Copyrighted by Gabriel Tang B.Ed., B.Sc. Page 1.

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1 Alger Prerequisites Chpter: Alger Review P-: Modelig the Rel World Prerequisites Chpter: Alger Review Model: - mthemticl depictio of rel world coditio. - it c e formul (equtios with meigful vriles), properly drw grph, clerly lelled digrm with qutittive mesuremets. Modellig: - the process of discoverig the mthemticl model. Emple : To covert temperture mesuremets from degree Celsius to Fhreheit, we c use the formul, T F TC +.. Wht is the temperture i Fhreheit whe the outside temperture is 0 C?. Wht is the temperture i degree Celsius for ptiet with temperture of 0 F? c. At wht temperture whe its umericl vlue of degree Celsius is equivlet to tht of Fhreheit? c. At the sme umericl vlue,. T F TC + T F (0) + we c set T F T C T F 8 + T F F T F TC +. We c mipulte the formul first efore sustitutio. T F TC + T F TC + (TF ) T C ((0) ) TC T C 0. C 0 F 0 C Emple : A rectgulr o hs width mesured twice its height d its legth is three times its width.. Fid the volume of the o if it hs height of 8 cm.. Write formul for the volume V of this o i terms of its height. c. Wht re the dimesios of this o if it hs volume of 78 cuic feet? Copyrighted y Griel Tg B.Ed., B.Sc. Pge.

2 Prerequisites Chpter: Alger Review Alger Emple : Four ideticl circles re eclosed y squre s show elow. Determie the cut out re A i terms of r s represets y the shded re. r P-: Rel Numers Set: - group of ojects (clled elemets of the set). - we commoly use fcy rckets, { }, to iclude elemets of set. Nturl Numers (N): - coutig umers. Whole Numers (W): - coutig umers with 0. Itegers (I): - positive d egtive whole umers. N {,,,,, } W {0,,,,,, } Set Nottio ( ): - symol to idicte oject elogs i the prticulr set. I {,,,,,, 0,,,,,, } Emple: 0 W ut 0 N (0 elogs to i set of whole umers ut ot i set of turl umers.) Set-Buildig Nottio: - set ottio tht ivolves series of umer. Emple: Z {,,,,, 7} c e writte s Z { 7 d N } (Z is set such tht the elemets, represeted y, re etwee to 7 d they re turl umers) (Note: whe set-uildig ottio does ot iclude the type of umers it is ssumed R rel umers) Rtiol Numers (Q): - umers tht c e tured ito frctio, where, I, d 0. - iclude ll Termitig or Repetig Decimls. - iclude ll Nturl Numers, Whole Numers d Itegers. - iclude y perfect roots (rdicls). 7. Termitig Decimls: - decimls tht stops. Emples: Repetig Decimls: - decimls tht repets i ptter d goes o. Emples: 0..7 c. Perfect Roots: - rdicls whe evluted will result i either Termitig or repetig decimls, or frctios, where, I, d 0. Emples: 0. ± ± 0... ± ± Pge. Copyrighted y Griel Tg B.Ed., B.Sc.

3 Alger Prerequisites Chpter: Alger Review To Covert Deciml ito Frctio usig TI-8 Plus Emple: Covert 0. ito frctio. () MATH Select Optio ENTER Repet eterig to the edge of the scree Irrtiol Numers (Q ): - umers tht CANNOT e tured ito frctio, where, I, d 0. - iclude ll o-termitig, o-repetig decimls. - iclude y o-perfect roots (rdicls).. No-termitig, No-repetig Decimls: - decimls tht do ot repet ut go o d o. Emples: π. 0.. No-Perfect Roots: rdicls whe evluted will result i No-Termitig, No-Repetig decimls. Emples: ± ± Rel Numers (R): - y umers tht c e put o umer lie. - iclude ll turl umers, whole umers, itegers, rtiol d irrtiol umers.. / π 0 Rel Numers Q I W N Q Uio ( ): - the comied elemets of two sets. - for A B, it mes ll elemets i A or B (or i oth). Itersectio ( ): - icludes ll elemets tht re i oth sets. - for A B, it mes ll elemets i A d B. A A B A B B Copyrighted y Griel Tg B.Ed., B.Sc. Pge.

