Ch.5 PolynomialandRationalFunctions
5.1 PolynomialFunctionsandModels
1 IdentifyPolynomialFunctionsandTheirDegree
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Statewhetherthefunctionisapolynomialfunctionornot.Ifitis,giveitsdegree.Ifitisnot,tellwhynot.
1) f(x)=4x+2x5
A) Yes;degree5 B) Yes;degree1 C) Yes;degree2 D) Yes;degree4
2) f(x)=–11x4+7x3+4
A) Yes;degree4 B) Yes;degree7
C) Yes;degree8 D) No;thelasttermhasnovariable
3) f(x)=8–x4
7
A) Yes;degree4 B) Yes;degree1
C) No;itisaratio D) No;xisanegativeterm
4) f(x)=1
2
–1
5x
A) Yes;degree1 B) Yes;degree5
C) Yes;degree0 D) No;xhasafractionalcoefficient
5) f(x)=7
A) Yes;degree0 B) No;itisaconstant
C) No;itcontainsnovariables D) Yes;degree1
6) f(x)=1+8
x
A) No;xisraisedtoanegativepower B) Yes;degree0
C) Yes;degree8 D) Yes;degree1
7) f(x)=x(x–12)
A) Yes;degree2 B) Yes;degree0
C) No;itisaproduct D) Yes;degree1
8) f(x)=8–2
x6
A) No;xisraisedtothenegative6 power B) Yes;degree6
C) Yes;degree–6 D) Yes;degree1
6
9) f(x)=x4–6
x6
A) No;itisaratioofpolynomials B) Yes;degree4
C) Yes;degree6 D) Yes;degree–6
Page1
10) f(x)=x4
/
3–x5–2
A) No;xisraisedtonon–integer4
/
3 power B) Yes;degree5
C) Yes;degree4
/
3 D) Yes;degree4
11) 6(x–1)11(x+1)4
A) Yes;degree15 B) Yes;degree11 C) Yes;degree66 D) Yes;degree6
12) f(x)=x(x–5)
A) No;xisraisedtonon–integerpower B) Yes;degree1
C) Yes;degree2 D) No;itisaproduct
13) f(x)=–6x4+πx3–7
2
A) Yes;degree4 B) Yes;degree7
C) Yes;degree8 D) No;x3hasanon–integercoefficient
Page2
2 GraphPolynomialFunctionsUsingTransformations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Usetransformationsofthegraphofy=x4ory=x5tographthefunction.
1) f(x)=(x+5)4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page3
2) f(x)=x4–5
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page4
3) f(x)=–1
3x4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page5
4) f(x)=3x4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page6
5) f(x)=(x–4)4+3
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page7
6) f(x)=1
2(x–3)4+3
x
-5 5
y
10
-10
x
-5 5
y
10
-10
A)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
B)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
C)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
D)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
Page8
7) f(x)=–2(x+4)4+4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page9
8) f(x)=3–(x+3)4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page10
9) f(x)=(x+4)5
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page11
10) f(x)=x5+5
x
-5 5
y
10
-10
x
-5 5
y
10
-10
A)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
B)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
C)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
D)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
Page12
11) f(x)=1
5x5
x
-5 5
y
10
-10
x
-5 5
y
10
-10
A)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
B)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
C)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
D)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
Page13
12) f(x)=5x5
x
-5 5
y
10
-10
x
-5 5
y
10
-10
A)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
B)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
C)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
D)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
Page14
13) f(x)=(x+3)5+2
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page15
14) f(x)=1
2(x+2)5+3
x
-5 5
y
10
-10
x
-5 5
y
10
-10
A)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
B)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
C)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
D)
x
-5 5
y
10
-10
x
-5 5
y
10
-10
Page16
15) f(x)=–2(x+3)5+2
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page17
16) f(x)=2–(x–3)5
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
3 IdentifytheRealZerosofaPolynomialFunctionandTheirMultiplicity
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Formapolynomialwhosezerosanddegreearegiven.
1) Zeros:–3
,
–1
,
2;degree3
A) f(x)=x3+2x2–5x–6fora=1 B) f(x)=x3–2x2–5x+6fora=1
C) f(x)=x3+2x2+5x+6fora=1 D) f(x)=x3–2x2+5x–6fora=1
Page18
2) Zeros:0,–7
,
6;degree3
A) f(x)=x3+x2–42xfora=1 B) f(x)=x3+x2+42xfora=1
C) f(x)=x3+x2+x–42fora=1 D) f(x)=x3+x2+x+42fora=1
3) Zeros:–1,1,–7;degree3
A) f(x)=x3+7x2–x–7fora=1 B) f(x)=x3–7x2+x–7fora=1
C) f(x)=x3–7x2–x+7fora=1 D) f(x)=x3+7x2+x+7fora=1
4) Zeros:–5
,
–3
,
3;degree3
A) f(x)=x3–9x+5x2–45fora=1 B) f(x)=x3–9x–5x2+45fora=1
C) f(x)=x3+9x+5x2+45fora=1 D) f(x)=x3+9x–5x2–45fora=1
5) Zeros:2
multiplicity2;–2
,
multiplicity2;degree4
A) f(x)=x4–8x2+16 B) f(x)=x4+4x3–8x2+8x–16
C) f(x)=x4+8x2+16 D) f(x)=x4–4x3+8x2–8x+16
6) Zeros:–4
,
multiplicity2;–3
,
multiplicity1;degree3
A) x3+11x2+40x+48 B) x3–11x2+40x–48
C) x3+8x2+40x+48 D) x3–11x2+24x–48
7) Zeros:–5
,
–3
,
–1
,
2;degree4
A) x4+7x3+5x2–31x–30 B) x4–7x3+5x2+31x–30
C) x4+13x2–30 D) x4+7x3+5x2–30x–30
Forthepolynomial,listeachrealzeroanditsmultiplicity.Determinewhetherthegraphcrossesortouchesthe
x–axisateachx–intercept.
