Polynomial Functions - mathematicsvisionproject.org€¦ · SECONDARY MATH III // MODULE 3 POLYNOMIAL FUNCTIONS Mathematics Vision Project Licensed under the Creative Commons Attribution

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MODULE 3

Polynomial Functions

SECONDARY

MATH THREE

An Integrated Approach

SECONDARY MATH III // MODULE 3

POLYNOMIAL FUNCTIONS

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

MODULE 3 - TABLE OF CONTENTS


3.1 Scott’s March Madness – A Develop Understanding Task Introduce polynomial functions and their rates of change (F.BF.1, F.LE.3, A.CED.2) Ready, Set, Go Homework: Polynomial Functions 3.1

3.2 You-mix Cubes – A Solidify Understanding Task Graph ! = #$ with transformations and compare to ! = #%. (F.BF.3, F.IF.4, F.IF.5, F.IF.7) Ready, Set, Go Homework: Polynomial Functions 3.2

3.3 It All Adds Up – A Develop Understanding Task Add and subtract polynomials. (A.APR.1, F.BF.1, F.IF.7, F.BF.1b) Ready, Set, Go Homework: Polynomial Functions 3.3

3.4 Pascal’s Pride – A Solidify Understanding Task Multiply polynomials and use Pascal’s to expand binomials. (A.APR.1, A.APR.5) Ready, Set, Go Homework: Polynomial Functions 3.4

3.5 Divide and Conquer – A Solidify Understanding Task Divide polynomials and write equivalent expressions using the Polynomial Remainder Theorem. (A.APR.1, A.APR.2) Ready, Set, Go Homework: Polynomial Functions 3.5

3.6 Sorry, We’re Closed – A Practice Understanding Task Compare polynomials and integers and determine closure under given operations. (A.APR.1, F.BF.1b) Ready, Set, Go Homework: Polynomial Functions 3.6

3.7 Building Strong Roots – A Solidify Understanding Task Understand the Fundamental Theorem of Algebra and apply it to cubic functions to find roots. (A.SSE.1, A.APR.3, N.CN.9) Ready, Set, Go Homework: Polynomial Functions 3.7




3.8 Getting to the Root of the Problem – A Solidify Understanding Task Find the roots of polynomials and write polynomial equations in factored form. (A.APR.3, N.CN.8, N.CN.9) Ready, Set, Go Homework: Polynomial Functions 3.8

3.9 Is This the End? – A Solidify Understanding Task Examine the end behavior of polynomials and determine whether they are even or odd. (F.LE.3, A.SSE.1, F.IF.4, F.BF.3) Ready, Set, Go Homework: Polynomial Functions 3.9

3.10 Puzzling Over Polynomials – A Practice Understanding Task Analyze polynomials, determine roots, end behavior, and write equations (A.APR.3, N.CN.8, N.CN.9, A.CED.2) Ready, Set, Go Homework: Polynomial Functions 3.10


POLYNOMIAL FUNCTIONS– 3.1


3.1 Scott’s March Madness

A Develop Understanding Task

Eachyear,Scottparticipatesinthe“MachoMarch”promotion.The

goalof“MachoMarch”istoraisemoneyforcharitybyfinding

sponsorstodonatebasedonthenumberofpush-upscompleted

withinthemonth.Lastyear,Scottwasproudofthemoneyhe

raised,butwasalsodeterminedtoincreasethenumberofpush-

upshewouldcompletethisyear.

PartI:RevisitingthePast

BelowisthebargraphandtableScottusedlastyeartokeeptrackofthenumberofpush-upshe

completedeachday,showinghecompletedthreepush-upsondayoneandfivepush-ups(fora

combinedtotalofeightpush-ups)ondaytwo.Scott

continuedthispatternthroughoutthemonth.

1 2 3 4

1. Writetherecursiveandexplicitequationsforthenumberofpush-upsScottcompletedonany

givendaylastyear.Explainhowyourequationsconnecttothebargraphandthetableabove.

?Days A(?)

Push-ups

eachday

E(?)

Totalnumberof

pushupsinthe

month

1 3 32 5 83 7 154 9 245 11 35… … ?

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2. Writetherecursiveandexplicitequationfortheaccumulatedtotalnumberofpush-ups

Scottcompletedbyanygivendayduringthe“MachoMarch”promotionlastyear.

PartII:MarchMadness

Thisyear,Scott’splanistolookatthetotalnumberofpush-upshecompletedforthemonthlastyear

(g(n))anddothatmanypush-upseachday(m(n)).

3. Howmanypush-upswillScottcompleteondayfour?Howdidyoucomeupwiththisnumber?

Writetherecursiveequationtorepresentthetotalnumberofpush-upsScottwillcompletefor

themonthonanygivenday.

4. Howmanytotalpush-upswillScottcompleteforthemonthondayfour?

?Days A(?)

Push-upseach

daylastyear

E(?)

Totalnumber

ofpushupsin

themonth

N(?)

Push-upseach

daythisyear

T(n)

Totalpush-ups

completedfor

themonth

1 3 3 3 2 5 8 8 3 7 15 15 4 9

24 5 … … ?

2




5. Withoutfindingtheexplicitequation,makeaconjectureastothetypeoffunctionthatwould

representtheexplicitequationforthetotalnumberofpush-upsScottwouldcompleteonany

givendayforthisyear’spromotion.

6. Howdoestherateofchangeforthisexplicitequationcomparetotheratesofchangeforthe

explicitequationsinquestions1and2?

7. Testyourconjecturefromquestion5andjustifythatitwillalwaysbetrue(seeifyoucanmove

toageneralizationforallpolynomialfunctions).

3


POLYNOMIAL FUNCTIONS – 3.1


3.1

Needhelp?Visitwww.rsgsupport.org

READY Topic:Completinginequalitystatements

Foreachproblem,placetheappropriateinequalitysymbolbetweenthetwoexpressionstomakethestatementtrue.If! > #, %ℎ'(: *+- > 10, %ℎ'(: *+0 < - < 1

1. 3!____3#

4.-3____25

7.-____-3

2. # − !____! − #

5.√-____-3

8.√-____-

3.! + -____# + - 6.-3____-9 9.-____3-

SET Topic:Classifyingfunctions

Identifythetypeoffunctionforeachproblem.Explainhowyouknow.10

- +(-) 1 3 2 6 3 12 4 24 5 48

11.- +(-) 1 3 2 6 3 9 4 12 5 15

12.- +(-) 1 3 2 9 3 18 4 30 5 45

13.- +(-)1 72 93 134 215 37

14.- +(-)1 -262 -193 04 375 98

15.- +(-)1 -42 33 184 415 72

16.WhichoftheabovefunctionsareNOTpolynomials?

READY, SET, GO! Name PeriodDate

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3.1


GO Topic:Recallinglongdivisionandthemeaningofafactor

Findthequotientwithoutusingacalculator.Ifyouhavearemainder,writetheremainderas

awholenumber.Example: remainder217.

18.

19.Is30afactorof510?Howdoyouknow?


21.

22.



25.

26.

27.Is92afactorof3405? 28.Is27afactorof3564?

21 1497

30 510 13 8359

22 14857 952 40936

92 3405 27 3564

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3.2 You-mix Cubes A Solidify Understanding Task

InScott’sMarchMadness,thefunctionthatwasgeneratedby

thesumoftermsinaquadraticfunctionwascalledacubicfunction.Linearfunctions,

quadraticfunctions,andcubicfunctionsareallinthefamilyoffunctionscalledpolynomials,

whichincludefunctionsofhigherpowerstoo.Inthistask,wewillexploremoreaboutcubic

functionstohelpustoseesomeofthesimilaritiesanddifferencesbetweencubicfunctionsand

quadraticfunctions.

Tobegin,let’stakealookatthemostbasiccubicfunction,!(#) = #& .Itistechnicallyadegree3polynomialbecausethehighestexponentis3,butit’scalledacubicfunctionbecausethese

functionsareoftenusedtomodelvolume.Thisislikequadraticfunctionswhicharedegree2

polynomialsbutarecalledquadraticaftertheLatinwordforsquare.Scott’sMarchMadness

showedthatlinearfunctionshaveaconstantrateofchange,quadraticfunctionshavealinearrate

ofchange,andcubicfunctionshaveaquadraticrateofchange.

