polynomial function for thesis
TRANSCRIPT
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WELCOME TO THEART OF
POLYNOMIAL
EQUATIONS
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POLYNOMIAL EQUATIONS
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Definitionofthe Polynomial
Letn beanonnegativeintegerandletan,an-1,a2,
a1 anda0 berealnumbers withan 0. Thefunction
defined by
f(x) = anxn +an-1x
n-1 ++a2x2+a1x+a0
iscalleda polynomialfunctionofn. Thenumberan,thecoefficientofthevariabletothehighest power,
iscalledtheleadingcoefficient.
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Question 3: Is quadraticfunction,f(x) = ax2 +
bx +c, wherea0,a polynomial
function?
Answer: Yes. Itisclassifiedasa polynomial
function withdegree2.
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Characteristicsofthe Graphsof
Polynomial Functions
Smooth Itmeansthatthegraphof polynomial
functionscontainonlyroundedcurves with
nosharp corners.
C
ontinuous
Itmeansthatthegraphsof polynomial
functionhaveno breaksandcan be
drawn withoutliftingthe pencilonthe
rectangularcoordinatesystem. Thus,its
domainisallrealnumbersas wellas
range [Domain:(,), Range:(,)].
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Example
Whichofthefollowinggraphsareclassifiedas
Polynomial Functions?
A CB D
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Answer
The polynomialfunctionsare A andBsincethegraphs
havesmoothandroundedcorners,andcontinuous,
thatfollowsthecharacteristicsofa polynomial
graph.
CandDarenotclassifiedas polynomialfunctions
sincegraphCisdiscontinuousandgraphDhassharp corners.
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The Importanceofthe Leading
Coefficient
As wedefinedthe polynomialfunctionas:
f(x) = anxn
+an-1xn-1
++a2x2
+a1x+a0,(a0n0)
Whenthevalueofx increasesordecreases without
bound,thegraphof polynomialfunctioneventually
risesorfalls.
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The Importanceofthe Leading
Coefficient
Ifthehighestdegree
(n)isoddandtheleadingcoefficientis
positive(+),the
graphof polynomial
willfallstotheleftandrisestotheright.
falls
rises
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The Importanceofthe Leading
Coefficient
Ifthehighestdegree
(n)isoddandtheleadingcoefficientis
negative(-),the
graphof polynomial
willfallstotherightandrisestotheleft.
fallsrises
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The Importanceofthe Leading
Coefficient
Ifthehighestdegree
(n)isevenandtheleadingcoefficientis
positive(+),the
graphof polynomial
risestotheleftandtotheright.
risesrises
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The Importanceofthe Leading
Coefficient
Ifthehighestdegree
(n)isoddandtheleadingcoefficientis
negative(-),the
graphof polynomial
fallstotheleftandtotheright.
fallsfalls
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Zeros/Rootsof Polynomial Functions
Iff isa polynomialfunction,thenthevaluesofx for
whichf(x) isequalto0areallcalled zerosoff.
Thesevaluesofx arerootsorsolutionsofthe
polynomialequationf(x) = 0. Eachrealrootofpolynomialappearsasanx-interceptofthegraph
ofthe polynomialfunction.
Insimple wayitisthevalueofx thatmakesf(x) = 0.Tofinditletf(x) = 0.
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Example
1. Findtherootsoff(x) = x3 +2x2 x 2.
Solution:
0 = x3 +2x2 x 2
= x2(x+2) (x+2)
= (x2-1)(x+2)
x+2 = 0 andx2 1 = 0
x = -2 x2 = 1
x2=1
The zeros/rootsoffare-2,-1,and1. Thisindicatesthatthegraphpassedthroughthe points(-2,0),(-1,0)and(1,0) whicharethexinterceptsofthe polynomial.
Let f(x) = 0
Factorx2 fromthefirsttwoterms and-1fromthelasttwo
terms
A commonfactorofx+2isfactoredfromtheexpression.
Seteachfactorequalto0.
Solveforx
Rememberthat if x2 = d,thenx =
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Example
2. Findthe zerooff(x) = - x4 +2x3 x2.
Solution:
0 = - x4 +2x3 x2
-1(0 = - x4 +2x3 x2)-1
1. = x4 2x3 +x2
= x2(x2 2x+1)
= x2(x-1)2
x2 = 0 andx-1 = 0
x = 0 x = 1
Therootsoffare0and2. Thisindicatesthatthegraphsx-interceptsare(0,0)and(2,0) whereinthegraph passesthrough.
Letx = 0
Multiply bothsides by 1.
Factorcompletely
Factoroutx2
Seteachfactorequalto0.
Solveforx
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Example
3. Findtherootsoff(x) = x3 +4x2 4x 16.
Solution:
0 = x3 +4x2 4x 16
= x2(x+4) 4(x+4)
= (x2 4)(x+4)
x2 4 = 0 andx+4 = 0
x = 2 x = -4
Therootsoffare(-4,0),(-2,0)and(2,0). Thisindicates
thatthegraph passesthroughthethese points.