8) f(x)=2(x–2)(x–1)4
A) 2
,
multiplicity1,crossesx–axis;1
,
multiplicity4
,
touchesx–axis
B) –2
,
multiplicity1,crossesx–axis;–1
,
multiplicity4
,
touchesx–axis
C) 2
,
multiplicity1,touchesx–axis;1
,
multiplicity4
,
crossesx–axis
D) –2
,
multiplicity1,touchesx–axis;–1
,
multiplicity4
,
crossesx–axis
9) f(x)=3(x–1)(x+4)3
A) 1
,
multiplicity1,crossesx–axis;–4
,
multiplicity3,crossesx–axis
B) –1
,
multiplicity1,crossesx–axis;4
,
multiplicity3,crossesx–axis
C) 1
,
multiplicity1,touchesx–axis;–4
,
multiplicity3
D) –1
,
multiplicity1,touchesx–axis;4
,
multiplicity3
10) f(x)=2(x2+1)(x–4)2
A) 4
,
multiplicity2,touchesx–axis
B) –1
,
multiplicity1,crossesx–axis;4
,
multiplicity2,touchesx–axis
C) –1
,
multiplicity1,touchesx–axis;4
,
multiplicity2,crossesx–axis
D) 4
,
multiplicity2,crossesx–axis
Page19
11) f(x)=x+1
3
4(x+1)5
A) –1
3,multiplicity4,touchesx–axis;–1,multiplicity5,crossesx–axis
B) –1
3,multiplicity4,crossesx–axis;–1,multiplicity5,touchesx–axis
C) 1
3,multiplicity4,touchesx–axis;1,multiplicity5,crossesx–axis
D) 1
3,multiplicity4,crossesx–axis;1,multiplicity5,touchesx–axis
12) f(x)=x+1
3
2(x2+9)3
A) –1
3,multiplicity2,touchesx–axis
B) –1
3,multiplicity2,touchesx–axis;–9,multiplicity3,crossesx–axis
C) 1
3,multiplicity2,touchesx–axis;9,multiplicity3,crossesx–axis
D) –1
3,multiplicity2,crossesx–axis
13) f(x)=1
3x(x2–3)
A) 0,multiplicity1,crossesx–axis;
3,multiplicity1,crossesx–axis;
–3,multiplicity1,crossesx–axis
B) 0,multiplicity1,touchesx–axis;
3,multiplicity1,touchesx–axis;
–3,multiplicity1,touchesx–axis
C) 0,multiplicity1D)3
,multiplicity1,touchesx–axis;
–3,multiplicity1,touchesx–axis
14) f(x)=1
2x4(x2–5)
A) 0,multiplicity4
,
touchesx–axis;
5,multiplicity1,crossesx–axis;
–5,multiplicity1,crossesx–axis
B) 0,multiplicity4
,
crossesx–axis;
5,multiplicity1,touchesx–axis;
–5,multiplicity1,touchesx–axis
C) 0,multiplicity4
,
touchesx–axis D) 0,multiplicity4
,
crossesx–axis
15) f(x)=4(x2+2)(x2+6)2
A) Norealzeros
B) –2
,
multiplicity1,crossesx–axis;–6
,
multiplicity2,touchesx–axis
C) –2
,
multiplicity1,touchesx–axis;–6
,
multiplicity2,crossesx–axis
D) 2,multiplicity1,crossesx–axis;–2,multiplicity1,crossesx–axis;
6,multiplicity2,touchesx–axis;–6,multiplicity2,touchesx–axis
Page20
16) f(x)=1
2x4(x2–3)(x+6)
A) 0,multiplicity4
,
touchesx–axis;
–6,multiplicity1,crossesx–axis;
3,multiplicity1,crossesx–axis;
–3,multiplicity1,crossesx–axis
B) 0,multiplicity4
,
crossesx–axis;
–6,multiplicity1,touchesx–axis;
3,multiplicity1,touchesx–axis;
–3,multiplicity1,touchesx–axis
C) 0,multiplicity4
,
touchesx–axis;
–6,multiplicity1,crossesx–axis
D) 0,multiplicity4
,
crossesx–axis;
