S i m p l i f y i n g A l g e b r a SIMPLIFYING ALGEBRA.

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Terence Newton
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1 S i m p l i y i n g A l g r SIMPLIFYING ALGEBRA

3 Simpliying SIMPLIFYING Algr ALGEBRA Algr is mthmtis with mor thn just numrs. Numrs hv ix vlu, ut lgr introus vrils whos vlus n hng. Ths r rprsnt y lttrs. Although thy work similrly to numrs it is importnt to wr o how to, sutrt, multiply n ivi xprssions ontining vrils. Answr ths qustions, or working through th hptr. I us to think: Wht is th irn twn lik n unlik trms? Writ n xmpl xplining th istriution proprty Wht is inomil prout? Answr ths qustions, tr working through th hptr. But now I think: Wht is th irn twn lik n unlik trms? Writ n xmpl xplining th istriution proprty Wht is inomil prout? Wht o I know now tht I in t know or? 100% Simpliying Algr J 5 1 Mthltis 100% P Lrning SERIES TOPIC

4 Simpliying Algr Bsis Lik Trms Trms with th sm vrils n inis r ll lik trms. I inis or vrils ir thy r ll unlik trms. Exmpls o lik n unlik trms Lik Trms p n p xy n 7xy Unlik Trms 5m n n n Dirnt virls Dirnt inis Only lik trms n or sutrt. Simpliy th ollowing s muh s possil Lik trms Lik trms This n't simplii nymor us ths r unlik trms. Multiplying Trms It s sy to s tht # 6 n # 6, so it osn't mttr in whih orr n r multipli. This is lwys tru! In ny sitution # #. This is vn tru i mor thn two trms r multipli: # # # # # #. This mns xy yx n thy r lik trms. r ll qul n r lso lik trms. Th tims sign ^# h is ropp in multipltion o irnt trms. So # x x, # n x # y # z xyz Th oiints r lwys multipli sprtly. Hr r som mor xmpls. Writ th ollowing prouts using lgr m # n mn p # q # r Coiints multipli sprtly ^ # h # p # q # r 8pqr 6xy # x y ^6 # h # ^x # x h^y # y h 18x y Coiints multipli sprtly J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

5 Simpliying Algr Bsis Ruls or Multiplition Whn trms hv th sm sign thir prout is positiv, whn thy hv irnt signs thir prout is ngtiv. + # + + Positiv # Positiv Positiv + # - - Positiv # Ngtiv Ngtiv - # + - Ngtiv # Positiv Ngtiv - # - + Ngtiv # Ngtiv Positiv Hr r som xmpls o multiplying with signs Simpliy th ollowing -7 # ^-7 # h # ^ # h -1 -x #-5y ^- #-5h # ^x # yh 15xy -m # n #-5p -5t #-u #-v ^- # #-5h # ^m # n # ph ^-5 #- #-1h # ^t # u # vh 0mnp -0tuv Th mil stps o n n skipp i you rmmr this rul: I thr r n vn mount o ngtivs thn th prout is positiv. I thr r n o mount o ngtivs thn th prout is ngtiv. Orr o Oprtions Rmmr tht rkts n multiplition (n ivision) r lwys on or ition n sutrtion. Simpliy th ollowing s muh s possil xy + x #-y -5 # - 6 #-7 Multiplition FIRST Multiplition FIRST Multiplition FIRST xy - 8xy (-) Simpliy lik trms -(-) -5xy % Simpliying Algr J 5 Mthltis 100% P Lrning SERIES TOPIC

6 Simpliying Algr Bsis 1. Sy i ths r lik or unlik trms n n pq n qp n. Complt th ollowing # # yz ghi. Us ition n sutrtion to simpliy th ollowing s muh s possil x + 5x - x jk + j + 7k + 5jk + 6k x y + xy - x + 8xy + yx J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

7 Simpliying Algr Bsis. Us multiplition n th orr o oprtions to simpliy th ollowing s muh s possil # x # 6y -5t # 5u 10p #-q 7 # #- -g # 6h #-i g -8ux #-v #-w h p #-q # r #-s i x + x # y j x #- y + xy k 5 # + #-10 l -7x # w - w #- z + wz + zw 100% Simpliying Algr J 5 5 Mthltis 100% P Lrning SERIES TOPIC

