Algebra 2 Unit 1: Algebraic Essentials Review Algebra 2 ... · Algebra 2 Unit 1: Algebraic...

Algebra2Unit1:AlgebraicEssentialsReview

Ms.Talhami 1

Algebra2Unit1:AlgebraicEssentials

Review

Name_________________


Ms.Talhami 2

VARIABLES, TERMS, AND EXPRESSIONS COMMONCOREALGEBRAII

Mathematicshasdevelopedalanguagealltoitselfinordertoclarifyconceptsandremoveambiguityfromtheanalysisofproblems.Toachievethis,though,wehavetoagreeonbasicdefinitionssothatwecanallspeakthissamelanguage.So,westartourcourseinAlgebraIIwithsomebasicreviewofconceptsthatyousawinAlgebraI.Exercise#1:Considertheexpression 22 3 7x x+ − .

(c) What is the sum of this expressionwith the

expression 25 12 2x x− + ?Exercise#2:Most students learn that toadd two like terms they simplyadd the coefficientsand leave thevariables and powers unchanged. But,why does thiswork? Below is an example of the technical steps tocombinetwoliketerms.Whatrealnumberpropertyjustifiesthefirststep?

( )( )

2 2 2

2 2

4 6 4 6

10 10

x y x y x y

x y x y

+ = +

= =

SOMEBASICDEFINITIONS

Variable:Aquantitythatisrepresentedbyaletterorsymbolthatisunknown,unspecified,orcanchangewithinthecontextofaproblem.Terms:Asinglenumberorcombinationofnumbersandvariablesusingexclusivelymultiplicationordivision.Thisdefinitionwillexpandwhenweintroducehigher-levelfunctions.Expression:Acombinationoftermsusingadditionandsubtraction.

(a) Howmanytermsdoesthisexpressioncontain? (b) Evaluate this expression, without yourcalculator, when . Show yourcalculations.

LIKETERMS

LikeTerms:Twoormoretermsthathavethesamevariablesraisedtothesamepowers.Inliketerms,onlythecoefficients(themultiplyingnumbers)candiffer.

Justification?


Ms.Talhami 3

Exercise#3:Theprocedureforsimplifyingthelinearexpression ( ) ( )8 2 3 5 3 1x x+ + + isshownbelow.Statetherealnumberpropertythatjustifieseachstep.( ) ( )

( ) ( )

( )

( )

8 2 3 5 3 1 8 2 8 3 5 3 5 1

8 2 24 5 3 5 16 24 15 5

16 15 24 5

16 15 24 5

31 24 5

31 29

x x x x

x x x x

x x

x

x

x

+ + + = ⋅ + ⋅ + ⋅ + ⋅

= ⋅ + + ⋅ + = + + +

= + + +

= + + +

= + +

= +

Exercise#4:Becauseweusedrealnumberpropertiestotransformtheexpression ( ) ( )8 2 3 5 3 1x x+ + + intoasimplerform 31 29x + ,thesetwoexpressionsareequivalent.Howcanyoutestthisequivalency?Showworkforyourtest.

REALNUMBERPROPERTIES

If areanyrealnumbersthenthefollowingpropertiesarealwaystrue:

1.TheCommutativePropertiesofAdditionandMultiplication:

and 2.TheAssociativePropertiesofAdditionandMultiplication:

and

3.TheDistributivePropertyofMultiplicationandDivisionOverAdditionandSubtraction:

and


Ms.Talhami 4

VARIABLES, TERMS, AND EXPRESSIONS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Foreachofthefollowingexpressions,statethenumberofterms.

