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Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge and Skills for Mathematics TAC §111.42(c). 2.01 Quadratic Functions Precalculus (1)(A) Precalculus (1)(B) Precalculus (1)(C) Precalculus (1)(G) Precalculus (2)(F) Precalculus (2)(G) Precalculus (2)(I) Precalculus (2)(J) Precalculus (2)(N) 2.02 Complex Numbers Precalculus (2)(I) Precalculus (2)(N) 2.03 Polynomial and Power Functions Precalculus (1)(G) Precalculus (2)(F) Precalculus (2)(G) Precalculus (2)(I) Precalculus (2)(J) Precalculus (2)(N) Precalculus (5)(J) 2.04 Long Division Precalculus (5)(J) 2.05 Rational Functions Precalculus (1)(G) Precalculus (2)(F) Precalculus (2)(I) Precalculus (2)(J) Precalculus (2)(K) Precalculus (2)(L) Precalculus (2)(M) Precalculus (2)(N) Copyright 2017 Licensed and Authorized for Use Only by Texas Education Agency 1

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Page 1: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

Section2:PolynomialandRationalFunctionsThefollowingmapsthevideosinthissectiontotheTexasEssentialKnowledgeandSkillsforMathematicsTAC§111.42(c).2.01QuadraticFunctions

• Precalculus(1)(A)• Precalculus(1)(B)• Precalculus(1)(C)• Precalculus(1)(G)• Precalculus(2)(F)• Precalculus(2)(G)• Precalculus(2)(I)• Precalculus(2)(J)• Precalculus(2)(N)

2.02ComplexNumbers

• Precalculus(2)(I)• Precalculus(2)(N)

2.03PolynomialandPowerFunctions

• Precalculus(1)(G)• Precalculus(2)(F)• Precalculus(2)(G)• Precalculus(2)(I)• Precalculus(2)(J)• Precalculus(2)(N)• Precalculus(5)(J)

2.04LongDivision

• Precalculus(5)(J)2.05RationalFunctions

• Precalculus(1)(G)• Precalculus(2)(F)• Precalculus(2)(I)• Precalculus(2)(J)• Precalculus(2)(K)• Precalculus(2)(L)• Precalculus(2)(M)• Precalculus(2)(N)

Copyright 2017 Licensed and Authorized for Use Only by Texas Education Agency 1

Page 2: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

2.06Inequalities

• Precalculus(5)(K)• Precalculus(5)(L)

Note:Unlessstatedotherwise,anydataisfictitiousandusedsolelyforthepurposeofinstruction.

Copyright 2017 Licensed and Authorized for Use Only by Texas Education Agency 2

Page 3: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

2.01QuadraticFunctions

Quadraticfunction–Apolynomialofdegreetwowhosegraphisaparabola

• Quadraticform–𝑦 = 𝑎𝑥& + 𝑏𝑥 + 𝑐

Vertexis ℎ, 𝑘 ,whereℎ = − .&/and𝑘 = 𝑓(ℎ)

• Vertexform–𝑦 = 𝑎 𝑥 − ℎ & + 𝑘

Vertexis ℎ, 𝑘

• Axisofsymmetry–The𝑥-coordinateofthevertex

• Leadingcoefficient–Representedby𝑎

o If𝑎 > 0,thenthegraphopens_______,andthevertexisa________. as𝑥 → ∞,𝑓(𝑥) → andas𝑥 → −∞,𝑓(𝑥) →

o If𝑎 < 0,thenthegraphopens_______,andthevertexisa________. as𝑥 → ∞,𝑓(𝑥) → andas𝑥 → −∞,𝑓(𝑥) →

1. Findthevertex,domain,range,axisofsymmetry,andvertexformof𝑓 𝑥 = 𝑥& − 4𝑥 + 3.

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Page 4: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

2. Findtheequationofaparabolawithavertexof(5, −8)thatpassesthroughthepoint(2, 1).

Whenaskedtofindthemaximumorminimumvalueofafunctioninawordproblemthatinvolvesquadraticfunctions,youarefindingthevertexofthatequation.

3. SupposethepriceandcostofbuyingTexasLonghornsfootballjerseysinbulkcanbeexpressedas𝑝 𝑥 = 400 − ?

&𝑥and𝐶 𝑥 = 20𝑥 + 300respectively.Findthenumberof

jerseysthatneedtobesoldtomaximizeprofit.

4. Therearetwopositivenumbers;thesumofthefirstnumberandtwicethesecondnumberis40.Determinethetwonumbersthatmaximizetheirproduct.

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Page 5: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

2.02ComplexNumbers

Imaginaryunit–Thenumberrepresentedby𝑖suchthat𝑖 = −1

Complexnumbers–Thecombinationofrealandimaginarynumbers

Intheform𝑎 + 𝑏𝑖,𝑎istherealpartand𝑏𝑖istheimaginarypart.

Noticethatpowersoftheimaginaryunitrepeateveryfourterms.

𝑖? = 𝑖& = 𝑖B = 𝑖C =

𝑖D = 𝑖E = 𝑖F = 𝑖G =

Whentheimaginaryunithasalargeexponent,youcandividethelasttwodigitsby4.Theremainderwillbeequivalenttothenewpowerof𝑖.

