Chapter2-FactorTheoremand

Inequalities

LessonPackage

MHF4U

Chapter2OutlineUnitGoal:Bytheendofthisunit,youwillbeabletofactorandsolvepolynomialsuptodegree4usingthefactortheorem,longdivision,andsyntheticdivision.Youwillalsolearnhowtosolvefactorablepolynomialinequalities.

Section Subject LearningGoals CurriculumExpectations

L1 LongDivision -dividepolynomialexpressionsusinglongdivision-understandtheremaindertheorem C3.1

L2 SyntheticDivision -dividepolynomialexpressionsusingsyntheticdivision 3.1

L3 FactorTheorem-beabletodeterminefactorsofpolynomialexpressionsbytestingvalues

C3.2

L4 SolvingPolynomialEquations

-solvepolynomialequationsuptodegree4byfactoring-makeconnectionsbetweensolutionsandx-interceptsofthegraph

C3.2,C3.3,C3.4,C3.7

L5 FamiliesofPolynomialFunctions

-determinetheequationofafamilyofpolynomialfunctionsgiventhex-intercepts

C1.8

L5 SolvingPolynomialInequalities

-Solvefactorablepolynomialinequalities C4.1,C4.2,C4.3

Assessments F/A/O MinistryCode P/O/C KTACNoteCompletion A P PracticeWorksheetCompletion F/A P

Quiz–FactorTheorem F P PreTestReview F/A P Test–FactorTheoremandInequalities O

C1.8C3.1,3.2,3.3,3.3,3.7

C4.1,4.2,4.3P K(21%),T(34%),A(10%),

C(34%)

L1–2.1–LongDivisionofPolynomialsandTheRemainderTheoremLessonMHF4UJensenInthissectionyouwillapplythemethodoflongdivisiontodivideapolynomialbyabinomial.Youwillalsolearntousetheremaindertheoremtodeterminetheremainderofadivisionwithoutdividing.Part1:DoYouRememberLongDivision(divide,multiply,subtract,repeat)?107 ÷ 4canbecompletedusinglongdivisionasfollows:Everydivisionstatementthatinvolvesnumberscanberewrittenusingmultiplicationandaddition.Wecanexpresstheresultsofourexampleintwodifferentways:

OR

Example1:Uselongdivisiontocalculate753 ÷ 22

4doesnotgointo1,sowestartbydetermininghowmanytimes4goesinto10.Itgoesin2times.Soweputa2inthequotientabovethe10.Thisisthedivisionstep.

Then,multiplythe2by4(thedivisor)andputtheproductbelowthe10inthedividend.Thisisthemultiplystep.

Now,subtract8fromthe10inthedividend.Thenbringdownthenextdigitinthedividendandputitbesidethedifferenceyoucalculated.Thisisthesubtractstep.

Youthenrepeatthesestepsuntiltherearenomoredigitsinthedividendtobringdown.

Part2:UsingLongDivisiontoDivideaPolynomialbyaBinomialThequotientof 3𝑥* − 5𝑥, − 7𝑥 − 1 ÷ 𝑥 − 3 canbefoundusinglongdivisionaswell…Theresultinquotientformis:Theexpressionthatcanbeusedtocheckthedivisionis:

Focusonlyonthefirsttermsofthedividendandthedivisor.Findthequotientoftheseterms.

Since3𝑥* ÷ 𝑥 = 3𝑥,,thisbecomesthefirsttermofthequotient.Place3𝑥,abovethetermofthedividendwiththesamedegree.

Multiply3𝑥,bythedivisor,andwritetheanswerbelowthedividend.Makesuretolineup‘liketerms’.3𝑥,(𝑥 − 3) = 3𝑥* − 9𝑥,.Subtractthisproductfromthedividendandthenbringdownthenextterminthedividend.

Now,onceagain,findthequotientofthefirsttermsofthedivisorandthenewexpressionyouhaveinthedividend.Since4𝑥, ÷ 𝑥 = 4𝑥,thisbecomesthenextterminthequotient.