(a) 23 1x − (b) 2 38 7 2x x x+ − + (c) 2 2 417 22

xy x y xy− +

2. Simplifyeachofthefollowingexpressionsbycombiningliketerms.Becarefultoonlycombinetermsthat

havethesamevariablesandpowers. (a) 2 22 8 1 5 2 8x x x x+ − + − − (b) 2 25 2 10 7 5x x x x− − + − + + (c) 2 2 2 24 2 9x y xy xy x y− + − (d) 2 2 2 3 2 2 37 2 4 2 9 4x x y xy y x x y y− + − + + +

3. Giventhealgebraicexpression 2

12 121

xx+−

dothefollowing:

(c) Ninabelievesthatthisexpressionisequivalenttodividing12byonelessthanx.Doyourresultsfrom(a)

and(b)supportthisassertion?Explain.

(a) Evaluatetheexpressionforwhen . (b) Evaluatetheexpressionforwhen .


Ms.Talhami 5

4. Classifyeachofthefollowingaseitheramonomial(singleterm),abinomial(twoterms)oratrinomial(threeterms).

(a) 24x (b) 23 2 1x x− + − (c) 216 x−

(d) 2 2 25x y + (e)553x (f) 216 10 4t t+ −

5. Usethedistributivepropertyfirstandthencombineeachofthefollowinglinearexpressionsintoasingle,

equivalentbinomialexpression.

(a) ( ) ( )5 2 3 2 4 1x x+ + − (b) ( ) ( )2 10 1 3 4 5x x+ − − 6. Whichofthefollowingisequivalenttotheexpression ( ) ( )2 6 4 2 1 3x x− + + + ? (1) ( )8 2x− (3) ( )4 2 3x+ (2) ( )5 2 1x− (4) ( )10 1x− REASONING7. Eachstepinsimplifyingtheexpressionsyouworkedwithin5and6canbejustifiedusingoneofthemajor

propertiesofrealnumbersreviewedinthelesson.Justifyeachstepbelowwitheitherthecommutative,associativeordistributivepropertieswhensimplifyingtheexpression ( ) ( )8 3 1 2 5 7x x+ + + .

( ) ( )

( ) ( )

( )

8 3 1 2 5 7 24 8 10 14

24 10 8 14

24 10 8 14

24 10 22

34 22

x x x x

x x

x x

x

x

+ + + = + + +

= + + +

= + + +

= + +

= +


Ms.Talhami 6

SOLVING LINEAR EQUATIONS COMMONCOREALGEBRAII

WewilllearnmanynewequationsolvingtechniquesinAlgebraII,butthemostbasicofallequationsarethosewherethevariable,sayx,isonlyraisedtothefirstpower.Theseareknownaslinearequations.YouneedtohavegoodfluencywithsolvingtheseequationsinordertobesuccessfulinthebeginningportionsofAlgebraII.Let'sstartwithsomepractice.Exercise#1:Solveeachofthefollowinglinearequationsforthevalueofx.(a)3 5 26x + = (b)8 7 4 5x x− = −

(c) 8 62x + = − (d) ( ) ( )6 4 2 1 2 20x x x+ − − = +

Itisimportanttounderstandthateachstepinsolvingoneoftheseequationscanbejustifiedbyeitherusingoneofthepropertiesofrealnumbers(fromthelastlesson)orapropertyofequality(suchastheadditiveormultiplicativeproperties).Exercise #2: Justify each step in solving ( )2 7 4 44x x+ + = using either a property of real numbers(commutative,associative,ordistributive)orapropertyofequality(additiveormultiplicative).

( )

( )

2 7 4 442 14 4 442 4 14 442 4 14 446 14 44

6 14 14 44 146 306 306 6

5

x xx xx x

xx

xxx

x

+ + =+ + =+ + =+ + =

+ =+ − = −

=

=

=


Ms.Talhami 7

Strangethingscansometimeshappenwhensolvinglinear(andother)equations.Sometimeswegetnosolutionsatall,inwhichcasetheequationisknownasinconsistent.Othertimes,anyvalueofxwillsolvetheequation,inwhichcaseitisknownasanidentity.

Exercise#3: Try to solve the followingequation. Statewhether theequation is an identity or inconsistent.Explain.

