CPAlgebra2

Unit3B:Polynomials

Name:_____________________________________Period:_____

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.

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!

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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.

There’sanothermethodofsolvingthatshouldbeprettyfamiliartoyoubynow!☺SolvingByFactoring(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 −

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#:_________

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.

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

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= = − = + .

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