8Algebraic UNCORRECTED SAMPLE PAGES. techniques. What you will learn. Australian curriculum. Chapter 8A 8B 8C 8D 8E 8F

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1 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K Chptr Wht you will lrn 8Algri thniqus Epning inomil prouts Prt squrs n irn o prt squrs Ftorising lgri prssions Ftorising th irn o two squrs Ftoristion y grouping Ftorising qurti trinomils (Etning) Ftorising trinomils o th orm + + (Etning) Simpliying lgri rtions: multiplition n ivision Simpliying lgri rtions: ition n sutrtion Furthr simpliition o lgri rtions (Etning) Equtions with lgri rtions (Etning) Austrlin urriulum NUMBER AND ALGEBRA Pttrns n lgr Apply th istriutiv lw to th pnsion o lgri prssions, inluing inomils, n ollt lik trms whr pproprit Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

2 Fr lling Th istn ( units) o n ojt rom th top o uiling tr it hs n ropp (whr ir rsistn is ngligil) n oun using th ormul ut + _ 1 t whr u is th initil vloity o th ojt, t th tim sin th ojt hs n ropp n th lrtion u to grvity, whih is pproimtly qul to 9.8 m/s. Whn n ojt is ropp it hs n initil vloity o 0 m/s, so th istn th ojt hs lln oms.9t. Using lgr, th istn rom th uiling tr t sons n oun or th tim tkn to rh groun lvl oul lult. I th ojt is inst ropp rom hot ir lloon sning t 10 m/s, th ojt i rst trvls in n upwr irtion. Its istn ( mtrs) ov or low th hight o th lloon rom whn th ojt is ropp n oun using 10t.9t. Knowing th tim tkn or th ojt to rh th groun, w oul gin us lgr to i n tors, suh s th hight o th lloon, th grtst hight rh y th ojt n th tim tkn or th ojt to rturn to th hight rom whih it ws rls. Onlin rsours Chptr pr-tst Vios o ll work mpls Intrtiv wigts Intrtiv wlkthroughs Downlol HOTshts Ass to HOTmths Austrlin Curriulum ourss Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

3 88 Chptr 8 Algri thniqus 8A Ky is Epning inomil prouts A inomil is n prssion with two trms suh s + or +. You will rll rom Chptr tht w look t th prout o singl trm with inomil prssion,.g. ( ) or ( 1). Th prout o two inomil prssions n lso pn using th istriutiv lw. This involvs multiplying vry trm in on prssion y vry trm in th othr prssion. Lt s strt: Rtngulr pnsions Epning th prout o two inomil prssions n ppli to prolms involving th pnsion o rtngulr rs suh s rmr s pok. I ( + 1) n ( + ) r th si lngths o rtngl s shown, th totl r n oun s n prssion in two irnt wys. Writ n prssion or th totl r o th rtngl using lngth ( + ) n with ( + 1). Now in th r o h o th our prts o th rtngl n omin to giv n prssion or th totl r. Compr your two prssions ov n omplt this qution: ( + )( ) + +. Cn you plin mtho or pning th lt-hn si to giv th right-hn si? Epning inomil prouts uss th istriutiv lw. ( + )( + ) ( + ) + ( + ) Digrmmtilly ( + )( + ) For mpl: ( + 1)( + ) Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

4 Numr n Algr 89 Empl 1 Epning inomil prouts Epn th ollowing. ( + )( + ) ( )( + 7) ( 1)( 6) ( )( + 7) SOLUTION ( + )( + ) ( )( + 7) ( 1)( 6) ( )( + 7) Eris 8A Th givn igrm shows th r ( + )( + ). + + EXPLANATION Us th istriutiv lw to pn th rkts n thn ollt th lik trms n. Atr pning to gt th our trms, ollt th lik trms 7 n. Rmmr n 1 ( 6) 6. Rll 1. Writ own n prssion or th r o h o th our rgions insi th rtngl. Copy n omplt: ( + )( + ) Th givn igrm shows th r ( + )( + 1). 1 (½) Writ own n prssion or th r o h o th our rgions insi th rtngl. Copy n omplt: ( + )( ) UNDERSTANDING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

5 90 Chptr 8 Algri thniqus 8A Empl 1 Empl 1,, Copy n omplt ths pnsions. ( + 1)( + ) ( )( + ) ( )(7 + ) + 6 ( 1)( ) + + (½) (½) Epn th ollowing. ( + )( + ) ( + )( + ) (t + 8)(t + 7) (p + 6)(p + 6) ( + 9)( + 6) ( + 1)( + ) g ( + 1)( + 7) h (y + 10)(y + ) i (m + )(m + 1) Epn th ollowing. ( + )( ) ( + )( ) ( + )( 8) ( 6)( + ) ( 1)( + 10) ( 7)( + 9) g ( )( + 7) h ( 1)( ) i ( )( ) j ( + )( + ) k ( + )( + 1) l ( + 1)( + ) m ( )( + ) n (8 )( + ) o ( )( + 1) p ( + )( 7) q ( + )( ) r ( + 1)( ) s ( )(6 ) t ( )( 1) u (7 )( ) 6 Epn ths inomil prouts. ( + )( + ) ( )( + ) ( )( + ) ( y)(y z) (y )(z y) (1 )(1 + y) g ( + y)( y) h ( + )( ) i ( y)( + y) j ( )( + ) k ( y)( y) l (y yz)(z + ) 7 A room in hous with imnsions m y m is to tn. Both th lngth n with r to inrs y m. Fin n pn prssion or th r o th nw room. I : i in th r o th nw room ii y how muh hs th r inrs? 7 7, 8 6(½) 8, 9 UNDERSTANDING FLUENCY PROBLEM-SOLVING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

6 Numr n Algr 91 8 A pitur rm m wi hs lngth whih is twi th with m. Fin n prssion or th totl r o th rm n pitur. Fin n prssion in pn orm or th r o th pitur only. Pitur m m 9 Th outsi g o pth roun rtngulr swimming pool is 1 m long n 10 m wi. Th pth is mtrs wi. m Fin n prssion or th r o th pool in pn orm. Pool Fin th r o th pool i. 1 m 10(½) 10(½), Writ th missing trms in ths pnsions. ( + )( + ) ( + )( + ) ( + 1)( + ) ( + )( + 9) ( + )( ) + ( )( + ) g ( + 1)( + ) + + h ( )( 1) 9 + i ( + )( + ) j ( )( 1) Consir th inomil prout ( + )( + ). Fin th possil intgr vlus o n i: ( + )( + ) ( + )( + ) + 6 ( + )( + ) + 6 ( + )( + ) 6 Trinomil pnsions 1 Using th istriutiv lw ( + )( + + ) Us this knowlg to pn n simpliy ths prouts. Not:. ( + 1)( + + 1) ( )( + ) ( 1)( + ) ( + 1)( + ) ( + )( ) ( + 7)( 7) g ( + )( + ) h ( )( ) i ( + )( + ) j ( )( + + ) 1 Now try to pn ( + 1)( + )( + ). 10(½), 11 m 10 m 1, 1 PROBLEM-SOLVING REASONING ENRICHMENT 8A Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