( ) ( )6 2 4 3 2 5x x x x− + = + + − Exercise#4:Anidentityisanequationthatistrueforallvaluesofthesubstitutionvariable.Tryingtosolvethemcanleadtoconfusingsituations.Considertheequation:

( )2 6 1 3 3 2x x x− + − = − +

(a) Testthevaluesof 5x = and 3x = inthisequation.Showthattheyarebothsolutions.(b) Attempttosolvetheequationuntilyouaresurethisisanidentity.Exercise#5:Whichofthefollowingequationsareidentities,whichareinconsistent,andwhichareneither?

(a) ( ) ( )8 2 3 5 1x x x x− + = − + (b) 4 2 8 2 92x x+ + = +

(c) ( ) ( )2 8 7 2 2 3x x x+ − − = − (d) ( ) 16 42 1 2 14xx x −+ + − =


Ms.Talhami 8

SOLVING LINEAR EQUATIONS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Solveeachofthefollowinglinearequations.Iftheequationisinconsistent,stateso.Iftheequationisan

identity,alsostateso.Reduceanynon-integeranswerstofractionsinsimplestform.

(a)7 5 2 35x x+ = − (b) 7 53x − = − (c) 4 5 4 1x x+ = −

(d) ( )5 31 14

2x −

− = (d) ( )3 1 2 9x x− + = + (e) ( )4 2 1 5 6x x x x− − = + + −

(f) ( ) ( )5 2 6 2 4 3 8 9x x x− + + = − (g) 2 56 18x x+ = (Crossmultiplytobegin)

(h) ( )10 4 7 5 12x x− + = + (i) ( ) 8 2018 2 7 2

2xx −− + = −


Ms.Talhami 9

APPLICATIONS2. Lauraisthinkingofanumbersuchthatthesumofthenumberandfivetimestwomorethanthenumberis

26morethanfourtimesthenumber.DeterminethenumberLauraisthinkingof.3. Asif#2wasn'tconfusingenough,Lauraisnowtryingtocomeupwithanumberwherethreelessthan8

timesthenumberisequaltohalfof16timesthenumberafteritwasincreasedby1.Shecan'tseemtofindanumberthatworks.Explainwhy.

4. When finding the intersection of two lines frombothAlgebra I andGeometry, you first "set the linear

equationsequal"toeachother.Findthe intersectionpointofthetwo lineswhoseequationsareshownbelow.Besuretofindboththexandycoordinates.

5 1 and 2 11y x y x= + = −

REASONING5. Explainwhyyoucannotfindtheintersectionpointsofthetwolinesshownbelow.Givebothanalgebraic

reasonandagraphicalreason.4 1 and 4 10y x y x= + = +


Ms.Talhami 10

COMMON ALGEBRAIC EXPRESSIONS COMMONCOREALGEBRAII

InAlgebraIIwewillspendalotoftimeevaluatingandsimplifyingalgebraicexpressions.Justtobeclear:Itisimportanttobeabletoevaluatealgebraicexpressionsforvaluesofthevariablescontainedinthem.Exercise#1:Considerthealgebraicexpression 24 1x + .

Exercise#2:Considerthemorecomplexalgebraicexpression(knownasarationalexpression) 3

4 37

xx

+−

.

Expressionscancontainmorecomplexoperators,suchasthesquareandcuberootsaswellastheabsolutevalue.Wewillneedeachoftheseoverthespanofthiscourse,sosomepracticewithallofthemiswarranted.Exercise#3:Istheabsolutevalueexpression 8 2x− + equivalentto 10x + ?Howcanyoucheckthis?

ALGEBRAICEXPRESSION

Algebraic expressions are just combinations of constants and variables using the typical operations ofaddition,subtraction,multiplication,anddivisionalongwithexponentsandroots(squareroots,cuberoots,etcetera).

(a) Describe theoperationsoccurringwithin thisexpressionandtheorderinwhichtheyoccur.

