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The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius
© 2018 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Office of Education
This work is licensed under the Creative Commons Attribution CC BY 4.0
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
POLYNOMIAL FUNCTIONS
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
SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS– 3.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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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SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS– 3.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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 … … ?
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SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS– 3.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5. Withoutfindingtheexplicitequation,makeaconjectureastothetypeoffunctionthatwould
representtheexplicitequationforthetotalnumberofpush-upsScottwouldcompleteonany
givendayforthisyear’spromotion.
6. Howdoestherateofchangeforthisexplicitequationcomparetotheratesofchangeforthe
explicitequationsinquestions1and2?
7. Testyourconjecturefromquestion5andjustifythatitwillalwaysbetrue(seeifyoucanmove
toageneralizationforallpolynomialfunctions).
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POLYNOMIAL FUNCTIONS – 3.1
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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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POLYNOMIAL FUNCTIONS – 3.1
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GO Topic:Recallinglongdivisionandthemeaningofafactor
Findthequotientwithoutusingacalculator.Ifyouhavearemainder,writetheremainderas
awholenumber.Example: remainder217.
18.
19.Is30afactorof510?Howdoyouknow?
20.Is13afactorof8359?Howdoyouknow?
21.
22.
23.Is22afactorof14587?Howdoyouknow?
24.Is952afactorof40936?Howdoyouknow?
25.
26.
27.Is92afactorof3405? 28.Is27afactorof3564?
21 1497
30 510 13 8359
22 14857 952 40936
92 3405 27 3564
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SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.2
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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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SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.2
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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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SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.2
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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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POLYNOMIAL FUNCTIONS – 3.2
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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(#).
READY, SET, GO! Name PeriodDate
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POLYNOMIAL FUNCTIONS – 3.2
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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
10
SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.2
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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
11
SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.2
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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
12
SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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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POLYNOMIAL FUNCTIONS – 3.3
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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)
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SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.3
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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
SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.3
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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)
16
SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.3
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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)
READY, SET, GO! Name PeriodDate
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POLYNOMIAL FUNCTIONS – 3.3
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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
SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.4
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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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SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.4
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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
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SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.4
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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
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SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.4
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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.
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POLYNOMIAL FUNCTIONS – 3.4
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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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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( )
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(# + ,)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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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
SECONDARY MATH III // MODULE 3
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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
+ + + +
- +
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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
+ ++ + + +
+ - -
+ +
- +
+- +
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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
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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
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c)(6:G − 11:; − 4: + 5) ÷ (2: − 1) d)(:R − 23:G + 49: + 4) ÷ (:; + : + 2)
Multiplicationstatements:
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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)
Multiplicationstatement(s)
x +1( ) x3 − 3x2 + 6x +11 x − 5( ) x3 − 9x2 + 23x −15
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12. 13.
Multiplicationstatement(s)
Multiplicationstatement(s)
14. 15.
Multiplicationstatement(s)
Multiplicationstatement(s)
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
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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:
increasing:decreasing:
y-intercept:x-intercept(s):
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19.3(") = 89:." − 1domain:range:
increasing:decreasing:
y-intercept:x-intercept(s):
Identifythekeyfeaturesofthegraphedfunctions.
20.
domain:range:
increasing:decreasing:
y-intercept:x-intercept(s):
21.
domain:range:
increasing:decreasing:
y-intercept:x-intercept(s):
36
SECONDARY MATH III // MODULE 3
POLYNOMIAL FUNCTIONS – 3.6
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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.
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6.Thesumofalinearfunctionandanexponentialfunctionisapolynomial.7.Thesetofpolynomialsisclosedunderaddition.8.Thesetofwholenumbersisclosedundersubtraction.9.Thesetofintegersisclosedundersubtraction.10.Aquadraticfunctionsubtractedfromacubicfunctionisacubicfunction.11.Alinearfunctionsubtractedfromalinearfunctionisapolynomialfunction.12.Acubicfunctionsubtractedfromacubicfunctionisacubicfunction.13.Thesetofpolynomialfunctionsisclosedundersubtraction.
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14.Theproductoftwolinearfunctionsisaquadraticfunction.15.Thesetofintegersisclosedundermultiplication.16.Thesetofpolynomialsisclosedundermultiplication.17.Thesetofintegersisclosedunderdivision.18.Acubicfunctiondividedbyalinearfunctionisaquadraticfunction.19.Thesetofpolynomialfunctionsisclosedunderdivision.20.Writetwoclaimsofyourownaboutpolynomialsanduseexamplestodemonstratethattheyaretrue.Claim#1:Claim#2:
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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
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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.
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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
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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?
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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.
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Here’sanotherexampleofacubicfunction.
%(') = '4 + 7'* + 8' − 16
10.Usethegraphtofindtherootsofthecubicfunction.
11.Write%(')infactoredform.Verifythatthefactoredformisequivalenttothestandardform.Makeanycorrectionsneeded.
12.Aretheresultsyoufoundin#10consistentwiththeFundamentalTheoremofAlgebra?
Explain.
13.We’veseenthemostbasiccubicpolynomialfunction,ℎ(') = '4andweknowitsgraphlookslikethis:
Explainhowℎ(') = '4isconsistentwiththeFundamentalTheoremofAlgebra.
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14.Hereisonemorecubicpolynomialfunctionforyourconsideration.Youwillnoticethatitis
giventoyouinfactoredform.Usetheequationandthegraphtofindtherootsof;(').
;(') = (' + 3)('* + 4)
15.Usetheequationtoverifyeachroot.Showyourworkbelow.
16.Aretheresultsyoufoundin#14consistentwiththeFundamentalTheoremofAlgebra?
Explain.