Multiply4𝑥bythedivisor,andwritetheanswerbelowthelastlineinthedividend.Makesuretolineup‘liketerms’.4𝑥(𝑥 − 3) = 4𝑥, − 12𝑥 .Subtractthisproductfromthedividendandthenbringdownthenextterm.

Now,findthequotientofthefirsttermsofthedivisorandthenewexpressioninthedividend.Since5𝑥 ÷ 𝑥 = 5,thisbecomesthenextterminthequotient.Multiply5(𝑥 − 3) = 5𝑥 − 15.Subtractthisproductfromthedividend.

Theprocessisstoppedoncethedegreeoftheremainderislessthanthedegreeofthedivisor.Thedivisorisdegree1andtheremainderisnowdegree0,sowestop.

Theresultofthedivisionof𝑃(𝑥)byabinomialoftheform𝑥 − 𝑏is:

𝑃(𝑥)𝑥 − 𝑏 = 𝑄(𝑥) +

𝑅𝑥 − 𝑏

Where𝑅istheremainder.Thestatementthatcanbeusedtocheckthedivisionis:

𝑃(𝑥) = (𝑥 − 𝑏)𝑄(𝑥) + 𝑅

Note:youcouldcheckthisanswerbyFOILingtheproductandcollectingliketerms.

Example2:Findthefollowingquotientsusinglongdivision.Expresstheresultinquotientform.Also,writethestatementthatcanbeusedtocheckthedivision(thencheckit!).a)𝑥, + 5𝑥 + 7dividedby𝑥 + 2b)2𝑥* − 3𝑥, + 8𝑥 − 12dividedby𝑥 − 1c)4𝑥* + 9𝑥 − 12dividedby2𝑥 + 1

Theresultinquotientformis:Theexpressionthatcanbeusedtocheckthedivisionis:

Theresultinquotientformis:Theexpressionthatcanbeusedtocheckthedivisionis:

Theresultinquotientformis:Theexpressionthatcanbeusedtocheckthedivisionis:

Example3:Thevolume,incubiccm,ofarectangularboxisgivenby𝑉 𝑥 = 𝑥* + 7𝑥, + 14𝑥 + 8.Determineexpressionsforpossibledimensionsoftheboxiftheheightisgivenby𝑥 + 2.ExpressionsforthepossibledimensionsoftheboxarePart3:RemainderTheoremWhenapolynomialfunction𝑃(𝑥)isdividedby𝑥 − 𝑏,theremainderis𝑃(𝑏);andwhenitisdividedby𝑎𝑥 − 𝑏,theremainderis𝑃 9

:,where𝑎and𝑏areintegers,and𝑎 ≠ 0.

Example3:Applytheremaindertheorema)Usetheremaindertheoremtodeterminetheremainderwhen𝑃 𝑥 = 2𝑥* + 𝑥, − 3𝑥 − 6isdividedby𝑥 + 1

Dividingthevolumebytheheightwillgiveanexpressionfortheareaofthebaseofthebox.

Factortheareaofthebasetogetpossibledimensionsforthelengthandwidthofthebox.

b)VerifyyouranswerusinglongdivisionExample4:Usetheremaindertheoremtodeterminetheremainderwhen𝑃 𝑥 = 2𝑥* + 𝑥, − 3𝑥 − 6isdividedby2𝑥 − 3Example5:Determinethevalueof𝑘suchthatwhen3𝑥> + 𝑘𝑥* − 7𝑥 − 10isdividedby𝑥 − 2,theremainderis8.

L2–2.1–SyntheticDivisionLessonMHF4UJensenInthissectionyouwilllearnhowtousesyntheticdivisionasanalternatemethodtodividingapolynomialbyabinomial.Syntheticdivisionisanefficientwaytodivideapolynomialbyabinomialoftheform𝑥 − 𝑏.

Part1:Syntheticdivisionwhenthebinomialisoftheform𝒙 − 𝒃Divide3𝑥' − 5𝑥) − 7𝑥 − 1by𝑥 − 3.Inthisquestion,𝑏 = 3.Don’tforgetthattheanswercanbewrittenintwoways…3𝑥' − 5𝑥) − 7𝑥 − 1

𝑥 − 3 =OR3𝑥' − 5𝑥) − 7𝑥 − 1 =

Listthecoefficientsofthedividendinthefirstrow.Totheleft,writethe𝑏value(thezeroofthedivisor).Placea+signabovethehorizontallinetorepresentadditionanda×signbelowthehorizontallinetoindicatemultiplicationofthedivisorandthetermsofthequotient.