4 Prerequisites Chpter: Alger Review Alger Empty Set ( ): - whe the set cosists of o elemets. Emple : If F {,, 0,,,, }, G {0,, }, d H {, 7, 8}, fid. F G. F G c. G H Ifiity ( ): - use to deote tht the ptters go o d o i specific directio of the rel umer lie. - positive ifiity ( ) mes ifiity towrds the right of the umer lie. - egtive ifiity ( ) mes ifiity towrds the left of the umer lie. + 0 Ope Itervl: - whe the oudry umers re ot icluded (eclusive). - we use orml rckets for ope itervls. - o the umer lie, we use ope circles t the edpoits. Emple: (, ) mes ll umers etwee d eclusively (ot icludig d ) 0 Closed Itervl: - whe the oudry umers re icluded (iclusive). - we use squre rckets for ope itervls. - o the umer lie, we use closed (filled i) circles t the edpoits. Emple: [, ] mes ll umers etwee d iclusively (icludig d ) 0 Iequlities d Itervls Nottio Meig d Set Descriptio Grphs > or (, ) Greter th { > } < or (, ) Less th { < } or [, ) Greter th or equl to { } or (, ] Less th or equl to { } Pge. Copyrighted y Griel Tg B.Ed., B.Sc.

5 Alger Prerequisites Chpter: Alger Review Nottio Meig d Set Descriptio Grphs ( lower, upper ) is etwee the lower d upper oudries (eclusive). { lower < < upper } lower is etwee the lower d upper [ lower, upper ] oudries (iclusive). { lower upper } lower is etwee the lower (ope) ( lower, upper ] d upper (closed) oudries. { lower < upper } lower is etwee the lower (closed) [ lower, upper ) d upper (ope) oudries. { lower < upper } lower is less th the lower oudry (, lower ] [ upper, ) d is greter th the upper oudry (iclusive). lower { lower upper } is less th the lower oudry (, lower ) ( upper, ) d is greter th the upper oudry (eclusive). { < lower > upper } lower upper upper upper upper upper upper Emple : Epress ech itervl i terms of iequlities (set descriptios), d the grph the itervls.. [, ). (, ) [, ) Emple : Grph ech set.. (, 8] [, ). (, 8] [, ) P- Assigmet: pg. 70 #,,,, 8 d ; Hoours: # P- Assigmet: pg. #,, 7,,,, 7,,, 7 d 7; Hoour: #77 Copyrighted y Griel Tg B.Ed., B.Sc. Pge.

6 Prerequisites Chpter: Alger Review Alger P-: Iteger Epoets Iteger Epoet: - epoet tht elogs i iteger set. - epoet idictes how my fctors the se is multiplyig itself. Note: The epoet oly pplies to the immedite umer, vrile or rcket precedig it. power se to the th power epoet ()()()() () fctors Emple : Evlute the followigs.. () () ()()()(). () Lws of Epoets Multiply Powers of the Sme Bse Addig Epoets Divide Powers of the Sme Bse Sutrctig Epoets Power Rule Multiplyig Epoets ( m )( ) m + m m ( m ) m Zero Epoet 0 Distriutio of Epoet with Multiple Bses Negtive Epoet Reciprocl ()()()() Distriutio of Negtive Epoet with Multiple Bses () m m () Note tht the epoet oly pplies to the immedite umer precedig it d eclude the egtive sig. Pge. Copyrighted y Griel Tg B.Ed., B.Sc.

7 Alger Prerequisites Chpter: Alger Review Emple : Simplify. Epress ll swers i positive epoets oly.. (7c d )(c 8 d 0 ). c. ( y ) c + d + 0 () ( )(y ) c d 7 y 0 0 ( y ) ( y ) d. e. (m 7 ( y ) 7 ) (m ) p q f. 7 p q ( )( ) ( 7 ) 0 8 m Whe reciproctig 7 y y 8 ( y ) ( ) etire rcket, do p q m NOT mess with its p q y m cotet. ( 8) p q 0 m y m 0 y p q () g. ( ) ( ) + ( ) () ( ) + ( ) () ( ) 8 ( ) + 8 ( ) 7 7 ( h k ) h. ( h k ) ( h k ) ( h k ) ( h k ) ( h k ) 0 ( 8h k ) 8 ( h k )( 8h k ) ( ) ( ) ( 8) h k Scietific Nottio: - commoly used to stte very ig or very smll umers. ( to. ) 0 where is iteger If < 0, the the ctul umer ws etwee 0 d If > 0, the the ctul umer ws greter th 0 Emple : Covert the followig stdrd ottios to scietific ottios or vice vers. h k p p 8. Speed of Light 0 km/s 00,000 km/s (moved deciml plces to the right). Mss of Electro. 0 kg kg (moved deciml plces to the left) c. Dimeter of Red Blood Cell m 7. 0 m (moved deciml plces to the right) d. 00 US Det $,80,000,000,000 $.80 0 (moved deciml plces to the left) q q p q Copyrighted y Griel Tg B.Ed., B.Sc. Pge 7.

