In Class 3.4 Multiplicity Graphing Polynomials · calculator or graphing utility to graph the...
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In Class 3.4, Algebra II Name: Block: Unit 3, Multiplicity and Zeros of Polynomials © 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Warm Up: Solve each polynomial by factoring: 1. 9 ! − 3 ! − 3 + 1 = 0 2. ! − 2 ! − 24 ! = 0 3. 3 ! + 18 ! − 21 ! = 0 4. − ! + 2 ! + 8 ! = 0 Example: Suppose we know that the polynomial equation 4 ! − 12 ! + 3 + 5 = 0 has three real solutions and that one of the factors of 4 ! − 12 ! + 3 + 5 is ( − 1). How can we find all three solutions to the given equation?
Transcript of In Class 3.4 Multiplicity Graphing Polynomials · calculator or graphing utility to graph the...
InClass3.4,AlgebraII Name: Block:Unit3,MultiplicityandZerosofPolynomials
©2014CommonCore,Inc.Somerightsreserved.commoncore.orgThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-ShareAlike3.0UnportedLicense.
WarmUp:Solveeachpolynomial byfactoring:
1. 9𝑥! − 3𝑥! − 3𝑥 + 1 = 0 2.𝑥! − 2𝑥! − 24𝑥! = 0
3.3𝑥! + 18𝑥! − 21𝑥! = 0 4.−𝑥! + 2𝑥! + 8𝑥! = 0
Example:Supposeweknowthatthepolynomialequation4𝑥! − 12𝑥! + 3𝑥 + 5 = 0hasthreerealsolutionsandthatoneofthefactorsof4𝑥! − 12𝑥! + 3𝑥 + 5is(𝑥 − 1).Howcanwefindallthreesolutionstothegivenequation?
InClass3.4,AlgebraII Name: Block:Unit3,MultiplicityandZerosofPolynomials
©2014CommonCore,Inc.Somerightsreserved.commoncore.orgThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-ShareAlike3.0UnportedLicense.
Considerthepolynomialfunctions 𝑝 𝑥 = 𝑥 − 2 𝑥 + 3 !
𝑞 𝑥 = 𝑥 − 2 ! 𝑥 + 3 !
and𝑟 𝑥 = 𝑥 − 2 ! 𝑥 − 3 !
Whatdoyounoticeaboutthethezerosofthefunctionsandthedegreeofthefunction?
Ifwewriteoutallofthefactorsfor𝑝,𝑞,and𝑟, weseethat
𝑝 𝑥 = 𝑥 − 2 𝑥 + 3 𝑥 + 3 𝑞 𝑥 = 𝑥 − 2 𝑥 − 2 𝑥 + 3 (𝑥 + 3)(𝑥 + 3)(𝑥 + 3) 𝑟 𝑥 = (𝑥 − 2)(𝑥 − 2)(𝑥 − 2)(𝑥 − 2)(𝑥 + 3)(𝑥 + 3)(𝑥 + 3)(𝑥 + 3)(𝑥 + 3).
§ Wenoticethat 𝑥 − 2 isafactorof𝑝________,and(𝑥 + 3)isafactorof𝑝__________.Thus,
wesaythat2isazeroof𝑝ofmultiplicity1,and−3isazeroof𝑝ofmultiplicity2.Zerosof
multiplicity1areusuallyjustreferredtoaszeros,withoutmentioningthemultiplicity.
Multiplicityisthe____________________ofthenumberoftimesafactorappearsinafactored
polynomialexpression.
Multiplicityalsotellsusboutthebehaviorofthegrapharoundthex-intercepts.
Foroddmultiplicity: Forevenmultiplicity:
𝑦 = 𝑝(𝑥) 𝑦 = 𝑞(𝑥) 𝑦 = 𝑟(𝑥)
InClass3.4,AlgebraII Name: Block:Unit3,MultiplicityandZerosofPolynomials
©2014CommonCore,Inc.Somerightsreserved.commoncore.orgThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-ShareAlike3.0UnportedLicense.
1. Findtherealzerosofthefollowingpolynomialfunctions,withtheirmultiplicities.a. 𝑓 𝑥 = (𝑥 + 1)(𝑥 − 1)(𝑥! + 1) b.𝑔 𝑥 = 𝑥 − 4 ! 𝑥 − 2 !
