Sixth Form Algebra Induction Booklet

Transcription

1 Sith Frm Algebr Inductin Bklet Mthemtics Deprtment St. Olve s Grmmr Schl

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3 Cntents Intrductin... A. Epnding brckets... 5 B. Fctrising epressins... 6 Cmmn fctrs... 6 Fur-term epressins... 6 Qudrtics... 6 Using the difference f tw squres... 6 When the cefficient f is ne... 6 When the cefficient f is nt ne... 6 C. Qudrtic equtins... 7 Fctristin... 7 The qudrtic frmul... 7 D. Mnipulting frmule... 8 Single ccurrences... 8 Multiple ccurrences... 8 E. Indices... 9 Definitins... 9 Inde lws... 9 F. Surds... 0 Surd lws... 0 Rtinlising the denmintr... 0 G. Cmpleting the squre... The cse when =... The cse where... H. Algebric frctins... Stndrd pertins... Slving equtins invlving lgebric frctins... I. Simultneus equtins... Elimintin... Substitutin... 5 Nn-liner simultneus equtins... 5 J. Inequlities... 6 Liner inequlities... 6 Qudrtic inequlities... 6 Assessment questins... 8 Slutins... 0

4 Intrductin This bklet hs been designed t id the trnsitin f students entering the Sith Frm. Mny res f AS nd A Mthemtics require the cnfident use f lgebric cncepts. Our eperience hs tught us tht withut such cnfidence, mny students will struggle nd perhps nt mke the prgress tht they shuld. It is hped tht the cmpletin f this bklet will prvide severl benefits, The student s new techers will be ble t btin n verview f the student s res f strength nd wekness; Mthemticl skill levels will be mintined during the summer hlidy; Students will cnslidte nd perhps enhnce their knwledge f key lgebric cncepts. At the bck f this bklet yu will find the eercises tht need t be cmpleted befre September, Single Mthemticins shuld ttempt sectins A t E (inclusive); whilst Duble Mthemticins shuld ttempt ll sectins. All wrk shuld be cmpleted n A lined pper, with yur nme clerly written t the tp. Slutins need t be well-structured nd presented in net, rderly mnner. The slutins t mst f the eercises pper t the bck f the bklet; cnsequently, it is epected tht students will prduce full slutins nd shw ll wrkings. Filure t d s my result in students being sked t repet the wrk. It is ls epected tht students tick slutins tht they cmplete successfully; thus llwing their techers t fcus n the res f difficulty when the wrk is cllected in. The cmpleted wrk will be cllected in yur first Mthemtics lessn in September. There will be n ssessment f yur lgebric skills in the first cuple f weeks f term. Plese nte tht the successful cmpletin f this bklet is requirement fr entry int Mthemtics t St Olve s. There re ntes nd emples cntined within this bklet t help yu. Sme f the eercises re demnding. D nt pnic. Persevere. Yu will feel the benefits lter in the curse. Gd luck, Jmes Dvis Hed f Mthemtics

5 A. Epnding brckets We ften need t epnd (multiply ut) brckets in rder t simplify n epressin. Vrius methds my be emplyed, mny peple use the FOIL methd (shwn belw), but the key pint t remember is tht everything n the inside needs t be multiplied by everything n the utside. ( + y ) = + y Epnd ( + ) ( ) using the FOIL methd ( First, Outside, Inside, Lst ) ( + ) ( ) = + 6 = 6 With cre, we cn epnd brckets cntining ny number f terms, ( b c)( b c) ( b c) b( b c) c( b c) 6b 9c b b bc c bc c b c 5b 5bc 8c Certin results re imprtnt nd it is wrth the effrt t lern them ( b) b b b b b ( b) b b b b b ( b)( b) b b b b Therefre, ( y) ( ) ( )( y) ( y) 9 y y ( y)( y) ( ) ( y) 9 y 5