8 Prerequisites Chpter: Alger Review Alger Emple : I stroomy, oe light yer is the distce light c trvel i oe yer. Light hs costt speed of 0 km/s i the vcuum of spce.. Clculte the distce of oe light yer.. The closest str to the Su, Alph Ceturi, is.78 0 km. How my light yers is it to our su?. Oe Light Yer ( 0 km/s)( dys/yr)( hr/dy)(0 mi/hr)(0 s/mi) d EE, Oe Light Yer.08 0 km/yr km.08 0 km yr light yers P- Assigmet: pg. 78 #,, 7,, 7,,, 7,,,, 80; Hoours: #8 P-: Rtiol Epoets d Rdicls Rdicls: - the result of umer fter root opertio. Rdicl Sig: - the mthemticl symol. Rdicd: - the umer iside rdicl sig. rdicl sig rdicd Ide: - the smll umer to the left of the rdicl sig idictig how my times umer (swer to the rdicl) hs to multiply itself to equl to the rdicd. squre root cue root fourth root fifth root To cll up the cue root or Choose Choose Optio Optio for cue root for cue root higher root fuctios, press MATH Choose Optio for higher Choose root. Optio But e sure for to higher eter the root. umer But e for sure the to eter the ide umer first! for the ide first! Emple : Evlute... c. d. ± ()() () ()() () ()()() ± ide ()()()() () ()()()() A rdicl with eve ide lwys hs two swers (±), d c oly hve rdicd greter th or equl to 0 iside rdicl sig. A rdicl with odd ide lwys hs oe swer oly d c hve egtive rdicd iside the rdicl sig. () ()()()()() Pge 8. Copyrighted y Griel Tg B.Ed., B.Sc.

9 Alger Prerequisites Chpter: Alger Review Emple : A formul v f v i + d c e used to fid the fil velocity (speed) of ccelerted oject, where v f fil velocity, v i iitil velocity, ccelertio, d d distce trvelled. A pple is throw from the tll uildig 00 m high with iitil velocity of m/s. The ccelertio due to grvity is.8 m/s. Wht is the fil velocity of the pple s it reches the groud? Solve for v f : v f? v i m/s d 00 m.8 m/s + d Emple : Evlute usig oly positive roots. v v f f vi + d v () + (.8)( 00) v i c. d. v v f f f v f 7. m/s Emple : Evlute usig oly positive roots. Verify y usig clcultor () + () 0 + 7() Properties of Rdicls Distriutio of Rdicls of the Sme Ide (where 0 d 0 if is eve) ( )( ) m Power Rule of Rdicls Multiplyig Epoets ( m ) Reverse Opertios of Rdicls d Epoets (if is odd) (if is eve) Etire Rdicls: - rdicls tht hve o coefficiet i frot of them. Emples: d 8 Mied Rdicls: - rdicls tht hve coefficiets i frot of them. Emples: d - the coefficiet is the th root of the rdicd s perfect th fctor. To covert etire rdicl to mied rdicl, fid the lrgest perfect th fctor of the rdicd d write its root s coefficiet follow y the rdicd fctor tht remis. Copyrighted y Griel Tg B.Ed., B.Sc. Pge.