1. Useatabletoverifythat!(#) = #&hasaquadraticrateofchange.

2. Graph!(#) = #& .

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3. Describethefeaturesof!(#) = #&includingintercepts,intervalsofincreaseordecrease,domain,range,etc.4. Usingyourknowledgeoftransformations,grapheachofthefollowingwithoutusingtechnology.

a) !(#) = #& − 3 b) !(#) = (# + 3)&

c) !(#) = 2#& d) !(#) = −(# − 1)& + 2

5. Usetechnologytocheckyourgraphsabove.Whattransformationsdidyougetright?What

areasdoyouneedtoimproveonsothatyourcubicgraphsareperfect?

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6. Sincequadraticfunctionsandcubicfunctionsarebothinthepolynomialfamilyoffunctions,

wewouldexpectthemtosharesomecommoncharacteristics.Listallthesimilaritiesbetween

!(#) = #&and,(#) = #- .

7. Asyoucanseefromthegraphof!(#) = #& ,therearealsosomerealdifferencesincubicfunctionsandquadraticfunctions.Eachofthefollowingstatementsdescribeoneofthose

differences.Explainwhyeachstatementistruebycompletingthesentence.

a) Therangeof!(#) = #&is(−∞,∞),buttherangeof,(#) = #-is[0, ∞)because:_______________________________________________________________________________________________________

b) For# > 1, !(#) > ,(#)because:_______________________________________________________________________________________________________________________________________________________________

c) For0 < # < 1, ,(#) > !(#)because:__________________________________________________________________________________________________________________________________________________________

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3.2


READY Topic:Addingandsubtractingbinomials

Addorsubtractasindicated.

1. (6# + 3) + (4# + 5)

2. (# + 17) + (9# − 13) 3. (7# − 8) + (−2# + 9)

4. (4# + 9) − (# + 2)

5. (−3# − 1) − (2# + 5) 6. (8# + 3) − (−10# − 9)

7. (3# − 7) + (−3# − 7)

8. (−5# + 8) − (−5# + 7) 9. (8# + 9) − (7# + 9)

10. Usethegraphsof0(#)and1(#)tosketchthegraphsof0(#) + 1(#)and0(#)– 1(#).

0(#) + 1(#) 0(#)– 1(#).


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3.2


SET Topic:Comparingsimplepolynomials

11. Completethetablesbelowfor5 = #7895 = #:7895 = #;

x 5 = #--−1

0

1

x 5 = #:

-1—1

0

1

x 5 = #;

-1—1

0

1

12. Whatassumptionmightyoubetemptedtomakeaboutthegraphsof

5 = #, 5 = #:7895 = #;basedonthevaluesyoufoundinthe3tablesabove?

13. Whatdoyoureallyknowaboutthegraphsof5 = #7895 = #:7895 = #;despitethe

valuesyoufoundinthe3tablesabove?

14.Completethetableswiththeadditionalvalues.

x 5 = #−1

−12Q

0

12Q

1

x 5 = #:−1

−12Q

0

12Q

1

x 5 = #;−1 −1

2Q

0

12Q

1

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3.2


15.Graph5 = #7895 = #:7895 = #;ontheinterval[−1, 1],usingthesamesetofaxes.

16.Completethetableswiththeadditionalvalues.

x 5 = #−2

−1

−12Q

0

12Q

1

2

x 5 = #:−2

−1

−12Q

0

12Q

1

2

x 5 = #;−2

−1 −1

2Q

0

12Q

1

2

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3.2


17.Graph5 = #7895 = #:7895 = #;ontheinterval[-2,2],usingthesamesetofaxes.

GO Topic:UsingtheexponentrulestosimplifyexpressionsSimplify.

18.#T :Q ∙ #TVQ ∙ #

TWQ 19.7X ;Q ∙ 7

:TYQ ∙ 7

XT;Q 20.ZW

[Q ∙ Z:TWQ ∙ Z

;X\Q

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3.3 It All Adds Up A Develop Understanding Task

Wheneverwe’rethinkingaboutalgebraandworkingwithvariables,itisusefultoconsiderhowitrelatestothenumbersystemandoperationsonnumbers.Rightnow,polynomialsareonourminds,solet’sseeifwecanmakesomeusefulcomparisonsbetweenwholenumbersandpolynomials.Let’sstartbylookingatthestructureofnumbersandpolynomials.Considerthenumber132.Thewaywewritenumbersisreallyashortcutbecause:

132 = 100 + 30 + 21.Compare132tothepolynomialCD + 3C + 2.Howaretheyalike?Howaretheydifferent?2.Writeapolynomialthatisanalogoustothenumber2,675.Whentwonumbersaretobeaddedtogether,manypeopleuseaprocedurelikethis: 132+ 451 5833.Writeananalogousadditionproblemforpolynomialsandfindthesumofthetwopolynomials.4.Howdoesaddingpolynomialscomparetoaddingwholenumbers?

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5.Usethepolynomialsbelowtofindthespecifiedsumsina-f.E(C) = CH + 3CD − 2C + 10 J(C) = 2C − 1 ℎ(C) = 2CD + 5C − 12 M(C) = −CD − 3C + 4 a)ℎ(C) + M(C) b)J(C) + E(C) c)E(C) + M(C)______________________________ _____________________________________________________________d) O(C) +P(C) e)P(C) + Q(C) f)O(C) + R(C)

R(C) Q(C)

O(C) P(C)

14




6.Whatpatternsdoyouseewhenpolynomialsareadded?Subtractionofwholenumbersworkssimilarlytoaddition.Somepeoplelineupsubtractionverticallyandsubtractthebottomnumberfromthetop,likethis: 368

-157211

7.Writetheanalogouspolynomialsandsubtractthem.8.Isyouranswerto#7analogoustothewholenumberanswer?Ifnot,whynot?9.Subtractingpolynomialscaneasilyleadtoerrorsifyoudon’tcarefullykeeptrackofyourpositiveandnegativesigns.Onewaythatpeopleavoidthisproblemistosimplychangeallthesignsofthepolynomialbeingsubtractedandthenaddthetwopolynomialstogether.Therearetwocommonwaysofwritingthis:

(CH + CD − 3C − 5) − (2CH − CD + 6C + 8)Step1: = (CH + CD − 3C − 5) + (−2CH + CD − 6C − 8)Step2: = (−CH + 2CD − 9C − 13)

Or,youcanlineupthepolynomialsverticallysothatStep1lookslikethis:Step1: CH + CD − 3C − 5 +(−2CH + CD − 6C − 8)

Step2: −CH + 2CD − 9C − 13

Thequestionforyouis:Isitcorrecttochangeallthesignsandaddwhensubtracting?Whatmathematicalpropertyorrelationshipcanjustifythisaction?

15




10.Usethegivenpolynomialstofindthespecifieddifferencesina-d.E(C) = CH + 2CD − 7C − 8 J(C) = −4C − 7ℎ(C) = 4CD − C − 15 M(C) = −CD + 7C + 4 a)ℎ(C) − M(C) b)E(C) − ℎ(C) c)E(C) − J(C)______________________________ _____________________________________________________________d)M(C) − E(C) e)O(C) −P(C)

______________________________

11.Listthreeimportantthingstorememberwhensubtractingpolynomials.

O(C) P(C)

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3.3


READY Topic:Usingthedistributiveproperty

Multiply.

1. 2"(5"% + 7) 2. 9"(−"% − 3) 3. 5"%(", + 6".)

4. −"("% − " + 1) 5. −3".(−2"% + " − 1) 6. −1("% − 4" + 8)

SET Topic:Addingandsubtractingpolynomials

Add.Writeyouranswersindescendingorderoftheexponents.(Standardform)7.(3", + 5"% − 1) + (2". + ") 8.(4"% + 7" − 4) + ("% − 7" + 14)

9.(2". + 6"% − 5") + ("4 + 3"% + 8" + 4) 10.(−6"4 − 2" + 13) + (4"4 + 3"% + " − 9)

Subtract.Writeyouranswersindescendingorderoftheexponents.(Standardform)

11.(5"% + 7" + 2) − (3"% + 6" + 2) 12.(10", + 2"% + 1) − (3", + 3" + 11)

13. (7". − 3" + 7) − (4"% − 3" − 11) 14. (", − 1) − (", + 1)


17




3.3


Graph.15. 6 = ". − 2

16. 6 = ". + 1

17. 6 = (" − 3).