(b) Evaluate this expression for the replacementvalue . Show each step in yourcalculation.Donotuseacalculator.

(a) Withoutusingyourcalculator,findthevalueofthis expression when . Reduce youranswertosimplestterms.Showyoursteps.

(b) If a studententered the followingexpressioninto their calculator, it would give them theincorrectanswer.Why?


Ms.Talhami 11

Exercise#4:Considerthealgebraicexpression 225 x− ,whichcontainsasquareroot.

(c) Maxthinksthatthesquarerootoperationdistributesoverthesubtraction.Inotherwords,hebelievesthe

followingequationisanidentity:

225 5x x− = −

Showthatthisisnotanidentity.Algebraicexpressionscanbecomequitecomplicated,butifyouconsiderorderofoperationsandworkgenerallyfrominsidetooutsidethenyoucanevaluateanyexpressionforreplacementvalues.

Exercise#5:Considertherathercomplicatedexpression 2

85 4xx−+

.

Exercise#6:Whichofthefollowingisthevalueof24 9

3

x x+ −when 10x = ?

(1)31 (3)18(2)24 (4)84

(a) Evaluatethisexpressionfor . (b)Whycanyounotevaluatetheexpressionfor ?

(a) Whatoperationcomeslastinthisexpression? (b) Evaluate theexpression for . Simplify itcompletely.


Ms.Talhami 12

COMMON ALGEBRAIC EXPRESSIONS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Whichofthefollowingexpressionshasthegreatestvaluewhen 5x = ?Showhowyouarrivedatyourchoice.

22 7x + 3 53

x − 10 23

xx

−−

2. Azeroofanexpressionisavalueoftheinputvariablethatresultsintheexpressionhavingavalueofzero

(catchyandappropriatename).Is 3x = azeroofthequadraticexpressionshownbelow?Justifyyouryes/noanswer.

24 8 12x x− −

3. Whichofthefollowingisthevalueoftherationalexpression22 3

6 4x

x−+

when 2x = − ?

(1) 1

22− (3) 141

(2) 58

− (4) 27

4. If 5x = and 2y = − then 2 2

x yx y

+−

is

(1) 17 (3) 3

29

(2)133 (4) 7

19


Ms.Talhami 13

5. Whatisthevalueof 10 3x x− − + if 2x = ?

(1)7 (3)3 (2)5 (4)17

6. If 2x = then24 2 510

x x+ + hasavalueof

(1) 52 (3) 2

5

(2) 75 (4) 1

2

APPLICATIONS7. Therevenue,indollars,thateMathInstructionmakesoffitsvideosinagivendaydependsonhowmany

viewstheyreceive.Ifxrepresentsthenumberofviews,inhundreds,thentheprofitcanbefoundwiththeexpression:

21 6 10

2x x+ −

Howmuchrevenuewouldtheymakeiftheirvideoswereviewed600times?REASONING8. Sameerbelievesthatthetwoexpressionsbelowareequivalent.Testvaluesandseeifyoucanbuildevidence

fororagainstthisbelief. ( )( )3 8x x− + 2 5 24x x+ −


Ms.Talhami 14

BASIC EXPONENT PROPERTIES COMMONCOREALGEBRAII

Exponents,attheirmostbasic,representrepeatedmultiplication.Thewaytheycombine,ordon'tcombine,isdictatedbythissimplepremise.Exercise#1:Thefollowingfourstepsaregiventofindtheproductofthemonomials 52x− and 24x .

Students (and teachers) can forget the basic properties used in simplifying the product of twomonomialsbecausewetendtopickuponthepatternofmultiplyingthenumericalcoefficientsandaddingthepowerswithoutthinkingaboutthecommutativeandassociativepropertiesthatjustifyourmanipulations.Exercise#2:Findtheproductofeachofthefollowingmonomials.