–6,multiplicity1,touchesx–axis
4 AnalyzetheGraphofaPolynomialFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthex–andy–interceptsoff.
1) f(x)=(x+12)2
A) x–intercept:–12;y–intercept:144 B) x–intercept:12;y–intercept:144
C) x–intercept:–12;y–intercept:0D)x
–intercept:12;y–intercept:0
2) f(x)=4x2(x–8)3
A) x–intercepts:0,8;y–intercept:0B)x
–intercepts:0,8;y–intercept:4
C) x–intercepts:0,–8;y–intercept:4D)x
–intercepts:0,–8;y–intercept:0
3) f(x)=(x+4)(x–5)(x+5)
A) x–intercepts:–4
,
–5
,
5;y–intercept:–100 B) x–intercepts:–5
,
5
,
4;y–intercept:100
C) x–intercepts:–4
,
–5
,
5;y–intercept:100 D) x–intercepts:–5
,
5
,
4;y–intercept:–100
4) f(x)=2x–x3
A) x–intercepts:0,2,–2;y–intercept:0B)x
–intercepts:0,2,–2;y–intercept:2
C) x–intercepts:0,–2;y–intercept:0D)x
–intercepts:0,–2;y–intercept:2
5) f(x)=(x+1)(x–9)(x–1)2
A) x–intercepts:–1,1,9;y–intercept:–9B)x
–intercepts:–1,1,9;y–intercept:9
C) x–intercepts:–1,1,–9;y–intercept:–9D)x
–intercepts:–1,1,–9;y–intercept:9
6) f(x)=–x2(x+7)(x2–1)
A) x–intercepts:–7
,
–1,0,1;y–intercept:0B)x
–intercepts:–1,0,1,7;y–intercept:0
C) x–intercepts:–7
,
0,1;y–intercept:–7D)x
–intercepts:–7
,
–1,0,1;y–intercept:–7
7) f(x)=–x2(x+3)(x2+1)
A) x–intercepts:–3
,
0;y–intercept:0B)x
–intercepts:–3
,
–1,0,1;y–intercept:0
C) x–intercepts:–3
,
–1,0;y–intercept:3D)x
–intercepts:–3
,
–1,0;y–intercept:–3
8) f(x)=(x–5)(x–6)
A) x–intercepts:5
,
6;y–intercept:30 B) x–intercepts:–5
,
–6;y–intercept:30
C) x–intercepts:5
,
6;y–intercept:–11 D) x–intercepts:–5
,
–6;y–intercept:–11
9) f(x)=x2(x–3)(x–4)
A) x–intercepts:0,3
,
4;y–intercept:0B)x
–intercepts:0,–3
,
–4;y–intercept:0
C) x–intercepts:0,3
,
4;y–intercept:12 D) x–intercepts:0,–3
,
–4;y–intercept:12
Page21
10) f(x)=(x–3)2(x2–25)
A) x–intercepts:–5
,
3
,
5;y–intercept:–225 B) x–intercepts:–5
,
3
,
5;y–intercept:225
C) x–intercepts:3
,
25;y–intercept:75 D) x–intercepts:–3
,
–25;y–intercept:75
Findthepowerfunctionthatthegraphoffresemblesforlargevaluesof|x|.
11) f(x)=(x+9)2
A) y=x2B) y=x81 C) y=x18 D) y=x9
12) f(x)=(x+5)3
A) y=x3B) y=x15 C) y=x125 D) y=x5
13) f(x)=(x+6)4(x+8)6
A) y=x10 B) y=x24 C) y=x4D) y=x6
14) f(x)=–x2(x+3)3(x2–1)
A) y=–x7B) y=x3C) y=x2D) y=x7
15) f(x)=7x–x3
A) y=–x3B) y=x4C) y=x2D) y=x3
Determinethemaximumnumberofturningpointsoff.
16) f(x)=–x2(x+5)3(x2–1)
A) 6 B) 7 C) 5 D) 2
17) f(x)=8x–x3
A) 2 B) 3 C) 1 D) 4
18) f(x)=(x–2)2(x+3)2
A) 3 B) 4 C) 2 D) 1
Usethex–interceptstofindtheintervalsonwhichthegraphoffisaboveandbelowthex–axis.
19) f(x)=(x+5)2
A) abovethex–axis:(–∞
,
–5),(–5
,
∞)
belowthex–axis:nointervals
B) abovethex–axis:nointervals
belowthex–axis:(–∞,–5),(–5,∞)
C) abovethex–axis:(–∞
,
–5)
belowthex–axis:(–5,∞)
D) abovethex–axis:(–5
,
∞)
belowthex–axis:(–∞,–5)
20) f(x)=(x+5)3
A) abovethex–axis:(–5
,
∞)
belowthex–axis:(–∞,–5)
B) abovethex–axis:(–∞
,
–5)
belowthex–axis:(–5,∞)
C) abovethex–axis:(–∞
,
–5),(–5
,
∞)
belowthex–axis:nointervals
D) abovethex–axis:nointervals
belowthex–axis:(–∞,–5),(–5,∞)
21) f(x)=(x–2)2(x+5)2
A) abovethex–axis:(–∞
,
–5),(–5
,
2),(2
,
∞)
belowthex–axis:nointervals
B) abovethex–axis:nointervals
belowthex–axis:(–∞,–5),(–5,2),(2,∞)
C) abovethex–axis:(–∞
,
–5),(2
,
∞)
belowthex–axis:(–5,2)
D) abovethex–axis:(–5
,
2)
belowthex–axis:(–∞,–5),(2,∞)
Page22
22) f(x)=x+1
4
2(x–5)3
A) abovethex–axis:(5
,
∞)
belowthex–axis:–∞,–1
4,–1
4,5
B) abovethex–axis:–∞,–1
4,–1
4,5
belowthex–axis:(5,∞)
C) abovethex–axis:–∞,–1
4,(5,∞)
belowthex–axis:–1
4,5
D) abovethex–axis:–1
4,5
belowthex–axis:–∞,–1
4,(5,∞)
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Analyzethegraphofthegivenfunctionfasfollows:
(a) Determinetheendbehavior:findthepowerfunctionthatthegraphoffresemblesforlargevaluesof|x|.