10 Prerequisites Chpter: Alger Review Alger Emple : Simplify. (Covert them to mied rdicls.) y.. ( ) y y ( ) y ( y ) Emple : Evlute usig oly positive roots. 8 c... ( ) y y 7c d 7 8p q p q Emple 7: Write the followigs s etire rdicls. Emple 8: Order,, d from lest to gretest. ( ) () 8 ( )() 8 00 Addig d Sutrctig Rdicls: - Rdicls c e dded or sutrcted if d oly if they hve the sme ide d rdicd. - Covert y etire rdicls ito mied rdicls first. The, comie like terms (rdicls with the sme rdicd) y ddig or sutrctig their coefficiets. Emple : Simplify. ( ) 8 ( ) ( ) ( ) 7c d 7c d 7 c d d c To covert mied rdicl to etire rdicl, rise the coefficiet to the ide, th power, d multiply the result to the rdicd. ( )( 7 ) ( )( 7 ) ( ) + ( ) p q p q ( p ) p q 7 pq + ( ) 8p q q ( 7 pq ) 7 < < 08 < < Pge 0. Copyrighted y Griel Tg B.Ed., B.Sc.

11 Alger Prerequisites Chpter: Alger Review Rtioliztio: - turig rdicl deomitor ito turl umer deomitor. For m <, Emple 0: Simplify. m m ( m) ( ) m ( m) ( ) ( ) Rtiol Epoets m m The ide of the rdicl is the deomitor of the frctiol epoet. Emple : Evlute usig clcultor.. ().. c. 7 8 Emple : Simplify usig oly positive epoets... ( ) y c. ( ) ( ) y ( ) d. ( 8 ) e. ( ) ( ) ( ) ( ) ( ) 8 ( ) ( ) y y ( ) 8 8 ( ) ( ) ( ) 8 8 ( + ) 0 d 7 ANS () P- Assigmet: pg. #,,, 7,, 7,,,,,, 7,, ; Hoours: #7 Copyrighted y Griel Tg B.Ed., B.Sc. Pge.

12 Prerequisites Chpter: Alger Review Alger P-: Algeric Epressios Epressios: - mthemticl seteces with o equl sig. Emple: + Equtios: - mthemticl seteces tht re equted with equl sig. Emple: Terms: - re seprted y dditio or sutrctio sig. - ech term egis with the sig precedig the vrile or coefficiet. Numericl Coefficiet Moomil: - oe term epressio. Emple: Epoet Vrile Biomil: - two terms epressio. Emple: + Triomil: - three terms epressio. Emple: + + Polyomil: - my terms (more th oe) epressio with whole umer epoets where 0,,, re rel umer coefficiets, d is whole umer epoets to the th degree. Degree: - the term of polyomil tht cotis the lrgest sum of epoets Emple: th Degree Polyomil Emple : Fill i the tle elow. Polyomil Numer of Terms Clssifictio Degree Clssified y Degree moomil 0 costt moomil lier + iomil lier + triomil qudrtic + + polyomil cuic + triomil qurtic Like Terms: - terms tht hve the sme vriles d epoets. Emples: y d y re like terms y d y re NOT like terms To Add d Sutrct Polyomils: - Comie like terms y ddig or sutrctig their umericl coefficiets. Emple : Simplify ( y + y ) + ( y 0 y ) c. ( y + y ) ( y 0 y ) y + y + y 0 y y + y y + 0 y y + 7 y y + y (drop rckets d switch sigs i the rcket tht hd sig i frot of it) Pge. Copyrighted y Griel Tg B.Ed., B.Sc.

13 Alger Prerequisites Chpter: Alger Review Multiplyig Moomils with Polyomils Emple : Simplify.. ( + ). ( + ) ( ) c. 8( + ) ( + 7) ( + ) + 8 Multiplyig Polyomils with Polyomils Emple : Simplify. ( + ) ( ) (oly multiply rckets right fter the moomil). ( + )( ). ( + )( + ) ( + ) ( ) ( + ) ( + ) c. ( + )( + ) ( )( + ) d. ( + )( + ) 8 ( + ) ( + 7) ( + ) ( + ) ( ) ( + ) ( ) ( + ) ( ) ( + ) ( + ) ( + ) Emple : Simplify. Specil Products (A + B)(A B) A B (A + B) A + AB + B (A + B) A + A B + AB + B (A B) A AB + B (A B) A A B + AB B. ( + ). ( ) Let A d B (A + B) A + AB + B ( + ) () + ()() + () + + ( + ) ( + ) Let A d B (A B) A A B + AB B ( ) () () () + ()() () P- Assigmet: pg. 0 #7,, 7,,, 7,, 7, 7, ; Hoours: #0 Copyrighted y Griel Tg B.Ed., B.Sc. Pge.