8 Simpliying Algr Knowing Mor Algri Frtions (Diviing Trms) Whn lgri trms r ivi, lgri rtions r orm. Ths n simplii y nlling lik trms. Writ th ivision s rtion. Alwys simpliy th oiints n nl th vrils whr nssry. Simpliy th ollowing s muh s possil: m ' 6m 1x ' xy Writ s rtion Writ s rtion m 6m # 6 1 m m m is in oth th numrtor n nomintor, so it n nll 1x xy 1 # y x xy y is only in th numrtor, so it is not nll. x is in th numrtor n nomintor, so x is nll Ruls or Division Algri ivision with signs hv th sm ruls s multiplition: I th trms hv th sm sign thir quotint is positiv, whn thy hv irnt signs thir quotint is ngtiv. + ' + + or + ' - - or - ' + - or - ' - + or Positiv ' Positiv Positiv Positiv ' Ngtiv Ngtiv Ngtiv ' Positiv Ngtiv Ngtiv ' Ngtiv Positiv Rmmr, lwys writ th ivision s rtion irst. Thn simpliy th oiints n nl th vrils. Simpliy th ollowing s muh s possil: 15x '-10x -0 '- Writ s rtion Writ s rtion 15x - 10x - Th oiints hv n simplii x is in oth th numrtor n nomintor, so it n nll Th nswr is ngtiv ^+ ' -h Th oiints hv n simplii n r in oth th numrtor n nomintor, so thy n nll nnot nll us it only pprs in th numrtor Th nswr is positiv ^- ' -h 6 J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

9 Simpliying Algr Knowing Mor Orr o Oprtions Rmmr tht rkts n ivision (n multiplition) r lwys on or ition n sutrtion. Simpliy 6x - xy ' y 6pqr ' 1r + 7p # q xy 6x - y ivision hs n on irst 6pqr + 8pq 1r multiplition n ivision hv n on irst 6x - x pq + 8pq x 1pq Rvising Inx Lws Hr is rminr o inx lws. m n m n 1. Multiplition with inis: # +. m n Division with inis: m ' n -. m n mn Rising n inx to n inx: ^ h. Prouts ris to inis: ^h 5. Quotints ris to inis: m ` j 0 6. Th zro inx: 1 or! 0 -n 7. Ngtiv inis: 1 or -n n ` j ` j m n m n m 8. Frtionl inis: n ^ h m m m m m m n n Inx lws r us to simpliy lgri rtions. Mor thn on lw might nssry to simpliy rtion. Simpliy ^ pq h - ' ^p q h # q 7 x y # x ' y # ^ h x x y y 7-7 x 1 m y 7 x y 6 p q ' 1 # 1 q 6 p q q p q 8 # 7 100% Simpliying Algr J 5 7 Mthltis 100% P Lrning SERIES TOPIC

10 Simpliying Algr Knowing Mor Comining Inx Lws n Algri Frtions Th nxt logil stp is to mix inis n lgri rtions. Algri rtions with inis work with th sm two stps. Stp 1: Simpliy th numrtor n nomintor sprtly. Stp : Simpliy th rtion n nl vrils whr nssry. Hr r som xmpls o lgri rtion ontining inis. Simpliy s muh s possil: y # y # y y + y + y Simpliy numrtor n nomintor sprtly ^6x y zh 5 18x y z 16y 8y 6 6x y z 5 18x y z Simpliy numrtor n nomintor sprtly y - 1 Simpliy rtion using inx lws or ivision x y z Simpliy rtion using inx lws or ivision y x -1 y z y z x x is in th nomintor us its inx is ngtiv 6 8 1x y x y ^p qh # ^p q h -p q 6 8 1x y ^16x y h Frtionl inis 6 8p q # 16p q -p q 6 8 1x y x y Simpliy numrtor n nomintor sprtly p q - p q Simpliy numrtor n nomintor sprtly x y Simpliy rtion using inx lws or ivision -18p q Simpliy rtion using inx lws or ivision x y 7-18p q 8 J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

11 Simpliying Algr Qustions Knowing Mor 1. Fin th quotint o th ollowing: 8 ' ' -pqr ' pr -100mn ' -0n 18wx ' -9xy -5st ' 0stu. Answr ths qustions out orr o oprtions: Simpliy # Simpliy # Is this sttmnt tru or ls: # # Is th orr o th trms in multiplition importnt? Simpliy ' Simpliy ' g Is this sttmnt tru or ls: ' ' Is th orr o th trms in ivision importnt? 100% Simpliying Algr J 5 9 Mthltis 100% P Lrning SERIES TOPIC