(b)Findthex–andy–interceptsofthegraph.
(c)Determinewhetherthegraphcrossesortouchesthex–axisateachx–intercept.
(d)Graphfusingagraphingutility.
(e)Usethegraphtodeterminethelocalmaximaandlocalminima,ifanyexist.Roundturningpointstotwo
decimalplaces.
(f) Usetheinformationobtainedin(a)–(e)todrawacompletegraphoffbyhand.Labelallinterceptsand
turningpoints.
(g)Findthedomainoff.Usethegraphtofindtherangeoff.
(h)Usethegraphtodeterminewherefisincreasingandwherefisdecreasing.
23) f(x)=x2(x+2)
24) f(x)=(x+2)(x–2)2
25) f(x)=–2(x–3)(x+1)3
26) f(x)=(x–3)(x–1)(x+2)
27) f(x)=–x2(x–1)(x+3)
28) f(x)=x2(x2–4)(x+4)
Analyzethegraphofthegivenfunctionfasfollows:
(a)Determinetheendbehavior:findthepowerfunctionthatthegraphoffresemblesforlargevaluesof|x|.
(b)Graphfusingagraphingutility.
(c)Findthex–andy–interceptsofthegraph.
(d)Usethegraphtodeterminethelocalmaximaandlocalminima,ifanyexist.Roundturningpointstotwo
decimalplaces.
(e)Usetheinformationobtainedin(a)–(d)todrawacompletegraphoffbyhand.Labelallinterceptsandturning
points.
(f)Findthedomainoff.Usethegraphtofindtherangeoff.
(g)Usethegraphtodeterminewherefisincreasingandwherefisdecreasing.
29) f(x)=x3–0.4x2–2.5861x+3.0912
Page23
Solvetheproblem.
30) Forthepolynomialfunctionf(x)=2x4–7x3+11x–4
a) Findthex–andy–interceptsofthegraphoff.Roundtotwodecimalplaces,ifnecessary.
b) Determinewhetherthegraphcrossesortouchesthex–axisateachx–intercept.
c) Endbehavior:findthepowerfunctionthatthegraphoffresemblesforlargevaluesof|x|.
d) Useagraphingutilitytographthefunction.Approximatethelocalmaximaroundedtotwodecimal
places,ifnecessary.Approximatethelocalminimaroundedtotwodecimalplaces,ifnecessary.
e) Determinethenumberofturningpointsonthegraph.
f) Putalltheinformationtogether,andconnectthepointswithasmooth,continuouscurvetoobtainthe
graphoff.
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
31) Whichofthefollowingpolynomialfunctionsmighthavethegraphshownintheillustrationbelow?
A) f(x)=x2(x–2)(x–1) B) f(x)=x(x–2)(x–1)2
C) f(x)=x(x–2)2(x–1) D) f(x)=x2(x–2)2(x–1)2
Page24
5 BuildCubicModelsfromData
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Solvetheproblem.
1) Theprofits(inmillions)foracompanyfor8yearswasasfollows:
Year,xProfits
1993,1
1994,2
1995,3
1996,4
1997,5
1998,6
1999,7
2000,8
1.1
1.7
2.0
1.4
1.3
1.5
1.8
2.1
Findthecubicfunctionofbestfittothedata.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
2) Theamountofwater(ingallons)inaleakybathtubisgiveninthetablebelow.Usingagraphingutility,fit
thedatatoathirddegreepolynomial(oracubic).Thenapproximatethetimeatwhichthereismaximum
amountofwaterinthetub,andestimatethetimewhenthewaterrunsoutofthetub.Expressallyour
answersroundedtotwodecimalplaces.
