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CPAlgebra2Unit3A:Polynomials

Name:_____________________________________Period:_____

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LearningTargets

Polynomials:TheBasics

1.Icanclassifypolynomialsbydegreeandnumberofterms.

2.Icanusepolynomialfunctionstomodelreallifesituationsandmakepredictions.

3.Icanidentifythecharacteristicsofapolynomialfunction,suchastheintervalsofincrease/decrease,intercepts,domain/range,relativeminimum/maximum,andendbehavior.

FactorsandZeros

4.Icanwritestandardformpolynomialequationsinfactoredformandviceversa.

5.Icanfindthezeros(orx-interceptsorsolutions)ofapolynomialinfactoredformandidentifythemultiplicityofeachzero.

6.Icanwriteapolynomialfunctionfromitsrealroots.

DividingPolynomials

7.Icanuselongdivisiontodividepolynomials.

8.Icanusesyntheticdivisiontodividepolynomials.

9.IcanusesyntheticdivisionandtheRemainderTheoremtoevaluatepolynomials.

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Polynomials:TheBasics Afterthislessonandpractice,Iwillbeableto…

¨ classifypolynomialsbydegreeandnumberofterms.(LT1)

¨ usepolynomialfunctionstomodelreallifesituationsandmakepredictions.(LT2)

¨ identifythecharacteristicsofapolynomialfunction,suchastheintervalsofincrease/decrease,intercepts,domain/range,relativeminimum/maximum,andendbehavior.(LT3)

--------------------------------------------------------------------------------------------------------------------------------------------------Quadraticfunctions:Functionswhosegreatestexponentonanyvariableis_________Polynomialfunctions:Functionswhosegreatestexponentonanyvariableisgreaterthan_________ClassifyPolynomials(LT1)Itwillbeeasiertotalkaboutpolynomialsifwesharesomecommonvocabulary…Monomial–A_________________,__________________,or____________ofnumbersandvariables.Examples:Polynomial–A___________________or____________ofmonomials.

StandardForm: where arethe______________ofthepowers

ofx,andnisanonnegativeinteger.

• Theexponentsofthevariablesaregivenin_______________orderwhenwritteningeneralform.• Eachofthemonomialsbeingaddediscalleda_________.

Examples:DegreeofaPolynomial–The_______________ofthetermwiththegreatest_______________.LeadingCoefficient–The___________________ofthetermwiththegreatest_______________.

y = anxn + an−1xn−1 + ...+ a1x + a0 an ,an−1,...,a1,a0

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Therearealsospecialnamesthatwegivetopolynomialsbasedontheirdegreeandtheirnumberofterms.

Example1:Writeeachpolynomialinstandardform.Thenclassifyeachpolynomialbyitsdegreeandnumberofterms.Finally,nametheleadingcoefficientofeachpolynomial.a. b. c. d. ModelRealLifeSituations(LT2)WewilluseDesmostoexploresomecharacteristicsofpolynomials.Fornow,let’sgetanideaofthegeneralshapeofpolynomialsofdifferentdegrees…

𝑦 = 𝑥! − 𝑥 − 1 𝑦 = 𝑥! + 6𝑥! + !!𝑥 − 1 𝑦 = 𝑥! + 3𝑥! + 𝑥! − 𝑥 − 1

!!−7x +5x4 !!x2 −4x +3x3 +2x

!!4x −6x +5 !!6−3x5

NumberofTerms

Name Example

Degree Name Example

0

1

2

3

4

5

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Arelativeminimumormaximumisapointthatisthemin.ormax.__________________toothernearbyfunctionvalues.Example 2: Graph the equation !!y =3x

3 −5x +5 in your calculator. Then determine the coordinates of allrelativeminimumsandmaximums(roundedto3decimalplaces).Herearethedirections:

Example3:Determinethecoordinatesofallrelativeminimumsandmaximums(roundedto3decimalplaces).a. !!0.5x4 −3x2 +3 b. !!−x3 +6x2 − x −1 Youhavealreadyusedyourcalculatortomodeldatausing______________regressionand_______________regression.Let’sexplorehowtouseotherregressionmodelstodevelopmodelsforreal-lifedata.Example4:a.Graphthefollowingdatainthecalculator.Thendeterminewhetheralinear,quadratic,orcubicregressionmodelbestrepresentsthedata.-Enteryourdata(STAT,Edit…)-TurnonStatPlot1(2nd,STATPLOT)-Graphthedata(ZOOM,9)-STAT,right,thenchooseyourdesiredregression.-Yourcommandshouldlooklike____Reg,Y1(VARS,right,ENTER,ENTER)b.Writetheequationthatbestrepresentsthedata:c.Predictthevalueofywhenxis12.

x 0 5 10 15 20y 10.1 2.8 8.1 16.0 17.8

- Enter the equation in Y1. Press GRAPH. - If necessary, adjust the window so that you can see all relative max/min. - For finding relative maximums/minimums: - Press 2nd CALC, 3 (minimum) or 4 (maximum)

- Give the calculator a left and right bound for the maximum/minimum and then press ENTER again when the calculator asks for a guess.