1. Findtheequivalentexpressionforeachofthefollowing:i. 𝑖?BF

ii. 𝑖HCD

iii. 𝑖&?II

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Page 6: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

Complexconjugatesdifferinthesignbetweentherealandimaginarynumber.

Theconjugateof𝑎 + 𝑏𝑖is𝑎 − 𝑏𝑖.Multiplyingthemtogethergivesus

2. Findthecomplexconjugateforeachofthefollowing:i. 5 − 4𝑖

ii. 7 + 3𝑖

iii. 3𝑖

iv. −7

Copyright 2017 Licensed and Authorized for Use Only by Texas Education Agency 6

Page 7: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

2.03PolynomialandPowerFunctions

Polynomialfunctionsareintheform𝑓 𝑥 = 𝑎K𝑥K + 𝑎KH?𝑥KH? + ⋯+ 𝑎&𝑥& + 𝑎?𝑥 + 𝑎I,where𝑛isawholenumberand𝑎sarerealnumbers.

CharacteristicsofPolynomials

• Turn–Thecoordinatewhereapolynomialhasalocalmaximumorminimum

Apolynomialofdegree𝑛hasatmost𝑛 − 1turnsinitsgraph.

Examples:

second-degreepolynomial: third-degreepolynomial:

fourth-degreepolynomial: fifth-degreepolynomial:

• Endbehavior–Wherethe“arrows”ofthegraphofthepolynomialarepointingas𝑥approachesinfinityandnegativeinfinity

o Foreven-degreepolynomials:§ Iftheleadingcoefficientispositive,theendbehavioristhatbothsidesof

thegraphareup.Meaningas𝑥 → ∞,𝑓(𝑥) → andas𝑥 → −∞,𝑓(𝑥) →§ Iftheleadingcoefficientisnegative,theendbehavioristhatbothsides

ofthegrapharedown.Meaningas𝑥 → ∞,𝑓(𝑥) → andas𝑥 → −∞,𝑓(𝑥) →

o Forodd-degreepolynomials:§ Iftheleadingcoefficientispositive,theendbehaviorisdowntotheleft

anduptotheright.Meaningas𝑥 → ∞,𝑓(𝑥) → andas𝑥 → −∞,𝑓(𝑥) →§ Iftheleadingcoefficientisnegative,theendbehaviorisdowntotheright

anduptotheleft.Meaningas𝑥 → ∞,𝑓(𝑥) → andas𝑥 → −∞,𝑓(𝑥) →

• Degreeofapolynomial

o Ifallliketermsofapolynomialarecombined,thedegreeisthehighestpower.o Ifthepolynomialiscompletelyfactored,thenthedegreeisfoundbyaddingup

themultiplicitiesofthezeroes.§ Zeroes–Thesolution(s)totheequation;alsoknownasthe𝑥-intercepts§ Multiplicities–Thepowersonthefactorsofeachzero

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Page 8: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

1. Listthezeroesandmultiplicitiesofeachpolynomial.Findthedegreeofeachpolynomialanditsendbehavior.

i. 𝑓 𝑥 = 2𝑥 + 3 𝑥 − 2 B

ii. 𝑔 𝑥 = −2𝑥B 𝑥 − 5 2𝑥 − 1 D

HowtoGraphaPolynomialFunction

1. Findthezeroes,multiplicitiesofthezeroes,andendbehavior.

2. Graphthezeroes,andthengraphtheendbehavior.

3. Usemultiplicitiestodetermineifthegraphtouchesorcrossesthezero.

o Ifthemultiplicityiseven,ittouchesthezero.o Ifthemultiplicityisodd,itcrossesthezero.

AquickacronymtousewhengraphingisZEM:graphbasedontheZeroes,Endbehavior,andMultiplicities.

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Page 9: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

2. Graphthepolynomialfunction𝑓 𝑥 = 2 𝑥 − 1 𝑥 + 3 &.

3. Graphthepolynomialfunction𝑓 𝑥 = 𝑥C − 4𝑥&.

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Page 10: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

2.04LongDivision

Longdivisionissimilartodividingintegers,butinthecaseofpolynomialdivision,youwilluse0asaplaceholderforanypowermissingbeforethehighestpowerofthepolynomial.

1. Uselongdivisiontowriteanequivalentexpressionfor 9𝑥B − 3𝑥& + 5 ÷ 3𝑥& + 1 .

Syntheticdivisionisashortcutoflongdivisionthatcanbeusedwhenthedivisorisabinomialintheform𝑥 − 𝑘.Onlythecoefficientsareusedforsyntheticdivision.

2. Usesyntheticdivisiontowriteanequivalentexpressionfor 2𝑥B − 4𝑥 + 1 ÷ 𝑥 + 2 .

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Page 11: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

RemainderTheorem–Ifyoudivideapolynomial,𝑓(𝑥),byafactor𝑥 − 𝑘,theremainderis𝑓(𝑘).

FactorTheorem–If𝑥 − 𝑘isafactorof𝑓(𝑥),then𝑓(𝑘) = 0.

