cp algebra 2 unit 3b: polynomials - state college area
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CPAlgebra2
Unit3B:Polynomials
Name:_____________________________________Period:_____
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LearningTargets
Solving Polynomials
10. I can use the fundamental theorem of algebra to find the expected number of roots.
11. I can solve polynomials by graphing (with a calculator).
12. I can solve polynomials by factoring.
Finding and Using Roots
13. I can find all of the roots of a polynomial.
14. I can write a polynomial function from its complex roots.
Graphing 15. I can graph polynomials.
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SolvingPolynomials Afterthislessonandpractice,Iwillbeableto…
¨ usethefundamentaltheoremofalgebratofindtheexpectednumberofroots.(LT10)
¨ solvepolynomialsbygraphing(withacalculator).(LT11)
¨ solvepolynomialsbyfactoring.(LT12)--------------------------------------------------------------------------------------------------------------------------------------------------Inthequadraticsunit,youlearnedfivestrategiesforsolvingquadraticequations.Let’sseehowmanyyoucanremember!1) 4)
2) 5)
3)
Todaywe’regoingtosolvepolynomials,whichwillseemverysimilartosolvingquadratics.There’sonethingweshouldlearnfirstthatwillhelpusaswesolve…FindtheExpectedNumberofRoots(LT10)Lookbackatthechartyoufilledoutatthebeginningofthisunit.Howdoesthedegreeofthepolynomialrelatetothenumberofx-intercepts?Thenumberof___________toapolynomialfunctionisequaltothe___________ofthepolynomial.Thisobservationisaveryimportantfactinalgebra…(Corollaryto)TheFundamentalTheoremofAlgebra–Everypolynomialinonevariableofdegree hasexactly______zeros,including_____________and____________zeros.Thistheoremmakesitpossibletoknowthenumberandtypeofzerosinagivenfunction,whichcanbehelpfulinfindingallzerosofapolynomial.Example1:Determinethenumberofzerosofthepolynomial.a.!!f (x)= x
3 −2x2 +4x −8 b.𝑦 = 15𝑥!" + 3𝑥! − 9Youcanalwaysusethattricktofigureouthowmanyzerosyoushouldexpectfromapolynomial.Nowlet’ssolve!
0n >
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Todaywe’llbesolvingbyfactoringandgraphing.Let’sstartwithgraphs,sinceit’sbasicallythesameprocessaswhenwesolvedquadraticsbygraphing.SolvingbyGraphing(LT11)Ourgraphingcalculatorswillhelpusfindzerosofapolynomialfunction.Let’suse𝑦 = 𝑥! + 12𝑥! + 𝑥 − 1
1)EntertheequationinyourcalculatorasY1=.PressGRAPH.2)Tomakesurewecanseethegraph,clickZOOMandZStandardorZoomFit
Youshouldseeaskinnyparabolathatlookslikeithastwozeros.Butlet’suseourFundamentalTheoremofAlgebratricktomakesurethereareonlytwozeros…BasedontheFundamentalTheoremofAlgebra,howmanyzerosshouldthispolynomialhave?_________Let’seditthewindowuntilwecanseeall____zeros.Thencontinueonwithstep3.
3)Press2ndTRACE,thenpress2:ZERO.
4)Moveyourcursorjusttothe“left”ofthefirstpointofintersection.PressENTER.5)Moveyourcursorjusttothe“right”ofthefirstpointofintersection.PressENTER.6)Thescreenwillshow“Guess”.PressENTERagain.Thecalculatorwilldisplaythezero.
7)Repeatsteps3-6toobtaintherestofthezeros.
Example2:Findtheexpectednumberofzeros,thenuseyourgraphingcalculatortofindthezerosofthefunction𝑦 = 𝑥! + 12𝑥! + 𝑥 − 1(Hint:Youmayneedtozoomout!)Example3:Findtheexpectednumberofzeros,thenuseyourgraphingcalculatortofindthezerosofthefunction𝑦 = 2𝑥! + 𝑥 − 7.Example4:Findtheexpectednumberofzeros,thenuseyourgraphingcalculatortofindthezerosofthefunction𝑦 = 𝑥! + 2𝑥! − 6𝑥! − 2.
