Review Factoring Polynomials:
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1 Chpter 4 Mth 0 Review Fctoring Polynomils:. GCF e. A) 5 5 A) Difference of Squres b = ( + b)( b) e. A) 9 6 B) C) 98y. Trinomils e. A) B) + C) Solving Polynomils:. A) ( 5)( ) = 0 B) 4 6 = 0 C) 6 = 0
2 Chpter 4 Mth 0 Prctice Problems: Fctor Completely w u m n 64p b b B. Solve. 5 = = 0. 4 = = 0
3 Chpter 4 Mth 0 Answers:. 9( + ). ( 6). (w + u)(w u) 4. (5 + )(5 ) 5. (8 + )(8 ) 6. (mn + 8p)(mn 8p) 7. ( + )( ) 8. 4( + 4)( 4) 9. b( + )( ) 0. ( + 9)( + )( ). ( 5)( 5). ( + 8)( + ). ( 6)( + ) 4. ( )( ) 5. ( + 0)( ) 6. ( + 7)( + 7) 7. ( + 9)( ) 8. ( + )( + ) 9. ( + )( ) 0. 4( 0)( + ). ( + 7)( + ) B.. = 5. = 0, = 4. =, =, 4. = 7, = 4.: EQUIVALENT RATIONAL EXPRESSIONS Lerning Outcomes: To compre the strtegies for writing equivlent forms of rtionl epressions to writing equivlent forms of rtionl numbers To be ble to eplin why given vlue is non-permissible for given rtionl epression To determine the non-permissible vlues for rtionl epression To determine rtionl epression tht is equivlent to given rtionl epression by multiplying the numertor nd denomintor by the sme fctor, nd stte the non-permissible vlues of the equivlent rtionl epression A Rtionl Epression: n lgebric frction with numertor nd denomintor tht re polynomils. + is n emple of rtionl epression Whenever you re working with lgebric frctions, it is importnt to determine ny vlues tht must be ecluded. You cn write n unlimited number of rithmetic frctions, or rtionl numbers, of the form, where nd b re integers. Wht integer cnnot be used for b? b Wht hppens in ech of the following epressions when =? i) 7 ii) 7 9 iii) 7 4+
4 Chpter 4 Mth 0 4 Wht vlue(s) cnnot be used for in ech of the following lgebric functions? i) 6 ii) 7 iii) 4 ( )(+) Write rule tht eplins how to determine ny vlues tht vrible cnnot be, for ny lgebric frction. Note: Whenever you use rtionl epression, you must identify ny vlues tht must be ecluded or re considered non-permissible vlues. Non-permissible vlues re ll vlues tht mke the denomintor zero. Emple : Determine the non-permissible vlue(s) for ech rtionl epression:. 4 bc b. (+)( ) c. 6 d. 5 6
5 Chpter 4 Mth 0 5 Emple :. Write rtionl number tht is equivlent to 8 Rtionl epressions re like frctions. To crete n equivlent rtionl epression, you cn use the sme strtegy used for rtionl numbers. Note, you cnnot introduce fctor tht required new restriction, becuse the epressions would not hve been equivlent.(cnnot dd ny new NPV) b. Write rtionl epression tht is equivlent to In order to mke n equivlent epression, wht process do we need to follow? Emple : For ech of the following, determine if the rtionl epressions re equivlent.. 9 nd 8 6 b. 4 nd Substitution cn be used to determine if two rtionl epressions re not equivlent, but it cnnot be used to determine if two rtionl epressions re equivlent. Questions pge -4 #,5,7,8,9,0,6
6 Chpter 4 Mth 0 6 Lerning Outcomes: 4. SIMPLIFYING RATIONAL EXPRESSIONS To be ble to simplify rtionl epression To eplin why the non-permissible vlues of given rtionl epression nd its simplified form re the sme To identify nd correct errors in given simplifiction of rtionl epression, nd eplin the resoning Eplore: Use fctoring nd elimintion to write rtionl epression tht is equivlent to 8 +8 Note: A rtionl epression is in simplified form when the numertor nd denomintor hve no common fctors Alwys stte the non-permissible vlues of the vribles s restrictions before simplifying rtionl epression. Otherwise, if you eliminte vrible or fctor s you simplify, you will lose the informtion bout the non-permissible vlue of the vrible. Simplifying rtionl epressions is similr to simplifying frctions. Common fctors in the numertor nd denomintor cn be cncelled.
