SA2$ Algebraic$Methods$$$ $$Unit$3$$$ $ $ $ Standard$Algebra$2$ Unit$3$ Algebraic$Methods$ $ $ $ $ Name$ $$Pd.$ $
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1 StandardAlgebra2 Unit3 AlgebraicMethods Name Pd. 5
2 Weneedtoreviewcombiningliketerms.Youhavealreadyusedthisskillthisyear,butwewill useitinamorecomplicatedwayduringthisunit. Pleasesimplifythefollowingalgebraicexpressions. 1.5x 2 x+4 2.K8yK3x+9 4xK11 3.5x 3 2x 7x+4x 2 4.K2x 2 +6x 3 +4x 2 K9x 3 xk10x 2 5.x+4) 3x 5) 6.x+2y)+y+6) 7.Whatistheperimeterofthefollowing? a. 4x + 5 b. 2x + 1 2x + 1 3x - 2 6x - 4 7x + 8 4x + 5 6
3 7 8.2x 2 +4xK1)+7x 2 K5x 3) 9.4x 2 K8x+6)Kx 2 K2x+9) 10.Astopsignhasasideoflength2a+1.Whatistheperimeterofthestopsign? 11.2xK1)+3x+4) 12.K62x+1)K55x+4) 13.x8xK3)+2x9x+10) 14.Kx2x+5)K4x3xK1) 15.x3x 2 +2xK1)+x4x 2 K4x 3) 16.xx 2 K8x+4)Kx9x 2 Kx+5)
4 RulesofExponents Name Rule Examples ProductofPowers PowerofaPower PowerofaProduct Let sreviewsomedefinitionsyoumightrecognize!pleaseusethevocabularytermslistedbelow tocompleteeachstatement. binomial polynomial monomial liketerms trinomial 1. A isapolynomialwiththreeterms. 2. A isapolynomialwithoneterm. 3. havethesamevariableandexponent.youneedtocombine allofthesetosimplifyanalgebraicexpression. 4. A isapolynomialwithtwoterms. 5. Anyexpressionthatisformedbyaddingtermsoftheformax n whereaisarealnumber andnisanonnegativeinteger,iscalleda expression. Pleasesimplifyeachexpressioncompletely. 6. x 4 x 7 7. x x x 5 3x x 3 x x 4 x 5 9 8
5 12. 6x 2 ) 4 x 2 ) 13. x 6 y 2 ) x 4 y 7 ) 14. 2gh 3 ) 4g 2 h 5 ) 15. x 2 ) y 3 ) n 4 ) x 7 ) x) y 8 ) xy) x 3 ) x 3 y 4 ) 2 9
6 MoreRulesofExponents Name Rule Examples QuotientofPowers PowerofaQuotient Let s practice. 2. a 8 a x 2 8 x 2 5. x 10 x 6. 12g7 h 4 4 g 3 h 7. 4 p 6 8p 2 8. c 9 6c x 5 y 8 2x 5 y x 14 y 6 18x 2 y 10
7 11. 9b 5 15b " x% ' # y& " 2 # x% " # x 2 y 3 % ' & " 2x 4 % ' # 3y & a8 b 5 30a 2 b 5 MoreRulesofExponents Name Rule Examples NegativePower orexponent) ZeroPower orexponent) Let s practice. 1. x 9 x 2 2. x 2 x x 10 y x 4 y g3 h 4 g 3 h x 6 6x 11 11
8 7. x 3 4x x 4 y 3 12x 6 y x y 3 y x 5 x x 7 x x 3 y 8 2x 5 y " 8x 0 % # 20x 4 ' & 16. " # 3x 9 4y % ' & x 6 y 2 x 2 y a 2 x 2 9a 0 x m 2 n 3 n 4 p " a # % b x 2 y 6 z 4 15x 3 y 0 z 1 12