(a) ( )( )2 65 3x x (b) ( )( )42 6x x− − (c) ( )4 103 62x x⎛ ⎞

⎜ ⎟⎝ ⎠ (d) ( )234x

Remember,monomials(orterms)canhavemorethanonevariable,justaslongastheyareallcombinedusingmultiplicationanddivisiononly.Multiplyingmonomialsthatcontainmorethanonevariablestilljustinvolvesapplicationofexponentlawsandrepeateduseoftheassociativeandcommutativeproperties.Exercise #3: Find each of the following products, which involvemonomials ofmultiple variables. Carefullyconsiderwhatyouaredoingbeforeapplyingpatterns.

(a) ( )( )3 2 54 5x y xy (b) ( )( )7 3 2 62 4x y x y− − (c) 2 51 52 2xy x y⎛ ⎞⎛ ⎞

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(1)

(2)

(3)

(4)

(a) Forsteps(1)through(3),writetherealnumberpropertythatjustifieseachmanipulation.

(b) Explainwhythefinalexponentonthevariablexis7.


Ms.Talhami 15

Oneofthekeyskillswewillneedthisyearwillbefactoringexpressions,especially factoringoutacommonfactor.Tobuildsomeskillswiththis,considerthefollowingproblem.Exercise#4:Fillinthemissingblankineachofthefollowingequationsinvolvingaproductsuchthattheequationisthenanidentity.(a) ( )( )5 26 2 ______x x= (b) ( )( )8 312 4 ______x x= (c) ( )( )2 4 320 2 _________x y xy= −

Thefinalskillwewillreviewinthislessonisusingthedistributivepropertyofmultiplication(anddivision)overaddition(andsubtraction).Exercise#5:Usethedistributivepropertytomultiplythefollowingmonomialsandpolynomials.(a) ( )2 5 3x x+ (b) ( )3 25 2 3 6x x x− + (c) ( )2 27 2 3x x x− − +

(d) ( )2 2 2xy x y− (e) ( )2 4 2 2 33 2 4x y x y xy y+ −

Now,tobuildourwayuptofactoringinlaterunits,let'smakesurewecanfillinmissingportionsofproducts.Exercise#6:SimilartoExercise#4,fillinthemissingportionofeachproductsothattheequationisanidentity.(a) ( )28 12 4 ___________________x x x− = (b) ( )4 3 2 27 21 28 7 _______________x x x x− − = (c) ( )3 2 2 3 5 210 20 35 5 ________________x y x y xy xy− + = (d) ( ) ( ) ( )( )24 2 9 2 2 _______________x x x x− − − = −


Ms.Talhami 16

BASIC EXPONENT PROPERTIES COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Thesteps in finding theproductof ( )2 53x y and ( )5 27x y areshownbelow.Fill ineither theassociative

propertyorthecommutativepropertytojustifyeachstep.

( )( )

( )( )( )

( )( )( )

( )( )

( )( )

( )( )( )

2 4 5 2

2 4 5 2

2 4 5 2

2 4 5 2

2 5 4 2

2 5 4 2

7 6

3 7

3 7

3 7

3 7

3 7

3 7

21

x y x y

x y x y

x y x y

x y x y

x x y y

x x y y

x y

= ⋅

=

= ⋅

=

= ⋅

=

2. Findeachofthefollowingproductsofmonomials.

(a) ( )( )2 43 10x x (b) ( )( )52 9x x− − (c) ( )( )2 5 34 8x y x y (d) ( )245x

(e) ( )( )2 54 15t t− − (f) ( )( )47 5x xy (g) ( )42 123x x⎛ ⎞

⎜ ⎟⎝ ⎠ (h) ( )( )( )2 42 5 6x x x−

3. Fillinthemissingportionofeachproducttomaketheequationanidentity.

(a) ( )6 218 3 ________x x= (b) ( )2 7 240 8 ________x y xy= (c) ( )490 15 __________x y xy= (d) ( )6 224 3 ________x x= − (e) ( )4 10 2 248 16 _______x y x y− = − (f) ( )8 6 4 349 7 ________x y x y=