Letf(x) = 0
Factorx2 fromthefirsttwoterms
and-4fromthelasttwoterms
A commonfactorofx+4isfactored
fromtheexpression.
Seteachfactorequalto0.
Solveforx
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Example
4. Findtherootsoff(x) = x4 9x2.
Solution:
0 = x4 9x2
= x2 (x2 9)
x2 = 0andx2 9 = 0
x = 0andx = 3
Therootsoffare(-3,0),(0,0)and(3,0). Thisindicatesthatthegraph passesthroughthese points
Letf(x) = 0
Factoroutx2
Seteachfactorequalto0.
Solveforx
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MultiplicityofZerosof Polynomial
Functions
Ifr isa zeroofevenmultiplicity,thenthegraphtouchesx axisandturnsaroundatr. Ifr isa zeroofoddmultiplicity,thenthegraphcrossesthex-axisatr. Regardlessof whetherthemultiplicityofazeroisevenorodd,graphstendtop flattenoutat zeros withmultiplicitygreaterthanone(1).
Say, f(x) = -x4 +4x3 4x2
-1(0= -x4 +4x3 4x2)-1
= x4 - 4x3 +4x2.
= x2(x2 - 4x+4)
=x2(x-2)2
Observethateachfactoroccurstwice. Inthefollowing,theequationforthe polynomialfunctionf,ifthesamefactorx - r occursk times(x-r)k, butnotk + 1 times, wecallthisasa zero/root withmultiplicityk.Thus,forthe polynomialfunctionf(x) = x2(x-2)2,0and2are bothzeros/roots withmultiplicity2. Therefore,thegraphoff touches(0,0)and(2,0) pointsandturnaround.
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Example
Findtherootsoff(x) = (x+2)(x-3)2andgiveitsmultiplicity. Stateifthegraphcrossesthex-axisortouchesthex-axisandturnsaroundateach zero.
Solution:
(x+2)(x-3)2= 0
x+2 = 0andx3 = 0
x = -2and x = 3
The zeros/rootsoffare-2and 3, withmultiplicityof1
and2,respectively.Becausethemultiplicityof-2isodd,thegraphcrossesthex axisatthisroot.Becausethemultiplicityof 3 even,thegraphtouchesthex-axisandturnaroundatthisroot.
Letf(x) = 0
Seteachfactorequalto0.
Solveforx
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Whento Usethe Intermediate Value
Theorem?
Definition. Letfbea polynomialfunction withrealcoefficients. If f(a) andf(b) haveoppositesigns,thenthereisatleastonevalueofc betweena andb for whichf(c)= 0. Equivalently,theequationf(x) = 0hasatleastonerealroot betweena andb.
Thistheoremisveyusefulifthe polynomialfunctionhasnofactor butthex-intercept/sexistsonthegraph.
b
a
c
(b,f(b))
f(b)>0
(a,f(a))
f(a)
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Example
Show thatthe polynomialhasareal zero between2and 3.
given:f(x) = x3 2x 5
Solution:
Evaluatefat2andat 3. Iff(2) andf(3) haveoppositesigns,thenthereisatleastonereal zero between2and 3. Usingf(x) = x3 2x 5,theobtainvalue
f(2) = (2)3 2(2) 5 = -1
and
f(3
)=
(3
)
3
2(3
) 5
=16
Becausef(2) = -1andf(3) = 16,thesignchangeshowsthatthe polynomialfunctionhasreal zero between2and 3. Thiszeroisactuallyirrationalandisapproximated.
f(2)isnegative
f(3)is positive
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Turning Pointsof Polynomial Function
Ineachturning pointofthe polynomialgraph,the
changesitsdirectionfromincreasingtodecreasing
orviceversa. Andtodeterminethe possiblenumber
ofturning points withoutgraphingthefunction weneedtofollow thecondition, iff isa polynomial
functionofdegreen,thenthegraphoffhasat
mostn-1 turning points.
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Easy Graphingof Polynomial Function
withoutusingany Graphing Utility
1. Usetheleadingcoefficienttesttodeterminethe
graphsend behavior.
2.Determinethex-intercept bysettingf(x) = 0and
solvingtheresulting polynomialequation. Ifthereis
anx-interceptatr asaresultof(x-r)k inthe
completefactorizationoff(x),thenthefollowing
must beconsidered:
a. Ifk iseven,thegraphtouchesthex-axisatr and
turnsaround.
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Easy Graphingof Polynomial Function
withoutusingany Graphing Utility
b. Ifk isodd,thegraphcrossesthex-axisatr.
c. If k>1,thegraphflattensoutat(r,0).
3.Determinethey-intercept bycomputingf(0) orlet
x = 0.
4. Usesymmetry,ifapplicable,tohelp draw the
graph:
a.y-axissymmetry:f(-x) = f(x)
b.originsymmetry:f(-x) = -f(x)
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Easy Graphingof Polynomial Function
withoutusingany Graphing Utility
5. Usethefactthatthemaximumnumberofturning
pointsofthegraphisn-1 tocheck whetheritis
drawncorrectly.
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Exampleof Graphing Polynomial
Functions
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