t(inminutes) 01234567
V(ingallons) 20 26 45 63 86 94 90 67
A) maximumamountofwaterafter5.31minutes;waterrunsoutafter8.23minutes
B) maximumamountofwaterafter5.37minutes;waterrunsoutafter11.06minutes
C) maximumamountofwaterafter8.23minutes;waterrunsoutafter19.73minutes
D) maximumamountofwaterafter5.31minutes;waterneverrunsout
5.2 PropertiesofRationalFunctions
1 FindtheDomainofaRationalFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthedomainoftherationalfunction.
1) g(x)=5x
x–1
A) {x|x≠1} B) {x|x≠–1} C) {x|x≠0} D) allrealnumbers
2) h(x)=5x
(x–4)(x–7)
A) {x|x≠4
,
x≠7} B) {x|x≠–4
,
x≠–7}
C) {x|x≠4
,
x≠7
,
x≠–5} D) allrealnumbers
Page25
3) g(x)=x+5
x2–64
A) {x|x≠–8
,
x≠8} B) {x|x≠–8
,
x≠8
,
x≠–5}
C) {x|x≠0,x≠64} D) allrealnumbers
4) g(x)=x+4
x2+64
A) allrealnumbers B) {x|x≠–8
,
x≠8
,
x≠–4}
C) {x|x≠0,x≠–64} D) {x|x≠–8
,
x≠8}
5) h(x)=x+6
x2+9x
A) {x|x≠0,x≠–9} B) {x|x≠–3
,
x≠3
,
x≠–6}
C) allrealnumbers D) {x|x≠–3
,
x≠3}
6) R(x)=–3x2
x2+10x–24
A) x x≠–12
,
2 B) x x≠12
,
2 C) x x≠12
,
–2 D) x x≠–24
,
1
7) f(x)=2x2–4
3x2+6x–45
.
A) {x|x≠3,x≠–5} B) {x|x≠–3,x≠5}
C) {x|x≠3,x≠–3,x≠–5} D) allrealnumbers
8) f(x)=–2x(x+2)
2x2–3x–5
A) x x≠5
2,–1 B) x x≠–5
2,1 C) x x≠2
5,–1 D) x x≠–2
5,1
9) f(x)=x(x–1)
9x2+18x+5
A) x x≠–1
3,–5
3B) x x≠–1
9,–5
9C) x x≠1
3,5
3D) x x≠–5
9,10
9
10) g(x)=x
x3–216
A) x x≠6 B) x x≠–6
,
6 C) x x≠36 D) x x≠–6
Page26
Usethegraphtodeterminethedomainandrangeofthefunction.
11)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) domain:{x|x≠2}
range:{y|y≠4}
B) domain:{x|x≠4}
range:{y|y≠2}
C) domain:{x|x≠–2}
range:{y|y≠4}
D) domain:{x|x≠4}
range:{y|y≠–2}
12)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) domain:{x|x≠–4}
range:{y|y>0}
B) domain:{x|x≠–4}
range:{y|y≥0}
C) domain:{x|x>0}
range:{y|y≠–4}
D) domain:{x|x≥0}
range:{y|y≠–4}
Page27
13)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) domain:{x|x≠0}
range:allrealnumbers
B) domain:allrealnumbers
range:allrealnumbers
C) domain:allrealnumbers
range:{y|y≠0}
D) domain:{x|x≠0}
range:{y|y≠0}
14)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) domain:{x|x≠0}
range:{y|y≤–8ory≥8}
B) domain:allrealnumbers
range:{y|y≤–8ory≥8}
C) domain:{x|x≠0}
range:allrealnumbers
D) domain: {x|x≤–8orx≥8}
range:{y|y≠0}
Page28
15)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) domain:{x|x≠–1
,
x≠1}
range:allrealnumbers
B) domain:allrealnumbers
range:{y|y≠–1,y≠1}
C) domain:{x|x≠–1
,
x≠1}
range:{y|y≠0}
D) domain:allrealnumbers
range:allrealnumbers
16)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) domain:{x|x≠–4
,
x≠4}
range:{y|y≤0ory>1}
B) domain:{x|x≤0orx>1}
range:{y|y≠–4,y≠4}
C) domain:{x|x≠–4
,
x≠4}
range:{y|y≤0ory≥1}
D) domain:allrealnumbers
range:allrealnumbers
2 FindtheVerticalAsymptotesofaRationalFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheverticalasymptotesoftherationalfunction.
1) h(x)=6x
x+5
A) x=–5B)x=5C)x=6 D) none
2) g(x)=3x
(x–4)(x–7)
A) x=4
,
x=7B)x=–4
,
x= –7C)x=4
,
x=7
,
x= –3D)x= –3
Page29
3) h(x)=x+6
x2–64
A) x=–8
,
x=8B)x= –8
,
x=8
,
x= –6
C) x=0,x=64 D) x=64
,
x= –6
4) g(x)=x+7
x2+36
A) none B) x= –6
,
x=6
,
x= –7
C) x=–6
,
x=6D)x= –6
,
x= –7
5) h(x)=x+11
x2+4x
A) x=0,x=–4B)x=–4
,
x= –11 C) x=0,x= –2
,
x=2D)x= –2
,
x=2
6) f(x)=x(x–1)
x3+25x
A) none B) x=0,x= –25 C) x=0,x= –5
,
x=5D)x= –5
,
x=5
7) R(x)=–3x2
x2+5x–14
A) x=–7
,
x=2B)x=7
,
x= –2
C) x=–7
,
x=2
,
x=–3D)x= – 14
8) f(x)=–2x(x+2)
5x2–3x–8
A) x=8
5,x=–1B)x=–8
5,x=1C)x=5
8,x=–1D)x=–5
8,x=1
9) f(x)=x(x–1)
25x2+40x+7
A) x=–1
5,x=–7
5B) x=–1
25,x=–7
25 C) x=1
5,x=7
5D) x=–7
25,x=14
25
10) g(x)=x
x3–343
A) x=7B)x=–7
,
x=7C)x=49 D) x= –7
11) f(x)=x–4
16x–x3
A) x=0,x=–4B)x=0,x= –4
,
x=4C)x=0,x=4D)x= –4
,
x=4
12) f(x)=–x2+16
x2+5x+4
A) x=–1B)x=–1,x= –4C)x= –1,x=4D)x=1,x= –4
Page30
Usethegraphtofindtheverticalasymptotes,ifany,ofthefunction.