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IdentifyCharacteristics(LT3)Insteadofavideo,you’llbedoingaDesmosexplorationforthisLT!FollowthedirectionsonCanvas.

QuickSketchofFunction

Isthefunctionalwaysincreasing,alwaysdecreasing,someofboth,or

neither?

Whatisthelargestnumberof

x-interceptsthatthefunctioncanhave?

Whatisthesmallestnumberof

x-interceptsthatthefunctioncanhave?

Domain

ConstantFunction

1stDegree𝑓(𝑥) = 𝑥

2ndDegree𝑓(𝑥) = 𝑥!

3rdDegree𝑓(𝑥) = 𝑥!

4thDegree𝑓(𝑥) = 𝑥!

Question1:HowdoestheLEADINGCOEFFICIENTaffectthegraphofapolynomial?

1. Whatdoallgraphswithapositiveleadingcoefficienthaveincommon?

2. Whatdoallgraphswithanegativeleadingcoefficienthaveincommon?Question2:HowdoestheDEGREEaffectthegraphofapolynomial?

1. Whatisthecommonbehaviorofallgraphswithanevendegree?

2. Whatisthecommonbehaviorofallgraphswithanodddegree?

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EndBehavior…let’sexaminetheendbehaviorofthepolynomials.Theendbehavioristhebehaviorofthegraphasxapproaches_________andasxapproaches_______.

Example7:Describetheendbehaviorofthegraphofeachpolynomialfunctionbycompletingthestatements.(NOTE–youneedtobeabletodothiswithoutlookingatthegraph!)A) 6 2( ) 4 2f x x x= − + + B) 3 2( ) 2 2 5 10f x x x x= + − − ( ) _____f x → as x→∞ ( ) _____f x → as x→∞ ( ) _____f x → as x→−∞ ( ) _____f x → as x→−∞

C) f (x) = 2x5 + x2 −1 D) f (x) = −x4 −5x +10

as as as as

( ) _____f x → x→∞ ( ) _____f x → x→∞

( ) _____f x → x→−∞ ( ) _____f x → x→−∞

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Example8:Determinetheintervalsofincreaseanddecrease,theintercepts,thedomainandrange,andthecoordinatesofallrelativeminimumsandmaximums.Roundallanswerstothreedecimalplaces.

Intervalsofincrease:_________________________Intervalsofdecrease:_________________________Intercepts:_________________________________ Domain:___________Range:_______________RelativeMinimum(s):________________________RelativeMaximum(s):________________________

Intervalsofincrease:_________________________Intervalsofdecrease:_________________________Intercepts:_________________________________ Domain:___________Range:_______________RelativeMinimum(s):________________________RelativeMaximum(s):________________________

Intervalsofincrease:_________________________

Intervalsofdecrease:_________________________Intercepts:_________________________________ Domain:___________Range:_______________RelativeMinimum(s):________________________RelativeMaximum(s):________________________

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Polynomials:FactorsandZeros Afterthislessonandpractice,Iwillbeableto…

¨ writestandardformpolynomialsinfactoredformandviceversa.(LT4)

¨ findthezeros(orx-interceptsorsolutions)ofapolynomialinfactoredformandidentifythemultiplicityofeachzero.(LT5)

¨ writeapolynomialfunctionfromitsrealroots.(LT6)--------------------------------------------------------------------------------------------------------------------------------------------------Inthelastunit,youused___________________________tohelpyousimplifysquareroots.Inthisprocess,youexpressednumbersas_______________of___________numbers.Justasanynumbercanbewrittenasaproductofprimefactors,polynomialscanbewrittenas_____________of___________factors.A linear factor is similar to a ___________ number in that it cannot be _____________ any further. Apolynomialiscompletelyfactoredifitisexpressedasaproductof___________factors.WriteStandardinFactoredFormandViceVersa(LT4)WritingPolynomialsfromFactoredFormtoStandardFormExample1:Writeeachexpressioninstandardform.a. !!(x +1)(x −2)(x +3) b. !!(x −3)(x +2)(x −4) c. !!(x +5)(x −1)(x +2)

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WritingPolynomialsfromStandardFormtoFactoredForm

Example2:Write!!2x3 +10x2 +12x infactoredform.