Ms.Talhami 17

4. Usethedistributivepropertytowriteeachofthefollowingproductsaspolynomials. (a) ( )4 5 2x x+ (b) ( )5 10x x− − (c) ( )26 4 8x x x− +

(d) ( )2 210 2 8x x x− + − (e) ( )3 2 57 2 5xy x y y− (f) ( )2 2 3 2 2 38 2 5x y x x y xy y− + −

(g) ( )3 27 4 2 1x x x− + − (h) ( )216 2 2 3t t t− − + (i) ( )2 212 2xy x xy y− +

5. Fillinthemissingpartofeachproductinordertomaketheequationintoanidentity.

(a) ( )5 3 310 35 5 ____________x x x− = (b) ( )3 2 2 38 2 10 2 ________________x y x y xy xy− + − = − (c) ( )2 5 218 45 9 _______________t t t− + = − (d) ( )4 3 2 245 30 15 15 ___________________x x x x− + = (e) ( ) ( ) ( )( )5 6 5 5 __________x x x x+ + + = + (f) ( ) ( ) ( )( )2 3 3 3 ______________x x x x− − − = − REASONINGAnotherveryimportantexponentpropertyoccurswhenwehaveamonomialwithanexponentthatisthenraisedtoyetanotherpower.Seeifyoucancomeupwithageneralpattern.6. Writeeachofthefollowingoutasextendedproductsandthensimplify.Thefirstisdoneasanexample. (a) ( )32 2 2 2 6x x x x x= ⋅ ⋅ = (b) ( )23x = (c) ( )45x = (d) ( )34x = 7. So,whatisthepattern?Forpositiveintegersaandb: ( ) __________

bax =


Ms.Talhami 18

MULTIPLYING POLYNOMIALS COMMONCOREALGEBRAII

Polynomialsareexpressionsthataremainlycombinationsoftermswithbothadditionandsubtractionthatcanhaveonlyconstantsandpositiveintegerpowers.Theyaretrulyjustanextensionofourbase-10numbersystem.Exercise#1:Giventhepolynomial 3 22 5 3 4x x x+ + + ,whatisitsvaluewhen 10x = ?Howcanyoudeterminethiswithouttheuseofyourcalculator? Ifyoucannot,useyourcalculatortohelpandthenexplainwhytheanswerturnsoutasitdoes.We'vealreadyreviewedhowtomultiplypolynomialsbymonomialsinthelastlesson.Inthislessonwewilllookatmultiplyingpolynomialsbythemselves.Thekeyhereisthedistributiveproperty.Let'sstartbylookingattheproductofbinomials.Exercise#2:Considertheproductof ( )3 2x+ with ( )2 5x+ .

Exercise#3:Findtheproductofthebinomial( )4 3x+ withthetrinomial( )22 5 3x x− − .Representyourproduct

usinganareaarray.Eventhoughtheresulthasan 3x term,theareaarraycanstillhelpuskeeptrackoftheproducttomakesurewearedistributingcorrectly.

(a) Find this product using the distributivepropertytwice(orpossibly"foiling.")

(b) Represent this product on the area modelshownbelow.


Ms.Talhami 19

Itiscriticaltounderstandthatwhenwemultiplytwopolynomialsthenourresultisequivalenttothisproductandthisequivalencecanbetested.Exercise#4:Considertheproductof ( )2x− and ( )2 5x− .

Exercise #5: The product of three binomials, just like the product of two, can be found with repeatedapplicationsofthedistributiveproperty.(a) Findtheproduct: ( )( )( )2 4 7x x x− + − .Useareaarraystohelpkeeptrackoftheproduct.(b) Forwhatthreevaluesofxwillthecubicpolynomialthatyoufoundinpart(a)haveavalueofzero?What

famouslawisthisknownas?(c) Testoneofthethreevaluesyoufoundin(b)toverifythatitisazeroofthecubicpolynomial.