2. Identifytherootsodeachequation.Statethemultiplicityofeachroot.
a.𝑥! + 3𝑥! + 3𝑥 + 1 = 0 b.𝑥! + 5𝑥! − 8𝑥 − 48 = 0
c.𝑥! + 10𝑥! + 17𝑥 = 28 d.3𝑥! + 10𝑥! − 27𝑥 = 10
InClass3.4,AlgebraII Name: Block:Unit3,MultiplicityandZerosofPolynomials
©2014CommonCore,Inc.Somerightsreserved.commoncore.orgThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-ShareAlike3.0UnportedLicense.
Wecanalwaysfindrelativemaximaorrelativeminimaofquadraticfunctions—theyoccuratthe𝑥-valueofthevertex.
§ Wecallthenumber𝑐arelativemaximum.Therelativemaximumvalue,𝑓(𝑐),maynotbethegreatestoverallvalueofthefunctionbutitisthehighestpointinaparticularsectionofagraph.
§ Wecallthenumber𝑑arelativeminimum.Therelativemaximumvalue,𝑓(𝑑),maynotbethelowestoverallvalueofthefunctionbutitisthelowestpointinaparticularsectionofagraph.
§ Therearevaluesofthefunctionthatarelargerthan𝑓(𝑐),suchas𝑓(9),but𝑓(𝑐)isthehighestvalueamongthe“neighbors”of𝑐.
Relativemaximumorminimumactuallyreferstothe𝒙-value,nottheorderedpaironthegraphofthefunctionorthevalueofthefunctionattherelativemaximumorminimum.The𝒚-valueiscalledtherelativemaximum(orminimum)valueofthefunction.
1. Foreachfunctionbelow,identifythelargestpossiblenumberof𝑥-interceptsandthelargestpossiblenumberofrelativemaximumandminimumpointsbasedonthedegreeofthepolynomial.Thenuseacalculatororgraphingutilitytographthefunctionandfindtheactualnumberof𝑥-interceptsand
InClass3.4,AlgebraII Name: Block:Unit3,MultiplicityandZerosofPolynomials
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relativemaximum/minimumpoints.
a) 𝒇 𝒙 = 𝟒𝒙𝟑 − 𝟐𝒙 + 𝟏b) 𝒈 𝒙 = 𝒙𝟕 − 𝟒𝒙𝟓 − 𝒙𝟑 + 𝟒𝒙c) 𝒉 𝒙 = 𝒙𝟒 + 𝟒𝒙𝟑 + 𝟐𝒙𝟐 − 𝟒𝒙 + 𝟐
FunctionLargestnumberof𝑥-
interceptsLargestnumberofrelativemax/mins
Actualnumberof𝑥-intercepts
Actualnumberofrelativemax/mins
a) 𝑓(𝑥)
b) 𝑔(𝑥)
c) ℎ 𝑥
2. Considerthefunction𝒇 𝒙 = 𝒙𝟑 − 𝟏𝟑𝒙𝟐 + 𝟒𝟒𝒙 − 𝟑𝟐.
a. Usethefactthat𝑥 − 4isafactorof𝑓tofactorthispolynomial.
b. Findthe𝑥-interceptsforthegraphof𝑓.Findthedegreeandleadcoefficient.
c. Tosketchagraphof𝑓,weneedtoconsiderwhetherthefunctionispositiveornegativeonthefourintervals𝑥 < 1,1 < 𝑥 < 4,4 < 𝑥 < 8,and𝑥 > 8.Whyisthat?
d. For𝑥 < 1,isthegraphaboveorbelowthe𝑥-axis?Howcanyoutell?
e. For1 < 𝑥 < 4,isthegraphaboveorbelowthe𝑥-axis?Howcanyoutell?
f. For4 < 𝑥 < 8,isthegraphaboveorbelowthe𝑥-axis?Howcanyoutell?
g. For𝑥 > 8,isthegraphaboveorbelowthe𝑥-axis?Howcanyoutell?
h. Graph𝑦 = 𝑓(𝑥)ontheintervalfrom[0,9]usingagraphingutility,findthemaximaandminimaandgraphthefunctionsingthisinformation.Compareyoursketchwiththegraphgeneratedbythegraphingutility.
