Unit1-Algebra

Chapter3–PolynomialsLESSONS

MPM1D

Chapter3OutlineUnitGoal:Bytheendofthisunit,youwillbeabletodemonstrateanunderstandingofexponentrulesandusethemtosimplifyexpressions.Youwillunderstandwhatapolynomialisandbeabletosimplifypolynomialexpressionsusingthedistributivepropertyandbycollectingliketerms.

Section Subject LearningGoals CurriculumExpectations

3.2 WorkWithExponents

-understandthecomponentsofapower-beabletosubstituteintoandevaluatealgebraicexpressionsinvolvingexponents(includingpowerswitharationalbase)

A1.1

3.3a ExponentLaws-understandexponentlawsthroughinvestigation-beabletoapplytheproduct,quotient,andpowerofapowerlaws

A1.3,A1.4

3.3b ExponentLaws-beabletosimplifyexpressionscontainingpowerswithnegativeexponent

A1.3,A1.4

3.4 CommunicatewithAlgebra

-understandwhatatermandapolynomialare-beabletoclassifypolynomialsbynameandbydegree

A1.2

3.5 CollectingLikeTerms-understandwhat'liketerms'are-beabletosimplifyexpressionsbycollectingliketerms

A2.4

3.6AddingandSubtractingPolynomials

-beabletoaddandsubtractpolynomialsandthensimplifytheexpressionbycollectingliketerms

A2.4

3.7

TheDistributive

Property

-beabletomultiplyapolynomialbyamonomialusingthedistributiveproperty-beabletosimplifyexpressionsinvolvingthedistributiveproperty

A2.5,A2.6

Assessments F/A/O MinistryCode P/O/C KTACNoteCompletion A P PracticeWorksheetCompletion F/A P

SkillBuilder#1–ExponentLaws F/A P/O

Quiz–ExponentLaws F P Assignment-Polynomials O A1.2,A1.3,A1.4,A2.4,A2.5,

A2.6 P K(26%),T(9%),A(32.5%),C(32.5%)

SkillBuilder#2-Polynomials F/A P/O GroupProblemSolvingTask–PolynomialCSD O A1.3,A1.4,A2.4,A2.5,A2.6 P/O/C K(30%),T(20%),A(30%),

C(20%)PreTestReview F/A P Test-Polynomials O A1.1,A1.2,A1.3,A1.4,A2.4,

A2.5,A2.6 P K(21%),T(34%),A(10%),C(34%)

Section3.2–WorkWithExponentsMPM1DJensenPart1:ExponentsInvestigationOnedaySammydecidedtotryanewplaceforlunch.HewenttoanewrestaurantcalledBarney’sBurgers.Helovedthefoodsomuchthatwhenhegotbacktoschoolhetoldtwoofhisfriends.SupposethatthistrendcontinuesandeverydayeachnewcustomertellstwonewfriendsatschoolaboutBarney’sBurgers.HowmanynewcustomerswillBarneygeteachday?a)Completethechartusingyourknowledgeofexponents

Day NewCustomers ExpandedForm Power1

2

2

21

2

4

2x2

22

3

4

b)UsethismodeltodeterminehowmanynewcustomersBarneyshouldexpectonDay7.Showyourwork.c)UsethismodeltodeterminehowmanynewcustomersBarneyshouldexpectonDay14.Isthisanswerrealistic?Whyorwhynot?d)Supposethateachnewcustomertoldthreefriendsinsteadoftwo,andthatthistrendcontinued

i) How many new customer should Barney expect after 2 days?

ii) How many new customer should Barney expect after 4 days?

BrainTeaser:Arectangularsheetofpapermeasures25cmby9cm.Thedimensionsofasquaresheetofpaperwiththesameareaare...Part2:ExponentsRepeatedmultiplicationofthesamenumberbyitselfcanbeexpressedasapower.Thenumberissaidtobeinexponentialform.Expresseachofthefollowinginexponentialform:1)3 × 32)5 × 5 × 5 × 53)(−2)(−2)(−2)(−2)(−2)4)ℎ × ℎ × ℎWriteeachexpressioninexpandedformandthenevaluate:5)2!6)(−3)"7)−3"

8)*#!+!