3. Whichexpressionisafactorof𝑓 𝑥 = 𝑥B − 2𝑥& + 1?

i. 𝑥 − 1ii. 𝑥 + 1iii. 𝑥 − 2

NumberofZeroesTheorem–Apolynomialofdegree𝑛hasatmost𝑛distinctzeroes. Example:Apolynomialofdegreefourhasatmostfourdifferentzeroes.

ConjugateZeroesTheorem–Ifoneofthezeroesofapolynomialis𝑎 + 𝑏𝑖,then𝑎 − 𝑏𝑖isanotherzero.Forexample,if5 − 4𝑖isazero,then________isanotherzero.

4. Findapolynomialoflowestdegreewithzeroesof3,2𝑖,0,and𝑓 2 = 8.

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Page 12: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

2.05RationalFunctions

Rationalfunction–Aquotientoftwopolynomialfunctionsintheform𝑓 𝑥 = R(S)T(S)

,where

𝑞(𝑥)isnottheconstantpolynomial0.Whencomparedtopolynomialfunctions,rationalfunctionsmayhaveadditionaltraits,likeholesandasymptotes.

Hole–Asinglepointwherethegraphisundefinedandindicatedvisuallybyanopencircle

Todetermineholes:Whenafactor(𝑥 − 𝑘)ispresentinboththenumeratoranddenominator,you“cross”thesefactorsouttowriteanequivalentequation,sothesamefactorisnolongerinthedenominator,givingyouaholeat𝑥 = 𝑘.

Verticalasymptotes

• 𝑥 = 𝑎willbeaverticalasymptotewhen𝑓(𝑥) → ∞,or𝑓 𝑥 → −∞as𝑥 → 𝑎fromtheleftsideorrightsideof𝑥 = 𝑎.

• Tofindtheverticalasymptotes,setthedenominatorequaltozeroafteryousimplifytheequationforholes.

Horizontalasymptotes

• 𝑦 = 𝑏willbeahorizontalasymptotewhen𝑥 → ∞or𝑥 → −∞as𝑓(𝑥) → 𝑏.• Tofindhorizontalasymptotes,comparethedegreeofthepolynomialinthenumerator

tothedegreeofthepolynomialinthedenominator.o Case1:Ifthedegreesarethesame,thehorizontalasymptoteistheratioofthe

leadingcoefficients.o Case2:Ifthedegreeofthedenominatorislarger,thehorizontalasymptotewill

be𝑦 = 0.o Case3:Ifthedegreeofthenumeratorislarger,thereisnohorizontal

asymptote. Rather,thereisanobliqueasymptote,whichcanbefoundusinglongdivision.Theequationoftheobliqueasymptoteistheequationbeforetheremainder.

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Page 13: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

1. Findtheholesandasymptotesofeachfunctionbelow.

i. 𝑓 𝑥 = SVBSWHX

ii. 𝑔 𝑥 = &SWHCSSWHSH&

iii. ℎ 𝑥 = SV?SWV?

iv. 𝑗 𝑥 = SZV?SWV?

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Page 14: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

Afunctioncancrossortouchitshorizontalasymptote.Tofindthe𝑥valuewherethisoccurs,set𝑓 𝑥 =horizontalasymptote,andsolvefor𝑥.

2. Findwherethefunctionintersectsitshorizontalasymptote.Ifitdoesnotintersect,thenstateso.

i. 𝑓 𝑥 = SVBSWVX

ii. 𝑔 𝑥 = SVCSH&

3. Giventhefunction𝑓 𝑥 = BSWHESSWVSHE

,find(a)thedomain,(b)theintercepts,(c)theasymptotes,(d)theholes,and(e)wherethefunctioncrossesthehorizontalasymptote.Then,graphthefunction.

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Page 15: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

2.06Inequalities

Tosolvepolynomialorrationalinequalities:

Step1: Moveeverythingtooneside,equaltheothersidetozero,andfactortheentireexpression.

Step2: Findthedomainandthenreducetothelowestterms.Step3: Setthenumeratoranddenominatorequaltozero.Step4: PlacethevaluesfromStep3andthedomainfromStep2onanumberline.Step5: Assignopenandclosedcircles.

• Inthecaseof>or<,allareopen.• Inthecaseof≥or≤,allareclosedexceptfornumbersnotinthedomain,

whichareopen.Step6: Todeterminethesignbetweeneachnumberonyournumberline:

• Pluginavaluefor𝑥betweeneachintervaltodetermine+or–𝑦value,or• Pluginonenumbertostart,thenusemultiplicities.

o Evenpowerskeepthesamesign.o Oddpowerschangesign.

Step7: Writethesolutiontotheinequalityinintervalnotation.

1. Findtheinterval(s)wheretheinequality𝑥& − 12 > 𝑥istrue.

2. Findtheinterval(s)wheretheinequality H&S[ SHD W

SV? &SH?WZ< 0istrue.

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Page 16: Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

3. Findthedomainofthefunction𝑓 𝑥 = &SHD \(BHS)Z

BSW SV?.

4. Findtheinterval(s)forwhichtheinequality BSHD

≤ CSVB

istrue.

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