to subtrct ever term in tht epression. When one denomintor is the dditive inverse of the other, first multipl either rtionl epressions b to obtin common denomintor. Answers: ML-

4 Mini Lecture 8. Adding nd Subtrcting Rtionl Epressions with Different Denomintors Lerning Objectives:. Find the lest common denomintor (LCD).. Add nd subtrct rtionl epressions with different denomintors. Emples: Find the lest common denomintor for the rtionl numbers or rtionl epressions. Fctor the denomintors first, then build the lest common denomintor from those fctors... nd 7 nd nd Rewrite ech of the following s n equivlent epression with the given denomintor ( )( ) Add or subtrct g. h i. 6 Teching Notes: Students m need to be reminded of fctoring steps. It is ver importnt to be ble to fctor quickl nd completel. Students will find this concept es if the cn relte dding nd subtrcting rtionl epressions to dding nd subtrcting frctions. Wtch the signs when subtrcting! Students need to understnd tht the LCD is build with the fctors of the denomintors. Answers: ( )( )( ) g. 6 0 ( )( ) h. i. ( )( ) ( )( ) ( )( ) ML-

5 Mini Lecture 8. Comple Rtionl Epressions Lerning Objectives:. Simplif comple rtionl epressions b dividing.. Simplif comple rtionl epressions b multipling b the LCD. Emples: Simplif b dividing; simplif, if possibl Simplif b the LCD method; simplif, if possibl Teching Notes: Comple rtionl epressions re clled comple frctions. The hve numertors or denomintors contining one or more rtionl epressions. One method for simplifing comple rtionl epression is to combine its numertor into single epression nd combine its denomintor into single epression, then divide b multipling b the reciprocl of the term in the denomintor. A second method for simplifing comple rtionl epression is to multipl ech term in the numertor nd denomintor b the lest common denomintor (LCD). This will produce n equivlent epression tht does not contin frctions in the numertor or denomintor. Both methods for simplifing comple rtionl epressions produce the sme nswer. See which method ou prefer. Answers: ML-

6 Mini Lecture 8.6 Solving Rtionl Equtions Lerning Objectives:. Solve rtionl equtions.. Solve problems involving formuls with rtionl epressions. Emples: Wht is the LCD in ech problem?.. Solv Cler ech eqution of frctions first Teching Notes: Students m need etr prctice finding the LCD nd m need to be reminded to Fctor First. When ll denomintors re in fctored form list the restricted vlues. Restricted vlues is n numbers tht would mke n denomintor zero. Some rtionl equtions cn be solved using cross products, but students need to be wre tht the method cn onl be used when there is onl one rtionl epression on ech side of the eqution. Students need to constntl be reminded to multipl ech term or epression on both sides of the eqution b the LCD to get ride of the frctions. Students must check solutions for restricted vlues. Answers:.. ( ) ,, 0 ML-

7 Mini Lecture 8.7 Applictions Using Rtionl Equtions nd Vrition Lerning Objectives:. Solve problems involving motion.. Solve problems involving work.. Solve problems involving similr tringles.. Solve problems involving vrition. Emples:. A bot trvels km upstrem in the sme mount of time tht the bot covers km downstrem. The current in the strem moves t speed of km/h. Wht is the speed of the bot in still wter?. John working lone cn pint room in hours. His helper, Luke, would need 6 hours to do the job b himsel If the work together, how long will the pint job tke to complete?. A tree csts shdow of 7. ft. At the sme time, Elizbeth mesures the length of her shdow which is ft. If Elizbeth is. feet tll, how tll is the tree?. If vries directl s nd 00 when is 00, find when.. If vries indirectl s nd when 0, find when 700. Teching Notes: Time in motion eqution: d t time trveled r distnce trveled rte of trvel Work problem eqution: frctionl prt of job done frctionl prt of job job completed b one person done b the second person Similr Figures: Corresponding ngles hve the sme mesure nd the rtios of the lengths of the corresponding sides re equl. In corresponding figures, the lengths of the corresponding sides re proportionl. When tringles ABC nd DEF re ll similr then: M A M D, M B M E, M C M F b d e Direct Vrition Eqution: k Where k is the constnt of vrition, we s tht vries directl s. k Inverse Vrition Eqution: Where k is the constnt of vrition, we s tht vries indirectl s. With vrition questions, write the eqution from the English sttements. Solve for k nd then substitute the vlue of k bck into the originl eqution to nswer the originl question. Answers:., km/h. 6,. hours.. 7. c f,.7 feet. 0 ML-6