FindtheTrendEvaluateeachofthefollowing:(−2)# (−2)! (−2)" (−2)$Ifthebaseofthepowerisnegative: -andtheexponentisaneven#,theanswerwillbe_________________________. -andtheexponentisanodd#,theanswerwillbe__________________________.Part3:SubstituteandEvaluateEvaluatetheexpressionforthegivenvaluesofthevariables:9)6𝑥#for𝑥 = 510)6𝑥# − 2𝑥 − 24for𝑥 = −6

𝑁𝑜𝑡𝑒: *𝑎𝑏+

%=𝑎%

𝑏% Therefore:

7238

!

=

Section3.3a–ExponentLawsMPM1DJensenPart1:ExponentLawsInvestigation

ProductofPowersRule:Completethefollowingtable

Describeanytrendsyousee:QuotientofPowersRule:Completethefollowingtable

Describeanytrendsyousee:

PowerofaPowerRule:CompletethefollowingtableDescribeanytrendsyousee:SummaryofExponentLaws:

Part2:SummaryofExponentLawsProductofPowersRuleWhenmultiplyingpowerswiththesamebase,keepthesame_________________and__________________theexponents.GeneralRule:

𝑥! ∙ 𝑥" =QuotientofPowersRuleWhendividingpowerswiththesamebase,keepthesame_________________and__________________theexponents.GeneralRule:

𝑥! ÷ 𝑥" =PowerofaPowerRuleApowerofapowercanbewrittenasasinglepowerby_______________________theexponents.GeneralRule:

(𝑥!)" =PowerofaQuotientWhenyouhaveasinglepowerwitharationalbase,youcanevaluateitbyapplyingtheexponenttothe______________________________andthe_______________________________.Rule:

*𝑎𝑏+

%=

PowerofaProductWhenyouhaveasinglepowerwithabasethatisaproduct,theexponentgetsputontoeach__________________________inthebrackets.Pleasenoticethatthisonlyworkswheninsidethebracketsisasingleterm(no+or–signsseparatingterms)Rule:(𝑎𝑏)% =Part3:ApplytheProductRuleWriteeachproductasasinglepower.Then,evaluatethepowerwherepossible.1)3# × 3! 2)5# × 5 ×5#3)(𝑥#)(𝑥&) 4)(𝑎")(𝑎")(𝑎$)

5)(−2)" × (−2)! 6)*'#+!×*'

#+#

Part4:ApplytheQuotientRuleWriteeachquotientasasinglepower.Then,evaluatethepowerwherepossible.7)8& ÷ 8$ 8)4& ÷ 4 ÷ 4!9)𝑥&( ÷ 𝑥"( ÷ 𝑥#) 10)%

!

%"

11)(+(.$)#

(+(.$)" 12)

."$/"×."$/

%

."$/& 13)1

&1%

1#1'

Note:Anexponentofzeroalwaysgivestheanswerof__________

Part5:ApplythePowerofaPowerRuleWriteeachpowerofapowerasasinglepower.Then,evaluatethepowerwherepossible.14)(3#)" 15)[(−2)!]"

16)=*#!+#># 17)(3𝑎𝑏&)#

ProductofPowersRule 𝑥! ∙ 𝑥" = 𝑥!#"

QuotientofPowersRule 𝑥! ÷ 𝑥" = 𝑥!$"

PowerofaPowerRule (𝑥!)" = 𝑥!×"

PowerofaQuotient '!"(&= !!

"!

PowerofaProduct (𝑎𝑏)& = 𝑎& ∙ 𝑏&

ZeroExponentRule 𝑥' = 1

NegativeExponentRule 𝑥$! =1𝑥!

Note:for#16youwillneedthepowerofaquotientruleand#17youwillneedthepowerofaproductrule.

Section3.3b–ExponentLawsMPM1DJensenPart1:DoItNowSimplifyandevaluateeachofthefollowingexpressions:1)(3#)(3$)2)2

'(

2

3)(𝑦!)"4)3𝑥𝑦 − 2𝑥𝑦Completethefollowingtable:Part2:NegativeExponentsAnynon-zeronumberraisedtoanegativeexponentisequaltoits________________________raisedtotheoppositepositivepower1)𝑥+! 2)5+# 3)%

"

%&

ProductofPowersRule 𝑥! ∙ 𝑥" =

QuotientofPowersRule 𝑥! ÷ 𝑥" =

PowerofaPowerRule (𝑥!)" =

PowerofaQuotient '𝑎𝑏(

&=

PowerofaProduct (𝑎𝑏)& =

ZeroExponentRule 𝑥' =

NegativeExponentRule 𝑥$! =

Note:#4isasubtractionquestion.Simplifyingthisiscalled‘collectingliketerms’.