Bringthefirsttermdown,thisisthecoefficientofthefirsttermofthequotient.Multiplyitbythe𝑏valueandwritethisproductbelowthesecondtermofthedividend.

Nowaddthetermstogether.

Multiplythissumbythe𝑏valueandwritetheproductbelowthethirdtermofthedividend.Repeatthisprocessuntilyouhaveacompletedthechart.

Thelastnumberbelowthechartistheremainder.Thefirstnumbersarethecoefficientsofthequotient,startingwithdegreethatisonelessthanthedividend.

IMPORTANT:WhenusingPolynomialORSyntheticdivision…

• Termsmustbearrangedindescendingorderofdegree,inboththedivisorandthedividend.• Zeromustbeusedasthecoefficientofanymissingpowersofthevariableinboththedivisor

andthedividend.

Example1:Usesyntheticdivisiontodivide.Thenwritethemultiplicationstatementthatcouldbeusedtocheckthedivision.a)(𝑥1 − 2𝑥' + 13𝑥 − 6)÷(𝑥 + 2)

b)(2𝑥' − 5𝑥) + 8𝑥 + 4) ÷ (𝑥 − 3)

=

Note:sincetheremainderiszero,boththequotientanddivisorare________________ofthedividend.

Part2:Syntheticdivisionwhenthebinomialisoftheform𝒂𝒙 − 𝒃Divide6𝑥' + 5𝑥) − 16𝑥 − 15by2𝑥 + 32𝑥 + 3 = 2 𝑥 + '

)à𝑏 = − '

)

Checkanswerusinglongdivision

Tousesyntheticdivision,thedivisormustbeintheform𝑥 − 𝑏.Re-writethedivisorbyfactoringoutthecoefficientofthe𝑥.

Wecannowdivide6𝑥' + 5𝑥) − 16𝑥 − 15by9𝑥 + '):using

syntheticdivisionaslongasyouremembertodividethequotientby2after.

Note:Syntheticdivisioncanonlybeusedwithalineardivisor.Itismostusefulwithadivisoroftheform𝑥 − 𝑏.Ifthedivisoris𝑎𝑥 − 𝑏,itcanbeusedbutlongdivisionmaybeeasier.

Example2:Findeachquotientbychoosinganappropriatestrategy.a)Divide𝑥' − 4𝑥) + 2𝑥 + 3by𝑥 − 3b)Divide12𝑥1 − 56𝑥' + 59𝑥) + 9𝑥 − 18by2𝑥 + 1

c)Divide𝑥1 − 2𝑥' + 5𝑥 + 3by𝑥) + 2𝑥 + 1d)Divide𝑥1 − 𝑥' − 𝑥) + 2𝑥 + 1by𝑥) + 2

L3–2.2–FactorTheoremLessonMHF4UJensenInthissection,youwilllearnhowtodeterminethefactorsofapolynomialfunctionofdegree3orgreater.Part1:RemainderTheoremRefreshera)Usetheremaindertheoremtodeterminetheremainderwhen𝑓 𝑥 = 𝑥$ + 4𝑥' + 𝑥 − 6isdividedby𝑥 + 2b)Verifyyouranswertoparta)bycompletingthedivisionusinglongdivisionorsyntheticdivision.FactorTheorem:𝑥 − 𝑏isafactorofapolynomial𝑃(𝑥)ifandonlyif𝑃 𝑏 = 0.Similarly,𝑎𝑥 − 𝑏isafactorof𝑃(𝑥)ifandonlyif𝑃 1

2= 0.

RemainderTheorem:Whenapolynomialfunction𝑃(𝑥)isdividedby𝑥 − 𝑏,theremainderis𝑃(𝑏);andwhenitisdividedby𝑎𝑥 − 𝑏,theremainderis𝑃 31

24,

where𝑎and𝑏areintegers,and𝑎 ≠ 0.