14 Prerequisites Chpter: Alger Review Alger P-: Fctorig (Prt ) Fctorig: - reverse opertio of epdig (multiplyig). - i essece, we re dividig, with the eceptio tht the fctors c e polyomils. Commo Fctors: - fctors tht re commo i ech term of polyomil.. Numericl GCF: - Gretest Commo Fctor of ll umericl coefficiets d costt.. Vrile GCF: - the lowest epoet of prticulr vrile. After otiig the GCF, use it to divide ech term of the polyomil for the remiig fctor. Fctor y Groupig (Commo Brckets s GCF) (c + d) + (c + d) (c + d) ( + ) Commo Brckets Tke commo rcket out s GCF Emple : Fctor ech epressio ( ) + ( ) ( + ) GCF ( )( + ) GCF ( ) c. + c + + c d. y + y ( + c) + ( + c) ( + c) + ( + c) GCF ( + c) ( + c)( + ) Fctorig + + c (Ledig Coefficiet is ) ( y ) + ( y ) ( y ) + ( y ) Try gi fter rerrgig terms! + y y ( + ) (y + y ) ( + ) y ( + ) ( + )( y ) Brckets re NOT the sme! We might hve to first rerrge terms. Switch Sig i Secod Brcket! We hve put mius sig i frot of ew rcket! + + c Wht two umers multiply to give c, ut dd up to e? Emple : Completely fctor ech epressio.. Fctor Pirs of 0: ( 0) ( 0) ( + )( ) ( ) ( ) ( )( ) Fctor Pirs of : ( ) ( ) ( ) ( ) ( + ) sum of ( + ) sum of 8 Pge. Copyrighted y Griel Tg B.Ed., B.Sc.

15 Alger ( 0) Tke out GCF ( + )( ) (+)() 0 (+) + () Fctorig + + c (Ledig Coefficiet is ot, ) Emple : Fctor + + Emple : Fctor completely. Prerequisites Chpter: Alger Review c. 7y + y Fctor Pirs of : ( ) ( ) d. w w ( y)( y) ( ) ( ) w w + Rerrge i Descedig Degree. ( ) ( ) (w + w ) Tke out s commo fctor. () + () sum of 7 (+7) + () (+7)() (w + 7)(w ) e. 0 f. + (+)() ( + )( ) (+) + () For fctorig triomil with the form + + c, we will hve to fctor y groupig. + + Assume + + c s the sme s + + c d fctor. The swer will e ( ) ( ). Multiply d c. Fctor Pirs of : ( ) ( ) ( ) ( ) ( 8) ( 8) ( ) ( ) Split the term ito two seprte terms. ( + ) + sum (8 of + ) Group y rckets ( + ) + ( + ) Tke out GCF for ech rcket. ( + )( + ) Fctor y Commo Brcket! ( + 8) sum of. +. 8m m ( 7 + ) GCF 8m + m m ( + ) ()() (8m + m) (m + ) [ ( ) ( ) ] () + () 7 m (m + ) (m + ) [ ( ) ( ) ] switch sig! (m + )(m ) ( sig i frot ( )( ) of rcket) c. ( ) + ( ) + d. 8 7 y + y ( ) + ( ) + Let A ( ) A + A + A + A + A + (A + A) + (A + ) ()() A(A + ) + (A + ) () + () (A + )(A + ) [( ) + ] [( ) + ] Sustitute ( ) ( )( + ) ck ito A 8 y y + y (8 y) ( y y ) ( y) y ( y) ( y)( y) P- (Prt ) Assigmet: pg. 7 #,,,, 7, 8 7 ()() 7 () + () switch sig! ( sig i frot of rcket) 8 7 ()() 7 () + () 7 switch sig! ( sig i frot of rcket) Copyrighted y Griel Tg B.Ed., B.Sc. Pge.