(a) Evaluatethisproductfor .Showtheworkthatleadstoyourresult.

(b) Findatrinomialthatrepresentstheproductofthesetwobinomials.

(c) Evaluate the trinomial for . Is itequivalenttotheansweryoufoundin(a)?

(d)Whatisthevalueofthetrinomialwhen?Canyouexplainwhythismakessensebasedonthetwobinomials?


Ms.Talhami 20

MULTIPLYING POLYNOMIALS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Multiplythefollowingbinomialsandexpresseachproductasanequivalenttrinomial.Useanareamodelto

helpfindyourproduct,ifnecessary. (a) ( )( )5 8x x+ + (b) ( )( )3 2 2 7x x+ − (c) ( )( )5 2 2 3x x− − (d) ( )( )2 24 10x x− + (e) ( )( )3 32 1 5 4x x+ + (f) ( )( )2 21 9x x− −

2. Findeachofthefollowingproductsinequivalentform.Useanarraymodeltohelpfindyourfinalanswersif

youfindithelpful. (a) ( )( )25 3 2x x x+ + + (b) ( )( )22 3 4 5 7x x x− + −

(c) ( )32 5x+


Ms.Talhami 21

APPLICATIONS3. A square of an unknown side length x inches has one side length increased by 4 inches and the other

increasedby7inches. (a) If the original square is shown below with side lengths marked as x, label the second diagram to

representthenewrectangleconstructedbyincreasingthesidesasdescribedabove. (b) Labeleachportionoftheseconddiagramwiththeirareas intermsofx (whenapplicable).Statethe

productof ( )4x+ and ( )7x+ asatrinomialbelow.

REASONING4. Thinkabouttheexpression ( )( )8 4x x− + .

(c) Showthatthistrinomialisalsoequaltozeroatthelargervalueofxfrompart(a).

x

x

x

x

(c) Iftheoriginalsquarehadasidelengthof inches, then what is the area of the secondrectangle? Show how you arrived at youranswer.

(d) Verifythatthetrinomialyoufoundinpart(b)hasthesamevalueas(c)for .

(a) For what values of x will this expression beequaltozero?Showhowyouarrivedatyouranswer.

(b)Writethisproductasanequivalenttrinomial.


Ms.Talhami 22

USING TABLES ON YOUR CALCULATOR COMMONCOREALGEBRAII

Thegraphingcalculatorisanamazingdevicethatcandomanythings.Onefunctionthatitisparticularlygoodatisevaluatingexpressionsfordifferentinputvalues.Wewillbelookingattwotoolsonthecalculatortoday,theSTOREfeatureandTABLES.Firstlet'slookathowtouseSTORE.Exercise#1:FindthevalueofeachofthefollowingexpressionsbyusingtheSTOREfeatureonyourcalculator.

(a) 2 2 7x x− + for 5x = (b) 2 63 5xx+−

for 10x = − (c) 27 20x x− + for 2x =

Sometimes the calculator can even tell us useful information evenwhen it has a hard time evaluating anexpression.Exercise #2: Consider the expression 6 2x− . What happens when you try to use STORE to evaluate thisexpressionfor 5x = ?Evaluatetheexpressionbyhandtohelpexplainwhatthecalculatoristryingtotellus.Exercise#3:Let'sworkwiththeproductoftwobinomialsagain,specifically ( )3 2x+ and ( )5x + .

(c) UsetheSTOREcommandtoevaluatethetrinomialfrom(a)for 5x = − .Whydoesthevalueofthetrinomial

turnouttobethisspecificvalueat 5x = − ?Explain.

(a) Findtheirproductintrinomialform. (b) Evaluate both the trinomial and the originalproductfor .Whatdoyounotice?