Example3:Writeeachexpressioninfactoredform.a. !!3x3 −18x2 +24x b. !!6x3 −15x2 −36x FindZerosinFactoredForm(LT5)OpenDesmosandgraphthefollowingpolynomialwritteninfactoredform:!!y = (x +1)(x −2)(x +3) Inthelastunit,welearnedthatthex-interceptsarealsocalledthe____________.Whydoesthistermmakesensebasedonwhatyouseeinthegraph?Whatarethezerosofthatfunction?Example4:Findthezerosofeachfunction.Thensketchagraphofthefunction,showingxandyintercepts.a. !!y = (x −1)(x +2)(x +1) b. !!y = (x −4)(x −1)(x +2)

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Example5:Findthezerosofthefunction𝑦 = (𝑥 − 1)(𝑥 + 2)(𝑥 + 2).InExample5,noticethatwhilethefunctionhasthreetotalzeros,only____ofthenwere_________(___islistedasazerotwice).Whenazero(andthusitslinearfactor)isrepeated,itiscalleda______________zero.Inthisexample,-2issaidtohavemultiplicity____sinceitoccursasazero_________.Ingeneral,the_______________________ofazeroisequaltothenumberoftimesthezerooccurs.Example6:Findallzerosof!!f (x)= x

4 +6x3 +8x2 andstatethemultiplicityofeachzero.WriteaPolynomialFromitsRealRoots(LT6)Example7:Writeapolynomialequationinstandardformwithzerosat-2,3,and3.

Example8:Writeapolynomialequationinstandardformwithzerosat2,1,and-3.

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DividingPolynomials Afterthislessonandpractice,Iwillbeableto…

¨ uselongdivisiontodividepolynomials.(LT7)

¨ usesyntheticdivisiontodividepolynomials.(LT8)

¨ usesyntheticdivisionandtheRemainderTheoremtoevaluatepolynomials.(LT9)--------------------------------------------------------------------------------------------------------------------------------------------------Inthepreviouslesson,youexpressedpolynomialsinfactoredform.Whileyourpreviouslylearnedfactoringtechniquescanhelpyoutowritepolynomialsinfactoredform,youcanalsousedivisiontohelpyoufactorpolynomialsandthusdetermineitsremainder.Polynomialdivisionissimilartolongdivision.UseLongDivisiontoDividePolynomials(LT7)Dothelongdivisionproblemalongwiththerapvideo…

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ThispatternholdstruealsoforPOLYNOMIALdivision.

Consider!!x 2x2

Example1:Divide!!x2 +3x −12 by!!x −3 usinglongdivision.

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Example2:Divideusinglongdivision.a. !! x

2 +2x −30( )÷ x −5( ) b. !! x2 +10x +16( )÷ x +2( )

Example3:Determinewhether!!x +4 isafactorofeachpolynomial. a. !!x2 +6x +8 b. !!x3 +3x2 −6x −7 AdvancedPolynomialDivisionLet’stakealookatsomestrategiesfordividingmorecomplexpolynomials.

Example4:Divide!! 12x4 −5x2 −3( )÷ x −2( )

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Example5:Divide!! x4 +3x3 −5x2 −8x +6( )÷ x2 +2x −6( )

Example6:Divide!! x

4 − x3 −2x2 −2x −8( )÷ x2 +2( ) DividePolynomialsUsingSyntheticDivision(LT8)ThereisasimplifiedversionofpolynomialdivisioncalledSyntheticDivision…Example7:Divideeachpolynomialby 2x + usingsyntheticdivision.Decideif(x–r)isafactor.Writetheresultofthedivision.

a. !!6x3 +11x2 −17x −30 b. !!4x3 +2x −7 Important:Whenusingsyntheticdivision,thenumberyouusetodivideisthe_______correspondingtothegivenlinearbinomial.Example8:Divideusingsyntheticdivision.Decideif(x–r)isafactor.Writetheresultofthedivision.a. 3 2( 4 1) ( 3)x x x x+ − − ÷ + b. ( )3 22 14 72 ( 6)x x x− + ÷ −

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c. (9x4 −18x3 +10x −16) ÷ (x − 2) d. 9x4 +14x3 −14x2 −13x − 2( ) ÷ (x + 2)

Recall, thepurpose for learning todividepolynomials is todiscover the_________of thepolynomial. Let’sshowhowsyntheticdivisioncanhelpyoufindallzerosofcertainpolynomials.Example9:Findallzerosofeachpolynomialusingsyntheticdivisionwiththegivenfactor. a. !!x3 −13x −12 ;onefactor=!!x +1 b. !!x3 + x2 −24x +36 ;onefactor=!!x −3 EvaluateUsingSyntheticDivision(LT9)InExample8c,youdividedapolynomialbyalinearbinomialanddiscoveredthatitwasnotafactorsincetheremainderwasnotequaltozero.Infact,youdeterminedthattheremainderwas______.Let’slookwhathappenswhenwetryevaluatingtheexpressionat_____.RemainderTheorem–IfapolynomialP(x)ofdegreeofatleastoneisdividedbya,whereaisaconstant,thentheremainderisP(a).

**Thistheoremprovidesafastwayofevaluatingsomecomplexpolynomialsquicklyandfordeterminingifagivenexpressionisafactorofthepolynomial.

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Example10:Find!!P(−4) for!!P(x)= x4 −5x2 +4x +12 .Thendetermineif-4isazeroofP(x).

Example11:Find!!P(−2) for!!P(x)=5x3 +12x2 + x −9 .Thendetermineif-2isazeroofP(x).