6 B. Fctrising epressins Whilst it is imprtnt t be ble t epnded brckets, it is pssibly mre-imprtnt t be ble t reverse the prcess; tht is, t be ble t fctrise n epressin. Cmmn fctrs Sme epressins cn be fctrised by identifying cmmn fctrs. y = ( y ) = ( y ) this epressin hs tw cmmn fctrs Fur-term epressins Sme epressins cn be fctrised by gruping in pirs. + y b 6by = ( + y) b ( + y) = ( b) ( + y) ( + y) is nw cmmn fctr Qudrtics Depending n the prticulr qudrtic, the prcess f fctristin my be esy r difficult. Using the difference f tw squres Be n the lk ut fr these situtins, y = ()² (y)² = ( + y)( y) 8 50 = ( 5 ) = ( + 5 )( 5) When the cefficient f is ne Simply find tw numbers tht multiply t give the cnstnt nd sum t give the cefficient f + 6 = ( + ) ( ) multiply t give 6 nd dd t give +; i.e. nd When the cefficient f is nt ne This is mre-difficult. Fr emple, if we needed t fctrise 5, the slutin culd be f the frm ( +?)( +??) r ( +?)( +??). If yu re lucky yu might be ble t spt the crrect fctristin, but mst peple wuld hve t resrt t the fllwing lgrithm.. Multiply the cefficient f by the cnstnt term 5 = 60. Find fctrs f 60 tht sum t give the cefficient f (i.e. ) =. Split the middle term using these numbers Fctrise the first tw terms nd then the lst tw terms ( + ) 5 ( + ) 5. Cmplete the fctristin esy! ( 5) ( + ) 6

7 C. Qudrtic equtins The bility t clculte the rts f qudrtic equtin is etremely useful. Qudrtic equtins ccur in the mst unlikely res f mthemtics the flight f prjectile, fr emple. Plese nte tht the prblem my require yu t rerrnge n equtin int the frm + b + c = 0 befre ttempting t slve it Fctristin We cn use fctristin (see bve fr detils) t slve b c 0. Slve 6 0 As we hve seen bve, 6 0 ( )( ) 0 Nw if the left-hnd side is equl t zer, either ( + ) = 0 r ( ) = 0 Therefre, the rts f the equtins re = nd =. The slutin t the equtin is the set {,}; i.e. ll rts t the equtin. Slve 5 0 As we hve seen bve, 5 0 ( 5)( ) 0 Emplying the sme lgic s befre we see tht the rts re = ½ nd = ½ The qudrtic frmul This cn be used t slve qudrtic equtins by inputting the cefficients f fllwing equtin: b c 0 int the b b c Slve + = 0 Creful Here =, b = nd c = b b c ( ) r.56 ( sf ) 7

8 D. Mnipulting frmule In the frmul A = r², A is the subject f the frmul. Single ccurrences Mke the subject f s = ut + ½ t² s ut t [ nly ccurs in ne term islte it] s ut t Mke h the subject f S r h r S r h r S r h r S h r r h S r h r S r r [islte the ] [squre bth sides] Multiple ccurrences With multiple ccurrences, cllect ll ccurrences f the relevnt vrible n ne side f the equtin nd fctrise. Mke the subject f All the s re n ne side; nw we cn fctrise y y y( ) y y y y y y ( y ) y y y 8

9 E. Indices Definitins In m, is the bse nd m is the inde. Plese nte tht the plurl f inde is indices, nt indicies. Inde lws If tw quntities re in the sme bse then the fllwing rules pply: m n ( mn) m n mn ( ) m m n ( mn) n 0 m n n m D nt cnfuse these tw rules Questins my require yu t cnvert ll quntities t the sme bse nd/r cmbine severl f the rules bve. Find the vlue f () 8, (b) 8, (c) Evlute ( 6) We cn either cube 6 nd then find the furth-rt; r we cn find the furth-rt f 6 nd cube the nswer. Obviusly, ne ptin is much esier thn the ther. 9

10 F. Surds A surd is n irrtinl number. Often it includes the psitive rt f nn-squre number; fr emple, nd ( 5 ) re surds, but is nt surd, s =. Surd lws b b b b When simplifying surds it is imprtnt t try nd identify squre fctrs. Emples: Simplify the fllwing Squre number () (b) 7 (c) ( ) ( ) 7 9 ( ) ( ) 8 ( ) ( ) (6 ) ( ) (8 8 ) ( ) Mny peple re cnfused by this simplifictin, but 8 + = whether is rtinl r nt Rtinlising the denmintr It is preferble t hve rtinl denmintr; therefre, if the denmintr is irrtinl we must rtinlise it. Rtinlise the denmintrs in the fllwing qutients By multiplying tp nd bttm by, we rtinlise the denmintr