18. 6 = (" + 1).

GO Topic:Usingexponentrulestocombineexpressions

Simplify.

19. "8 9: ∙ "< ,: ∙ "=< %: 20.". <>: ∙ "=8 9: ∙ ". ,: 21.", 8: ∙ "% ?: ∙ "=< .:

18




3.4 Pascal’s Pride A Solidify Understanding Task

Multiplyingpolynomialscanrequireabitofskillinthe

algebradepartment,butsincepolynomialsare

structuredlikenumbers,multiplicationworksvery

similarly.Whenyoulearnedtomultiplynumbers,youmayhavelearnedtouseanarea

model.Tomultiply12 × 15theareamodelandtherelatedprocedureprobablylookedlikethis:

Youmayhaveusedthissameideawithquadraticexpressions.Areamodelshelpusthinkabout

multiplying,factoring,andcompletingthesquaretofindequivalentexpressions.Wemodeled

(' + 2)(' + 5) = '+ + 7' + 10astheareaofarectanglewithsidesoflength' + 2and' + 5.Thevariouspartsoftherectangleareshowninthediagrambelow:

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' + 5

'+ ' ' ' ' '

1

' 1

'

1 1 1 1

1 111'+2

12

15

10

50

+20

100

180

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Somepeopleliketoshortcuttheareamodelalittlebittojusthavesectionsofareathatcorrespond

tothelengthsofthesides.Inthiscase,theymightdrawthefollowing.

= '+ + 7' + 10

1.Whatisthepropertythatallofthesemodelsarebasedupon?

2.Nowthatyou’vebeenremindedofthehappypast,youarereadytousethestrategyofyour

choicetofindequivalentexpressionsforeachofthefollowing:

a) (' + 3)(' + 4) b) (' + 7)(' − 2)

Maybenowyouremembersomeofthedifferentformsforquadraticexpressions—factoredform

andstandardform.Theseformsexistforallpolynomials,althoughasthepowersgethigher,the

algebramaygetalittletrickier.Instandardformpolynomialsarewrittensothatthetermsarein

orderwiththehighest-poweredtermfirst,andthenthelower-poweredterms.Someexamples:

Quadratic: '+ − 3' + 8 or '+ − 9

Cubic: 2'4 + '+ − 7' − 10 or '4 − 2'+ + 15

Quartic: '5 + '4 + 3'+ − 5' + 4

Hopefully,youalsorememberthatyouneedtobesurethateachterminthefirstfactoris

multipliedbyeachterminthesecondfactorandtheliketermsarecombinedtogettostandard

form.Youcanuseareamodels,boxes,ormnemonicslikeFOIL(first,outer,inner,last)tohelpyou

organize,oryoucanjustcheckeverytimetobesurethatyou’vegotallthecombinations.Itcanget

morechallengingwithhigher-poweredpolynomials,buttheprincipalisthesamebecauseitis

baseduponthemightyDistributiveProperty.

' +5

' '+ 5'

+2 2' 10

20




3.Tia’sfavoritestrategyformultiplyingpolynomialsistomakeaboxthatfitsthetwofactors.She

setsituplikethis:(' + 2)('+ − 3' + 5)

TryusingTia’sboxmethodtomultiplythesetwofactorstogetherandthencombiningliketermsto

getapolynomialinstandardform.

4.Trycheckingyouranswerbygraphingtheoriginalfactoredpolynomial,(' + 2)('+ − 3' + 5)andthengraphingthepolynomialthatisyouranswer.Ifthegraphsarethesame,youareright

becausethetwoexpressionsareequivalent!Iftheyarenotthesame,gobackandcheckyourwork

tomakethecorrections.

5.Tehani’sfavoritestrategyistoconnectthetermsheneedstomultiplyinorderlikethis:

(' − 3)('+ + 4' − 2)

TrymultiplyingusingTehani’sstrategyandthencheckyourworkbygraphing.Makeany

correctionsyouneedandfigureoutwhytheyareneededsothatyouwon’tmakethesamemistake

twice!

6.Usethestrategyofyourchoicetomultiplyeachofthefollowingexpressions.Checkyourwork

bygraphingandmakeanyneededcorrections.

a) (' + 5)('+ − ' − 3) b) (' − 2)(2'+ + 6' + 1)

c) (' + 2)(' − 2)(' + 3)

'+ −3' +5

'

+2

21




Whengraphing,itisoftenusefultohaveaperfectsquarequadraticoraperfectcube.Sometimesit

isalsousefultohavethesefunctionswritteninstandardform.Let’stryre-writingsomerelated

expressionstoseeifwecanseesomeusefulpatterns.

7.Multiplyandsimplifybothofthefollowingexpressionsusingthestrategyofyourchoice:

a) N(') = (' + 1)+ b) N(') = (' + 1)4

Checkyourworkbygraphingandmakeanycorrectionsneeded.

8.SomeenterprisingyoungmathematiciannoticedaconnectionbetweenthecoefficientsofthetermsinthepolynomialandthenumberpatternknownasPascal’sTriangle.Putyouranswersfromproblem5intothetable.CompareyouranswerstothenumbersinPascal’sTrianglebelowanddescribetherelationshipyousee.

(' + 1)U 1 1

(' + 1)V ' + 1 11

(' + 1)+ 121

(' + 1)4 1331

(' + 1)5

9.ItcouldsavesometimeonmultiplyingthehigherpowerpolynomialsifwecouldusePascal’sTriangletogetthecoefficients.First,wewouldneedtobeabletoconstructourownPascal’sTriangleandaddrowswhenweneedto.LookatPascal’sTriangleandseeifyoucanfigureouthowtogetthenextrowusingthetermsfromthepreviousrow.Useyourmethodtofindthetermsinthenextrowofthetableabove.

10.NowyoucancheckyourPascal’sTrianglebymultiplyingout(' + 1)5andcomparingthecoefficients.Hint:Youmightwanttomakeyourjobeasierbyusingyouranswersfrom#7insomeway.Putyouranswerinthetableabove.

22




11.Makesurethattheansweryougetfrommultiplying(' + 1)5andthenumbersinPascal’sTrianglematch,sothatyou’resureyou’vegotbothanswersright.ThendescribehowtogetthenextrowinPascal’sTriangleusingthetermsinthepreviousrow.12.CompletethenextrowofPascal’sTriangleanduseittofindthestandardformof(' + 1)b.Writeyouranswersinthetableon#6.13.Pascal’sTrianglewouldn’tbeveryhandyifitonlyworkedtoexpandpowersof' + 1.Theremustbeawaytouseitforotherexpressions.ThetablebelowshowsPascal’sTriangleandtheexpansionof' + d.(' + d)U 1 1

(' + d)V ' + d 11

(' + d)+ '+ + 2d' + d+ 121

(' + d)4 '4 + 3d'+ + 3d+' + d4 1331

(' + d)5 '5 + 4d'4 + 6d+'+ + 3d4' + d5 14641

Whatdoyounoticeaboutwhathappenstothedineachofthetermsinarow?

14.UsethePascal’sTrianglemethodtofindstandardformfor(' + 2)4.Checkyouranswerbymultiplying.15.Useanymethodtowriteeachofthefollowinginstandardform:a) (' + 3)4 b) (' − 2)4 c) (' + 5)5

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3.4


READY Topic:Recallingthemeaningofdivision

1. Given:!(#) = (# + 7)(2# − 3),-./(#) = (# + 7).Find .

2. Given:!(#) = (5# + 7)(−3# + 11),-./(#) = (−3# + 11).Find

3. Given:!(#) = (# + 2)(#3 + 3# + 2),-./(#) = (# + 2)Find

4. Given:!(#) = (5# − 3)(#3 − 11# − 9),-./(#) = (5# − 3),-.ℎ(#) = (#3 − 11# − 9).

a.)Find b.)Find

5. Given:!(#) = (5# − 6)(2#3 − 5# + 3),-./(#) = (# − 1),-.ℎ(#) = (2# − 3).

a.)Find b.)Find

SET Topic:MultiplyingpolynomialsMultiply.Writeyouranswersinstandardform.