Note:Ichosesyntheticsinceitisalineardivisoroftheform𝑥 − 𝑏.

Example1:Determineif𝑥 − 3and𝑥 + 2arefactorsof𝑃 𝑥 = 𝑥$ − 𝑥' − 14𝑥 + 24𝑃 3 =Sincetheremainderis___,𝑥 − 3dividesevenlyinto𝑃(𝑥);thatmeans𝑥 − 3_______________________of𝑃(𝑥).𝑃 −2 =Sincetheremainderisnot____,𝑥 + 2doesnotdivideevenlyinto𝑃(𝑥);thatmeans𝑥 + 2

____________________________of𝑃(𝑥).

Part2:HowtodetermineafactorofaPolynomialWithLeadingCoefficient1Youcouldguessandcheckvaluesof𝑏thatmake𝑃 𝑏 = 0untilyoufindonethatworks…OryoucanusetheIntegralZeroTheoremtohelp.IntegralZeroTheoremIf𝑥 − 𝑏isafactorofapolynomialfunction𝑃(𝑥)withleadingcoefficient1andremainingcoefficientsthatareintegers,then𝒃isafactoroftheconstanttermof𝑃(𝑥).Note:Onceoneofthefactorsofapolynomialisfound,divisionisusedtodeterminetheotherfactors.

Example2:Factor𝑥$ + 2𝑥' − 5𝑥 − 6fully.Let𝑃 𝑥 = 𝑥$ + 2𝑥' − 5𝑥 − 6Findavalueof𝑏suchthat𝑃 𝑏 = 0.Basedonthefactortheorem,if𝑃 𝑏 = 0,thenweknowthat𝑥 − 𝑏isafactor.Wecanthendivide𝑃(𝑥)bythatfactor.Theintegralzerotheoremtellsustotestfactorsof_____Test________________________________.Onceonefactorisfound,youcanstoptestingandusethatfactortodivide𝑃(𝑥).𝑃 1 =Since________________,weknowthat________________________afactorof𝑃(𝑥).𝑃 2 =Since________________,weknowthat____________________afactorof𝑃(𝑥).YoucannowuseeitherlongdivisionorsyntheticdivisiontofindtheotherfactorsMethod1:Longdivision Method2:SyntheticDivision

Example3:Factor𝑥: + 3𝑥$ − 7𝑥' − 27𝑥 − 18completely.

Let𝑃 𝑥 = 𝑥: + 3𝑥$ − 7𝑥' − 27𝑥 − 18

Findavalueof𝑏suchthat𝑃 𝑏 = 0.Basedonthefactortheorem,if𝑃 𝑏 = 0,thenweknowthat𝑥 − 𝑏isafactor.Wecanthendivide𝑃(𝑥)bythatfactor.Theintegralzerotheoremtellsustotestfactorsof________Test________________________________________________.Onceonefactorisfound,youcanstoptestingandusethatfactortodivide𝑃(𝑥).Since________________,thistellusthat____________isafactor.Usedivisiontodeterminetheotherfactor.Wecannowfurtherdivide𝑥$ + 2𝑥' − 9𝑥 − 18usingdivisionagainorbyfactoringbygrouping.

Method1:Division

Method2:FactoringbyGrouping

Therefore,𝑥: + 3𝑥$ − 7𝑥' − 27𝑥 − 18 =Example4:TryFactoringbyGroupingAgain𝑥: − 6𝑥$ + 2𝑥' − 12𝑥

Note:Factoringbygroupingdoesnotalwayswork…butwhenitdoes,itsavesyoutime!

Groupthefirst2termsandthelast2termsandseparatewithanadditionsign.CommonfactorwithineachgroupFactoroutthecommonbinomial

Part3:HowtodetermineafactorofaPolynomialWithLeadingCoefficientNOT1Theintegralzerotheoremcanbeextendedtoincludepolynomialswithleadingcoefficientsthatarenot1.Thisextensionisknownastherationalzerotheorem.RationalZeroTheorem:Suppose𝑃 𝑥 isapolynomialfunctionwithintegercoefficientsand𝑥 = 1

2isazeroof𝑃(𝑥),where𝑎and

𝑏areintegersand𝑎 ≠ 0.Then,

• 𝑏isafactoroftheconstanttermof𝑃(𝑥)• 𝑎isafactoroftheleadingcoefficientof𝑃(𝑥)• (𝑎𝑥 − 𝑏)isafactorof𝑃(𝑥)

Example5:Factor𝑃 𝑥 = 3𝑥$ + 2𝑥' − 7𝑥 + 2Wemuststartbyfindingavalueof1

2where𝑃 1

2= 0.