7 9 Chptr 8 Algri thniqus 8B Prt squrs n irn o prt squrs W know tht, 1, n ( + ) r ll mpls o prt squrs. To pn ( + ) w multiply ( + ) y ( + ) n us th istriutiv lw: ( + ) ( + )( + ) A similr rsult is otin or th squr o ( ): ( ) ( )( ) + + Anothr typ o pnsion involvs th s tht ls with th prout o th sum n irn o th sm two trms. Th rsult is th irn o two prt squrs: ( + )( ) + Binomil prouts n us to lult th most iint wy to ut th shps rquir or rition out o mtl sht. (sin, th two mil trms nl h othr out.) Lt s strt: Sing th pttrn Using ( + )( + ) + + +, pn n simpliy th inomil prouts in th two sts low. St A ( + 1)( + 1) ( + )( + ) ( )( ) St B ( + 1)( 1) + 1 ( )( + ) ( )( + ) Dsri wht pttrns you s in oth sts o pnsions ov. Gnrlis your osrvtions y omplting th ollowing pnsions. A ( + )( + ) B ( + )( ) + ( )( ) + + Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

8 Numr n Algr 9 9,, (y), ( 1) n ( y) r ll mpls o prt squrs. Epning prt squrs ( + ) ( + )( + ) ( ) ( )( ) + + Dirn o prt squrs (DOPS) ( + )( ) + ( )( + ) lso pns to Th rsult is irn o two prt squrs. Empl Epning prt squrs Epn h o th ollowing. ( ) ( + ) SOLUTION ( ) ( )( ) + + Altrntiv solution: ( ) + + ( + ) ( + )( + ) Altrntiv solution: ( + ) () EXPLANATION Writ in pn orm. Us th istriutiv lw. Collt lik trms. Epn using ( ) + whr n. Writ in pn orm. Us th istriutiv lw. Collt lik trms. Epn using ( + ) + + whr n. Rll (). Ky is Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

9 9 Chptr 8 Algri thniqus Empl Forming irn o prt squrs Epn n simpliy th ollowing. ( + )( ) ( y)( + y) SOLUTION ( + )( ) + Altrntiv solution: ( + )( ) () () EXPLANATION Epn using th istriutiv lw ( + )( ). Hr n. ( y)( + y) 9 + 6y 6y y Epn using th istriutiv lw. 9 y 6y 6y 0. Altrntiv solution: ( y)( + y) () (y) Eris 8B ( + )( ) with n y 9 y hr. 1 Complt ths pnsions. ( + )( + ) ( + )( + ) ( )( ) + ( 7)( 7) 7 + Sustitut th givn vlu o into + + n simpliy. i ii 11 iii 1 Sustitut th givn vlu o into + n simpliy. i ii 9 iii 0 Complt ths pnsions. ( + )( ) + ( 10)( + 10) + 10 ( 1)( + 1) + ( + )( ) (½), (½) UNDERSTANDING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

10 Numr n Algr 9 Empl Empl Empl Empl 7(½) 8(½) 8(½) Epn h o th ollowing prt squrs. ( + 1) ( + ) ( + ) ( + ) ( + ) ( + 9) g ( + 7) h ( + 10) i ( ) j ( 6) k ( 1) l ( ) m ( 9) n ( 7) o ( ) p ( 1) Epn h o th ollowing prt squrs. ( + 1) ( + ) ( + ) ( + 1) ( + ) ( + ) g (7 + ) h ( + ) i ( ) j ( 1) k ( ) l ( 9) m ( + y) n ( + y) o (7 + y) p (6 + y) q ( 9y) r ( 7y) s ( 10y) t ( 6y) u (9 y) 6 Epn h o th ollowing prt squrs. ( ) ( ) (1 ) (6 ) (11 ) ( ) g (7 ) h (1 ) i (8 ) j ( ) k (9 ) l (10 ) 7 8 Epn n simpliy th ollowing to orm irn o prt squrs. ( + 1)( 1) ( + )( ) ( + 8)( 8) ( + )( ) ( + 1)( 1) ( + 11)( 11) g ( 9)( + 9) h ( )( + ) i ( 6)( + 6) j ( )( + ) k ( )( + ) l (7 )(7 + ) Epn n simpliy th ollowing. ( )( + ) ( )( + ) ( )( + ) (7 y)(7 + y) (9 y)(9 + y) (11 y)(11 + y) g (8 + y)(8 y) h (10 9y)(10 + 9y) i (7 y)(7 + y) j (6 11y)(6 + 11y) k (8 y)(8 + y) l (9 y)(9 + y) 9 Lr is yrs ol n hr two st rins r ( ) n ( + ) yrs ol. 9 9, 10 Writ n prssion or: i th squr o Lr s g ii th prout o th gs o Lr s st rins (in pn orm). Ar th nswrs rom prts i n ii qul? I not, y how muh o thy ir? 9, 10 FLUENCY PROBLEM-SOLVING 8B Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

11 96 Chptr 8 Algri thniqus 8B 10 A squr pi o tin o si lngth 0 m hs our squrs o si lngth m rmov rom h ornr. Th sis r ol up to orm try. Th ntr squr orms th try s. Writ n prssion or th si lngth o th s o th try. Writ n prssion or th s o th try. Epn your nswr. Fin th r o th try s i. Fin th volum o th try i. m m Try s 0 m 0 m 11 Four tnnis ourts r rrng s shown with squr storg r in th ntr. Eh ourt r hs th sm imnsions. Writ n prssion or th si lngth o th totl r. Writ n prssion or th totl r. Writ n prssion or th si lngth o th insi storg r. Writ n prssion or th r o th insi storg r. Sutrt your nswr to prt rom your nswr to prt to in th r o th our ourts. Fin th r o on ourt. Dos your nswr onirm tht your nswr to prt is orrt? 1 A squr o si lngth units hs on si ru y 1 unit n th othr inrs y 1 unit. 11 1, 1 Fin n pn prssion or th r o th rsulting rtngl. Is th r o th originl squr th sm s th r o th rsulting rtngl? Eplin why/why not? 1, 1 PROBLEM-SOLVING REASONING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

12 Numr n Algr 97 1 A squr o si lngth is rmov rom squr o si lngth. Using sutrtion writ own n prssion or th rmining r. Writ prssions or th r o th rgions: i A ii B iii C A ll th prssions rom prt to s i you gt your nswr rom prt. Etn pnsions 1 Epn n simpliy ths prssions. ( + ) ( 1) ( + )( ) ( + 1) ( + 1)( 1) ( + 1) ( 1) g ( )( + ) ( + ) h ( 1) ( + 1)( 1) i ( + y) ( y) + ( + y)( y) j ( ) + ( + ) k ( ) ( + ) l ( ) + ( ) m ( ) ( )( + ) n ( + y) ( y) C A B Th logil skills o lgr hv pplitions in omputr progrmming. 1 REASONING ENRICHMENT 8B Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

13 98 Chptr 8 Algri thniqus 8C Ky is Ftorising lgri prssions Th pross o toristion is ky stp in th simpliition o mny lgri prssions n in th solution o qutions. It is th rvrs pross o pnsion n involvs writing n prssion s prout o its tors. pning ( ) 6 torising Lt s strt: Whih toris orm? Th prout ( + 8) whn pn givs + 8. Ftorising is ky mthmtil skill rquir in mny ivrs ouptions, suh s in usinss, sin, thnology n nginring. Writ own thr othr prouts tht whn pn giv + 8. (Do not us rtions.) Whih o your prouts uss th highst ommon tor o n 8? Wht is this highst ommon tor? Whn torising prssions with ommon tors, tk out th highst ommon tor (HCF). Th HCF oul : numr For mpl: + 10 ( + ) pronumrl (or vril) For mpl: + ( + ) th prout o numrs n pronumrls For mpl: + 10 ( + ) A toris prssion n hk y using pnsion. For mpl: ( + ) Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