6.(, + 7)(, + 7) 7.(# − 3)(#3 + 3# + 9)

8.(# − 5)(#3 + 5# + 25) 9.(# + 1)(#3 − # + 1)

10.(# + 7)(#3 − 7# + 49) 11.(, − 7)(,3 + ,7 + 73)

g x( ) f x( )

g x( ) f x( )

g x( ) f x( )

g x( ) f x( ) h x( ) f x( )

g x( ) f x( ) h x( ) f x( )


24




3.4


(# + ,)9 1 1

(# + ,): # + , 11

(# + ,)3 #3 + 2,# + ,3 121

(# + ,); #; + 3,#3 + 3,3# + ,; 1331

(# + ,)< #< + 4,#; + 6,3#3 + 3,;# + ,< 14641

Usethetableabovetowriteeachofthefollowinginstandardform.

12.(# + 1)P

13.(# − 5); 14.(# − 1)<

15,(# + 4);

16.(# + 2)< 17.(3# + 1);

GO Topic:ExaminingtransformationsondifferenttypesoffunctionsGraphthefollowingfunctions.

18./(#) = # + 2

19.ℎ(#) = #3 + 2 20.!(#) = 2R + 2

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3.4


21./(#) = 3(# − 2)

22.ℎ(#) = 3(# − 2)3

23.!(#) = 3√# − 2

24./(#) = :

3(# − 1) − 2

25.ℎ(#) = :

3(# − 1)3 − 2

26.!(#) = |# − 1| − 2

26




3.5 Divide And Conquer A Solidify Understanding Task

We’veseenhownumbersandpolynomialsrelateinaddition,subtraction,andmultiplication.Nowwe’rereadytoconsiderdivision.Division,yousay?Like,longdivision?Yup,that’swhatwe’retalkingabout.Holdthejudgment!It’sactuallyprettycool.Asusual,let’sstartbylookingathowtheoperationworkswithnumbers.Sincedivisionistheinverseoperationofmultiplication,thesamemodelsshouldbeuseful.Theareamodelthatweusedwithmultiplicationisalsousedwithdivision.Whenwewereusingareamodelstofactoraquadraticexpression,wewereactuallydividing.Let’sbrushuponthatabit.1.Theareamodelfor:; + 7: + 10isshownbelow:Usetheareamodeltowrite:; + 7: + 10infactoredform.2.Wealsousednumberpatternstofactorwithoutdrawingtheareamodel.Useanystrategytofactorthefollowingquadraticpolynomials:a):; + 7: + 12 b):; + 2: − 15

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c):; − 11: + 24 d):; − 5: − 36

Factoringworksgreatforquadraticsandafewspecialcasesofotherpolynomials.Let’slookatamoregeneralversionofdivisionthatisalotlikewhatwedowithnumbers.Let’ssaywewanttodivide1452by12.Ifwewritetheanalogouspolynomialdivisionproblem,itwouldbe:(:G + 4:; + 5: + 2) ÷ (: + 2).Let’susethedivisionprocessfornumberstocreateadivisionprocessforpolynomials.(Don’tpanic—inmanywaysit’seasierwithpolynomialsthannumbers!)Step1:Startwithwritingtheproblemaslongdivision.Thepolynomialneedstohavethetermswrittenindescendingorder.Ifthereareanymissingpowers,it’seasierifyouleavealittlespaceforthem.

Step2:Determinewhatyoucouldmultiplythedivisorbytogetthefirsttermofthedividend.

Step3:Multiplyandputtheresultbelowthedividend.

Step4:Subtract.(Ithelpstokeepthesignsstraightifyouchangethesignoneachtermandaddonthepolynomial.)

12 1452 3 22 4 5 2x x x x+ + + +

12 14521 2

232 4 5 2x

x x x x+ + + +

0

11 12 452

120-

2

2

3 2

32 4 5 2

( 2 )

xx xx x

x x

+ + + +

- +

28




Step5:Repeattheprocesswiththenumberorexpressionthatremainsinthedividend.

Step6:Keepgoinguntilthenumberorexpressionthatremainsissmallerthanthedivisor.

Inthiscase,121dividedby12leavesnoremainder,sowewouldsaythat12isafactorof121.Similarly,since(:G + 4:; + 5: + 2)dividedby(: + 2)leavesnoremainder,wewouldsaythat(: + 2)isafactorof(:G + 4:; + 5: + 2).Polynomialdivisiondoesn’talwaysmatchupperfectlytoananalogouswholenumberproblem,buttheprocessisalwaysthesame.Let’stryit.

112 1452

1200252

-

2

3 2

3 2

2

2 4 5 2

( 2 )

2 5 2

xx x x x

x x

x x

+ + + +

+ - -

+ +

12 1452

12005224012

21

2-

-

2

2

3 2

3 2

2

2 4 5 2

( 2 )

5 2(

2

2

22 4 )

xx

xx x x

x

xx

x

x x

x

++ + + +

+ - -

+ +

- +

+

122 1452

12002522402120

11

1

-

-

-

2

3 2

3 2

2

2

22 4 5 2

( 2 )

2 5 2(2 4 )

2( 2)

0

1x xx x x x

x x

x x

x

xx

x

+ ++ + + +

+ - -

+ +

- +

+- +

29




3.Uselongdivisiontodetermineif(: − 1)afactorof(:G − 3:; − 13: + 15).Don’tworry:thestepsforthedivisionprocessarebelow:a)Writetheproblemaslongdivision.b)Whatdoyouhavetomultiply:bytoget:G?Writeyouranswerabovethebar.c)Multiplyyouranswerfromstepbby(: − 1)andwriteyouranswerbelowthedividend.d)Subtract.Becarefultosubtracteachterm.(Youmightwanttochangethesignsandadd.)e)Repeatstepsa-duntiltheexpressionthatremainsislessthan(: − 1).Wehopeyousurvivedthedivisionprocess.Is(: − 1)afactorof(:G − 3:; − 13: + 15)?_________4.Tryitagain.Uselongdivisiontodetermineif(2: + 3)isafactorof2:G + 7:; + 2: + 9.Nohintsthistime.Youcandoit!Whendividingnumbers,thereareseveralwaystodealwiththeremainder.Sometimes,wejustwriteitastheremainder,likethis:

because3(8) + 1 = 25

8 .13 25r

30




Youmayrememberalsowritingtheremainderasafractionlikethis:

because3 N8OGP = 25

Wedothesamethingswithpolynomials.Maybeyoufoundthat(2:G + 7:; + 2: + 9) ÷ (2: + 3) = (:; + 2: − 2)Q. 15.(Wesurehopeso.)Youcanuseittowritetwomultiplicationstatements:

(2: + 3)(:; + 2: − 2) + 15 =(2:G + 7:; + 2: + 9)

and

(2: + 3)(:; + 2: − 2 +15

2: + 3) = (2:G + 7:; + 2: + 9)

5.Divideeachofthefollowingpolynomials.Writethetwomultiplicationstatementsthatgowithyouranswersifthereisaremainder.Writeonlyonemultiplicationstatementifthedivisorisafactor.Usegraphingtechnologytocheckyourworkandmakethenecessarycorrections. a)(:G + 6:; + 13: + 12) ÷ (: + 3) b)(:G − 4:; + 2: + 5) ÷ (: − 2)

Multiplicationstatements:

138

3 25

31




c)(6:G − 11:; − 4: + 5) ÷ (2: − 1) d)(:R − 23:G + 49: + 4) ÷ (:; + : + 2)

Multiplicationstatements:

32




3.5


READY Topic:Solvinglinearequations

Solveforx.1.5" + 13 = 48 2.)* " − 8 = 0 3.−4 − 9" = 0

4.". − 16 = 0 5.". + 4" + 3 = 0 6.". − 5" + 6 = 0

7.(" + 8)(" + 11) = 0 8.(" − 5)(" − 7) = 0 9.(3" − 18)(5" − 10) = 0

SET Topic:Dividingpolynomials

Divideeachofthefollowingpolynomials.Writeonlyonemultiplicationstatementifthedivisorisafactor.Writethetwomultiplicationstatementsthatgowithyouranswersifthereisaremainder.10. 11.

Multiplicationstatement(s)


x +1( ) x3 − 3x2 + 6x +11 x − 5( ) x3 − 9x2 + 23x −15


33




3.5


12. 13.



14. 15.