16 Prerequisites Chpter: Alger Review Alger P-: Fctorig (Prt ) Specil Epressios Differece of Squres A B (A + B)(A B) Perfect Triomil Squres A + AB + B (A + B) Perfect Triomil Squres A AB + B (A B) Sum of Cues A + B (A + B)(A AB + B ) Differece of Cues A B (A B)(A + AB + B ) Emple : Fctor completely c. 8 (NOT Fctorle ( 00) GCF ( )( + ) Sum of Squres) ( 0)( + 0) ( )( + )( + ) d. y e. ( + ) ( ) ( 8y)( + 8y) [( + ) ( )] [( + ) + ( )] [ + ] [ + ] ( )( + ) Wtch Out! Sutrctig rcket! Look t ( + ) d ( ) s idividul items! Tke out egtive sig from the first rcket! Perfect Triomil Squre + + c ( + c ) + c ( c ) where, c re squre umers, d ( )( c ) Emple : Epd ( + ). ( + ) ( +)( + ) ( )( ) Emple : Fctor completely ( )( ) 0 ( + ) c ( )( ) 8 7 ( 7) Assumes + +c is the sme s + + c. But the swer will e i the form of ( + ) ( + ) ( )( 00 ) 0 ( 0) Pge. Copyrighted y Griel Tg B.Ed., B.Sc.

17 Alger Prerequisites Chpter: Alger Review Emple : Fctor completely y. 7 Let A 7 d B 8y Hece, A d B y A + B (A + B)(A AB + B ) 7 + 8y ( + y)(() ()(y) + (y) ) ( 8) ( + )( + + ) Fctorig No-Polyomil Epressios - lwys tke out the GCF with the lowest epoets of y commo vriles. - divide ech term y the GCF. Be creful with frctiol epoets. Emple : Fctor completely.. y y y ( + y)( y + y ) y (y y ) GCF y (lowest epoet) y (y 8)(y + ) Fctor form + + c. r ( r + ) (r + ) Let A (r + ) r ( + ) Sum of Cues A + B (A + B)(A AB + B ) Differece of Cues A B (A B)(A + AB + B ) r (r + ) r A A ( + ) ( + ) A 7 ( 8) Let A d B 8 Hece, A d B GCF A B (A B)(A + AB + B ) 8 ( + )(() + ()() + () ) y y y y ra ( ) r A ra ( ) y y ( ) y y A [ra ] GCF A (lowest epoet) r [r(r + ) ] Sustitute (r + ) ck ito A r [r + r ] Fctor form + + c A A ( ) A ( + ) r (r )(r + ) Fctorig Cuic Polyomils y Groupig - for cuic polyomils cosists of four terms, we c sometimes fctor them y groupig. Emple : Fctor + 0 completely. ( ) ( 0) switch sig! ( sig i frot of rcket) ( ) ( ) Fctor GCF from ech group ( )( ) GCF ( ) ( )( + )( ) Fctor Differece of Squres P- (Prt ) Assigmet: pg. 8 #7,,,,,, 7,,, 7,,,, 7,, 8 d 8c; Hoours: #7, 7 Copyrighted y Griel Tg B.Ed., B.Sc. Pge 7.

18 Prerequisites Chpter: Alger Review Alger P-7: Rtiol Epressios Frctiol Epressio: - quotiet of two lgeric epressios. - the vrile(s) c hve egtive d frctiol epoets (or i rdicl form). Emples: Rtiol Epressio: - frctiol epressios with polyomils s deomitor d / or umertor. Emples: Domi: - ll possile -vlues from lgeric epressio. - some lgeric epressios hve certi o go zoe. This might ivolve ot eig le to divide y zero or hs to e positive ecuse it is i eve ideed rdicl. Emples: Domi is { 0} Domi is { 0} Emple : Fid the domi of the followig epressios. Domi is { > 0} There is o restrictio o Sice there is polyomil s c e ythig i the epressio i the deomitor, rel umer set. Hece, the we eed to solve it whe it is ot domi is R. equl to zero y fctorig to fid the domi. 0 ( )( + ) 0 0 or + 0 Domi: or c. We eed to fid the domi of the umertor (rdicl) s well s the deomitor (polyomil). 0 For, 0 Comie Domi Domi: 0 d Simplifyig Rtiol Epressios: - fctor oth the umertor d deomitor d ccel out the commo fctors / rckets etwee them. - this is similr to reducig umericl frctio y ccellig out the commo fctors etwee the umertor d deomitor. - the fil domi is the domi of the origil rtiol epressio, ot the domi of the reduced form. Emples: Pge 8. This is ecuse relly mes ( ) ( + ) ( )( ) ( ) ( )( + ) ( + ) Note: we cot ccel Domi: or + Copyrighted y Griel Tg B.Ed., B.Sc. + d we hve do the prethesis first efore divisio.