𝑏mustbeafactoroftheconstantterm.Possiblevaluesfor𝑏are:____________𝑎mustbeafactoroftheleadingcoefficient.Possiblevaluesof𝑎are:____________Therefore,possiblevaluesfor1

2are:________________________________

Testvaluesof1

2for𝑥in𝑃(𝑥)tofindazero.

Since________________________________________________of𝑃(𝑥).Usedivisiontofindtheotherfactors.

Example6:Factor𝑃 𝑥 = 2𝑥$ + 𝑥' − 7𝑥 − 6Possiblevaluesfor𝑏are:________________________

Possiblevaluesof𝑎are:____________

Therefore,possiblevaluesfor12are:____________________________________

Part4:ApplicationQuestionExample7:When𝑓 𝑥 = 2𝑥$ − 𝑚𝑥' + 𝑛𝑥 − 2isdividedby𝑥 + 1,theremainderis−12and𝑥 − 2isafactor.Determinethevaluesof𝑚and𝑛.

Hint:Usetheinformationgiventocreate2equationsandthenusesubstitutionoreliminationtosolve.

L4–2.3–SolvingPolynomialEquationsLessonMHF4UJensenInthissection,youwilllearnmethodsofsolvingpolynomialequationsofdegreehigherthantwobyfactoring(usingthefactortheorem).Youwillalsoidentifytheconnectionbetweentherootsofpolynomialequations,thex-interceptsofthegraphofapolynomialfunction,andthezerosofthefunction.Part1:Investigation

a)Usetechnologytographthefunction𝑓 𝑥 = 𝑥$ − 13𝑥( + 36b)Determinethex-interceptsfromthegraphc)Factor𝑓(𝑥).Then,usethefactorstodeterminethezeros(roots)of𝑓(𝑥).d)Howarethe𝑥-interceptsfromthegraphrelatedtotheroots(zeros)oftheequation?

Remember:Thezerosofthefunctionarethevaluesof𝑥thatmake𝑓(𝑥) = 0.Ifthepolynomialequationisfactorable,thenthevaluesofthezeros(roots)canbedeterminedalgebraicallybysolvingeachlinearorquadraticfactor.

Example1:Statethesolutionstothefollowingpolynomialsthatarealreadyinfactoredforma)𝑥 2𝑥 + 3 𝑥 − 5 = 0 b)(2𝑥( − 3)(3𝑥( + 1)

Example2:Solveeachpolynomialequationbyfactoringa)𝑥0 − 𝑥( − 2𝑥 = 0 b)3𝑥0 + 𝑥( − 12𝑥 − 4 = 0

Methodsoffactoring:

• Longdivisionandsyntheticdivision• Factorbygrouping• Differenceofsquares𝑎( − 𝑏( = (𝑎 − 𝑏)(𝑎 + 𝑏)• CommonFactoring• Trinomialfactoring(sumandproduct)• Sumanddifferenceofcubes𝑎0 + 𝑏0 = (𝑎 + 𝑏)(𝑎( − 𝑎𝑏 + 𝑏()

𝑎0 − 𝑏0 = (𝑎 − 𝑏)(𝑎( + 𝑎𝑏 + 𝑏()

Solution(s):

Solution(s):

Example3:a)Usethefactortheoremtosolve𝑓 𝑥 = 2𝑥0 + 3𝑥( − 11𝑥 − 6Possiblevaluesof𝑏are:____________________________Possiblevaluesforare:____________________________Possiblevaluesfor4

5=____________________________________________

b)Whatdoyouranswerstoparta)represent?