12 Simpliying Algr Qustions Knowing Mor. Us th orr o oprtions to simpliy ths s muh s possil: 10-0 ' 10 6xyz ' xz + 6y 1mn ' n - 6m xz # xyz ' xz 0xy ' 5x + 0yz ' 6z 15 # - 00 ' J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

13 Simpliying Algr Qustions Knowing Mor. Simpliy ths lgri rtions s muh s possil y + y + 5y y # y # y ^x h ^x h # x ^5xy h # ^x yh ^x y h ^pqh # ^- p q h -pq 9 g # ^gh ^ gh # g -qr # q r q o ^ rh 100% Simpliying Algr J 5 11 Mthltis 100% P Lrning SERIES TOPIC

14 Simpliying Algr Qustions Knowing Mor 5. Answr ths qustions - n Us th inx lw 1 to prov n n -n ` j ` j (Hint: -n -1 n ` ) j `` j j Us this rsult to simpliy this rtion - ^xy h o ^x yh 1 J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

15 Simpliying Algr Using Our Knowlg Expning Brkts A trm outsi rkt is multipl with ll trms insi rkt, so tht ^ + h # + # + This proprty o multiplition with rkts is ll th 'istriution proprty'. Hr r som xmpls: Expn th ollowing ^ + xh x^5 + x + yh - # # x x # 5 x # x x # y 6 + x 5x + x + xy Thr is no irn i th numr on th outsi is ngtiv. Alwys multiply th trm outsi with ll th trms insi. Expn th ollowing -( y - 5) - p^q + 6p - 9h # y - #-5 -p # q -p # 6p -p #-9 - y + 0-6pq - 18p + 7p Somtims lik trms n simplii tr xpning rkts. Expn th ollowing ^x - x h + x^5 - xh - q^1 + p - pqh + q^p - 5pqh Lik trms Lik trms 8x - x + 15x - 6x -q - 8pq + 6pq + pq - 10pq Lik trms Lik trms x - 10x q pq pq % Simpliying Algr J 5 1 Mthltis 100% P Lrning SERIES TOPIC

16 Simpliying Algr Using Our Knowlg Multiplying Brkts Brkts n lso multipli togthr. Both trms in th irst rkt r multipli with oth trms in th son rkt. 1 ( + )( + ) Th prout o two rkts n lso thought o in this wy ( + )( + ) ( + ) + ( + ) Fin th ollowing prouts: 1 1 ( x + )( x + ) ( y + )( y - ) 1 1 ( x # x) + ( x + ) + ( # x) + ( # ) ( y # y) + ( y #- ) + ( # y) + ( #-) x + x + x + 6 6y - 8y + 9y - 1 Lik trms x + 5x + 6 6y + y - 1 Lik trms ^p + qh^ p - qh ^ - 5h^ + h ^p # ph + ^p #- qh + ^q # ph + ^q #-qh ^ # h + ^ # h + ^- 5 # h + ^-5 # h p - 8pq + pq - 1q p - 5pq - 1q 1 J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

17 Simpliying Algr Qustions Using Our Knowlg 1. Expn ths rkts: 5( y + ) 6( - t) -(- - 5m) -( 9 - x) 1 ( 8 x - y) - 1 ( 16 n - m) g x^x - yh h x ^xy - yh i 5p^q + pq + ph j xy^x - y + 1h 100% Simpliying Algr J 5 15 Mthltis 100% P Lrning SERIES TOPIC

18 Simpliying Algr Qustions Using Our Knowlg. Expn ths rkts: x^ + x - y + xyh -^ h. Expn thn simpliy using lik trms: ^x - x h + x^7x - 5h ^1 - + h + ^5 - - h 5p^ q - pq h - q^p q + p - h - ^ - + 6h - (8 + - ) + 16 J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

19 Simpliying Algr Qustions Using Our Knowlg. Expn n simpliy: ^x + h^x - 1h ^st + th^t - sh ^k - 1h^ - kh ^ - h^ + 6h ^x + yh^x - yh ^ - h^ + h g ^ x y - xy h^xy + xh h ^ - h^5 - h 100% Simpliying Algr J 5 17 Mthltis 100% P Lrning SERIES TOPIC