13)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) x=2B)x=2,y=5C)y=5D)x=2
,
x=0
14)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) x=–4B)y=–4C)x= –4
,
x=0 D) none
15)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) x=0B)y=0C)x=0,y=0 D) none
Page31
16)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) x=0B)y=–6
,
y=6C)x=0,y=0 D) none
17)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) x=–4
,
x=4B)x=–4
,
x=4
,
x=0C)x= –4
,
x=4
,
y=0 D) none
18)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) x=–3
,
x=3B)x= –3
,
x=3
,
x=0
C) x=–3
,
x=3
,
y=1D)x= –3
,
x=3
,
x=0,y=1
Page32
3 FindtheHorizontalorObliqueAsymptotesofaRationalFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Givetheequationofthehorizontalasymptote,ifany,ofthefunction.
1) h(x)=7x–6
x–2
A) y=7B)y=0
C) y=2D)nohorizontalasymptotes
2) g(x)=x2+2x–7
x–7
A) y=1B)y=7
C) y=0D)nohorizontalasymptotes
3) f(x)=7x2+2
7x2–2
A) y=1B)y=2
C) y=7D)nohorizontalasymptotes
4) h(x)=6x2–9x–4
3x2–2x+6
A) y=2B)y=0
C) y=9
2D) nohorizontalasymptotes
5) h(x)=8x3–6x–5
5x+5
A) y=8
5B) y=0
C) y=8D)nohorizontalasymptotes
6) g(x)=x+2
x2–36
A) y=0B)y=1
C) y=–6
,
y=6D)nohorizontalasymptotes
7) f(x)=x(x–1)
x3+4x
A) y=0B)x=0,x= –4
C) y=1D)nohorizontalasymptotes
8) R(x)=–3x2
x2+2x–120
A) y=–3B)y=0
C) y=–12
,
y=10 D) nohorizontalasymptotes
Page33
9) f(x)=x2–3
9x–x4
A) y=0B)y= –3
,
y=3
C) y=–1D)nohorizontalasymptotes
10) f(x)=49x5–7
x–x3
A) y=0B)y= –49
C) y=–1,y=1D)nohorizontalasymptotes
11) f(x)=–x2+16
x2+5x+4
A) y=–1B)y= –16
C) y=0D)nohorizontalasymptotes
Usethegraphtofindthehorizontalasymptote,ifany,ofthefunction.
12)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=5B)y=0,y=5C)y=0D)x=3
13)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) none B) y=–6C)y=6D)y=0
Page34
14)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=0B)y= –4
,
y=4
C) x=–4
,
x=4
,
y=0D)nohorizontalasymptotes
15)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=1B)y=0,y=1C)x= –2
,
x=2
,
y=1D)y= –2
,
y=2
Givetheequationoftheobliqueasymptote,ifany,ofthefunction.
16) f(x)=x2+4x–2
x–3
A) y=x+7B)y=x+1
C) x=y+7D)noobliqueasymptotes
17) h(x)=5x2–8x–5
6x2–3x+9
A) y=5
6B) y=5
6x
C) y=x+ 5
6D) noobliqueasymptote
18) f(x)=x2–6x+8
x+2
A) y=x–8B)y=x+14
C) x=y+6D)noobliqueasymptote
Page35
19) f(x)=x2+3x+3
x+8
A) y=x–5B)y=x–11
C) x=y–5D)noobliqueasymptotes
20) f(x)=2x3+11x2+5x–1
x2+6x+5
.
A) y=2x–1B)y=2x C) y=2x+1D)y=0
21) f(x)=x+5
x2–9
A) y=x+5B)y=0
C) y=5x D) noobliqueasymptote
22) f(x)=x2–3
9x–x4
A) y=0B)y=9x
C) y=x–3D)noobliqueasymptote
23) f(x)=6x3+25x2+22x+6
–3x–5
A) y=–2x2–5x+1B)y=0
C) y=–2x+1D)noobliqueasymptote
Usethegraphtofindtheobliqueasymptote,ifany,ofthefunction.
24)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=x+3B)y=4
C) y=4x+3D)noobliqueasymptote
Page36
25)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=xB)y= –x
C) y=x+1D)noobliqueasymptote
26)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=xB)y= –x
C) y=2x D) noobliqueasymptote
27)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=xB)y=1
C) y=x+1D)noobliqueasymptote
Page37
4 DemonstrateAdditionalUnderstandingandSkills
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Graphthefunctionusingtransformations.
1) f(x)=6
(2+x)2
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page38
2) f(x)=1
x
–3
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Page39
3) f(x)=–2
x–2
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Page40
4) f(x)=1
x–2
+1
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Page41
5) f(x)=1
x2
+4
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
6) f(x)=2–1
(x+3)2
x
-10 10
y
10
-10
x
-10 10
y
10
-10
A)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
B)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
C)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
D)
x
-10 10
y
10
-10
x
-10 10
y
10
-10
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Solvetheproblem.