Solution(s):

Example4:Findthezerosofthepolynomialfunction𝑓 𝑥 = 𝑥$ − 2𝑥0 − 7𝑥( + 8𝑥 + 12Possiblevaluesfor𝑏are____________________________________

Solution(s):

Example5:a)Findtherootsofthepolynomialfunction𝑓 𝑥 = 𝑥0 + 𝑥 − 3𝑥( − 3Startbyrearrangingindescendingorderofdegree:

b)Usetechnologytolookatthegraphofthefunction𝑓(𝑥).Commentonhow𝑥-intercept(s)ofthegrapharerelatedtotheREALandNON-REALrootsoftheequation.

Note:Sincethesquarerootofanegativenumberisnotarealnumber,theonlyREALrootis𝑥 = 3.𝑥 = ±√−1isconsideredaNON-REALroot.

Solution(s):

Example6:Findallrealrootsforeachpolynomialequationa)𝑓 𝑥 = 2𝑥0 − 3𝑥( − 𝑥 − 2Possiblevaluesof𝑏are:____________Possiblevaluesforare:____________Possiblevaluesfor4

5=________________

b)𝑔 𝑥 = 8𝑥0 + 125Hint:Thisisadifferenceofcubesà𝑎0 + 𝑏0 = (𝑎 + 𝑏) 𝑎( − 𝑎𝑏 + 𝑏(

Solution(s):

Solution(s):

L5–2.4–FamiliesofPolynomialFunctionsLessonMHF4UJensenInthissection,youwilldetermineequationsforafamilyofpolynomialfunctionsfromasetofzeros.Givenadditionalinformation,youwilldetermineanequationforaparticularmemberofthefamily.Part1:Investigation1)a)Howarethegraphsofthefunctionssimilarandhowaretheydifferent?

Same Different

b)Describetherelationshipbetweenthegraphsoffunctionsoftheform𝑦 = 𝑘(𝑥 − 1)(𝑥 + 2),where𝑘 ∈ ℝ

2)a)Examinethefollowingfunctions.Howaretheysimilar?Howaretheydifferent?

i) 𝑦 = −2(𝑥 − 1)(𝑥 + 3)(𝑥 − 2)ii) 𝑦 = −(𝑥 − 1)(𝑥 + 3)(𝑥 − 2)iii) 𝑦 = (𝑥 − 1)(𝑥 + 3)(𝑥 − 2)iv) 𝑦 = 2(𝑥 − 1)(𝑥 + 3)(𝑥 − 2)

b)Predicthowthegraphsofthefunctionswillbesimilarandhowtheywillbedifferent.c)Usetechnologytohelpyousketchthegraphsofallfourfunctionsonthesamesetofaxes.

A______________offunctionsisasetoffunctionsthathavethesamecharacteristics.Polynomialfunctionswiththesame______________aresaidtobelongtothesamefamily.Thegraphsofpolynomialfunctionsthatbelongtothesamefamilyhavethesame𝑥-interceptsbuthavedifferent𝑦-intercepts(unless0isoneofthe𝑥-intercepts).Anequationforthefamilyofpolynomialfunctionswithzeros𝑎/, 𝑎1, 𝑎2,…, 𝑎5is:

𝑦 = 𝑘(𝑥 − 𝑎/)(𝑥 − 𝑎1)(𝑥 − 𝑎2)… (𝑥 − 𝑎5),where𝑘 ∈ ℝ, 𝑘 ≠ 0

Part2:RepresentaFamilyofFunctionsAlgebraically1)Thezerosofafamilyofquadraticfunctionsare2and-3.a)Determineanequationforthisfamilyoffunctions.b)Writeequationsfortwofunctionsthatbelongtothisfamilyc)Determineanequationforthememberofthefamilythatpassesthroughthepoint(1,4).2)Thezerosofafamilyofcubicfunctionsare-2,1,and3.a)Determineanequationforthisfamily.b)Determineanequationforthememberofthefamilywhosegraphhasa𝑦-interceptof-15.

d)SketchagraphofthefunctionPart3:DetermineanEquationforaFunctionFromaGraph3)Determineanequationforthequarticfunctionrepresentedbythisgraph.