11 G. Cmpleting the squre The prcess f cmpleting the squre invlves re-writing ( + b + c ) s ( + p)² + q ; tht is, squre plus n djustment. The cse when = Fr emple, let s put the qudrtic equtin = 0 int cmpleted squre frm. Clerly, if we wish t end up with, we need t begin with ( )² ( )² = +, which is nerly the qudrtic required Hwever, we dn t wnt +, we wnt nd s we must subtrct 7 this is the djustment. = ( )² 7 Cmplete the squre fr 6 If the cefficient f is ne then the number in the brcket is hlf f the cefficient f The cse where In the cse where, we strt by tking ut s fctr; then we cmplete the squre fr the qudrtic inside the brcket; befre finlly multiplying ut. Cmplete the squre fr The first step is tke ut the cefficient f s fctr ( ) Nw we cmplete the squre s befre As befre, this number is hlf the cefficient f ; i.e. hlf f ½ Finlly, multiply ut t leve the qudrtic in cmpleted squre frm Rther thn trying t wrk ut the djustment in ne g, simply subtrct the cnstnt frm the squre.

12 H. Algebric frctins Algebric frctins my be delt with in the sme wy s numericl frctins. The key pints t remember re: Fctrise ll numertrs nd denmintrs befre prceeding; be n the lk ut fr qudrtics tht cn be fctrised using the difference f tw squres; Fr dditin / subtrctin, find the lwest cmmn multiple (LCM) fr the denmintr; Fr multiplictin, cncel cmmn fctrs in the numertr nd denmintr befre multiplying; Fr divisin, chnge t multiplictin sign, invert the secnd frctin nd prceed s fr multiplictin. Stndrd pertins ( ) ( )( ) [cncelling the cmmn fctr] 6 ( ) [Fctrise denmintr] ( ) ()( ) ( )( ) ( )( ) [LCM is ()( )] ( ) 6( ) ()( ) [cmbine frctins] 06 ()( ) [simplify] 8 8 ( )( ) ( )( ) ( ) ( 6) [Fctrise] ( )( ) ( 8)( ) [LCM is ( )( )( )] ( )( )( ) ( )( )( ) ( )( )( ) 89 ( )( )( ) ( )( 9) ( )( )( ) 9 ( )() [Multiply ut] [Fctrise] [Simplify]

13 ( )( ) ( )( ) 5 6 ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) [cncel cmmn fctrs] 6 9 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) [Fctrise] [Invert] [cncel] [simplify] Slving equtins invlving lgebric frctins When slving equtins invlving lgebric frctins, multiply thrugh by suitble fctrs t remve the denmintrs. [Multiply bth sides by ( ) nd ()] ( ) ( )( ) ( )( ) ( ) ( ) 5 0 [Frm the qudrtic] ( )( ) 0 r

14 I. Simultneus equtins A liner equtin in the frm y = m + c represents stright line with grdient m nd y-intercept (0, c). If we hve tw such liner equtins, which re simultneusly true, then there re three pssible utcmes. i). ii). iii). There my be unique slutin; the lines intersect t ne pint. There my be n infinite number f slutins; bth equtins refer t the sme line. There is n slutin; the lines re prllel. i) ii) iii) A simultneus equtin my be slved by either: Elimintin Multiply ne equtin t ensure tht yu hve the sme number f s r ys in ech equtin, then dd r subtrct s required. Fr emple, slve + y = nd + 5y = y (A) 5y (B) (A) 6 y 8 (C) (B) 65y (D) (D) (C) y y

15 Substitutin Rerrnge ne f the equtins t mke either r y the subject, then substitute this epressin int the ther equtin. Fr emple, slve + y = nd + 5y = y y 5y 8 y 5y y y y Nn-liner simultneus equtins The methd f substitutin is the methd t use t slve pir f simultneus equtins when ne f the equtins is nn-liner. Fr emple, slve the simultneus equtins + y = nd + y = Here we culd mke either r y the subject f the liner equtin. Obviusly, if we chse t mke the subject, it will result in much esier equtin. y y y ( y) y y y y 5y y y 0 y.7 r 0.7 ( sf ) Grphicl depictin f slutin Hving fund ne f the vribles it is VERY imprtnt tht yu substitute bck int the LINEAR equtin t find the crrespnding vlues f the ther vrible. When y.7.5 When y 0.7.9( sf ) Yu must stte which ges with which y. 5