20 Simpliying Algr Using Our Knowlg Aing n Sutrting Algri Frtions Lik ll rtions, lgri rtions n only or sutrt i thy hv th sm nomintor. I th rtions lry hv ommon nomintor, thn simpliy th numrtors only. Fin th ollowing prouts: 9h - h h - h 19 5h 19 Ths rtions lry hv ommon nomintor y y y y 9y y y 1 1 is th lowst ommon nomintor o, n is th lowst ommon nomintor o n 8p q p q p - q + 6p + q 0 5q 5q is th lowst ommon nomintor o 15, 0 n 10 Somtims its iiult to in ommon nomintor. In ths situtions, th ommon nomintor n oun y multiplying th nomintors togthr. Aing n sutrting rtions with omplit nomintors 5 x m + n - m - n Stp 1: Fin ommon nomintor Stp 1: Fin ommon nomintor 7^x + h This is th prout o th originl nomintors ( m + n)( m - n) This is th prout o th originl nomintors Stp : Writ oth rtions ovr this nomintor 1 5 x 7 x + + ^ + h ^ h 7^x + h 1 + 5^x + h 7^x + h Stp : Writ oth rtions ovr this nomintor ^m - nh ^m + nh - ^m + nh^m - nh ^m + nh^m - nh ^m - nh - ^m + nh ^m + nh^m - nh Stp : Simpliy Stp : Simpliy 1 + 5x x + 1 Us th istriutiv proprty to xpn th rkt m - n - m - n m + mn - mn - n Multiply th rkts to in this nomintor 5x + 9 7x n m - n 18 J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

21 Simpliying Algr Qustions Using Our Knowlg 5. Simpliy ths rtions: 7x + 5x x - 5x 1 + mn km x y y g 1 h y y - 1 p p p 100% Simpliying Algr J 5 19 Mthltis 100% P Lrning SERIES TOPIC

22 Simpliying Algr Using Our Knowlg Multiplying n Diviing Algri Frtions Multiplying n iviing lgri rtions is sy us ommon nomintor is not nssry. To multiply lgri rtions, simply multiply th numrtors n th nomintors sprtly. Fin ths prouts: y # x p # p q # y x # Multiply numrtors n nomintors sprtly p # p # q Multiply numrtors n nomintors sprtly y x p q t 5 # 10 t 7 # # t # 10 5 # t Multiply numrtors n nomintors sprtly 7 # # Multiply numrtors n nomintors sprtly 10t 5t 7 t t 6 To ivi lgri rtions, simply in th riprol o th son rtion n multiply. Fin ths quotints: x y ' ' 1 x # 16 8 y 'lip' th son rtion to in th riprol. - # 5 1 'lip' th son rtion to in th riprol. x # 16 8 # y - # 5 # 1 x 8y x y J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

23 Simpliying Algr Qustions Using Our Knowlg 6. Writ ths in simplst orm: p p # 10 mn # 10 5 n mn # 5 5 m n p q 6q # 7 p 8s t 9p 7p s # 16t # # g 5 # # 5 h -5wx y # 10 5wxy 100% Simpliying Algr J 5 1 Mthltis 100% P Lrning SERIES TOPIC

24 Simpliying Algr Qustions Using Our Knowlg 7. Writ ths in simplst orm: p q ' 6 # 5 # 9 9 ' s t 0s t ' 6 x y x y 6 7p q r ' 5 1p q r J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

25 Simpliying Algr Thinking Mor Shortuts or Multiplying Brkts A prout o two rkts it ll inomil prout. Som inomil prouts n oun mor sily thn othrs. Look t this xmpl: ^ + h^ - h Th mil trms nl h othr wy - Cn you s th shortut? Thir prout is th squr o th irst trm minus th squr o th son trm. Hr r som xmpls using this shortut: Fin ths prouts: ^x + h^x - h ^y - 5h^y + 5h ^xh -^h ^yh -^5h 1 st trm squr n trm squr 1 st trm squr n trm squr x - 9y - 5 Wht out th prt squr o rkt? Look t this xmpl: ^ + h ^ + h^ + h st trm squr # prout o trms n trm squr Cn you s th shortut? Hr r som xmpls using this shortut: Fin ths prouts: ^x + h ^x h + ^xh^h + ^h ^p - qh ^ph + ^ph^- qh + ^-qh 1 st trm squr # prout o trms n trm squr 1 st trm squr # prout o trms n trm squr x + 8x p - 16pq + q 100% Simpliying Algr J 5 Mthltis 100% P Lrning SERIES TOPIC

26 Simpliying Algr Thinking Mor Ths shortuts my sm silly or ths sir xmpls, ut thy r spilly usul or mor omplit xmpls. Fin ths prouts: x ` + - x x j` x j ^x y + pq h^x y - pq h ` -^x x j h ^x yh -^pq h 16 x - x 6 9x y - 16p q 5y - m y ^m n - nh ^5yh + ^5yh m m y y ^m n h + ^m n h^- nh + ^-nh 15y 5y + - m + 9 y y 5y y 6 m n + ^- 6m n h + 9n 6 m n - 1m n + 9n W n vn us ths ruls in lultions using tul numrs ^11 + 9h^11-9h ^0 + 6h ^0h^h 0 + ^0h^6h J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