GO Topic:Describingthefeaturesofavarietyoffunctions

Graphthefollowingfunctions.Thenidentifythekeyfeaturesofthefunctions.Includedomain,range,intervalswherethefunctionisincreasing/decreasing,intercepts,maximum/minimum,andendbehavior.

2x −1( ) 2x3 +15x2 − 34x +13 x + 4( ) x3 +13x2 + 26x − 25

34




3.5


16.3(") = ". − 9domain:range:increasing:decreasing:y-intercept:x-intercept(s):

17.3(4 − 1) = 3(4) + 3; 3(1) = 4domain:range:

increasing:decreasing:

y-intercept:x-intercept(s):

18.3(") = √" − 3+1domain:range:



35




3.5


19.3(") = 89:." − 1domain:range:



Identifythekeyfeaturesofthegraphedfunctions.

20.

domain:range:



21.

domain:range:



36




3.6 Sorry, We’re Closed A Practice Understanding Task

Nowthatwehavecomparedoperationsonpolynomialswithoperationsonwholenumbersit’stimetogeneralizeabouttheresults.Beforewegotoofar,weneedatechnicaldefinitionofapolynomialfunction.Hereitis:Apolynomialfunctionhastheform:

!(#) = &'#' + &')*#')*+. . . +&*# + &,where&', &')*, . . . &*, &,arerealnumbersandnisanonnegativeinteger.Inotherwords,apolynomialisthesumofoneormoremonomialswithrealcoefficientsandnonnegativeintegerexponents.Thedegreeofthepolynomialfunctionisthehighestvaluefor/where&'isnotequalto0.1.Thefollowingexamplesandnon-exampleswillhelpyoutoseetheimportantimplicationsofthedefinitionofapolynomialfunction.Foreachpair,determinewhatisdifferentbetweentheexampleofapolynomialandthenon-examplethatisnotapolynomial.Thesearepolynomials: Thesearenotpolynomials:

a)0(1) = 12 b)3(1) = 35 Howareaandbdifferent?

c)0(1) = 217 + 51 − 12 d)3(1) = 75;5;)25<7

Howarecandddifferent?

e)0(1) = −12 + 317 − 21 − 7 0)3(1) = 12 + 317 − 21 + 101)? − 7Howareeandfdifferent?

h)0(1) = ?7 1 i)g(1) = ?

75Howarehandidifferent?

j)0(1) = 17 k)3(1) = 1@;Howarejandkdifferent?

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2.Basedonthedefinitionandtheexamplesabove,howcanyoutellifafunctionisapolynomialfunction?Maybeyouhavenoticedinthepastthatwhenyouaddtwoevennumbers,theansweryougetisalwaysanevennumber.Mathematically,wesaythatthesetofevennumbersisclosedunderaddition.Mathematiciansareinterestedinresultslikethisbecauseithelpsustounderstandhownumbersorfunctionsofaparticulartypebehavewiththevariousoperations.3.Youcantryityourself:Isthesetofoddnumbersclosedundermultiplication?Inotherwords,ifyoumultiplytwooddnumberstogetherwillyougetanoddnumber?Explain.Ifyoufindanytwooddnumbersthathaveanevenproduct,thenyouwouldsaythatoddnumbersarenotclosedundermultiplication.Evenifyouhaveanumberofexamplesthatsupporttheclaim,ifyoucanfindonecounterexamplethatcontradictstheclaim,thentheclaimisfalse.Considerthefollowingclaimsanddeterminewhethertheyaretrueorfalse.Ifaclaimistrue,giveareasonwithatleasttwoexamplesthatillustratetheclaim.Yourexamplescanincludeanyrepresentationyouchoose.Iftheclaimisfalse,giveareasonwithonecounterexamplethatprovestheclaimtobefalse.4.Thesetofwholenumbersisclosedunderaddition.5.Thesumofaquadraticfunctionandalinearfunctionisacubicfunction.

38




6.Thesumofalinearfunctionandanexponentialfunctionisapolynomial.7.Thesetofpolynomialsisclosedunderaddition.8.Thesetofwholenumbersisclosedundersubtraction.9.Thesetofintegersisclosedundersubtraction.10.Aquadraticfunctionsubtractedfromacubicfunctionisacubicfunction.11.Alinearfunctionsubtractedfromalinearfunctionisapolynomialfunction.12.Acubicfunctionsubtractedfromacubicfunctionisacubicfunction.13.Thesetofpolynomialfunctionsisclosedundersubtraction.

39




14.Theproductoftwolinearfunctionsisaquadraticfunction.15.Thesetofintegersisclosedundermultiplication.16.Thesetofpolynomialsisclosedundermultiplication.17.Thesetofintegersisclosedunderdivision.18.Acubicfunctiondividedbyalinearfunctionisaquadraticfunction.19.Thesetofpolynomialfunctionsisclosedunderdivision.20.Writetwoclaimsofyourownaboutpolynomialsanduseexamplestodemonstratethattheyaretrue.Claim#1:Claim#2:

40




3.6


READY Topic:Connectingthezerosofafunctiontothesolutionoftheequation

Whenwesolveequations,weoftensettheequationequaltozeroandthenfindthevalueofx.Anotherwaytosaythisis“findwhen!(#) = &.“That’swhywecallsolutionstoequationsthezerosofanequation.Findthezerosforthegivenequations.Thenmarkthesolution(s)asapointonthegraphoftheequation.1.(()) = *

+ ) + 4

2..()) = − 0+ ) − 2

3.ℎ()) = 2) − 6

4.4()) = 5)* − 10) − 15


41




3.6


5.8()) = 4)* − 20)

6.9()) = −)* + 9

SET Topic:Exploringclosedmathematicalnumbersets

Identifythefollowingstatementsassometimestrue,alwaystrue,ornevertrue.Ifyouranswerissometimestrue,giveanexampleofwhenit’strueandanexampleofwhenit’snottrue.Ifit’snevertrue,giveacounter-example.7.Theproductofawholenumberandawholenumberisaninteger.

8.Thequotientofawholenumberdividedbyawholenumberisawholenumber.

9.Thesetofintegersisclosedunderdivision.

10.Thedifferenceofalinearfunctionandalinearfunctionisaninteger.

11.Thedifferenceofalinearfunctionandaquadraticfunctionisalinearfunction.

42




3.6


12.Theproductofalinearfunctionandalinearfunctionisaquadraticfunction.

13.Thesumofaquadraticfunctionandaquadraticfunctionisapolynomialfunction.

14.Theproductofalinearfunctionandaquadraticfunctionisacubicfunction.

15.Theproductofthreelinearfunctionsisacubicfunction.

16.Thesetofpolynomialfunctionsisclosedunderaddition.

GO Topic:Identifyingconjugatepairs

Aconjugatepairissimplyapairofbinomialsthathavethesamenumbersbutdifferbyhavingoppositesignsbetweenthem.Forexample(; + =)and(;– =)areconjugatepairs.You’veprobablynoticedthemwhenyou’vefactoredaquadraticexpressionthatisthedifferenceoftwosquares.Example:)* − 25 = () + 5)() − 5).Thetwofactors() + 5)() − 5)areconjugatepairs.

Thequadraticformula# = ?@±B@C?DEFCE cangeneratebothsolutionstoaquadraticequationbecause

ofthe±locatedinthenumeratoroftheformula.Whenthe√=* − 4;Hpartoftheformulageneratesanirrationalnumber(e.g.√2)oranimaginarynumber(e.g.2I),theformulaproducesapairofnumbersthatareconjugates.Thisisimportantbecausethistypeofsolutiontoaquadraticalwayscomesinpairs.Example:TheconjugateofJ3 + √2LIMJ3 − √2L.Theconjugateof(−2I)is(+2I).Thinkofitas(0 − 2I);N9(0 + 2I).Changeonlythesignbetweenthetwonumbers.Writetheconjugateofthegivenvalue.17.J8 + √5L 18.(11 + 4I) 19.9I 20.−5√721.(2 − 13I) 22.(−1 − 2I) 23.J−3 + 5√2L 24.−4I

43




3.7 Building Strong Roots A Solidify Understanding Task

Whenworkingwithquadraticfunctions,welearned

theFundamentalTheoremofAlgebra:

An"#$degreepolynomialfunctionhas"roots.Inthistask,wewillbeexploringthisideafurtherwithotherpolynomialfunctions.