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There’sanothermethodofsolvingthatshouldbeprettyfamiliartoyoubynow!JSolvingByFactoring(LT12)Recallourpreviousstrategiesforfactoringquadratics:1) 4)
2) 5)
3)
Let’saddtwoadditionalfactoringstrategiesbeforelearninghowtousethemtosolvepolynomialequations.PolynomialFactoringStrategy#1:SumsandDifferencesofCubesJustastherearepatternsforthedifferenceoftwo____________,therealsoexistpatternsforthesumsanddifferencesoftwocubes!SumofTwoCubesDifferenceofTwoCubesExample5:Factorcompletely.a. 3 64x + b. 5 216 250z z− Example6:Factorcompletely.a. 3 8x + b. 38 1x − c. 3512 216m −
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PolynomialFactoringStrategy#2:QuadraticFormsSomepolynomialsofhigher-degreecanbesolvedusingstrategiesyouusedwhenyoufactoredquadratics.Thekeyisinrecognizingifthepolynomialisin“quadraticform.”Example7:Factorcompletely.
a. !!x4 −2x2 −8 b. !!x4 +7x2 +6 Example8:Factorcompletely.
a. !!x4 − x2 −2 b. !!x4 +8x2 −9 Nowthatyouhavesomeadditionalfactoringstrategies,let’sutilizethesestrategiestosolvepolynomialequationsbyfactoring!Example9:Findtheexpectednumberofzeros,thensolveeachequationbyfactoring. a. !!27x3 +1=0 b. !!x4 − x2 =12 c.!!3x3 +2x2 −15x −10=0 Expected#:_________ Expected#:_________Expected#:_________
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FindingandUsingRoots Afterthislessonandpractice,Iwillbeableto…
¨ findalloftherootsofapolynomial.(LT13)
¨ writeapolynomialfunctionfromitscomplexroots.(LT14)--------------------------------------------------------------------------------------------------------------------------------------------------Todaywe’regoingtolearnafewothertechniquesforfindingrootsandusingthemtowriteequationsinfactoredform.IdentifyRoots(LT13)Youcanidentifyanyrationalrootsbygraphingapolynomialinyourcalculatorandusingthezerofunctiontofindaroot.Onceyouhavearoot,youcanusesyntheticdivisiontogetthepolynomialdowntoaquadratic.Seetheboxbelowforthesteps.Example3:Findallrootsofeachfunctionandwriteeachfunctioninfactoredformwithintegercoefficients.a. !!f (x)= x
3 −7x2 +2x +40 b. !!f (x)=2x
3 −5x2 −14x +8 Example4:Findallrootsofthefunction!!f (x)=2x
3 +3x2 −8x +3 andwriteitinfactoredformwithintegercoefficients.
Strategies for Finding All Roots of a Polynomial 1) List all possible rational roots. 2) Use your calculator to verify one rational root. 3) Use synthetic division until the expression is quadratic and then use other algebraic techniques to find the remaining zeros.
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Unfortunately,asyouhaveobserved,notallpolynomialshaveexclusively___________roots.Nevertheless,youcanuserationalrootstohelpyoufindallzerosofapolynomial.Example5:Findallrootsofeachfunctionandwriteeachfunctioninfactoredform.a. !!f (x)= x
4 −5x3 −11x2 +25x +30 b. !!f (x)=3x
3 + x2 − x +1 Theresultstotheseexamplesleadustotwoadditionalpolynomialtheorems:IrrationalRootTheorem–If___________isarootofapolynomialequationwithrationalcoefficients,thenthe_______________________isalsoarootoftheequation.ImaginaryRootTheorem–If___________isarootofapolynomialequationwithrealcoefficients,thenthe_______________________isalsoarootoftheequation.NAMETHATCONJUGATE!
1. !3− 7 2. !!1+2i 3. !!−12−5i 4. !− 15 5. !πi
Example6:Supposeapolynomialwithrationalcoefficientshasthefollowingroots:!5+ 10 and!−4− 2 .Findtwoadditionalroots.Example7:Aquarticpolynomialwithrealcoefficientshasrootsof-3and!!2−5i .Whichofthefollowingcannotbeanotherrootofthepolynomial?
A. 12 B. 0 C. ! 2 D. !!2+5i
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Example8:Findallrootsofthefunction!!f (x)= x3 −2x2 −3x +10 andwriteitinfactoredform.
WritePolynomialsFromComplexRoots(LT14)Nowwe’llexplorehowtowritepolynomialequationsusinginformationaboutitsroots.Example9:Findapolynomialfunctioninstandardformwhosegraphhasx-intercepts3,5,-4,andy-intercept180.Recall from the previous lesson, thatwhen polynomials have _____________ or ___________ zeros, theyalwaysappearas__________________________.Example10:Writeapolynomialfunctioninstandardformwithrealcoefficientsandzeros
2, 5, 3 4x x x i= = − = + .
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GraphingPolynomials Afterthislessonandpractice,Iwillbeableto…
¨ graphpolynomials.(LT15)--------------------------------------------------------------------------------------------------------------------------------------------------Let’scombineeverythingwe’velearnedtographsomepolynomials!Example1:Findallzerosof!!f (x)= x
3 +3x2 − x −3 .Thencompletetherequestedinformation:Zeros/Roots:____________ FactoredForm:!!f (x)= _______________________________f()=_____,f()=_____,f()=_____because_____________________________________
y-intercept = ( , ) x-intercept(s) = ( , ) ( , ) ( , ) End behavior:
as
as