16 J. Inequlities Liner inequlities A liner inequlity cn be treted s liner equtin with ne imprtnt eceptin if yu multiply / divide n inequlity by negtive quntity, the sign f the inequlity reverses. Fr emple, it is true tht > but it wuld nt be true t sy > ; this is why the sign must be reversed. Emple Slve + Remember, if yu begin with n r equls t inequlity then yu must end up with n r equls t inequlity, nt strict equlity (, > >, etc, BUT > ) Hpefully, yu will nt need reminding tht questin invlving n inequlity NEVER EVER ends with equls Nt = ½!!! Qudrtic inequlities A qudrtic inequlity my ls be treted s qudrtic equtin with the nrml eceptins. It is imprtnt t relise, hwever, tht the resulting nswer will ne f tw pssible frms: < < b < r > b N ther frm is cceptble!!! Often students invent their wn nttin such s > > 5, which suggests tht > 5, they men < nd > 5. Only the tw frms given bve re permissible. Emples Slve 5 > 0 Fctrising gives ( 5)( + ) > 0 Clerly = nd = 5 re criticl vlues 5 is psitive ( ) prbl. We require the sectin tht is greter thn zer; i.e. the sectin bve the -is Therefre, we require the utside re; viz. < nd > 5. The curve is > 0 in the shded regin 6

17 Lk t the inequlity sign Slve 0 0 Fctrising gives ( 5)( + ) 0 The criticl vlues re = 5 nd = This is psitive ( ) prbl We require the sectin beneth the -is Therefre, we chse the middle sectin Tht is, 5 Slve 9 0 Fctrising gives ( )( + ) 0 The criticl vlues re ± ½ This is negtive ( ) prbl We require the sectin bve the -is Therefre, we chse the middle sectin Tht is, ½ ½ Slve Rerrnge t give This cnnt be fctrised Therefre use b b c The criticl vlues re 5 7 This is psitive ( ) prbl We require the sectin bve the -is Therefre, 5 7 nd 5 7 As yu cn see, it is nt lwys cler which re yu require. It is very imprtnt t drw sketch f the grph t vid mking silly mistkes. 7

18 Assessment questins A. Epnding brckets Epnd nd simplify the fllwing. NB n nswers given.. ( + )( 5). ( y)( + y). ( + )( ). ( + b)² 5. ( 7)( + ) 6. ( 7)( ) 7. ( b + c)( + b c) 8. (5 9)² 9. ( + 7)( 7) 0. ( + + )( ) B. Fctrising epressins Fctrise the fllwing epressin. NB n nswers given t² + 5t 6. d + bd c bc y y + y. ( + )² y p² q² 5p + 5q C. Qudrtic equtins Slve the fllwing equtins. Give nswers t sf where pprprite. NB nswers nw given n finl pge = 0. 8 =. + = = = = = = 0 D. Mnipulting frmule Mke the letter in the [brcket] the subject f the frmule.. v u s [ ]. l [ ]. T [ l] g s ut t u n S ( n ) d [ d]. 5. S [] r 6. r E. Indices Evlute the fllwing y [ ] P p 7. e [ p] PT pt [ u] u v f

19 Duble Mthemticins nly, frm this pint frwrd. F. Surds Simplify the fllwing. Full wrkings re epected G. Cmpleting the squre Cmplete the squre fr the fllwing H. Algebric Frctins Simplify Q t Q5; Slve the equtins in Q6 nd Q I. Simultneus equtins Slve the fllwing equtins.. 5 y. y y9 6 y 5 y y y 0 y y7. y5 y 7. 9y 7 y. y 6 y 8. y y 7 J. Inequlities Slve the fllwing inequlities. ( ) > ( + ).. ( + )( )< < 0 6. > 9 9

20 Slutins A. Epnding brckets Nne given. B. Fctrising epressins Nne given. C. Qudrtic equtins.,.,. /,. 5/, 5. /, / , ,.9 8., / 9..9,. 0.,,, D. Mnipulting frmule v u s t gt ( S n).. u. l. d s t nn ( ) S y P( et ) fv 5. r p 8. u S y et v f E. Indices. /5.. / / / F. Surds ( ) / / G. Cmpleting the squre. ( )² 7. ( + 5)² 5. (.5)² ( )² 5. 5(.5)² 89 / 6. ( )² + H. Algebric frctins ( )( 5) 7.. (5)(6) , , /. ( )( ) I. Simultneus equtins. (8, ). ( 0.5, ). (.5,.5). (, 5) (.5, 6) 5. (, ) (9, /) 6. (, 5) (5, 6) 7. (, ) 8. (, ) ( 8, ) J. Inequlities. < < <., 5..5 < < 0 6. <.5, >.5 0