14 Numr n Algr 99 Empl Fining th HCF Dtrmin th HCF o th ollowing. 6 n 8 SOLUTION n 6y EXPLANATION HCF o 6 n 8 is. HCF o n is. HCF o n 6 is. Empl Ftorising prssions Ftoris th ollowing SOLUTION HCF o n y is. 8 1 EXPLANATION ( ) Th HCF o 0 n 16 is 8. Pl 8 in ront o th rkts n ivi h trm y ( + ) Th HCF o th trms is, inluing th ommon ngtiv. Pl th tor in ront o th rkts n ivi h trm y. Empl 6 Tking out inomil tor Ftoris th ollowing. ( + y) + ( + y) (7 ) (7 ) SOLUTION ( + y) + ( + y) ( + y)( + ) (7 ) (7 ) 1(7 ) (7 ) (7 )(1 ) EXPLANATION HCF ( + y). Th son pir o rkts ontins wht rmins whn ( + y) n ( + y) r ivi y ( + y). Insrt 1 in ront o th irst rkt. HCF (7 ). Th son rkt must ontin 1 tr iviing (7 ) n (7 ) y (7 ). Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

15 00 Chptr 8 Algri thniqus Eris 8C 1 (½), Empl Empl Empl Empl 6 1 Writ own th highst ommon tor (HCF) o ths pirs o numrs. 8, 1 10, 0, 60, 0, 100, 7 g 16, h 6, 7 Writ own th missing tor y 6y 1 6 g 6 h 0 i 7y 1 y Writ own th missing tor in h prt. i ( + ) iii ( + ) ii ( + ) Whih qution ov uss th HCF o 6 n 1? By looking t th trms lt in th rkts, how o you know you hv tkn out th HCF? 7(½) 8(½) Dtrmin th HCF o th ollowing. 6 n 1y 1 n 18 10m n 1y n 8 1t n 6s 1 n p g 9 n y h 6n n 1mn i 10y n y j 8 n 1 k y n 18y l n 1 Ftoris th ollowing y 10 + g 9 h 6 i 1 + j 6m + 6n k 10 8y l 0 m + n o y 7y p q p + p r 8 8 s + 1 t 6y 10y u 1 1 v 9m + 18m w 16y Ftoris th ollowing y toring out th ngtiv sign s prt o th HCF y y 8 g 10 1y h m 0n i 18 j 8 1 k 16y 6y l 10 m 6 0 n 6p 1p o 16 8 p 9 7 Ftoris th ollowing whih involv inomil ommon tor. ( + ) + ( + ) ( + 1) + ( + 1) 7(m ) + m(m ) ( 7) + ( 7) 8( + ) ( + ) ( + 1) ( + 1) g y(y + ) (y + ) h ( + ) ( + ) i t(t + ) + (t + ) j m(m ) + (m ) k y(y 1) (y 1) l (7 ) + (7 ) 8(½) UNDERSTANDING FLUENCY Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

16 Numr n Algr 01 8 Ftoris ths mi prssions y + 9y g y y + y h i m(m + ) + (m + ) j ( + ) ( + ) k ( ) + ( ) l ( + 1) ( + 1) m y( y) ( y) n ( + ) + ( + ) o (y + 1) (y + 1) 9 Writ own th primtr o ths shps in toris orm Th prssion or th r o rtngl is ( + 8) squr units. Fin n prssion or its with i th lngth is ( + ) units Th hight, in mtrs, o ll thrown in th ir is givn y t t, whr t is th tim in sons. Writ n prssion or th ll s hight in toris orm. Fin th ll s hight t ths tims: i t 0 ii t iii t How long os it tk or th ll s hight to rturn to 0 mtrs? Us tril n rror i rquir. 9 9, (½), 10, 11 FLUENCY PROBLEM-SOLVING 8C Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

17 0 Chptr 8 Algri thniqus 8C n vlut y irstly torising to 7(9 + ). This givs Us similr thniqu to vlut th ollowing Common tors n lso rmov rom prssions with mor thn two trms. For mpl: y ( + + y) Ftoris ths prssions y tking out th HCF z 10z + zy y + y y + 6 1y 8yz 0yz Somtims w n hoos to tor out ngtiv or positiv HCF. Both toristions r orrt. For mpl: ( ) (HCF is 1) OR ( + ) (HCF is 1) 1( ) 1 1, 1 Ftoris in two irnt wys: th irst y toring out ngtiv n th son y positiv HCF n m + m g + h y + y i 8n + 1n j 8y + 0 k 1mn + 10 l 1 + Ftoring out ngtiv 1 Using th t tht ( ) you n toris ( ) ( ) y ollowing ths stps. ( ) ( ) ( ) + ( ) ( )( + ) Us this i to toris ths prssions. ( ) + ( ) ( ) ( ) ( ) ( ) ( ) + ( ) ( ) + ( ) ( ) + ( ) g ( ) ( ) h ( ) + (10 ) i ( ) + (6 ) 1, 1 1 REASONING ENRICHMENT Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

18 Numr n Algr 0 8D Ftorising th irn o two squrs Rll tht irn o two prt squrs is orm whn pning th prout o th sum n irn o two trms. For mpl, ( + )( ). Rvrsing this pross mns tht irn o two prt squrs n toris into two inomil prssions o th orm ( + ) n ( ). Lt s strt: Epning to unrstn torising Complt th stps in ths pnsions thn writ th onlusion. ( + )( ) + ( )( + ) ( + )( ) ( + )( ) + ( + )( ) ( + )( ) Ftorising th irn o prt squrs (DOPS) uss th rul ( + )( ). 16 ( + )( ) () 10 ( + 10)( 10) y (y) ( + y)( y) First tk out ommon tors whr possil. 18 ( 9) ( + )( ) Empl 7 Ftorising DOPS Ftoris h o th ollowing y ( + 1) Ky is SOLUTION ( + )( ) EXPLANATION Writ s DOPS ( is th sm s ). Writ in toris orm ( + )( ). Hr n. Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

19 0 Chptr 8 Algri thniqus 9 () ( + )( ) Writ s DOPS. 9 is th sm s (). Writ in toris orm. Empl 7 81 y (9) y (9 + y)(9 y) ( 16) ( ) ( + )( ) ( + 1) ( + 1) Eris 8D ( )( + 1 ) ( + )( 1) 81 (9) Us ( + )( ) First, tor out th ommon tor o. Writ s DOPS n thn toris. Writ s DOPS. In hr, is th prssion + 1 n. Writ in toris orm n simpliy. 1 Epn ths inomil prouts to orm irn o prt squrs. ( + )( ) ( 7)( + 7) ( 1)( + 1) ( + y)( y) ( y)( + y) ( + )( ) Writ th missing trm. Assum it is positiv numr. ( ) 9 ( ) 11 ( ) 81 ( ) 00 ( ) ( ) 9 g ( ) h ( ) 9y Complt ths toristions ( ) ( + )( ) ( + 1)( ) 16 1 ( ) ( ) 9 ( ) ( ) ( + )( 1) 1 (½) (½) ( + )( ) 6(½) 7(½) Ftoris h o th ollowing. 9 y y g 81 h y i j 16 k l 1 m 6 y n 11 o 00 p 900 y 7(½) UNDERSTANDING FLUENCY Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