7 Chpter 4 Mth 0 7 Emple : Simplify. 76, y y, 0, y 0. 5 y 4 5y, 0, y , Emple : Simplify the following rtionl epressions:. 6m 8m m 4m Steps: Fctor numertor nd denomintor Determine NPV Simplify by cncelling like terms
8 Chpter 4 Mth 0 8 b. + **creful of negtive fctors** E. E. Emple : Identify nd correct the errors in the simplifiction of ech rtionl epression. (), (b) (c) , 0, Questions pge 9- #-9,
9 Chpter 4 Mth MULTIPLYING RATIONAL EXPRESSIONS Lerning Outcomes: To develop strtegies for multiplying rtionl epressions To determine the non-permissible vlues when performing multipliction on rtionl epressions Steps for multiplying rtionl epressions:. Fctor numertor nd denomintor. Identify ll non-permissible vlues. Simplify using common fctors 4. Multiply ll numertors together nd multiply ll denomintors together. Emple : Simplify ( 0 ) (4 9 ). Emple : Simply the following. Stte the restrictions on the vribles. (A) ( 5p ) 7n (4n) (B) (+) (+5) 5p ( )(+5) (+) Why it is importnt to identify ll non-permissible vlues before simplifying rtionl epressions
10 Chpter 4 Mth 0 0 Emple : Simplify the following products: b c (+) Questions pge 8-9 #c,b,d,4c,5,6,9, 5b
11 Chpter 4 Mth 0 Lerning Outcomes: Investigtion DIVIDING RATIONAL EXPRESSIONS To develop strtegies for dividing rtionl epressions To determine the non-permissible vlues when performing division on rtionl epressions Determine the vlue of 6. The sme strtegy is used to divide rtionl epressions. Emple : Simplify: 4y y 9 8 non-permissible vlues when dividing rtionl epressions: consider the division b c d where,b,c nd d re vribles. The non-permissible vlues of b nd c d re The first step to simplifying her is to multiply by the reciprocl b d c. This introduces nother non-permissible vlue:
12 Chpter 4 Mth 0 Steps for dividing rtionl epressions:. Fctor numertor nd denomintor. Stte ll restrictions, for b c d the NPV occur t b,c nd d. Tke the reciprocl of the second epression 4. Simplify using common fctors 5. Multiply ll numertors together nd multiply ll denomintors together. Emple : Simplify ech quotient. Stte the restrictions on the vrible b
13 Chpter 4 Mth 0 c d (+5) Questions pge 8-9 #bd,cd,bc,4bd,5b,7,9, 5
14 Chpter 4 Mth ADDING AND SUBTRACTING RATIONAL EXPRESSIONS Lerning Outcomes: To determine, in simplified form, the sum or difference of rtionl epressions tht hve the sme denomintor To determine, in simplified form, the sum or difference of rtionl epressions tht hve different denomintors To determine the non-permissible vlues when performing dding nd subtrcting on rtionl epressions Reminder: To ADD or SUBTRACT frctions, you need common denomintor Emple : Determine ech sum or difference. Give nswers in lowest terms b Emple : The sme strtegy is used to dd or subtrct rtionl epressions b. 9 c d
15 Chpter 4 Mth 0 5 e. 5 + Steps for dding or subtrcting rtionl epressions:. Find common denomintor (my need to fctor denomintor). Chnge to equivlent frctions with the common denomintor. Add or subtrct the numertors 4. Reduce if possible 5. Stte ny restrictions (use LCD) Emple : Determine ech sum or difference. Epress ech nswer in simplest form. Identify ll non-permissible vlues.. 0m 4m 8 m 4m b ( )(+) ( )(+) ( )(+)
16 Chpter 4 Mth 0 6 Emple 4: Determine ech sum or difference. Epress ech nswer in simplest form b m m+ c. 4 Questions pge #-9,, 4 omit 8
17 Chpter 4 Mth 0 7 Lerning Outcomes: 4.5 SOLVING RATIONAL EQUATIONS Identify non-permissible vlues in rtionl eqution Determine the solution to rtionl eqution lgebriclly Emples: Solve the eqution: = To eliminte frctions Multiply both sides of the eqution by the lowest common denomintor. + = 4
18 Chpter 4 Mth 0 8 Working with rtionl eqution is similr to working with rtionl epressions. A significnt difference occurs becuse in n eqution, wht you do to one side you must lso do to the other side. Look through the emple shown. Wht steps were used to solve the problem? (z )(z + )(6) [ + 0 = z 4 6z+ z + 0 = (z )(z+) 6(z+) z z ± + 0 (z )(z+) 6(z+) ] = (z )(z + )(6) [ Wht hppened during this step? Step? z ] Wht ws done during this step? Recll how to combine frctions. (6)() + (z )(0) = (z + )(6) + 0z 0 = 6z + 4z = 0 z = 5 Step 4? (5) + = = Finl Step?