9 WenowneedtolookatanewtypeofexpressioncalledaRationalExpression. Wealwayswanttosimplifyrationalexpressionsintotheirsimplestform.Todothat,weneedto reviewfactoring.let stryafewreviewexamples. 1. x x 4 +18x 3. x 2 5x x x 2 x x 3 15x 2 So,howdowesimplifyafractionintosimplestform?Let sstartwithsomeeasyonestoreview ARationalExpressionisnothingmorethanafractioninwhichthe numeratorand/orthedenominatorarepolynomials.herearesome examples: mn 12x 2 20mp 11. ***Wemustalwaysmakesurethatthedenominatordoesnot=0. 30x x 3 28x 2
10 Pleasesimplifyeachrationalexpression.Youmustalwaysfactorcompletelyfirst! x +1) x 4) x + 3) x + 4) 3x +1) 14. x +1) x + 3) ) x 2) 4x 8 ) 4x 1) 16. 5x x x x x 5x 2 +10x 3x 2 27x 3 15x x x 2 x 7 x 2 + 2x x 20. x 2 + 3x x +1) 2 5x x +1 6x + 24
11 x 2 36 x x +18 x 2 x 2 9 x 3 + 3x 2 10x x 2 + 7x x 2 8x 2x + 5) 2 3x 1) 3x 1) 2 2x + 5) 28. 2x x 2 x 2 9x + 20 x 2 + 2x x 4 + 8x 2 x
12 MultiplyingRationalExpressions AlwaysFactorFirst!!! Simplify x y 2 y 6x 6. 4a2 3b b xy 16x 2 y 32x 4 y 3 12x 2 y 8. 30xy 18abc 24ac 12x 2 y 3 16
13 9. x + 3) x +1 ) x 4) x + 3) 10. 2x +10 x 4 3x 12 x x + 8 x 1 3x 3 x x 15 2x +12 x x 13. x 2 4x 12 x 2 4 x + 2 x x 2 5x 6 6x + 6 2x 4 x x 2 4 x x 2 + 2x 3 x 2 + x 6 4 x 16. x 2 + x 2x 2 3x 4 x 2 9 2x 2 + 3x 17
14 Weneedtoreviewsolvinglinearequations.Tosolvetheseequations,weneedtoreviewinverse operations.pleasesolvethefollowingforx. 1.x 3=K7 2.x+9=2 Toundosubtraction,weuse.Toundoaddition,weuse.Theyareinverseoperations. 3.5x=45 4. x 4 = 6 Toundomultiplication,weuse.Toundodivision,weuse.Theyareinverseoperations. Youmustuseinverseoperationstosolvelinearequations.Hereisalistofstepstohelp! Pleaseusetheabovestepstosolvethefollowingequations. 5.3xK12=15 6. x 8 7 = 1 1. Removeanyparenthesisbyusingthedistributiveproperty. 2. Combine liketerms 3. Moveallofthevariablestotheleftside,usinginverseoperations. 4. Movealloftheconstantstotherightsside,usinginverseoperations. 5. Useinverseoperationstoeliminatethecoefficient.Younowhave yoursolutiontoyourequation. 18
15 x 7x= =8x 58 9.K23Kx)=14 10.xK7=K13Kx 11.K8+3x=2xK5) x=8+10x 13.K3x+1=K4x+8 14.K8x+3xK2)=K3x+2 15.K56 x)=2xk15) 16.2xK4) 3x+4)=K50
16 Now,wewillreviewsolvingquadraticequations.Remember StandardFormofaQuadraticEquation: So,thereisasquareinthesetypesofequations.Therefore,weneedtosolvetheminadifferent way!doyourememberhowtosolvethese? 1.xK3)x+1)=0 2.xx+5)=0 3.2x+4)x 6)=0 4.7x 7)3x+2)=0 5.x 2 K3xK40=0 6.x 2 +7x+10=0 Let sthinkbacktothestepstosolvingalinearequation.whatstepsshouldyoutaketosolvethis equation? 7.2x 2 +3xK10=x 2 +5x You should always put your quadratic equation in standard form. You should always have a positive coefficient in front of x 2.