20 Numr n Algr 0 Empl 7, Empl 7 Empl 7 Ftoris h o th ollowing. 9 9 m y 9 81 g 1 h 6 i 16 9y j 6 y k y l 6 9 m p q n 81m n o 9 p Ftoris h o th ollowing y irst tking out th ommon tor y m g y h y i 6 7 Ftoris h o th ollowing. ( + ) 9 ( + ) ( + 10) 16 ( ) ( 7) 1 ( ) 6 g 9 ( + ) h ( + ) i 81 ( + 8) 8 Th hight ov groun (in mtrs) o n ojt thrown o th top o uiling is givn y 6 t whr t is in sons. Ftoris th prssion or th hight o th ojt y irstly tking out th ommon tor. Fin th hight o th ojt: i initilly (t 0) ii t sons (t ). How long os it tk or th ojt to hit th groun? Us tril n rror i you wish. 9 This multisiz squr pitur rm hs si lngth 0 m n n hol squr pitur with ny si lngth lss thn 6 m. 8 8, 9 I th si lngth o th pitur is m, writ n prssion or: i th r o th pitur ii th r o th rm (in toris orm). Us your rsult rom prt ii to in th r o th rm i: i 0 ii th r o th pitur is m. m 0 m 8, 9 FLUENCY PROBLEM-SOLVING 8D Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

21 06 Chptr 8 Algri thniqus 8D 10 Initilly it my not ppr tht n prssion suh s + 9 is irn o prt squrs. Howvr, swpping th position o th two trms mks + 9 9, whih n toris to ( + )( ). Us this i to toris ths irn o prt squrs y g 16 + y z h Olivi toriss 16 to gt ( + )( ) ut th nswr sys ( + 1)( 1). Wht shoul Olivi o to gt rom hr nswr to th tul nswr? Wht shoul Olivi hv on initilly to voi this issu? 1 Fin n plin th rror in this working n orrt it. 9 ( 1) ( + 1)( 1) ( + )( ) Ftorising with rtions n powrs o 10 10, 11 1 Som prssions with rtions or powrs o n toris in similr wy. Ftoris ths g i y j k 1 1 h l y REASONING ENRICHMENT Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

22 Numr n Algr 07 8E Ftoristion y grouping Whn n prssion ontins our trms, suh s +, it my possil to toris it into prout o two inomil trms lik ( 1)( + ). In suh situtions th mtho o grouping is otn us. Lt s strt: Two mthos sm rsult Th our-trm prssion + is writtn on th or. Ftorising y grouping is it lik rrnging sttr ojts into som sort o orr. Tommy hooss to rrrng th trms to giv + thn toriss y grouping. Shron hooss to rrrng th trms to giv + thn lso toriss y grouping. Complt Tommy n Shron s toristion working. Tommy + ( ) + 1( ) ( )( ) Shron + ( ) ( ) Disuss th irns in th mthos. Is thr ny irn in thir nswrs? Whos mtho o you prr? ( + 1)( ) Ftoristion y grouping is mtho whih is otn us to toris our-trm prssion. Trms r group into pirs n toris sprtly. + 6 Th ommon inomil tor is thn tkn out to omplt th toristion. ( + ) ( + ) Trms n rrrng to ssist in th srh o ommon tor. ( + )( ) Empl 8 Ftorising y grouping Us th mtho o grouping to toris ths prssions SOLUTION ( + ) + ( + 6) ( + ) + ( + ) ( + )( + ) EXPLANATION Group th irst n son pir o trms. Ftoris h group. Tk th ommon tor ( + ) out o oth groups. Ky is + 1 ( + ) + ( 1) ( + ) ( + ) ( + )( ) Group th irst n son pir o trms. Ftoris h group. Tk out th ommon tor ( + ). Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

23 08 Chptr 8 Algri thniqus Empl 9 Rrrnging n prssion to toris y grouping Ftoris using grouping. Empl 8 SOLUTION Altrntivly: ( + 1) 9( + 1) ( + 1)( 9) Eris 8E ( 9) + 1( 9) ( 9)( + 1) EXPLANATION Rrrng so tht h group hs ommon tor. Ftoris h group thn tk out ( + 1). Altrntivly, you n group in nothr orr whr h group hs ommn tor. Thn toris. Th nswr will th sm. 1 (½) 1 Epn h prssion. ( 1) ( + ) (1 ) ( ) ( + ) ( ) g ( ) h y( y) i ( + 1) + ( + 1) j ( ) + ( ) k ( ) ( ) l (1 ) (1 ) Copy n thn ill in th missing inormtion. ( + 1) + ( + 1) ( + 1)( ) ( + ) ( + ) ( + )( ) ( + ) ( + ) ( + )( ) ( + 7) + ( + 7) ( + 7)( ) ( ) + ( ) ( )( ) ( + ) ( + ) ( + )( ) g ( ) ( ) ( )( ) h ( ) + ( ) ( )( ) Tk out th ommon inomil trm to toris h prssion. ( ) ( ) ( + ) + ( + ) ( 7) + ( 7) ( + 1) ( + 1) ( ) ( ) ( + ) ( + ) g ( ) + ( ) h ( + 1) ( + 1) i ( ) + ( ) (½) Us th mtho o grouping to toris ths prssions g h + 1 i j + k + 9 l + 1 Us th mtho o grouping to toris ths prssions. Th HCF or h pir inlus pronumrl y 8z + wy 1wz rs 10r + st t + 1y 9y + (½) 6(½) 6(½) UNDERSTANDING FLUENCY Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

24 Numr n Algr 09 Empl 9 6 Ftoris ths prssions. Rmmr to us tor o 1 whr nssry, or mpl, + ( ) + 1( ) , 8 Ftoris ths prssions y irst rrrnging th trms y + 8y g 6m n + mn h 1p 8r pr + i 16 y 8y Wht pn prssion toriss to th ollowing? ( )( + ) ( )( ) ( + y)( z) ( 1)( + ) ( )( ) ( y)(y + z) g ( + )( + ) h (m )(y + z) 9 Not tht whih n toris y grouping. Us similr mtho to toris th ollowing oul rrrng in two irnt wys or torising. Mtho ( + 7) ( ) Mtho ( ) + 7( ) Copy n omplt oth mthos or th ov prssion. Us irnt rrngmnts o th our trms to omplt th toristion o th ollowing in two wys. Show working using oth mthos. i 6 + ii y 8 + y iii m 1n + 6m 10mn iv m + n mn 6 v vi Mk up t lst thr o your own our-trm prssions tht toris to inomil prout. Dsri th mtho tht you us to mk up h our-trm prssion. Grouping with mor thn our trms 1 Ftoris y grouping. ( ) ( ) ( ) ( + 1) + ( + 1) ( + 1) ( + 1) ( + 1) ( ) ( ) + (1 ) + + ( ) + + g h 6y z + 10yz + y y i 8z y + + y 1 z j + y + + 6y Using CAS lultor 8E: Epning n torising This tivity is in th intrtiv ttook in th orm o printl PDF. 7, 9 10, 11 1 PROBLEM-SOLVING FLUENCY REASONING ENRICHMENT 8E Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

25 10 Chptr 8 Algri thniqus 8F Ftorising qurti trinomils EXTENDING Ky is An prssion tht tks th orm + +, whr n r onstnts, is n mpl o moni qurti trinomil whih hs th oiint o qul to 1. To toris qurti prssion, w n to us th istriutiv lw in rvrs. Consir th pnsion shown t right: I w min th pnsion ov w n s how h trm o th prout is orm. Prout o n is ( + )( ) 8 Prout o n is 8 ( ( ) 8, th onstnt trm) Lt s strt: So mny hois ( ) pning ( + )( ) 8 toris orm pn orm torising ( + )( ) 8 Mi sys tht sin 6 thn + 6 must qul ( )( + ). A n to giv th mil trm, ( +, th oiint o ) Epn ( )( + ) to s i Mi is orrt. Wht othr pirs o numrs multiply to giv 6? Whih pir o numrs shoul Mi hoos to orrtly toris + 6? Wht vi n you giv Mi whn trying to toris ths typs o trinomils? To toris qurti trinomil o th orm + +, in two numrs whih: multiply to giv n to giv. For mpl: 10 ( )( + ) hoos n + sin 10 n + Chk toristion stps y pning. hk: ( )( + ) Writ th tors in ny orr. writ ( )( + ) or ( + )( ) Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