17.Explainhowtofindthefactoredformofapolynomial,giventheroots.
18.Explainhowtofindtherootsofapolynomial,giventhefactoredform.
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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
READY, SET, GO! Name PeriodDate
x −5( ) 3x3 −15x2 +12x −60
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Belowarethegraphsofthepolynomialsfromthepreviouspage.Checkyourpredictions.Thenusethegraphtohelpyouwritethepolynomialinfactoredform.11.!(#) = #& + 3# − 10Factoredform:
12.,(#) = #- + #& − 9# − 9Factoredform:
13./(#) = −2# − 4Factoredform:
14.2(#) = #3 − #- − 4#& + 4#Factoredform:
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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
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POLYNOMIAL FUNCTIONS – 3.8
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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:
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8.Function:5(#) = #6 − #( + 4# − 4 Factor:(# − 2G) Rootsoffunction:Factoredform:9.Isitpossibleforapolynomialwithrealcoefficientstohaveonlyoneimaginaryroot?Explain.10.BasedontheFundamentalTheoremofAlgebraandthepolynomialsthatyouhaveseen,makeatablethatshowsallthenumberofrootsandthepossiblecombinationsofrealandimaginaryrootsforlinear,quadratic,cubic,andquarticpolynomials.
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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)
READY, SET, GO! Name PeriodDate
D(5)
((5)
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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
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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
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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?
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@ = B @ → ∞ @ → −∞
E(F) = FG
K(F) = FG m = FN
m = FL
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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:
Endbehavior:AsF → −∞, H(F) → ______
Endbehavior:AsF → ∞, H(F) → ______
3.H(F) = 3J
Functiontype:
Endbehavior:AsF →−∞, H(F) → ______
Endbehavior:AsF → ∞, H(F) → ______
4.H(F) = FL + 2FN − F + 5
Functiontype:
Endbehavior:AsF → −∞, H(F) → ______
Endbehavior:AsF → ∞, H(F) → ______
5.H(F) = −2FL + 2FN − F + 5
Functiontype:
Endbehavior:AsF →−∞, H(F) → ______
Endbehavior:AsF → ∞, H(F) → ______
6.H(F) = EvQNF
Functiontype:
Endbehavior:AsF →−∞, H(F) → ______
Endbehavior:AsF → ∞, H(F) → ______
Usethegraphsbelowtodescribetheendbehaviorofeachfunctionbycompletingthestatements.7. 8.
Endbehavior:AsF →−∞, H(F) → ______
Endbehavior:AsF → ∞, H(F) → ______
Endbehavior:AsF →−∞, H(F) → ______
Endbehavior:AsF → ∞, H(F) → ______
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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:
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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.
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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:
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6.Whatdoyouobserveaboutthecharacteristicsofanoddfunction?7.Thealgebraicdefinitionofanoddfunctionis: �(@)isanoddfunctionifandonlyif�(−@) = −�(@)forallvaluesof@inthedomainof�.Explainhoweachoftheexamplesofoddfunctionsabovemeetthisdefinition.8.Howcanyoutellifanodd-degreepolynomialisanoddfunction?9.Areallfunctionseitheroddoreven?
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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?
READY, SET, GO! Name PeriodDate
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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 → __________
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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(__________________)(__________________).
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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
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3.10 Puzzling Over Polynomials A Practice Understanding Task
Foreachofthepolynomialpuzzlesbelow,afewpiecesofinformationhavebeengiven.Yourjobistousethosepiecesofinformationtocompletethepuzzle.Occasionally,youmayfindamissingpiecethatyoucanfillinyourself.Forinstance,althoughsomeoftherootsaregiven,youmaydecidethatthereareothersthatyoucanfillin.
1.
Function(infactoredform)Function(instandardform)Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):-2,1,and1Valueofleadingco-efficient:-2Degree:3
Graph:
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2.Function(infactoredform)Function(instandardform)Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):
2 + /, 4, 0Valueofleadingco-efficient:1Degree:4
Graph:
3.
Function:)($) = 2($ − 1)($ + 3)5Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):Valueofleadingco-efficient:Domain:Range:AllRealnumbers
Graph:
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4.
Function:Endbehavior:!"$ → −∞, )($) → ∞!"$ → ∞, )($) → _____Roots(withmultiplicity):(3,0)m:1;(-1,0)m:2(0,0)m:2Valueofleadingco-efficient:-1Domain:Range:
Graph:
5.
Function:Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):Valueofleadingco-efficient:1Domain:Range:Other:)(0)=16
Graph:
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6.
Function(instandardform):)($) = $6 − 2$5 − 7$ + 2
Function(infactoredform):Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):-2Domain:Range:
Graph:
7.
Function(instandardform):)($) = $6 − 2$
Function(infactoredform):Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):Domain:Range:
Graph:
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READY Topic:Reducingrationalnumbersandexpressions
Reducetheexpressionstolowestterms.(Assumenodenominatorequals0.)
1.!"#"$
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:
READY, SET, GO! Name PeriodDate
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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) → ______
Roots(withmultiplicity):
Degree:
y-intercept:
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14.Graph:
Functioninstandardform:
Functioninfactoredform:Endbehavior:As6 → −∞, E(6) → _______As6 → ∞, E(6) → ______
Roots(withmultiplicity):
Degree:
Valueofleadingcoefficient:
15.Graph:
Functioninstandardform:F(6) = 6! + 26& + 6 + 2
Functioninfactoredform:
Endbehavior:As6 → −∞, F(6) → _______As6 → ∞, F(6) → ______
Roots(withmultiplicity):
6 = H
Degree:
y-intercept:
16.Finishthegraphifitis
anevenfunction.
17.Finishthegraphifit
isanoddfunction.
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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
________________________.
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