17 Pleasesolvethefollowingquadraticequations. 8.6=Kx 2 +3x+6 9.x 2 +49=14x x 2 =2x 2 KxK5 11.x 2 Kx=3x =Kx 2 K18x+4 13.K1+5x 2 K10x=3x 2 K7xK2 14. The length of a rectangle is 5 more than its width. Please find the dimensions of the rectangle if its area is 84 m 2. x + 5 x 21
18 Arationalequationisanequationinwhichoneormoreofthetermsarefractional. Inthissection,therearethreetypesofrationalequationstosolve.Thefirsttype hasafractionsetequaltoanonkfraction.tosolve,wemultiplybybothsidesby thedenominatorofthefraction. Solveforx.! 1.)!! 8 3x 7 = 4! 2.)!! 9 7 5x = 2! 3.)!! 3 2x + 5 = 3! 4.)!! 6 4x 8 = 3! 5.)!! 13x 8x 3 = 2! 6.)!! 10x 6x +1 = 5! 7.)!! 3x 2x + 7 = 3 22
19 Thesecondtypeofrationalequationhastwofractionssetequaltoeachother.To solve,wecrossmultiply.! 8.)!! 3 2x + 5 = 4 3x + 4! 9.)!! x 5x 2 = 1 3! 10.)!! 3 x = 10 x + 7! 11.)!! 4 2x + 3 = 5 x! 12.)!! 1 x 2 1 = 1 5x 7! 13.)!! x 5 = 1 x
20 ! 14.)!! 1 x +12 = x 2x 21! 15.)!! 8 x + 8 = x x +2! 16.)!! 8 x + 3 = x + 5 3! 17.)!! 2 x 3 = x 4 2x 9 24
21 Equationsthatcontainvariablesintheradicandarecalledradicalequations.Tosolve radicalequations: Isolatethevariableononesideoftheequation. Thensquareeachsideoftheequationtoeliminatetheradical. Solvetheremainingequationforthevariable. Power Property If you square both sides of an equation, the resulting equation is still true. If, then If, then **Extraneous solutions can be introduced when you raise both sides of an equation to a power. So always check solutions in the original equation. 1. Theequationv = 2.5r representshowfastyoucansafelydriveyourcaronan unbankedcurvewherevisthemaximumvelocityofyourcarinmphandristhe radiusofthecurveinfeet.ifyouaredriving65mph,whatisthemaximumradius ofthecurveandstilldrivesafely? 2. ThepowerPinwatts)thatacircularsolarcellproducesandtheradiusofthecell incentimetersarerelatedbythesquarerootequation r = muchpowerisproducedbyacellwitharadiusof10cm? P 0.02π.Abouthow 25
22 3. Supposethefunction S = π 9.8l 7,whereSrepresentsspeedinmeterspersecond andlistheleglengthofapersoninmeters,canapproximatethemaximumspeed thatapersoncanrun. a.whatisthemaximumrunningspeedofapersonwithaleglengthof1.1meters tothenearesttenthofameter? b.whatistheleglengthofapersonwitharunningspeedof2.7meterspersecond tothenearesttenthofameter? c.asaperson sleglengthincreases,doestheirspeedincreaseordecrease? Explain. Solvethefollowingradicalequations.Checkyoursolution. 4. x = 3 5. x + 5 = 4 6. a =12 7. c 3 2 = 4 26
23 h +1 = x 6 = x +1 = 3x x = x x 2 2x + 7 = x + 2 x + 4 = 0 27
24 18. Theformula S = 2π L 32 representstheswingofapendulum.sisthetimein secondstoswingbackandforth,andlisthelengthofthependuluminfeet. a.howlongdoesittakefora3footpendulumtoswingbackandforth?roundto threedecimalplaces) b.findthelengthofapendulumthatmakesoneswingin2.5seconds.roundto threedecimalplaces.) 19. Thespeedthatatsunamitidalwave)cantravelismodeledbytheequation S = 356 d wheresisthespeedinkilometersperhouranddistheaveragedepth ofthewaterinkilometers. a.whatisthespeedofthetsunamiwhentheaveragewaterdepthis0.512 kilometers?roundtonearesttenth) b.atsunamiisfoundtobetravelingat120kilometersperhour.whatisthe averagedepthofthewater?roundtothreedecimalplaces) 28
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