Ms.Talhami 23

TheSTOREfeatureisextremelyhelpfulwhenyouaretryingtodeterminethevalueofanexpressionatoneortwoinputvaluesofx.But,ifyouwanttoknowanexpression'svalueformultipleinputs,thenTABLESareamuchbettertool.Exercise#4:Theexpression 3 22 16 32x x x+ − − hasanintegerzerosomewhereontheinterval 0 10x≤ ≤ .UseaTABLEtofindthezeroonthisinterval.Showthetable.Table commands canbeparticularly goodat establishingproof that twoexpressionsareequivalent. This isparticularlyhelpfulwhenyou'vedoneanumberofmanipulationsandyouwanttohaveconfidencethatyou'veproducedanalgebraicallyequivalentexpression.Exercise#5:Considerthemorecomplexalgebraicexpressionshownbelow:

( )( ) ( )( )5 8 3 2x x x x+ + − + −

(a) Thisrelativelycomplexexpressionsimplifies intoa linearbinomialexpression.Determinethisexpressioncarefully.Showyourworkbelow.

(b) Setupatableusingtheoriginalexpressionandtheoneyoufoundin(a)overtheinterval0 5x≤ ≤ .Compare

valuestodetermineifyoucorrectlysimplifiedtheoriginalexpression.

x

0

1

2

3

4

5


Ms.Talhami 24

USING TABLES ON YOUR CALCULATOR COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. UsetheSTOREfeatureonyourcalculatortoevaluateeachofthefollowing.Noworkneedstobeshown. (a)7 18x + for 8x = − (b) 23 2 5x x− + for 3x = (c) 3 25 4 20x x x+ − − for 5x = −

(d) 2 2 8x x− − for 1x = (e) 2

5 34 5xx−+

for 2x = (f) 49x

x−+

for 5x = −

2. TheSTOREfeaturesisparticularlyhelpfulincheckingtoseeifavalueisasolutiontoanequation.Let'ssee

howthisworksinthisproblem.Considertherelativelyeasylinearequation:

6 3 4 9x x− = +

3. Twoofthefollowingvaluesofxaresolutionstotheequation: 2 4 12 10 4x x x+ − = + .Determinewhichthey

areandprovideajustificationforyouranswer. 2x = − 5x = − 6x = 8x =

(a) Solvethisequationforx. (b)Using STORE ,determine thevalueofboth theleft hand expression, , and the righthandexpression, ,atthevalueofxyoufoundin(a).

(c) Whydoeswhatyoufoundinpart(b)verifythatyoursolutioniscorrect(orpossiblyincorrectifyoumadeamistakein(a))?


Ms.Talhami 25

4. Thequadratic expression 2 8 10x x− + has its smallest value for some integer value of x on the interval0 10x≤ ≤ .SetupaTABLEtofindthesmallestvalueoftheexpressionandthevalueofxthatgivesthisvalue.Showyourtablebelow.

5. Considerthecomplexexpression ( )( ) ( )( )7 3 1 4x x x x+ + + − − . (a) Multiply the two sets of binomials and combine like terms in order to write this expression as an

equivalenttrinomialinstandardform.Showyourwork. (b) SetupaTABLEtoverifythatyouranswerinpart(a)isequivalenttotheoriginalexpression.Don'thesitate

topointoutthatitisnotequivalent(whichmeansyoueithermadeamistakeinyouralgebraorinyourtablesetup).Showyourtable.

6. Theproductofthreebinomialsisshownbelow.Writethisproductasapolynomialinstandardform.(Its

highestpowerwillbe 3x ).( )( )( )1 2 4x x x− + −

7. Setupatablefortheansweryoufoundin#6ontheinterval 5 5x− ≤ ≤ .Wheredoesthisexpressionhave

zeroes?

Algebra 2 Unit 1: Algebraic Essentials Review Algebra 2 ... · Algebra 2 Unit 1: Algebraic...

Documents

Transcript of Algebra 2 Unit 1: Algebraic Essentials Review Algebra 2 ... · Algebra 2 Unit 1: Algebraic...