7) Theaccelerationduetogravityg(inmeterspersecondpersecond)ataheighthmetersabovesealevelis
givenbyg(h)=3.99×1014
(6.374×106+h)2
where6.374×106istheradiusofEarthinmeters.DeathValleyin
Californiais86mbelowsealevel.
a) Findthevalueofg(h)atDeathValleytofourdecimalplaces.
b) Comparethevaluein(a)tothevalueofg(h)atsealevel.
Page43
8) Thedistanceformulastatesthatd=rt.Ifacardrives50miles,thefunctionr=50
t
isarationalfunction.
Findtheasymptotesofthisfunction.
9) Alenscanbeusedtocreateanimageofanobjectontheoppositesideofthelens,suchastheimage
createdonamoviescreen.Everylenshasameasurementcalleditsfocallength,f.Thedistances1ofthe
objecttothelensisrelatedtothedistances2ofthelenstotheimagebythefunction
s1=fs2
s2–f.
Foralenswithf=0.3m,whataretheasymptotesofthisfunction?
10) Whentwolensesareplacednexttoeachother,theircombinedfocallength(ameasurementthatcanbe
negativeorpositive)isdescribedbytheequation
f=f1f2
f1+f2.
Iff1=0.001,whataretheasymptotesofthisfunction?
5.3 TheGraphofaRationalFunction
1 AnalyzetheGraphofaRationalFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthedomainoftherationalfunction.
1) h(x)=7x
x+9
A) {x|x≠–9} B) {x|x≠9} C) {x|x≠0} D) allrealnumbers
2) h(x)=2x
(x+5)(x–9)
A) {x|x≠–5
,
9} B) {x|x≠5
,
–9} C) {x|x≠–5
,
9
,
–2} D) allrealnumbers
3) g(x)=x+8
x2–9
A) {x|x≠–3
,
3} B) {x|x≠–3
,
3
,
–8} C) {x|x≠0,9} D) allrealnumbers
4) h(x)=x+9
x2–9x
A) {x|x≠0,9} B) {x|x≠–3
,
3
,
–9} C) {x|x≠–3
,
3} D) allrealnumbers
5) R(x)=–3x2
x2+4x–96
A) {x|x≠–12
,
8} B) {x|x≠12
,
8} C) {x|x≠12
,
–8} D) {x|x≠–96
,
1}
6) f(x)=2x2–4
3x2+6x–24
A) {x|x≠–4
,
2} B) {x|x≠–2
,
4} C) {x|x≠–4
,
–2
,
2} D) allrealnumbers
Page44
7) f(x)=–2x(x+2)
5x2–3x–8
A) x x≠8
5,–1 B) x x≠–8
5,1 C) x x≠5
8,–1 D) x x≠–5
8,1
8) g(x)=x
x3–27
A) {x|x≠3} B) {x|x≠–3
,
3} C) {x|x≠9} D) {x|x≠–3}
Findtheindicatedintercept(s)ofthegraphofthefunction.
9) y–interceptoff(x)=x–6
3x–4
A) 0,3
2B) (0,6) C) 0,–2
3D) none
10) y–interceptoff(x)=7
x2–3x–23
A) 0,–7
23 B) (0,7) C) 0, 7
23 D) none
11) y–interceptoff(x)=11x
x2–19
A) (0,0) B) (0,11) C) 0,–11
19 D) none
12) y–interceptoff(x)=x–10
x2+14x–2
A) 0,5 B) (0,10) C) 0,–1
5D) none
13) y–interceptoff(x)= (5x–10)(x–4)
x2+8x–19
A) 0,–40
19 B) 0, 40
19 C) (0,2) D) (0,4)
14) y–interceptoff(x)=(x–5)2
(x+11)3
A) 0,25
1331 B) 0,–5
11 C) (0,5) D) 0,–25
1331
15) y–interceptoff(x)=23
(x+20)(x2–2)
A) 0,–23
40 B) 0,23 C) 0,23
40 D) none
Page45
16) y–interceptoff(x)=x
(x+15)(x–6)
A) (0,0) B) 0,–1
90 C) (0,6) D) none
17) y–interceptoff(x)=x2–8x
x2+5x–11
A) (0,0) B) 0,8
11 C) (0,8) D) 0,–11
8
18) y–interceptoff(x)=x2–7
x2+15x–8
A) 0,7
8B) (0,7) C) 0,–8
7D) none
19) y–interceptoff(x)=x2–8x+7
2x
A) 0,7
2B) (0,7) C) 0,–2
7D) none
20) y–interceptoff(x)=x2–4x+15
x2+13x–5
A) (0,–3) B) (0,15) C) (0,13) D) none
21) y–interceptoff(x)=x3+2
x2–1
A) (0,–2) B) (0,2) C) (0,3) D) none
22) y–interceptoff(x)=x+36
x
A) (0,6) B) (0,0) C) (0,36) D) none
23) x–interceptsoff(x)=2x+7
x–6
A) –7
2,0 B) (6
,
0) C) 7
2,0 D) (–6
,
0)
24) x–interceptsoff(x)=x–3
x2+9x–3
A) (3
,
0) B) (–3
,
0) C) (9
,
0) D) none
25) x–interceptsoff(x)=x2+6
x2+6x+6
A) (6
,
0) B) ( 6,0),(–6,0) C) (–6
,
0) D) none
26) x–interceptsoff(x)=5
x2–x–12
A) (3
,
0),(–4
,
0) B) (5,0) C) (4
,
0),(–3
,
0) D) none
Page46
27) x–interceptsoff(x)=8x
x2–25
A) (0,0) B) (–5
,
0),(5
,
0) C) (25
,
0) D) (8
,
0)
28) x–interceptsoff(x)=x2–25
8+x4
A) (–5
,
0),(5
,
0) B) (8
,
0) C) (25
,
0) D) none
29) x–interceptsoff(x)=x2+5x
x2+5x–2
A) (0,0),(–5
,
0) B) (–5
,
0) C) (0,0),(5
,
0) D) (5
,
0)
30) x–interceptsoff(x)=(x–2)(2x+5)
x2+2x–6
A) (2,0),–5
2,0 B) (–2,0),5
2,0 C) (2
,
0),(–5
,
0) D) none
31) x–interceptsoff(x)=x2–x–56
x2+5
.