26 Numr n Algr 11 Empl 10 Ftorising qurti trinomils Ftoris h o th ollowing qurti prssions SOLUTION EXPLANATION ( + )( + ) Ftors o 10 inlu: (10, 1) n (, ). Th pir tht s to 7 is (, ). + 8 ( + )( ) Ftors o 8 r ( 8, 1) or (8, 1) or (, ) or (, ) n + ( ) so hoos (, ) ( )( ) Ftors o 10 r: (10, 1) or ( 10, 1) or (, ) or (, ). Empl 11 Ftorising with ommon tor Ftoris th qurti prssion 1. SOLUTION 1 ( 6) Eris 8F ( )( + ) To to ngtiv ( 7), oth tors must thn ngtiv: + ( ) 7 so hoos (, ). EXPLANATION First tk out ommon tor o. Ftors o 6 r: ( 6, 1) or (6, 1) or (, ) or (, ). + 1 so hoos (, ). 1 (½) 1 Epn ths inomil prouts. ( + 1)( + ) ( + )( + 7) ( )( + 11) ( )( + 6) ( + 1)( ) ( + 1)( ) g ( )( 6) h ( 0)( 11) i ( 9)( 1) Di wht two numrs multiply to giv th irst numr n to giv th son numr. 6, 10, 7 1, 1 0, 9, 7, 6 g 1, h 0, 1 i 6, j 18, 11 k 0, 1 l 100, (½) UNDERSTANDING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

27 1 Chptr 8 Algri thniqus 8F Empl 10 Empl 10 Empl 10 Empl 11 Ftoris h o th ollowing qurti prssions g h i j k l Ftoris h o th ollowing qurti prssions g + 18 h i + 1 Ftoris h o th ollowing qurti prssions g h i Ftoris h o th ollowing qurti prssions g 1 h 11 1 i 1 7 6(½) 7(½) Ftoris h o th ollowing qurti prssions y irst tking out ommon tor g + 1 h i j + 90 k 6 0 l 6 7(½) 8 Fin th missing trm in ths trinomils i thy r to toris using intgrs. For mpl: th missing trm in oul 7 us toriss to ( + )( + ) n n r intgrs. Thr my mor thn on nswr in h s g 16 h + 9 A kyr, rtngulr in r, hs lngth mtrs mor thn its with ( mtrs). Insi th rtngl r thr squr pv rs h o r m s shown. Th rmining r is lwn. Fin n prssion or: i th totl kyr r ii th r o lwn in pn orm iii th r o lwn in toris orm. Fin th r o lwn i: i 10 ii (½), 9 ( + ) mtrs 8(½), 9 mtrs FLUENCY PROBLEM-SOLVING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

28 Numr n Algr 1 10 Th prssion toriss to ( )( ) ( ), whih is prt squr. Ftoris ths prt squrs g + + h 0 + i (½), Somtims it is not possil to toris qurti trinomils using intgrs. Di whih o th ollowing nnot toris using intgrs Complting th squr 10(½) 1 It is usul to l to writ simpl qurti trinomil in th orm ( + ) +. This involvs ing (n sutrting) spil numr to orm th irst prt squr. This prour is ll omplting th squr. Hr is n mpl. 6 ( ) ( )( ) 17 ( ) 17 10(½) Complt th squr or ths trinomils REASONING ENRICHMENT 8F Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

29 1 Chptr 8 Algri thniqus 8G Ftorising trinomils o th orm + + EXTENDING Ky is So r w hv toris qurti trinomils whr th oiint o is 1, suh s 0. Ths r ll moni trinomils. W will now onsir non-moni trinomils whr th oiint o is not qul to 1 n is lso not ommon tor to ll thr trms, suh s in Th mtho us in this stion uss grouping whih ws isuss in stion 8E. Lt s strt: How th grouping mtho works Consir th trinomil First writ thn toris y grouping. Not tht 9 ws split to giv + n th prout o n 10 is 0. Dsri th link twn th pir o numrs {, } n th pir o numrs {, 10}. Why ws 9 split to giv + n not, sy, + 6? Dsri how th 1 shoul split in so it n toris y grouping. Now try your mtho or 7 1. To toris trinomil o th orm + + y grouping, in two numrs whih sum to giv n multiply to giv. For mpl: ( 6) 0 so th two numrs r n sin 1 + ( ) 1 n 1 ( ) 0. ( + ) ( + ) ( + )( ) Mntlly hk your tors y pning your nswr. 6 ( + )( ) Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

30 Numr n Algr 1 Empl 1 Ftorising trinomils o th orm + + Ftoris SOLUTION ( + 1) + ( + 1) ( + 1)( + ) EXPLANATION Empl 1 Ftorising trinomils with ngtiv numrs Ftoris th qurti trinomils SOLUTION ( + ) ( + ) ( + )( ) Eris 8G ( ) ( ) ( )( ) 6 thn sk wht tors o this numr (6) to 7. Th nswr is 1 n 6, so split Thn toris y grouping. EXPLANATION 10 ( 9) 90 so sk wht tors o 90 to giv 9. Choos 1 n 6. Thn omplt th toristion y grouping so sk wht tors o 7 to giv 17. Choos 9 n 8. Complt mntl hk. ( )( ) , (½) (½) 1 List th two numrs whih stisy h prt. Multiply to giv 6 n to giv Multiply to giv 1 n to giv 8 Multiply to giv 10 n to giv Multiply to giv n to giv Multiply to giv 18 n to giv 9 Multiply to giv n to giv 1 g Multiply to giv 0 n to giv 7 h Multiply to giv 8 n to giv UNDERSTANDING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

31 16 Chptr 8 Algri thniqus 8G Empl 1 Empl 1 Copy n omplt ( ) + ( ) ( )( ) ( ) + ( ) ( )( ) ( ) ( ) ( )( ) ( ) 1( ) ( )( ) ( + ) + ( ) ( )( ) (½) ( ) + 1( ) ( )( ) (½) Ftoris ths qurti trinomils g h i Ftoris ths qurti trinomils g h i j 1 1 k l m 9 + n o 1 Ftoris ths qurti trinomils g h i 9 j k l Ftoris y irstly tking out ommon tor Ftoris ths trinomils (½) 6(½) (½) 6(½), 7 UNDERSTANDING FLUENCY PROBLEM-SOLVING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

32 Numr n Algr Whn splitting th in + 0, you oul writ: A or B Complt th toristion using A. Complt th toristion using B. Dos th orr mttr whn you split th? Ftoris ths trinomils twi h. Ftoris on y grouping thn rpt ut rvrs th orr o th two mil trms in th irst lin o working. i + 1 ii 1 iii Mk up iv non-moni trinomils with th oiint o not qul to 1 whih toris using th ov mtho. Eplin your mtho in ining ths trinomils. Th ross mtho 10 Th ross mtho is nothr wy to toris trinomils o th orm + +. It involvs ining tors o n tors o thn hoosing pirs o ths tors tht to. For mpl: Ftoris 6 1. Ftors o 6 inlu (, 6) n (, ). Ftors o 1 inlu (1, 1), ( 1, 1), (, ) n (, ). W rrng hosn pir o tors vrtilly thn ross-multiply n to gt ( ) ( 1) ( ) + 6 ( ) , 9 ( + ) You will n to ontinu until prtiulr omintion works. Th thir ross-prout givs sum o 1 so hoos th tors ( + ) n ( ) so: 6 1 ( + )( ) Try this mtho on th trinomils rom Qustions n. 10 REASONING ENRICHMENT 8G Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