19 Alger Prerequisites Chpter: Alger Review Emple : Simplify the followig epressios d stte their domis.. + ( + ) ( + ) 0 0 or + 0 Domi: 0 or ( )( ) ( )( ) ( ) ( ) + 0 ( )( ) 0 0 or 0 Domi: or Multiplyig d Dividig Rtiol Epressios: - much like multiplyig d dividig frctios, we fctor ll umertors d deomitors d reduce commo rcket(s) / fctors etwee them. - for divisio, we must flip (tke the reciprocl) of the frctio ehid the sig. - the fil domi is the domi of oth the origil rtiol epressios, ot the domi of the reduced swer. Emples: Emple : Perform the idicted opertios, simplify d stte their domis ( + )( ) ( )( + ) ( + ) ( ) ( + )( + ) ( + )( ) ( ) 0 or ( + ) 0 ( ) 0 or ( ) 0 Domi: or or or ( + )( + ) ( + )( + ) ( + )( ) ( )( + ) ( + )( + ) ( )( ) ( + ) 0 or ( ) 0 ( + ) 0 or ( + ) 0 ( ) 0 or ( + ) 0 Domi:,,, or Domi is tke from the umertor d the deomitor of the frctio fter the sig. Lowest Commo Deomitor (LCM) of Moomils: - LCD of moomil coefficiet, d the vrile(s) with its / their highest epoet(s). Emple: LCD of,, LCD 0 LCM of,, 0 Vrile with Highest Epoet Copyrighted y Griel Tg B.Ed., B.Sc. Pge.

20 Prerequisites Chpter: Alger Review Alger Lowest Commo Deomitor (LCM) of Polyomils: - commo fctor(s) (writte oce) log with y ucommo (leftover) fctor(s). Emple: LCD of d Fctors of ( ) ( + ) d Fctors of ( + ) ( ) Commo Fctor Leftovers Addig d Sutrctig Rtiol Epressios: - much like ddig d sutrctig frctios, we first fid the LCD of the deomitors. The, we covert ech frctio ito their equivlet frctios efore ddig or sutrctig the umertors. - the fil domi is the domi of oth the origil rtiol epressios, ot the domi of the reduced swer. Emple: + (LCD ) Emple : Perform the idicted opertios, simplify d stte their domis ( + ) ()() + ( + ) ( + ) ( + ). + Compoud Frctio: - frctio where the umertor d / or deomitor themselves coti frctio(s). Simplifyig Compoud Frctios: - simplify ech of the umertor d deomitor ito sigle frctios. The, divide the umertor s frctio y the deomitor s frctio. + Emple: Simplify LCD ( + ) + + ( + ) + ( + ) 0 Domi: LCD ( + ) ( ) ( ) ( )( + ) ( )( ) ( )( ) ( )( + ) ( )( )( + ) ( ) ( + ) ( + 0) ( ) ( + ) LCD ( ) ( ) ( + ) 0 ( ) ( + ) Commo Fctor Leftovers ( ) 0 or ( + ) 0 Domi: or + Pge 0. Copyrighted y Griel Tg B.Ed., B.Sc.

21 Alger Prerequisites Chpter: Alger Review Emple : Simplify ( y ) + y y + y y y + ( y ) ( y ) y + y y y +. y y ( y + )( y ) ( y )( y ) ( )( ) ( y )( y ) ( y + ) ( y ) Cojugtes: - iomils tht hve the ect sme terms y opposite sigs i etwee. Emples: ( + ) d ( ) ( + c d ) d ( c d ) Multiplyig Cojugte Rdicls: - multiplyig cojugte rdicls will lwys give Rtiol Numer (rdicl terms would ccel out). Emple : Simplify ( )( ) +. Rtiolizig Biomil Rdicl Deomitor: - multiply the rdicl epressio y frctio tht cosists of the cojugte of the deomitor over itself. Emple 7: Simplify. + 7 ( + )( ) () ( + 7) ( 7 ( ) ) ( 7) ( 7) ( 7) ( 7) 7 8 Notice the middle two rdicl terms lwys ccel out! P-7 Assigmet: pg. 7 #,, 7,,,,,,,,,, 77, 8 d ; Hoours: #7 Copyrighted y Griel Tg B.Ed., B.Sc. Pge.

SM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory

SM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.