Tosketchagraph:

• Ploty-intercept• Plotx-intercepts• Usedegreeandleading

coefficienttodetermineendbehaviour

L6–2.5–SolvingInequalitiesLessonMHF4UJensenInthissection,youwilllearnthemeaningofapolynomialinequalityandexaminemethodsforsolvingpolynomialinequalities.Part1:IntrotoInequalitiesTask:ReadthefollowingonyourownExaminethegraphof𝑦 = 𝑥$ + 4𝑥 − 12.The𝑥-interceptsare6and-2.Thesecorrespondtothezerosofthefunction𝑦 = 𝑥$ + 4𝑥 − 12.Notethatthefactoredformversionofthefunctionis𝑦 = (𝑥 +6)(𝑥 − 2).Bymovingfromlefttorightalongthe𝑥-axis,wecanmakethefollowingobservations:

• Thefunctionispositivewhen𝑥 < −6sincethe𝑦-valuesarepositive

• Thefunctionisnegativewhen−6 < 𝑥 < 2sincethe𝑦-valuesarenegative

• Thefunctionispositivewhen𝑥 > 2sincethe𝑦-valuesarepositive.

Thezeros-6and2dividethe𝑥-axisintothreeintervals.Ineachinterval,thefunctioniseitherpositiveornegative.Theinformationcanbesummarizedinatable:Interval 𝑥 < −6 −6 < 𝑥 < 2 𝑥 > 2SignofFunction + − +

-1 0 1 2 3 4 5 6 7 8 9 10 -1 0 1 2 3 4 5 6 7 8 9 10

PolynomialInequalitiesApolynomialinequalityresultswhentheequalsigninapolynomialequationisreplacedwithaninequalitysymbol.Therealzerosofapolynomialfunction,or𝑥-interceptsofthecorrespondinggraph,dividethe𝑥-axisintointervalsthatcanbeusedtosolveapolynomialinequality.Part1:InequalitiesandNumberLinesExample1:Writeaninequalitythatcorrespondstothevaluesof𝑥shownoneachnumberlinea) b)Part2:SolveanInequalitygiventheGraphExample2:Usethegraphofthefunction𝑓(𝑥)toanswerthefollowinginequalities…𝑓 𝑥 = 0.1(𝑥 − 1)(𝑥 + 3)(𝑥 − 4)

a)𝑓(𝑥) < 0b)𝑓(𝑥) ≥ 0

Part2:SolveLinearInequalitiesNote:Solvinglinear____________________isthesameassolvinglinear________________.However,whenbothsidesofaninequalityaremultipliedordividedbya________________number,theinequalitysignmustbe________________________.

Example3:Solveeachinequality

a)𝑥 − 8 ≥ 3 b)−4 − 2𝑥 < 12Part2:SolveInequalitiesofDegree2andHigher

Example4:Solveeachpolynomialinequalityalgebraicallya)2𝑥$ + 3𝑥 − 9 > 0Method1:Graphtheinequality

Stepsforsolvingpolynomialinequalitiesalgebraically:

1) Useinverseoperationstomovealltermstoonesideoftheinequality2) Factorthepolynomialtodeterminethezerosofthecorrespondingequation3) Findtheinterval(s)wherethefunctionispositiveornegativebyeither:

a. Graphingthefunctionusingthezeros,leadingcoefficient,anddegreeb. Makeafactortableandtestvaluesineachinterval

Method2:FactorTable(signchart)

Tomakeafactortable:

• Use𝑥-interceptsandverticalasymptotestodivideintointervals• Useatestpointwithineachintervaltofindthesignofeachfactor• Determinetheoverallsignoftheproductbymultiplyingsignsofeachfactor

withineachinterval.

b)−2𝑥6 − 6𝑥$ + 12𝑥 ≤ −16Method1:GraphtheinequalityMethod2:FactorTable(signchart)

c)𝑥6 + 4𝑥$ + 6𝑥 < −24Part2:ApplicationsofInequalities3)Theprice,𝑝,indollars,ofastock𝑡yearsafter1999canbemodeledbythefunction𝑝 𝑡 = 0.5𝑡6 −5.5𝑡$ + 14𝑡.Whenwillthepriceofthestockbemorethan$90?