33 18 Chptr 8 Algri thniqus Progrss quiz 8pt 8A 8pt 8B 8pt 8B 8pt 8C 8pt 8D 8pt 8E 8pt 8E 8pt 8F Et 8pt 8G Et Epn th ollowing. ( + )( + ) ( )( + 8) ( )( + 6) ( )( ) Epn h o th ollowing. (y + ) ( ) ( ) (7k + m) Epn n simpliy th ollowing. ( + )( ) (11 9y)(11 + 9y) Ftoris th ollowing ( + ) + ( + ) 7(8 + ) (8 + ) k(k ) (k ) Ftoris h o th ollowing y 0 1 y 1 (h + ) 6 Us th mtho o grouping to toris ths prssions Us grouping to toris ths prssions y irst rrrnging p 10 + p 1 Ftoris h o th ollowing qurti prssions m 11m + 0 k + k Ftoris ths qurti trinomils. k + 7k m 19m + 6 h + h Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

34 Numr n Algr 19 8H Simpliying lgri rtions: multiplition n ivision With numril rtion suh s 6, th highst ommon tor o 6 n 9 is, whih n 9 nll For lgri rtions th pross is th sm. I prssions r in toris orm, ommon tors n sily intii n nll. Lt s strt: Corrt nlling Consir this nlling ttmpt: Sustitut 6 into th lt-hn si to vlut Sustitut 6 into th right-hn si to vlut + 1. Wht o you noti out th two nswrs to th ov? How n you plin this? Di how you might orrtly nl th prssion on th lt-hn si. Show your stps n hk y sustituting vlu or. Simpliy lgri rtions y torising n nlling only ommon tors. Inorrt To multiply lgri rtions: toris prssions whr possil nl i possil multiply th numrtors n th nomintors. Corrt + 1 ( + ) 1 + To ivi lgri rtions: multiply y th riprol o th rtion ollowing th ivision sign ollow th ruls or multiplition tr onvrting to th riprol - Th riprol o is. Ky is Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

35 0 Chptr 8 Algri thniqus Empl 1 Simpliying lgri rtions Simpliy th ollowing y nlling. ( + )( ) 6( ) SOLUTION 1 ( + ) ( ) 1 6 ( ) ( ) 1 ( ) ( + )( ) ( + ) EXPLANATION Empl 1 Multiplying n iviing lgri rtions Simpliy th ollowing. ( 1) ( + ) ( + ) 9( 1)( 7) SOLUTION 1 ( 1) 1 ( ( + ) + 1 ) 1 9 ( 1) 1 ( 7) 1 1 ( 7) ( 7) ( )( + ) ( + ) ( + 7) + 7 ( ) ( + ) 1 ( + 7) ( + 7) 1 1 ( + ) 1 ( )( + ) ( + ) ( + 7) Cnl th ommon tors ( ) n. Ftoris th numrtor n nomintor thn nl ommon tor o ( ). Ftoris th irn o squrs in th numrtor thn nl th ommon tor. EXPLANATION + Et First, nl ny tors in th numrtors with ommon tor in th nomintors. Thn multiply th numrtors n th nomintors. Multiply y th riprol o th rtion tr th ivision sign. Cnl ommon tors n multiply rmining numrtors n nomintors. Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

36 Numr n Algr 1 Empl 1 Empl ( )( + ) 1 ( + 1) 1 1 ( ) ( + 1) 1 + Eris 8H 1 Simpliy ths rtions y nlling Ftoris ths y tking out ommon tors Copy n omplt. ( ) ( 1) ( + )( + ) Simpliy th ollowing y nlling. ( + ) ( + ) ( + )( ) ( + ) ( ) ( ) g 6( 1)( + ) 9( + ) First toris ll th lgri prssions. Not tht is irn o prt squrs. Thn nl s norml ( + 1) 6 7 Simpliy th ollowing y torising n thn nlling ( ) ( ) 8(½) 6 6 9(½) h 0( + 7) ( + 7) 1 ( ) ( ) ( )( + ) 1 6 9(½) UNDERSTANDING FLUENCY + 10 g + y + y h 8y 6y Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

37 Chptr 8 Algri thniqus 8H Empl 1 Empl 1 Empl 1 Et Empl 1 Et Simpliy th ollowing. Ths prssions involv irn o prt squrs ( 0) 00 Simpliy th ollowing y nlling. ( ) ( + 1) ( + 1) ( + )( + ) + + Simpliy th ollowing y nlling. ( + 1) ( + )( + 1) ( + ) ( 9)( + ) 9( + )( 9) ( + 6) ( + )( + 6) Simpliy y irstly torising ( 6) ( + )( ) + ( + )( + ) ( + 1) ( + 1)( ) ( + ) ( ) + 8 ( ) ( + )( ) ( + 7) Ths prssions involv omintion o trinomils, irn o prt squrs n simpl ommon tors. Simpliy y irstly torising whr possil (½) FLUENCY PROBLEM-SOLVING g h Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

38 Numr n Algr Et Et 11 Th prssion 1( + ) n writtn in th orm nll to 1. Us this i to simpliy ths lgri rtions ( ), whih n Just lik n nll to ( + ) +, nls to. Us this i to nl ths ( + ) rtions. ( + 1) ( + 1) ( 1)( + ) 18( 1)( + ) All in togthr ( ) ( ) Et 11(½) 11 7( + 7) 1( + 7) 11 + Us your knowlg o toristion n th is in Qustions 11 n 1 ov to simpliy ths lgri rtions. g i ( + ) (1 ) h j ( ) Et 11, 1 1 REASONING ENRICHMENT 8H Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

39 Chptr 8 Algri thniqus 8I Ky is Simpliying lgri rtions: ition n sutrtion Th pross rquir or ing or sutrting lgri rtions is similr to tht us or rtions without pronumrls. To simpliy +, or mpl, you woul in th lowst ommon multipl o th nomintors (1) thn prss h rtion using this nomintor. Aing th numrtors omplts th tsk. Lt s strt: Compr th working Hr is th working or th simpliition o th sum o pir o numril rtions n th sum o pir o lgri rtions Although lgri rtions, sm strt, prorming oprtions on thm n simpliying thm is ssntil to mny lultions in rl-li mthmtil prolms Wht typ o stps wr tkn to simpliy th lgri rtions tht r th sm s or th numril rtions? Writ own th stps rquir to (or sutrt) lgri rtions. To or sutrt lgri rtions: trmin th lowst ommon nomintor (LCD) prss h rtion using th LCD or sutrt th numrtors. Empl 16 Aing n sutrting with numrls in th nomintors Simpliy: SOLUTION EXPLANATION + + Dtrmin th LCD o n, i.. 0. Eprss h rtion s n quivlnt rtion with nomintor o Thn sutrt numrtors. Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

40 Numr n Algr ( + ) ( ) Not th LCD o n 6 is 6 not Simpliy 1 6 to in th inl stp. Th LCD o n is 10, writ s quivlnt rtions with nomintor 10. Epn th rkts n simpliy th numrtor y ing n ollting lik trms. Empl 17 Aing n sutrting with lgri trms in th nomintors Simpliy: SOLUTION Eris 8I + EXPLANATION Th LCD o n is, so rwrit th irst rtion in n quivlnt orm with nomintor lso o. Th LCD o n is so hng th irst rtion so its nomintor is lso, thn numrtors. 1 Fin th lowst ommon multipl o ths pirs o numrs. (6, 8) (, ) (11, 1) (1, 18) Writ quivlnt rtions y stting th missing prssion ( + 1) 1 UNDERSTANDING + 11 ( + ) ( + 1) Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