19 Chpter 4 Mth 0 9 Steps: To solve rtionl eqution:. Fctor ech denomintor. Identify the non-permissible vlues. Multiply both sides of the eqution by the lowest common denomintor 4. Solve by isolting the vrible on one side of the eqution 5. Check your nswers Emple : Solve the eqution. Wht re the non-permissible vlues? 5 4 Check: both sides equl
20 Chpter 4 Mth 0 0 Emple : Solve the eqution. Wht re the non-permissible vlues? + 5 = 5 6 Emple : 8 = 6 5 Questions pge #,5,6bd
21 Chpter 4 Mth APPLICATIONS OF RATIONAL EQUATIONS (Prt ) Emple : Two friends shre pper route. Sheen cn deliver the ppers in 40 min. Jeff cn cover the sme route in 50 min. How long, to the nerest minute, does the pper route tke if they work together? Orgnize informtion in tble: Time to deliver ppers (min) Frction of work done in min Frction of work done in t minutes Sheen 40 Jeff t 40 t 50 Together t t Emple : Sherry cn mow lwn in 5 hours. Mry cn mow the sme lwn in 4 hours. Determine how long it would tke to mow the lwn if Sherry nd Mry worked together. Time to mow lwn (hours) Frction of lwn mowed in hour Frction of lwn moved in t hours Sherry Mry Both
22 Chpter 4 Mth 0 Emple: Gerrd tkes 5 hours longer thn Hubert to ssemble ply set. If Gerrd nd Hubert worked together, they could ssemble the ply set in 6 hours. Determine how long it tkes ech person to ssemble the ply set if they worked lone. Time (hours) Frction in hour Frction in t hours Gerrd Hubert Both
23 Chpter 4 Mth 0 Emple 4: Cmeron nd Zch work t grge chnging tires on crs. It tkes Cmeron 0 minutes longer to chnge tires on cr. If both of them working together cn chnge the tires on cr in minutes, lgebriclly determine how long it tkes ech person to chnge tires if he ws working lone. Time (hours) Frction in hour Frction in t hours Cmeron Zch Both
24 Chpter 4 Mth 0 4 Emple 5: It tkes Sheryl twice s long to clen house s Krist. If they both work together, they would clen the house in four hours. Determine how long it tkes ech person to clen the house individully. Time (hours) Frction in hour Frction in t hours Sheryl Krist Both
25 Chpter 4 Mth 0 5 Note: For the remining questions, the eqution will be embedded in the question. Emple 6: A skiing club is going on skiing trip tht costs $500 totl for bussing. If 0 non-members re llowed to go, the price per person drops by $5. If represents the number of members nd the sitution is modelled by determine how mny members there re., lgebriclly Emple.7: Priddle Inc. is hving Christms prty for ll of its employees. Initilly, ll employees gree to ttend. The cost of ctering is $800, which is to be divided mongst ll people who ttend the prty. At the lst minute, 0 people decide not to come, incresing the cost per people by $. If represents the number of employees nd the sitution is modelled by people who re employed t Priddle Inc., lgebriclly determine the number of Questions pge 59 #0- nd ppliction prctice sheet
26 Chpter 4 Mth Appliction Prctice. A lrge field needs to be mowed. If Amber uses push mower nd Stephnie uses riding lwn mower, the lwn cn be finished in 4 hours. If Amber uses push mower only, it tkes her 5 hours longer to mow the lwn. Algebriclly determine the time it tkes to mow the lwn with the riding mower only.. Tyson cn wsh nd dry sink full of dishes in 0 minutes. It tkes Jeffrey three times s long to wsh nd dry the sme sink full of dishes. Algebriclly determine how long it tkes to wsh nd dry the dishes if they work together.. Tmr is n epert t crosswords, but Emily is novice. It tkes Emily n verge of 0 minutes longer to complete medium level puzzle thn Tmr. If they both work together, it tkes them totl of 8 minutes. Determine how long it tkes ech person to complete the crossword individully. 4. A group of students from Pyne Acdemy re going on field trip to St. John s. The bus hs fied cost of $00, which is to be divided eqully mongst ll the students. To help sve money, group of 7 students from Regulr High School re invited to go, which reduces the cost per student by $0. If represents the number of students t Pyne Acdemy nd the sitution is modelled by , lgebriclly determine the number of students from Pyne 7 Acdemy tht went on the trip. 5. Rim bought cse of concert T-shirts for $450. She kept two shirts for herself nd sold the rest for $560, mking profit of $0 on ech shirt. If represents the originl number of shirts in the cse, nd her profit is modelled by the equtions , lgebriclly determine how mny shirts were in the cse. 6. When they work together, Sturt nd Lucy cn deliver flyers to ll of the homes in the neighbourhood in 4 minutes. When Lucy works lone, she cn deliver the flyers in minutes less time thn Sturt cn when he works lone. Determine how long it tkes Sturt to deliver the flyers when he works lone. Answers:.5 hours. 5 min. Tmr 0 min Emily 40min 4. 5 students 5. 8 shirts 6. 9min
27 Chpter 4 Mth 0 7 Chpter 4 Review. Find ech product or quotient nd epress in simplest form. (A) (B) 6 5 (C) (D) (E) (F) 4 y y. Find the sum or difference nd epress in simplest form. (A) n n n 6 (B) 8 6 m m m m m (C) 6 5 (D). Completely simplify ech of the following comple rtionl epressions. (A) (B) 4. Solve. (Be sure to check for etrneous roots.) (A) (B) 5 y y y (C) (D) 5 5 ANSWERS:.() 5 (b) (c) 4 (d) 5 4 (e) 5) ( 5 (f) y.() (b). () ) ( n n n (b) 4) ( m m (c) (d) 4.() no solution (b) 5 (c) 7 (d)
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