StandardAlgebra2 Unit3 AlgebraicMethods Name Pd. 5
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
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)
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 3 8. 2x 5 3x 6 9. 2 3 2 4 10. 3 4 x 3 x 5 11. 5 2 x 4 x 5 9 8
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 ) 5 16. y 3 ) 6 17. n 4 ) 4 18. x 7 ) 2 19. 2x) 4 20. 5y 8 ) 2 21. xy) 3 22. 2x 3 ) 4 23. 3x 3 y 4 ) 2 9
MoreRulesofExponents Name Rule Examples QuotientofPowers PowerofaQuotient Let s practice. 2. a 8 a 3 3. 7 11 7 4. 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 4 9. 4x 5 y 8 2x 5 y 2 10. 3x 14 y 6 18x 2 y 10
11. 9b 5 15b 4 12. " x% ' # y& 7 13. " 2 # x% 4 14. " # x 2 y 3 % ' & 4 15. " 2x 4 % ' # 3y & 2 16. 12a8 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 3. 5 7 9 5 4 4. x 10 y x 4 y 6 5. 8g3 h 4 g 3 h 5 6. 2x 6 6x 11 11
7. x 3 4x 8. 53 5 8 9. 9x 4 y 3 12x 6 y 11 10. x 2 11. y 3 y 4 12. x 5 x 5 13. x 7 x 3 14. 4x 3 y 8 2x 5 y 8 15. " 8x 0 % # 20x 4 ' & 16. " # 3x 9 4y % ' & 2 17. x 6 y 2 x 2 y 6 18. 3a 2 x 2 9a 0 x 2 19. m 2 n 3 n 4 p 5 20. " a # % b 2 1 21. 25x 2 y 6 z 4 15x 3 y 0 z 1 12
WenowneedtolookatanewtypeofexpressioncalledaRationalExpression. Wealwayswanttosimplifyrationalexpressionsintotheirsimplestform.Todothat,weneedto reviewfactoring.let stryafewreviewexamples. 1. x 2 9 2.12x 4 +18x 3. x 2 5x 36 4. x 2 +16 5. 3x 2 x 2 6.10x 3 15x 2 So,howdowesimplifyafractionintosimplestform?Let sstartwithsomeeasyonestoreview 7. 3 9 ARationalExpressionisnothingmorethanafractioninwhichthe numeratorand/orthedenominatorarepolynomials.herearesome examples: 10. 10 36 8. 9. 50 12 8mn 12x 2 20mp 11. ***Wemustalwaysmakesurethatthedenominatordoesnot=0. 30x 3 12. 13 7x 3 28x 2
Pleasesimplifyeachrationalexpression.Youmustalwaysfactorcompletelyfirst! 13. 15. 17. 3x +1) x 4) x + 3) x + 4) 3x +1) 14. x +1) x + 3) ) x 2) 4x 8 ) 4x 1) 16. 5x x + 7 15x x 2 14 12 +16x 5x 2 +10x 3x 2 27x 3 15x 3 + 25x 2 18. 3x 2 x 7 x 2 + 2x 19. 7 x 20. x 2 + 3x + 2 21. x +1) 2 5x + 20 22. x +1 6x + 24
23. 25. 27. 29. x 2 36 x + 4 24. 3x +18 x 2 x 2 9 x 3 + 3x 2 10x x 2 + 7x +12 26. 4x 2 8x 2x + 5) 2 3x 1) 3x 1) 2 2x + 5) 28. 2x +14 49 x 2 x 2 9x + 20 x 2 + 2x 3 30. x 4 + 8x 2 x 2 + 8 15
MultiplyingRationalExpressions AlwaysFactorFirst!!! Simplify. 1. 3 5 5 3 2. 2 7 14 6 12 3. 20 25 18 4. 81 12 16 27 5. 3x y 2 y 6x 6. 4a2 3b b 2 7. 24xy 16x 2 y 32x 4 y 3 12x 2 y 8. 30xy 18abc 24ac 12x 2 y 3 16
9. x + 3) x +1 ) x 4) x + 3) 10. 2x +10 x 4 3x 12 x + 5 11. 2x + 8 x 1 3x 3 x + 4 12. 5x 15 2x +12 x + 6 3 x 13. x 2 4x 12 x 2 4 x + 2 x 6 14. x 2 5x 6 6x + 6 2x 4 x 2 36 15. 4x 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
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
19 7.13x 7x=24 8.14=8x 58 9.K23Kx)=14 10.xK7=K13Kx 11.K8+3x=2xK5) 12.17+7x=8+10x 13.K3x+1=K4x+8 14.K8x+3xK2)=K3x+2 15.K56 x)=2xk15) 16.2xK4) 3x+4)=K50
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+14 20 You should always put your quadratic equation in standard form. You should always have a positive coefficient in front of x 2.
Pleasesolvethefollowingquadraticequations. 8.6=Kx 2 +3x+6 9.x 2 +49=14x 10.15+x 2 =2x 2 KxK5 11.x 2 Kx=3x+12 12.85=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
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
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 + 4 23
! 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
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
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 + 5 + 7 =12 7. c 3 2 = 4 26
8. 4 + h +1 =14 9. 2 + 3x 6 = 6 10. 2x +1 = 3x + 4 11. 2x = x + 5 12. 3x 2 2x + 7 = 0 13. 3x + 2 x + 4 = 0 27
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