YouTry:a)𝑥$ ÷ 𝑥) b)3%

"

!%!

Part3:SimplifyExpressionsUsingExponentLaws4)5𝑥# ∙ 2𝑥& 5)2𝑎#𝑏! ∙ 3𝑎3𝑏"6)(5𝑥!)# 7)(𝑥"𝑦!)#8)'#4

%5)

"4&5& 9)+#67"∙96%7%

("67%)%

Hint:startbymultiplyingcoefficientstogether.Thenlooksforpowerswiththesamebaseandsimplifybywritingthemasasinglepowerbyfollowingtheproperexponentlaws.

Hint:Usethepowerofaproductrule.Theexponentoutsideofthebracketsmustbeappliedtoallcoefficientsandvariablesinsidethebracketsusingtheproperexponentlaws.

Hint:startbysimplifyingthenumeratoranddenominatorseparatelyasmuchaspossibleusingexponentlaws.Thenreducethecoefficientsifpossibleandusethequotientruletosimplifypowerswiththesamebase.

10) (!5%:)%

(#5:)(!5%:)

Youtry:a)$;

"<∙";%<%

(#;%<)% b)(!%2)

"

)%$2$

Section3.4–CommunicatewithAlgebraMPM1DJensenBrainTeaser:Inthediagram,whatistheareaofthetriangle?Part1:DoItNowAhockeyteamgets2pointsforawin,1pointforatie,and0pointsforaloss.a)Writeanequationfordeterminingtheamountofpointsateamhas.b)IfthePenguinswin54games,tie8,andlose20;howmanypointswilltheyget?Part2:TermsTerm:anexpressionformedbytheproductof____________________andor____________________. Exampleofaterm:

4𝑥! Thenumberinfrontofthevariableiscalledthe__________________________.Identifythecoefficientandthevariablefortheexpression4𝑥#: Coefficient:_______ Variable:_________

PracticewithTermsIdentifythecoefficientandthevariableofeachterm:a)Jimearns$7perhourathispart-timejob.Ifheworksfor𝑥hours,hisearnings,indollars,are7𝑥.b)Thedepth,inmeters,ofafallingstoneinawellaftertsecondsis−4.9𝑡# c)Theareaofatrianglewithbasebandheighthis'

#𝑏ℎ

d)Theareaofasquarewithsidelength𝑘is𝑘#Part3:PolynomialsPolynomial:analgebraicexpressionconsistingofoneormoretermsconnectedby_____________________or_____________________operators.Exampleofapolynomial:

3𝑥! + 2𝑥 Apolynomialcanbeclassifiedbythenumberoftermsithas: A_______________________________isapolynomialwithonlyoneterm. A_______________________________isapolynomialwithtwoterms. A_______________________________isapolynomialwiththreeterms. A_______________________________isapolynomialwithfourterms.

Classifyeachpolynomialbythenumberoftermsithas:

Polynomial NumberofTerms TypeofPolynomial

3𝑥" + 2𝑥

−2𝑚

4𝑥" − 3𝑥𝑦 + 𝑦"

𝑎 − 2𝑏 + 𝑐 − 3

Hint:Youcanfindthenumberoftermsbylookingfortheadditionandsubtractionoperatorsthatseparatetheterms.Part4:DegreeofaTermDegreeofaterm:thesumofthe________________________onthevariablesinaterm.Exampleofdeterminingthedegreeofaterm:Term:5𝑥#𝑦!Sumofexponentsonvariables:Degreeofterm:Findthedegreeofeachtermbyaddingtheexponentsofthevariables:

Term SumofExponents DegreeofTerm

𝑥"

3𝑦#

0.7𝑢𝑣

−2𝑎"𝑏

−5

Note: -avariablethatappearstohavenoexponentactuallyhasanexponentof_____ -aconstanthasadegreeof_____

Part5:DegreeofaPolynomialThedegreeofapolynomialisequaltothedegreeofthe_______________________________________________inthepolynomial.Example:Polynomial:3𝑥#𝑦" + 11𝑥#𝑦# + 𝑦$Highestdegreeterm:Degreeofhighest-degreeterm:Degreeofpolynomial:Findthedegreeofeachpolynomial:

Polynomial TermwithHighestDegree

DegreeofTermwithHighestDegree DegreeofPolynomial

𝑥 + 3

5𝑥" − 2𝑥

3𝑦$ + 0.2𝑦 − 1

7𝑥"𝑦# + 𝑥%𝑦

Part6:ApplyOurKnowledgeMr.Jensenworksparttimeasagolfinstructor.Heearns$125fortheseason,plus$20foreachchildren`slessonand$30foreachadultlessonthathegives. a)WriteanexpressionthatdescribesMr.Jensen'stotalearningsfortheseason.Identifythevariablesandwhattheystandfor. b)IfMr.Jensengave8children'slessonsand6adultlessons,whatwerehistotalearnings?

ReviewofTerms_______________________________:anexpressionformedbytheproductofnumbersand/orvariables _______________________________:analgebraicexpressionconsistingofoneormoretermsconnectedbyadditionorsubtractionsigns. _______________________________:thesumoftheexponentsonthevariablesinaterm _______________________________:equaltothedegreeofthehighest-degreeterminapolynomial

Section3.5–CollectingLikeTermsMPM1DJensenBrainTeaser:AtKing's,theratioofmalestofemaleswritingthePascalContestis3:7.Ifthereare21maleswritingtheContest,whatisthetotalnumberofstudentswriting?Part1:DoItNow1)Whatisthedegreeoftheterm:3𝑥#𝑦𝑧 2)Whatisthedegreeofthispolynomial:3𝑎#𝑏!𝑐 + 2𝑎𝑏"𝑐# − 7𝑎𝑏𝑐# 3)Classifythepolynomialfromquestion2)byname:Part2:LikeTermsLikeTermsaretermsthathavetheEXACTsame_________________________________withtheEXACTsame_________________________________.Identifytheliketermsinthispolynomial:3𝑥! − 5𝑥 + 2𝑥! + 3 − 1 + 4𝑥 + 12𝑥! − 120

Identifytheliketermsinthispolynomial:5𝑥#𝑦 − 9𝑥𝑦 + 6𝑥#𝑦 + 17.3𝑥 − 2𝑥𝑦 + 4𝑥#𝑦 + 92𝑥 − 133𝑥𝑦Part3:CollectingLikeTermsWhenadding/subtractingliketerms,keepthevariablesthesame,andadd/subtractonlythecoefficients.Example:6𝑥 + 4 + 8𝑥 + 3= Step1:Rearrangeliketermsintogroups= Step2:Add/SubtracttheliketermsPracticeCollectingLikeTerms1)3𝑥 + 4𝑥 2)3𝑥# + 5𝑥# + 33)2𝑏 − 𝑏 + 7 − 8 + 3𝑏 4)3𝑥# + 2 − 6𝑥 + 9𝑥 − 3𝑥#5)2𝑥# − 3𝑦# + 𝑥𝑦 + 2𝑦# − 8𝑥! 6)𝑎#𝑏 + 2𝑎𝑏 − 𝑎𝑏# + 2𝑎𝑏# − 3𝑎𝑏 + 𝑎#𝑏

Note: degree of terms should be in descending order (highest degree terms on the left).

Part4:ApplyourKnowledgea)Writeanexpressioninsimplestformfortheperimeterofthegivenshapeb)Evaluatetheexpressionif𝑥 = 5.(Whatistheperimeter?)

Section3.6–AddandSubtractPolynomialsMPM1DJensenPart1:DoItNowTheCoolPoolCompanymakespoolsandusesthefollowingdiagramtocalculatetheperimeterofpoolwithdifferentoptions:Option1–Bronzepackage(standardpoolonly) Option2–Silverpackage(standardpoolandshallowpool)Option3–Goldpackage(allsectionsofthepoolareincluded)a)InyourgroupchooseeithertheBronze,Silver,orGoldpackageandcreateasimplifiedexpressionfortheperimeterofyourpool.b)Whatistheperimeterofyourpoolif𝑥 = 4Part2:AddingPolynomialsPolynomial:analgebraicexpressionconsistingofoneormoretermsconnectedbyadditionorsubtractionoperatorsWhenaddingpolynomialsyoucansimply__________________thebracketsandcollecttheliketermsExample:(4𝑥 + 3) + (7𝑥 + 2)= Step1:RemovetheBrackets= Step2:Rearrangeliketermsintogroups= Step3:Collecttheliketerms