First,let’sbrushuponwhatwelearnedaboutquadratics.Theequationsandgraphsoffour

differentquadraticequationsaregivenbelow.Findtherootsforeachandidentifywhetherthe

rootsarerealorimaginary.

1.

a)%(') = '* + ' − 6

b)g(') = '* − 2' − 7

Roots: Roots:

Typeofroots: Typeofroots:

c)ℎ(') = '* − 4' + 4

d)2(') = '* − 4' + 5

Roots: Roots:

Typeofroots: Typeofroots:

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2.Didallofthequadraticfunctionshave2roots,aspredictedbytheFundamentalTheoremof

Algebra?Explain.

3.It’salwaysimportanttokeepwhatyou’vepreviouslylearnedinyourmathematicalbagoftricks

sothatyoucanpullitoutwhenyouneedit.Whatstrategiesdidyouusetofindtherootsofthe

quadraticequations?

4.Usingyourworkfromproblem1,writeeachofthequadraticequationsinfactoredform.When

youfinish,checkyouranswersbygraphing,whenpossible,andmakeanycorrectionsnecessary.

a)%(') = '* + ' − 6

b)g(') = '* − 2' − 7

Factoredform:

Factoredform:

c)ℎ(') = '* − 4' + 4

d)2(') = '* − 4' + 5

Factoredform:

Factoredform:

5.Basedonyourworkinproblem1,wouldyousaythatrootsarethesameas'-intercepts?Explain.

6.Basedonyourworkinproblem4,whatistherelationshipbetweenrootsandfactors?

45




Nowlet’stakeacloserlookatcubicfunctions.We’veworkedwithtransformationsof

%(') = '4,butwhatwe’veseensofarisjustthetipoftheiceberg.Forinstance,consider:

6(') = '4 − 3'* − 10'

7.Usethegraphtofindtherootsofthecubicfunction.Usetheequationtoverifythatyouare

correct.Showhowyouhaveverifiedeachroot.

8.Write6(')infactoredform.Verifythatthefactoredformisequivalenttothestandardform.

9.Aretheresultsyoufoundin#7consistentwiththeFundamentalTheoremofAlgebra?Explain.

46




Here’sanotherexampleofacubicfunction.

%(') = '4 + 7'* + 8' − 16

10.Usethegraphtofindtherootsofthecubicfunction.

11.Write%(')infactoredform.Verifythatthefactoredformisequivalenttothestandardform.Makeanycorrectionsneeded.

12.Aretheresultsyoufoundin#10consistentwiththeFundamentalTheoremofAlgebra?

Explain.

13.We’veseenthemostbasiccubicpolynomialfunction,ℎ(') = '4andweknowitsgraphlookslikethis:

Explainhowℎ(') = '4isconsistentwiththeFundamentalTheoremofAlgebra.

47




14.Hereisonemorecubicpolynomialfunctionforyourconsideration.Youwillnoticethatitis

giventoyouinfactoredform.Usetheequationandthegraphtofindtherootsof;(').

;(') = (' + 3)('* + 4)

15.Usetheequationtoverifyeachroot.Showyourworkbelow.

16.Aretheresultsyoufoundin#14consistentwiththeFundamentalTheoremofAlgebra?

Explain.

17.Explainhowtofindthefactoredformofapolynomial,giventheroots.

18.Explainhowtofindtherootsofapolynomial,giventhefactoredform.

48




3.7


READY Topic:Practicinglongdivisiononpolynomials

Divideusinglongdivision.(Theseproblemshavenoremainders.Ifyougetone,tryagain.)1. 2.

3.

4.

SET Topic:ApplyingtheFundamentalTheoremofAlgebra

Predictthenumberofrootsforeachofthegivenpolynomialequations.(RememberthattheFundamentalTheoremofAlgebrastates:Annthdegreepolynomialfunctionhasnroots.)5.!(#) = #& + 3# − 10 6.,(#) = #- + #& − 9# − 9 7./(#) = −2# − 48.2(#) = #3 − #- − 4#& + 4# 9.4(#) = −#& + 6# − 9 10.6(#) = #7 − 5#3 + 4#&

x + 3( ) 5x3 + 2x2 − 45x −18 x − 6( ) x3 − x2 − 44x + 84

x + 2( ) x4 + 6x3 + 7x2 − 6x − 8


x −5( ) 3x3 −15x2 +12x −60

49




3.7


Belowarethegraphsofthepolynomialsfromthepreviouspage.Checkyourpredictions.Thenusethegraphtohelpyouwritethepolynomialinfactoredform.11.!(#) = #& + 3# − 10Factoredform:

12.,(#) = #- + #& − 9# − 9Factoredform:

13./(#) = −2# − 4Factoredform:

14.2(#) = #3 − #- − 4#& + 4#Factoredform:

50




3.7


15.4(#) = −#& + 6# − 9Factoredform:

16.6(#) = #7 − 5#3 + 4#&Factoredform:

17.Thegraphsof#15and#16don’tseemtofollowtheFundamentalTheoremofAlgebra,butthere

issomethingsimilarabouteachofthegraphs.Explainwhatishappeningatthepoint(3,0)in#15andatthepoint(0,0)in#16.

GO Topic:SolvingquadraticequationsFindthezerosforeachequationusingthequadraticformula.18.4(#) = #& + 20# + 51

19.4(#) = #& + 10# + 25 20.4(#) = 3#& + 12#

21.4(#) = #& − 11

22.4(#) = #& + # − 1 23.4(#) = #& + 2# + 3

51




3.8 Getting to the Root of the Problem A Solidify Understanding Task

In3.7BuildingStrongRoots,welearnedtopredictthenumberofrootsofapolynomialusingtheFundamentalTheoremofAlgebraandtherelationshipbetweenrootsandfactors.Inthistask,wewillbeworkingonhowtofindalltherootsofapolynomialgiveninstandardform.Let’sstartbythinkingagainaboutnumbersandfactors.1.Ifyouknowthat7isafactorof147,whatwouldyoudotofindtheprimefactorizationof147?Explainyouranswerandshowyourprocesshere:2.Howisyouranswerlikeapolynomialwrittenintheform:!(#) = (# − 7)((# − 3)?Theprocessforfindingfactorsofpolynomialsisexactlyliketheprocessforfindingfactorsofnumbers.Westartbydividingbyafactorweknowandkeepdividinguntilwehaveallthefactors.Whenwegetthepolynomialbrokendowntoaquadratic,sometimeswecanfactoritbyinspection,andsometimeswecanuseourotherquadratictoolslikethequadraticformula.Let’stryit!Foreachofthefollowingfunctions,youhavebeengivenonefactor.Usethatfactortofindtheremainingfactors,therootsofthefunction,andwritethefunctioninfactoredform.3.Function:5(#) = #6 + 3#( − 4# − 12 Factor:(# + 3) Rootsoffunction:Factoredform:

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4.Function:5(#) = #6 + 6#( + 11# + 6 Factor:(# + 1) Rootsoffunction:Factoredform:5.Function:5(#) = #6 − 5#( − 3# + 15 Factor:(# − 5) Rootsoffunction:Factoredform:6.Function:5(#) = #6 + 3#( − 12# − 18 Factor:(# − 3) Rootsoffunction:Factoredform: 7.Function:5(#) = #F − 16 Factor:(# − 2) Rootsoffunction:

Factoredform:

53




8.Function:5(#) = #6 − #( + 4# − 4 Factor:(# − 2G) Rootsoffunction:Factoredform:9.Isitpossibleforapolynomialwithrealcoefficientstohaveonlyoneimaginaryroot?Explain.10.BasedontheFundamentalTheoremofAlgebraandthepolynomialsthatyouhaveseen,makeatablethatshowsallthenumberofrootsandthepossiblecombinationsofrealandimaginaryrootsforlinear,quadratic,cubic,andquarticpolynomials.

54




3.8


READY Topic:Orderingnumbersfromleasttogreatest

Orderthenumbersfromleasttogreatest.1.100$ √100 &'()100 100 2+,

2.2-+ −√100 &'() /181 0 (−2)+

3.2, √25 &'()8 2(5,), 5 ≠ 0 (2)-89

4.&'($3$ &'(;5-) &'(<6, &'(>4-+ &'()2$

Refertothegivengraphtoanswerthequestions.Insert>,<, 'B =ineachstatementtomakeittrue.