27 Simpliying Algr Qustions Thinking Mor 1. Prov tht ^x + yh^x - yh x - y. Simpliy ths prt squrs ^p + 1h ^5x - 7yh ^-m - nh ^p - pqh 1 ` + x x j y - m y 100% Simpliying Algr J 5 5 Mthltis 100% P Lrning SERIES TOPIC

28 Simpliying Algr Qustions Thinking Mor. Simpliy ths inomil prouts: ^x + h^x - h ^5 - mh^5 + mh ^ + h^ - h ^7x + yh^7x - yh ^xy + x y h^xy - x y h ^ 10 - h^10 + h g 1 m m 1 m m ` + j` - j h y ` - y 5x j` + 5x j 6 J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

29 Simpliying Algr Qustions Thinking Mor. Fin 9 n in h o th ollowing 9 ^x + h x + x + 9 ^x + h x + 1xy + 9y 9 9 ^9-5qh 16p - 0pq + 5q ^m + h m + 16mn + 5. Fin ths vlus using xpn orm: (Hint: ) 100% Simpliying Algr J 5 7 Mthltis 100% P Lrning SERIES TOPIC

30 Simpliying Algr Qustions Thinking Mor 6. A stg in stium is m up o rtngl n squr in th ollowing igrm. Th imnsions r in trms o x. x + x - Rtngl Squr x - Fin xprssion or th r o th squr in th orm o inomil prout Fin xprssion or th r o th rtngl in th orm o inomil prout Show tht th totl r o th stg is ( x - )( x + ) + ( x - ) Simpliy th xprssion or th r o th stg Fin th vlu o th r o th stg i x 5 8 J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

31 Simpliying Algr Answrs Bsis: Knowing Mor: 1. Lik trms Unlik trms. Lik trms Lik trms. # # yz z y # g Fls, thy r not qul. ghi g h i # # h Ys, th nswrs r irnt. # # #. 6 19y m. 6x xyz 9y 5 j + 1k + jk y 1 x 8x y + 10xyz - x 5x y 9p q. 1 1xy g r 81-5tu -10 pq ghi 5. ` j -n ` j n 8 9x y g i k -8uvwx h 1pqrs x + xy j -xy -0 l 11zw - 7xw 1. Using Our Knowlg: 10y t 1 0m + 7 1x + Knowing Mor: x - y - n + 6m q g 8x - xy h x y yx - 5m - w y - 1 u i j 5pq + 10p q + 15p 6x y - 6xy + xy. 8 8 Tru, thy r oth 8.. x + x - xy + x y No, th nswrs r th sm. 100% Simpliying Algr Mthltis 100% P Lrning J 5 9 SERIES TOPIC

32 Simpliying Algr Answrs Using Our Knowlg: Using Our Knowlg:. - x 6x 7. 9p q pq + 1q - 18 q tx s y 9 q p r. x - 8st - s t + 1t - 6st - 8k + 1k - + k x - y - g x y + 6x y - x y - 8x y 5 h Thinking Mor: p + p + 1 5x - 70xy + 9y 9m + 1mn + n 9p + 6p q + p q 1 x + x + y - 1y + 9 y 5. x x. x m - k - n kmn 17-10x 6-1x 9-9x - 9y x y - x y g 11-1y + ^ - 1h 5y^y + 1h y p - h ^y - 1h^y + 6h p g h m 6 - m y xy 5x x p 0 m 5 mn 18q 7p 5 p s t g 6 5 h - xy 0 J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

33 Simpliying Algr Answrs Thinking Mor: y 9 p n 9 16n ^x - h ^x - h^x + h 6x - 6x % Simpliying Algr Mthltis 100% P Lrning J 5 1 SERIES TOPIC

34 Simpliying Algr Nots J 5 SERIES TOPIC 100% Simpliying Algr Mthltis 100% P Lrning

# 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths.

# 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths. How os it work? Pl vlu o imls rprsnt prts o whol numr or ojt # 0 000 Tns o thousns # 000 # 00 Thousns Hunrs Tns Ons # 0 Diml point st iml pl: ' 0 # 0 on tnth n iml pl: ' 0 # 00 on hunrth r iml pl: ' 0