41 6 Chptr 8 Algri thniqus 8I Empl 16, Empl 16 Empl 17 Empl 17 Copy n omplt ths simpliitions ( + 1) + + Writ own th LCD or ths pirs o rtions , Simpliy: i m 7 + y 7 y 8 m 6 + m 7 9 Simpliy: g j y + + y t 1 + t 8 16 Simpliy: + Simpliy: + 8 7, j n y 8 + y h k , g k o m + m g 7 + +, 6 6(½) i l 7 10, 7(½) h l p 9 + m m 6 p 9 p y + y m 1 + m 6 h + 9 g 7 h + 7(½) 8(½) 8 9(½) 8 10(½) UNDERSTANDING FLUENCY PROBLEM-SOLVING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

42 Numr n Algr 7 9 Simpliy ths mi lgri rtions g + 10 Fin th missing lgri rtion. Th rtion shoul in simplst orm h Fin n sri th rror in h st o working. Thn in th orrt nswr , A stunt thinks tht th LCD to us whn simpliying is 8. 11, 1 Complt th simpliition using ommon nomintor o 8. Now omplt th simpliition using th tul LCD o. How os your working or prts n ompr? Whih mtho is prrl n why? Mor thn two rtions! 1 Simpliy y ining th LCD g + 1 j + m + h k n i l o PROBLEM-SOLVING REASONING ENRICHMENT 8I Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

43 8 Chptr 8 Algri thniqus 8J Furthr simpliition o lgri rtions EXTENDING Ky is Mor ompl ition n sutrtion o lgri rtions involvs prssions lik: 1 + n ( ) In suh mpls, r ns to tkn t h stp in th working to voi ommon rrors. Lt s strt: Thr ritil rrors Th ollowing simpliition o lgri rtions hs thr ritil rrors. Cn you in thm? ( + ) 6 6 Th orrt nswr is Fi th solution to prou th orrt nswr. Whn omining lgri rtions whih involv sutrtion signs, rll tht: th prout o two numrs o opposit sign is ngtiv numr th prout o two ngtiv numrs is positiv numr. For mpl: n ( 1) ( + ) (1 ) ( 1) A ommon nomintor n prout o two inomil trms. For mpl: ( 1) ( + )( 1) + ( + ) ( + )( 1) ( + )( 1) + 7 ( + )( 1) Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

44 Numr n Algr 9 Empl 18 Simpliying with mor ompl numrtors Simpliy: 1 + SOLUTION 1 + ( 1) ( + ) ( ) 6( ) EXPLANATION Empl 19 Simpliying with mor ompl nomintors Simpliy: SOLUTION ( ) ( + 1)( ) + ( + 1) ( + 1)( ) ( + 1)( ) 7 ( + 1)( ) ( 1) 1 ( 1) ( 1) ( 1) + ( 1) ( 1) Th LCD o n is 1. Insrt rkts roun h numrtor whn multiplying. Not: ( + ) 1 not + 1. Dtrmin th LCD n prss s quivlnt rtions. Insrt rkts. Epn th rkts, rll 6 ( ) 6 n thn simpliy th numrtor. ( 1) 1 EXPLANATION Th lowst ommon multipl o ( + 1) n ( ) is ( + 1)( ). Rwrit h rtion s n quivlnt rtion with this nomintor thn numrtors. Just lik th LCD o n is, th LCD o ( 1) n 1 is ( 1). Rmmr tht ( 1) +. Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

45 0 Chptr 8 Algri thniqus Eris 8J 1, (½) Empl 18 Empl 18 Empl 19 Empl 19 1 Epn th ollowing. ( + ) ( + 1) 7( + ) ( 1) 10( ) 16(1 ) Writ th LCD or ths pirs o rtions. 6 1, 9 1, + 1 Simpliy: g Simpliy: g Simpliy: g Simpliy: ( + 1) + 1 g + 8 ( ) 16, 1 8 7, + h h h ( + ) +,, 1, 1 g ( + 1), h + 1 (½) i i i (½) ( ) (½) ( + 1) h ( 6) 6(½) 9 ( + ) + i 6 7(½) ( ) (1 ) 1 6 7(½) UNDERSTANDING FLUENCY PROBLEM-SOLVING Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

46 Numr n Algr 1 7 Simpliy: ( 1) + 1 g ( ) h On o th most ommon rrors m whn sutrting lgri rtions is hin in this working shown on th right: 9 Simpliy: Wht is th rror n in whih stp is it m? By orrting th rror how os th nswr hng? 1 ( + )( + ) + ( + )( + ) ( 1)( ) 6 ( 1)( ) + 8 ( )( ) 10 Us th t tht 1( ) to hlp simpliy ths Ftoris irst + i 1 8 8, 9(½) ( ) ( + 1) 7 ( ) ( + 1)( + ) ( + 1)( + ) ( + 1)( ) ( + )( 1) ( + )( + ) Ftorising nomintor or urthr simpliition is usul stp. Simpliy ths y irstly torising th nomintors i possil g i h j ( + 1) , 10(½) PROBLEM-SOLVING REASONING ENRICHMENT 8J Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

47 Chptr 8 Algri thniqus 8K Equtions with lgri rtions EXTENDING Ky is For qutions with mor thn on rtion it is otn st to try to simpliy th qution y ling with ll th nomintors t on. This involvs ining n multiplying oth sis y th lowst ommon nomintor. Lt s strt: Why us th LCD? For this qution ollow h instrution Multiply vry trm in th qution y. Wht t os this hv on th rtions on th lt-hn si? Strting with th originl qution, multiply vry trm in th qution y. Wht t os this hv on th rtions on th lt-hn si? Strting with th originl qution, multiply vry trm in th qution y 1 n simpliy. Whih instrution ov os th st jo in simpliying th lgri rtions? Why? For qutions with mor thn on rtion multiply oth sis y th lowst ommon nomintor (LCD). Multiply vry trm on oth sis, not just th rtions. Simpliy th rtions n solv th qution using th mthos lrnt rlir. Altrntivly, prss h rtion using th sm nomintor thn simpliy y ing or sutrting n solv. Empl 0 Solving qutions involving lgri rtions Solv h o th ollowing qutions SOLUTION EXPLANATION Multiply h trm y th LCD (LCD o n is 6) n nl. Simpliy n solv or. Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

48 Numr n Algr OR ( ) 1 ( 1) ( ) ( 1) ( + 1)( + ) ( + 1) ( + ) ( + 1) ( + ) ( + ) ( + 1) OR ( + ) ( + 1) Altrntivly, writ h trm on th lt-hn si using th LCD 6. Simpliy y ing th numrtors n solv th rmining qution. Multiply h trm on oth sis y 1 (LCD o n is 1) n nl. Epn th rkts n simpliy y omining lik trms. Not: ( 1) + not. (Altrntivly, writ h trm using th LCD 1 thn omin th numrtors n ( ) ( 1) solv. 1) 1 1 LCD o n is 6. Multiply h trm y 6. Cnl n simpliy. Solv or lving th nswr in rtion orm. (Altrntiv solution: ) Multiply h trm y th ommon nomintor ( + 1)( + ). Epn th rkts. Sutrt rom oth sis to gthr trms on on si thn sutrt 1 rom oth sis. Sin h si is singl rtion you n ross-multiply : This givs th sm rsult s ov. Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