5.D(0)__________((0)6.D(2)__________((2)7.D(−1)__________((−1)8.D(1)__________((−1)

9.D(5)__________((5)

10.D(−2)__________((−2)


D(5)

((5)

55




3.8


SET Topic:FindingtherootsandfactorsofapolynomialUsethegivenroottofindtheremainingroots.Thenwritethefunctioninfactoredform.

Function Roots Factoredform11.D(5) = 5$ − 135) + 525 − 60 5 = 5

12.((5) = 5$ + 65) − 115 − 66 5 = −6

13.G(5) = 5$ + 175) + 925 + 150 5 = −3

14.J(5) = 5> − 65$ + 35) + 125 − 10 5 = √2

56




3.8


GO Topic:Usingthedistributivepropertytomultiplycomplexexpressions

Multiplyusingthedistributiveproperty.Simplify.Writeanswersinstandardform.15.K5 − √13LK5 + √13L 16.K5 − 3√2LK5 + 3√2L

17.(5 − 4 + 2M)(5 − 4 − 2M) 18.(5 + 5 + 3M)(5 + 5 − 3M)

19.(5 − 1 + M)(5 − 1 − M) 20.K5 + 10 − √2MLK5 + 10 + √2ML

57




3.9 Is This The End?

A Solidify Understanding Task

Inpreviousmathematicscourses,youhavecomparedand

analyzedgrowthratesofpolynomial(mostlylinearand

quadratic)andexponentialfunctions.Inthistask,weare

goingtoanalyzeratesofchangeandendbehaviorby

comparingvariousexpressions.

PartI:Seeingpatternsinendbehavior

1.Inasmanywaysaspossible,compareandcontrastlinear,quadratic,cubic,andexponentialfunctions.2.Usingthegraphprovided,writethefollowingfunctionsvertically,fromgreatesttoleastfor@ = B.Putthefunctionwiththegreatestvalueontopandthefunctionwiththesmallestvalueon

thebottom.Putfunctionswiththesamevaluesatthesamelevel.Anexample,E(F) = FG,hasbeen

placedonthegraphtogetyoustarted.

H(F) = 2J K(F) = FL + FN − 4 Q(F) = FN − 20

ℎ(F) = FT − 4FN + 1 V(F) = F + 30 X(F) = FY − 1

Z(F) = FT [(F) = \]

N^J _(F) = F`

3.WhatdeterminesthevalueofapolynomialfunctionatF = 0?Isthistrueforothertypesof

functions?

4.WritethesameexpressionsonthegraphinorderfromgreatesttoleastwhenFrepresentsa

verylargenumber(thisnumberissolarge,sowesaythatitisapproachingpositiveinfinity).Ifthe

valueofthefunctionispositive,putthefunctioninquadrant1.Ifthevalueofthefunctionis

negative,putthefunctioninquadrantIV.Anexamplehasbeenplacedforyou.

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5.WhatdeterminestheendbehaviorofapolynomialfunctionforverylargevaluesofF?

6.WritethesamefunctionsinorderfromgreatesttoleastwhenFrepresentsanumberthatis

approachingnegativeinfinity.Ifthevalueofthefunctionispositive,placeitinQuadrantII,ifthe

valueofthefunctionisnegative,placeitinQuadrantIII.Anexampleisshownonthegraph.

7.WhatpatternsdoyouseeinthepolynomialfunctionsforFvaluesapproachingnegativeinfinity?

Whatpatternsdoyouseeforexponentialfunctions?Usegraphingtechnologytotestthesepatterns

withafewmoreexamplesofyourchoice.

8.Howwouldtheendbehaviorofthepolynomialfunctionschangeiftheleadtermswerechanged

frompositivetonegative?

59




@ = B @ → ∞ @ → −∞

E(F) = FG

K(F) = FG m = FN

m = FL

60




PartII:Usingendbehaviorpatterns

Foreachsituation:

• Determinethefunctiontype.Ifitisapolynomial,statethedegreeofthepolynomialandwhetheritisanevendegreepolynomialoranodddegreepolynomial.

• Describetheendbehaviorbasedonyourknowledgeofthefunction.Usetheformat:AsF → −∞, H(F) → ______r[srtF → ∞H(F) → ______1. H(F) = 3 + 2F

Functiontype:

Endbehavior:AsF → −∞, H(F) → ______

Endbehavior:AsF → ∞, H(F) → ______

2.H(F) = FY − 16

Functiontype:



3.H(F) = 3J

Functiontype:

Endbehavior:AsF →−∞, H(F) → ______


4.H(F) = FL + 2FN − F + 5

Functiontype:



5.H(F) = −2FL + 2FN − F + 5

Functiontype:



6.H(F) = EvQNF

Functiontype:



Usethegraphsbelowtodescribetheendbehaviorofeachfunctionbycompletingthestatements.7. 8.





61




9.Howdoestheendbehaviorforquadraticfunctionsconnectwiththenumberandtypeofrootsforthesefunctions?Howdoestheendbehaviorforcubicfunctionsconnectwiththenumberandtypeofrootsforcubicfunctions?

PartIII:EvenandOddFunctionsSomefunctionsthatarenotpolynomialsmaybecategorizedasevenfunctionsoroddfunctions.Whenmathematicianssaythatafunctionisanevenfunction,theymeansomethingveryspecific.1.Let’sseeifyoucanfigureoutwhatthedefinitionofanevenfunctioniswiththeseexamples:Evenfunction:

H(F) = FN

Notanevenfunction:Q(F) = 2J

Differences:

Evenfunction:H(F) = FY − 3

Notanevenfunction:Q(F) = F(F + 3)(F − 2)

Differences:

62




Evenfunction:H(F) = −|F| + 4

Notanevenfunction:Q(F) = −|F + 4|

Differences:Evenfunction:

H(2) = 5r[sH(−2) = 5Notanevenfunction:

Q(2) = 3r[sQ(−2) = 5Differences:2.Whatdoyouobserveaboutthecharacteristicsofanevenfunction?3.Thealgebraicdefinitionofanevenfunctionis: �(@)isanevenfunctionifandonlyif�(@) = �(−@)forallvaluesof@inthedomainof�.Whataretheimplicationsofthedefinitionforthegraphofanevenfunction?4.Arealleven-degreepolynomialsevenfunctions?Useexamplestoexplainyouranswer.5.Let’strythesameapproachtofigureoutadefinitionforoddfunctions.

63




Oddfunction:H(F) = FL

Notanoddfunction:Q(F) = logN F

Differences:Oddfunction:

H(F) = −FT

Notanoddfunction:Q(F) = FL + 3F − 7

Differences:

Oddfunction:H(F) =]

J

Notanoddfunction:Q(F) = 2F − 3

Differences:Oddfunction:

H(2) = 3r[sH(−2) = −3Notanoddfunction:

Q(2) = 3r[sQ(−2) = 5Differences:

64




6.Whatdoyouobserveaboutthecharacteristicsofanoddfunction?7.Thealgebraicdefinitionofanoddfunctionis: �(@)isanoddfunctionifandonlyif�(−@) = −�(@)forallvaluesof@inthedomainof�.Explainhoweachoftheexamplesofoddfunctionsabovemeetthisdefinition.8.Howcanyoutellifanodd-degreepolynomialisanoddfunction?9.Areallfunctionseitheroddoreven?

65




3.9


READY Topic:Recognizingspecialproducts

Multiply.1.(" + 5)(" + 5) 2.(" − 3)(" − 3) 3.(( + ))(( + ))

4.Inproblems1–3theanswersarecalledperfectsquaretrinomials.Whatabouttheseanswersmakesthembeaperfectsquaretrinomial?

5.(" + 8)(" − 8) 6.+" + √3-+" − √3- 7.(" + ))(" − ))

8.Theproductsinproblems5–7endupbeingbinomials,andtheyarecalledthedifferenceoftwosquares.Whatabouttheseanswersmakesthembethedifferenceoftwosquares?

Whydon’ttheyhaveamiddletermliketheproblemsin1–3?9.(" − 3)(". + 3" + 9) 10.(" + 10)(". − 10" + 100) 11.(( + ))((. − () + ).)

12.Theworkinproblems9–11makesthemfeelliketheanswersaregoingtohavealotofterms.Whathappensintheworkoftheproblemthatmakestheanswersbebinomials?

