Sith Frm Algebr Inductin Bklet Mthemtics Deprtment St. Olve s Grmmr Schl
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
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
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
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. ) + 6 0 =. Split the middle term using these numbers + 6 0 5. Fctrise the first tw terms nd then the lst tw terms ( + ) 5 ( + ) 5. Cmplete the fctristin esy! ( 5) ( + ) 6
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. 67 9 6 5 5 76 0 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 ( ) 7 0.56 r.56 ( sf ) 7
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
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, 8 8 9 (b) 8, 8 8 7 (c) 8. 8 7 8 8 Evlute 6 6 6 6 8 ( 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
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 () 7 7 6 6 (b) 7 (c) ( ) ( ) 7 9 ( ) ( ) 8 ( ) ( ) (6 ) ( ) (8 8 ) ( ) 8 8 6 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. 6 6 6 0
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 6 0 9 0 8 0 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. 6 7 6 7 8 Rther thn trying t wrk ut the djustment in ne g, simply subtrct the cnstnt frm the squre.
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]
( )( ) ( )( ) 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
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
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 0 6 60 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
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
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 + 5 5 Rerrnge t give + 5 + 0 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
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.. + 6 + 5. 8 0. t² + 5t 6. d + bd c bc 5. + + 5 6. 5 7 + 6 7. 7 6 8. y 6 9. 5 + 0. 5 5. 9 y + y. ( + )² y. 8 9. 6 9 + 0 5. p² q² 5p + 5q C. Qudrtic equtins Slve the fllwing equtins. Give nswers t sf where pprprite. NB nswers nw given n finl pge.. + 5 + 6 = 0. 8 =. + = 0. + 5 = 0 5. 6 + + 6 = 0 6. + 7 + = 0 7. + = 8 8. 9. 0. 7 + 6 = 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. 5. 5. 5 6. y [ ] 6. 9. 6 0. P p 7. e [ p] PT pt 9 7. 8. 8. [ u] u v f 8. 0 0 0 8 6 8 5 8
Duble Mthemticins nly, frm this pint frwrd. F. Surds Simplify the fllwing. Full wrkings re epected.. 8. 5. 8. 99 5. 8 8 6. 7 7. 6 8. 9. 0. G. Cmpleting the squre Cmplete the squre fr the fllwing. 8 + 9. + 0. +. 5. 5 5 + 9 6. + H. Algebric Frctins Simplify Q t Q5; Slve the equtins in Q6 nd Q7. 5 5. 5 6. 9 8 6. 9 5. 5 7 6 9 6. 8 7. 6 I. Simultneus equtins Slve the fllwing equtins.. 5 y. y y9 6 y 5 y y 0 5. 6. y 0 y y7. y5 y 7. 9y 7 y. y 6 y 8. y y 7 J. Inequlities Slve the fllwing inequlities. ( ) > ( + ).. ( + )( )< 0. + 0 5. + < 0 6. > 9 9
Slutins A. Epnding brckets Nne given. B. Fctrising epressins Nne given. C. Qudrtic equtins.,.,. /,. 5/, 5. /, / 6. 6.70, 0.98 7..89,.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 6. 7. p 8. u S y et v f E. Indices. /5.. /. 000 000 5. / 6. 6 6 7. 8. 9 9 9. 8 0. / F. Surds. 7. 6.. 0 5. 6. 7 7. 8. ( ) / 9. 5 0. 7 / G. Cmpleting the squre. ( )² 7. ( + 5)² 5. (.5)² + 0.75. ( )² 5. 5(.5)² 89 / 6. ( )² + H. Algebric frctins. 5. 9 5 ( )( 5) 7.. (5)(6) 6. 0.5,.75 7. 0, /. ( )( ) I. Simultneus equtins. (8, ). ( 0.5, ). (.5,.5). (, 5) (.5, 6) 5. (, ) (9, /) 6. (, 5) (5, 6) 7. (, ) 8. (, ) ( 8, ) J. Inequlities. < 0... 0.5 < <., 5..5 < < 0 6. <.5, >.5 0