PracticeAddingPolynomials1)(3𝑦 + 5) + (7𝑦 − 4) 2)(2𝑝 − 2) + (4𝑝 − 7)3)(6𝑥 − 12) + (−9𝑥 − 4) + (𝑥 + 14) 4)(5𝑥 − 4𝑦 − 1) + (−2𝑥 + 5𝑦 + 13)Part3:SubtractingPolynomialsTosubtractpolynomials,addthe_____________________________________(switchthesignsofthetermsofthepolynomialbeingsubtracted)Example:(3𝑦 + 5) − (7𝑦 − 4)Theoppositeof(7𝑦 − 4)is:____________________∴ (3𝑦 + 5) − (7𝑦 − 4) =ORTosubtractpolynomials,subtract________________________________________________inthesecondpolynomial.∴ (3𝑦 + 5) − (7𝑦 − 4) =

5)(4𝑥 + 3) − (7𝑥 + 2) 6)(𝑎# − 2𝑎 + 1) − (−𝑎# − 2𝑎 − 5)7)(3𝑥 + 𝑦 − 4𝑧) − (7𝑥 + 3𝑦 − 2𝑧) 8)(6𝑥 − 12) − (−9𝑥 − 4) − (𝑥 + 14)Part4:ApplyOurKnowledgeTheBurghBirdsplayersgeta$𝑥bonusaddedtotheirbasesalaryforeverygoalthattheyscoreduringtheplayoffs.Herearethesalariesandgoalsscoredforthe3highestscoringplayersontheBurgBirdsduringtheplayoffslastseason.a)Writeandsimplifyanexpressionforeachplayer’syearendsalary. BardownJensen: Wayne-G: Timmy:b)Writeandsimplifyanexpressionforthetotalamountofmoneythattheowneroftheteamwillneedtopaythethreeplayersattheendoftheseason.c)If𝑥 = 25,whatisthetotalamountofmoneythattheownerwillneedtopaythethreeplayersattheendoftheseason?

Section3.7–TheDistributivePropertyMPM1DJensenPart1:DoItNowWriteasimplifiedexpressionfortheareaoftherectangle:𝐴𝑟𝑒𝑎𝑜𝑓𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 =

Beforewecansimplifytheexpressionweneedtolearnthedistributiveproperty!

DistributivePropertyWhenyouapplythedistributiveproperty,youaregettingridofthebracketsbymultiplyingeverythinginthebracketsbytheterminfrontofthebrackets.Example:5(4𝑥 + 2)= 20𝑥 + 10Part2:MultiplyaConstantbyaPolynomialExpandandsimplifythefollowing:1)2(5𝑥 + 3) 2)−2(7𝑥 − 4)

Remember:𝐴𝑟𝑒𝑎 = 𝑙𝑒𝑛𝑔𝑡ℎ × 𝑤𝑖𝑑𝑡ℎ

Toapplythedistributiveproperty,Imustmultiplybothtermsinthebracketby5.

Note:Makesuretoincludethenegativesignwhendistributingthe-2.Followintegerrulesformultiplication.

Note:Youcanalsoapplythedistributivepropertytotrinomials.

Remember:Youcancollectliketerms!Liketermshaveidenticalvariables(samelettersandexponents)

3)−3(2𝑥# − 5𝑥 + 4) 4)2(6𝑚 − 3) + 3(16 + 4𝑚)Part3:ApplyOurKnowledgeWriteanexpressionfortheareaoftherectangleinexpandedform:Whatistheareaoftherectangleif𝑥 = 5cm?Part4:DistributeVariablesExample:𝑥(𝑥# − 3)= 𝑥! − 3𝑥6)𝑥(𝑥 − 3) 7)– 𝑥(7𝑥 − 4)8)−3𝑥(2𝑥# − 5𝑥 + 4) 9)3𝑚(𝑚 − 5) − (2𝑚# −𝑚)

Remembertheexponentlaws:𝑥(𝑥#) = 𝑥'=# = 𝑥!

Forthisquestionyoucanmultiplythesecondpolynomialby-1orusethepropertiesforsubtractingpolynomials;bothgivethesameresult!

10)'#(2𝑤 − 6) − #

!(9𝑤 − 6)

Part5:NestedBracketsIfthereisabracketinsideofabracket,simplifytheinnermostbracketsfirstandthenworkyourwayout.11)3[2 + 5(2𝑘 − 1)]