49 Chptr 8 Algri thniqus Eris 8K 1 (½) (½) Empl 0 Empl 0 Empl 0 1 Writ own th lowst ommon nomintor o ll th rtions in ths qutions Simpliy th rtions y nlling g ( + ) 6( 7)( 1) 9( 7) h ( )(1 ) 9( ) Solv h o th ollowing qutions g 1 1 h m m 1 8 Solv h o th ollowing qutions g m + m i i ( + ) 7( + 1)( + ) 7( + 1) 8( )( 1) 8( ) y + y n + + n h Solv h o th ollowing qutions y y (½) i 7(½) y 1 y 6 1 n + n 1 m + m + 6 Solv h o th ollowing qutions. 1 1 m m y + 1 y g h 1 i 7 1 7(½) UNDERSTANDING FLUENCY Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

50 Numr n Algr Empl 0 7 Solv h o th ollowing qutions Hl o numr () plus on-thir o twi th sm numr is qul to. Writ n qution sriing th sitution. Solv th qution to in th numr Us your omin knowlg o ll th mthos lrnt rlir to solv ths qutions with lgri rtions g h Molly n Billy h hv th sm numr o omputr gms ( omputr gms h). Hzl tks on-thir o Molly s omputr gms n qurtr o Billy s omputr gms to giv hr totl o 77 omputr gms. Writ n qution sriing th totl numr o omputr gms or Hzl. Solv th qution to in how mny omputr gms Molly n Billy h h. 11 A ommon rror whn solving qutions with lgri rtions is m in this working. Fin th rror n plin how to voi it. 8 8, 9(½) i 1 + 9(½), ( + )( ) 11, 1 11, 1 1( 1) + 1 (LCD 1) ( 1) , 1 FLUENCY PROBLEM-SOLVING REASONING 8K 7 Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

51 6 Chptr 8 Algri thniqus 8K 1 Anothr ommon rror is m in this working. Fin n plin how to voi this rror. 1 1 (LCD 6) 6 6( 1) 6 ( 1) Som qutions with imls n y solv y irstly multiplying y powr o 10. Hr is n mpl Multiply oth sis y 10 to rmov ll imls. Solv ths iml qutions using th sm i. For prts you will n to multiply y Litrl qutions 1 Solv h o th ollowing qutions or in trms o th othr pronumrls. Hint: you my n to us toristion g + 1 h 1 1 j m + + k n i l o REASONING ENRICHMENT Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

52 Numr n Algr 7 Invstigtion Epning qurtis using rs Consir th pnsion o th qurti ( + )( + 6). This n rprsnt y ining th r o th rtngl shown. Totl r A 1 + A + A + A Thror: ( + )( + 6) Epning with positiv signs 6 Drw igrm n lult th r to trmin th pnsion o th ollowing qurtis. i ( + )( + ) ii ( + 7)( + 8) iii ( + ) iv ( + ) Using th sm thniqu stlish th rul or pning ( + ). Epning with ngtiv signs Consir th pnsion o ( )( 7). Ar rquir totl r (A + A + A ) Thror: [(A + A ) + (A + A ) A ] (7 + 8) ( )( 7) A 1 A A 1 A A A A A This r is ount twi whn w 7 +. Drw igrm n lult th r to trmin th pnsion o th ollowing qurtis. i ( )( ) ii ( 6)( ) iii ( ) iv ( ) Using th sm thniqu, stlish th rul or pning ( ). Dirn o prt squrs Using igrm to rprsnt ( )( + ), trmin th pproprit r n stlish rul or th pnsion o ( )( + ). 7 Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

53 8 Chptr 8 Algri thniqus Numril pplitions o prt squrs Th pnsion n toristion o prt squrs n irn o prt squrs n ppli to th mntl lultion o som numril prolms. Evluting prt squr Th prt squr n vlut using ( + ) + +. (0 + ) (Lt 0, ) 0 + (0)() Us th sm thniqu to vlut ths prt squrs. i ii 1 iii iv 1 v 1. vi. vii 6.1 viii 9.01 Similrly, th prt squr 9 n vlut using ( ) +. 9 (0 1) (Lt 0, 1) 0 (0)(1) Us th sm thniqu to vlut ths prt squrs. i 19 ii 9 iii 98 iv 87 v 1.9 vi.7 vii 8.8 viii.96 Evluting th irn o prt squrs Th irn o prt squrs 1 9 n vlut using ( + )( ). 1 9 (1 + 9)(1 9) (Lt 1, 9) 11 Us th sm thniqu to vlut ths irn o prt squrs. i 1 8 ii iii 1 iv 8 8 v vi.9.7 vii viii Th pnsion ( + )( ) n lso us to vlut som prouts. Hr is n mpl: 1 9 (0 + 1)(0 1) (Lt 0, 1) Us th sm thniqu to vlut ths prouts. i 1 19 ii 8 iii 6 7 iv 10 9 v vi vii 0 80 viii Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

54 Numr n Algr 9 Prolms n hllngs 1 Th irn twn th squrs o two onsutiv numrs is 97. Wht r th two numrs? Th irn twn th squrs o two onsutiv o numrs is 16. Wht r th two numrs? Th irn twn th squrs o two onsutiv multipls o is 81. Wht r th two numrs? I + y 6 n ( + y) 6, in th vlu o y. I + y 10 n y, in th vlu o y. Up or hllng? I you gt stuk on qustion, hk out th 'Working with Unmilir Qustions' postr t th n o th ook to hlp you. Fin th vlus o th irnt igits,, n i th our igit numr. Fin th qurti rul tht rlts th with n to th numr o mths in th pttrn low. n 1 n n Drw possil pttrn or ths ruls. i n + ii n(n 1) Ftoris n 1 n us th toris orm to plin why whn n is prim n grtr thn, n 1 is: i ivisil y ii iii ivisil y thus ivisil y 1. 6 Prov tht this prssion is qul to 1. 7 Prov tht or ll In r ovr km Ryn rn t onstnt sp. Sophi, howvr, rn th irst km t sp 1 km/h mor thn Ryn n rn th son km t sp 1 km/h lss thn Ryn. Who won th r? Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

55 0 Chptr 8 Algri thniqus Chptr summry Moni qurti trinomils (Et) Ths r o th orm: + + Rquir two numrs tht multiply to n to.g ( 9)( + ) sin 9 18 n Ftoristion Th rvrs pross o pnsion. Alwys rmov th highst ommon tor irst..g. + 8 HCF ( + ).g. ( + ) + y( + ) HCF + ( + )( + y) By grouping I thr r our trms, w my l to group into two inomil trms..g ( + ) ( + ) ( + )( ) DOPS ( )( + ).g. 16 ( )( + ).g. 9 () ( )( + ) Algri thniqus A/sutrt Must in lowst ommon nomintor (LCD) or pplying oprtion..g ( 1) Trinomils o th orm + + (Et) Cn toris using grouping lso.g , , Us 9 n ( + ) ( + ) sin ( + )( ) 9 ( ) 18 n 9 + ( ) 7 Epnsion Th pross o rmoving rkts..g. ( + ) + 10 ( + )( ) ( ) + ( ) Spil ss DOPS (irn o prt squrs) ( )( + ).g. ( )( + ) + Prt squrs ( + ) + + ( ) +.g. ( + ) ( + )( + ) Algri rtions Ths involv lgri prssions in th numrtor n/or nomintor..g.,, 7 7 ( 1) Multiply/ivi To ivi, multiply y th riprol. Ftoris ll prssions, nl n thn multiply..g ( + ) 1 ( )( + ) 1 1 ( ) Solving qutions with lgri rtions (Et) Fin lowst ommon nomintor (LCD) n multiply vry trm y th LCD..g. LCD o n is 6 LCD o ( 1) n ( + ) is ( 1)( + ) LCD o n is Unorrt r smpl pgs Cmrig Univrsity Prss Grnwoo t l., 01 ISBN Ph

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