Theseanswersarecalledthedifferenceoftwocubes(#9)andthesumoftwocubes(#10and

#11.)Whatabouttheseanswersmakesthembethesumordifferenceoftwocubes?


66




3.9


SET Topic:Determiningvaluesofpolynomialsatzeroandat±∞.(Endbehavior)

Statethey-intercept,thedegree,andtheendbehaviorforeachofthegivenpolynomials.13.5(") = "7 + 7"9 − 9": + ". − 13" + 8y-intercept:Degree:Endbehavior:As" → −∞, 5(") → __________As" → +∞, 5(") → __________

14.?(") = 3"9 + ": + 5". − " − 15y-intercept:Degree:Endbehavior:As" → −∞, ?(") → __________As" → +∞, ?(") → __________

15.ℎ(") = −7"A + ".y-intercept:Degree:Endbehavior:As" → −∞, ℎ(") → __________As" → +∞, ℎ(") → __________

16.B(") = 5". − 18" + 4y-intercept:Degree:Endbehavior:As" → −∞, B(") → __________As" → +∞, B(") → __________

17.D(") = ": − 94". − " − 20y-intercept:Degree:Endbehavior:As" → −∞, D(") → __________As" → +∞, D(") → __________

18.F = −4" + 12y-intercept:Degree:Endbehavior:As" → −∞, F → __________As" → +∞, F → __________

67




3.9


Topic:Identifyingevenandoddfunctions19.Identifyeachfunctionaseven,odd,orneither.a)5(") = ". − 3

b)5(") = ".

c)5(") = (" + 1).

d)5(") = ":

e)5(") = ": + 2

f)5(") = (" − 2):

GO Topic:Factoringspecialproducts

Fillintheblanksonthesentencesbelow.20.TheexpressionGH + HGI+ IHiscalledaperfectsquaretrinomial.Icanrecognizeitbecause

thefirstandlasttermswillalwaysbeperfect___________________________________________.

Themiddletermwillbe2timesthe______________________________and_______________________________.

Therewillalwaysbea__________________signbeforethelastterm.

Itfactorsas(__________________)(__________________).

68




3.9


21.TheexpressionGH − IHiscalledthedifferenceof2squares.Icanrecognizeitbecauseit’sa

binomialandthefirstandlasttermsareperfect_________________________________________________.

Thesignbetweenthefirsttermandthelasttermisalwaysa______________________________.

Itfactorsas(__________________)(__________________).

22.TheexpressionGJ + IJiscalledthesumof2cubes.Icanrecognizeitbecauseit’sabinomial

andthefirstandlasttermsare_____________________________________.Theexpression(: + ):factors

intoabinomialandatrinomial.Icanrememberitasashort(______)andalong(________________).

Thesignbetweenthetermsinthebinomialisthe_____________________asthesigninthe

expression.Thefirstsigninthetrinomialisthe_________________________ofthesigninthe

binomial.That’swhyallofthemiddletermscancelwhenmultiplying.

Thelastsigninthetrinomialisalways____________.

Itfactorsas(__________________)(___________________________________).

Factorusingwhatyouknowaboutspecialproducts.23.25". + 30 + 9

24.". − 16 25.": + 27

26.49". − 36 27.": − 1

28.64". − 240 + 225

69




3.10 Puzzling Over Polynomials A Practice Understanding Task

Foreachofthepolynomialpuzzlesbelow,afewpiecesofinformationhavebeengiven.Yourjobistousethosepiecesofinformationtocompletethepuzzle.Occasionally,youmayfindamissingpiecethatyoucanfillinyourself.Forinstance,althoughsomeoftherootsaregiven,youmaydecidethatthereareothersthatyoucanfillin.

1.

Function(infactoredform)Function(instandardform)Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):-2,1,and1Valueofleadingco-efficient:-2Degree:3

Graph:

CC

BY

Just

in T

aylo

r

http

s://f

lic.k

r/p/

4fU

zTo

70




2.Function(infactoredform)Function(instandardform)Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):

2 + /, 4, 0Valueofleadingco-efficient:1Degree:4

Graph:

3.

Function:)($) = 2($ − 1)($ + 3)5Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):Valueofleadingco-efficient:Domain:Range:AllRealnumbers

Graph:

71




4.

Function:Endbehavior:!"$ → −∞, )($) → ∞!"$ → ∞, )($) → _____Roots(withmultiplicity):(3,0)m:1;(-1,0)m:2(0,0)m:2Valueofleadingco-efficient:-1Domain:Range:

Graph:

5.

Function:Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):Valueofleadingco-efficient:1Domain:Range:Other:)(0)=16

Graph:

72




6.

Function(instandardform):)($) = $6 − 2$5 − 7$ + 2

Function(infactoredform):Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):-2Domain:Range:

Graph:

7.

Function(instandardform):)($) = $6 − 2$

Function(infactoredform):Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):Domain:Range:

Graph:

73




3.10


READY Topic:Reducingrationalnumbersandexpressions

Reducetheexpressionstolowestterms.(Assumenodenominatorequals0.)

1.!"#"$

2.&∙(∙"∙"∙"∙)!∙(∙"∙)∙)

3.*+,$

*+,$ 4.

(".&)("01)(".&)("01)

5.(!"0()(".2)("03)(!"0()

6.(&"033)(!".3*)(&"033)(!"0()

7.(4"0*)(".!)4"(".!)(&"0!)

8.!"(&".*)("03)(#"0()"(&".*)("03)(#"0()

9.Whyisitimportantthattheinstructionssaytoassumethatnodenominatorequals0?

SET Topic:Reviewingfeaturesofpolynomials

Someinformationhasbeengivenforeachpolynomial.Fillinthemissinginformation.10.Graph:Function:5(6) = 6!Functioninfactoredform:

Endbehavior:As6 → −∞, 5(6) → _______As6 → ∞, 5(6) → ______

Roots(withmultiplicity):

Degree:

Valueofleadingco-efficient:


74




3.10


11.Graph:Functioninstandardform:

Functioninfactoredform:=(6) = −6(6 − 2)(6 − 4)Endbehavior:As6 → −∞,=(6) → _______As6 → ∞,=(6) → ______Roots(withmultiplicity):Degree:Valueofleadingco-efficient:

12.Graph:Functioninstandardform:ℎ(6) = 6! − 26& − 36

Functioninfactoredform:

Endbehavior:As6 → −∞, ℎ(6) → _______As6 → ∞, ℎ(6) → ______Roots(withmultiplicity):Degree:ValueofB(C):

13.Graph:Functioninstandardform:Functioninfactoredform:

Endbehavior:As6 → −∞, 5(6) → _______As6 → ∞, 5(6) → ______


Degree:

y-intercept:

75




3.10


14.Graph:

Functioninstandardform:

Functioninfactoredform:Endbehavior:As6 → −∞, E(6) → _______As6 → ∞, E(6) → ______


Degree:

Valueofleadingcoefficient:

15.Graph:

Functioninstandardform:F(6) = 6! + 26& + 6 + 2

Functioninfactoredform:

Endbehavior:As6 → −∞, F(6) → _______As6 → ∞, F(6) → ______


6 = H

Degree:

y-intercept:

16.Finishthegraphifitis

anevenfunction.

17.Finishthegraphifit

isanoddfunction.

76




3.10


GO Topic:Writingpolynomialsgiventhezerosandtheleadingcoefficient

Writethepolynomialfunctioninstandardformgiventheleadingcoefficientandthezerosofthefunction.

18.Leadingcoefficient:2; JKKLM:2, √2,−√2

19.Leadingcoefficient:−1; JKKLM:1, 1 + √3, 1 − √3

20.Leadingcoefficient:2; JKKLM:4H, −4H

Fillintheblankstomakeatruestatement.21.If5(P) = 0,thenafactorof5(P)mustbe____________________________________.

22.Therateofchangeinalinearfunctionisalwaysa______________________________________.

23.Therateofchangeofaquadraticfunctionis_______________________________________________.

24.Therateofchangeofacubicfunctionis____________________________________________________.

25.Therateofchangeofapolynomialfunctionofdegreencanbedescribedbyafunctionofdegree

________________________.

77

Polynomial Functions - mathematicsvisionproject.org€¦ · SECONDARY MATH III // MODULE 3 POLYNOMIAL FUNCTIONS Mathematics Vision Project Licensed under the Creative Commons Attribution

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