A) (–7
,
0),(8
,
0) B) (–56
,
0) C) (–8
,
0),(0,0) D) (–8
,
0),(7
,
0)
32) x–interceptsoff(x)=x3–27
x2–25
A) (3
,
0) B) (–3
,
0),(3
,
0) C) (5,0) D) (–27
,
0)
33) x–interceptsoff(x)=x+ 64
x
A) (8
,
0) B) (–8
,
0),(8
,
0) C) (–64
,
0) D) none
Findtheverticalasymptotesoftherationalfunction.
34) f(x)=8x
x–3
A) x=3B)x=–3C)x=8 D) none
35) g(x)=x+4
x2–64
A) x=–8
,
x=8B)x= –8
,
x=8
,
x= –4
C) x=0,x=64 D) x=64
,
x= –4
36) f(x)=–2x(x+2)
4x2–3x–7
A) x=7
4,x=–1B)x=–7
4,x=1C)x=4
7,x=–1D)x=–4
7,x=1
Page47
Givetheequationofthehorizontalasymptote,ifany,ofthefunction.
37) g(x)=x2+4x–3
x–3
A) nohorizontalasymptote B) y=1
C) y=3D)y=0
38) h(x)=2x3–7x–2
5x+3
A) nohorizontalasymptote B) y=2
5
C) y=0D)y=2
Givetheequationoftheobliqueasymptote,ifany,ofthefunction.
39) h(x)=8x2–3x–4
2x2–5x+9
A) noobliqueasymptote B) y=4
C) y=4x D) y=x+4
40) f(x)=2x3+11x2+5x–1
x2+6x+5
A) y=2x–1B)y=2x C) y=2x+1D)y=0
Graphthefunction.
41) f(x)=3x
(x+3)(x–4)
x
-8 -4 4 8
y
40
20
-20
-40
x
-8 -4 4 8
y
40
20
-20
-40
A)
x
-8 -4 4 8
y
40
20
-20
-40
x
-8 -4 4 8
y
40
20
-20
-40
B)
x
-8 -4 4 8
y
40
20
-20
-40
x
-8 -4 4 8
y
40
20
-20
-40
Page48
C)
x
-8 -4 4 8
y
40
20
-20
-40
x
-8 -4 4 8
y
40
20
-20
-40
D)
x
-8 -4 4 8
y
40
20
-20
-40
x
-8 -4 4 8
y
40
20
-20
-40
Page49
42) f(x)=x+9
x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page50
43) f(x)=x2+4
x
x
-10 -5 5 10
y
50
25
-25
-50
x
-10 -5 5 10
y
50
25
-25
-50
A)
x
-10 -5 5 10
y
50
25
-25
-50
x
-10 -5 5 10
y
50
25
-25
-50
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
50
25
-25
-50
x
-10 -5 5 10
y
50
25
-25
-50
Page51
44) f(x)=x
x2–49
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
5
-5
x
-10 -5 5 10
y
5
-5
B)
x
-10 -5 5 10
y
5
-5
x
-10 -5 5 10
y
5
-5
C)
x
-10 -5 5 10
y
5
-5
x
-10 -5 5 10
y
5
-5
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page52
45) f(x)=x4–1
x2–81
x
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
y
300
240
180
120
60
-60
-120
-180
-240
-300
x
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
y
300
240
180
120
60
-60
-120
-180
-240
-300
A)
x
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
y
300
240
180
120
60
-60
-120
-180
-240
-300
x
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
y
300
240
180
120
60
-60
-120
-180
-240
-300
B)
x
-12 -10 -8 -6 -4 -2 2 4 6 8 10 12
y
300
240
180
120
60
-60
-120
-180
-240
-300
x
-12 -10 -8 -6 -4 -2 2 4 6 8 10 12
y
300
240
180
120
60
-60
-120
-180
-240
-300
C)
x
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
y
25
20
15
10
5
-5
-10
-15
-20
-25
x
-12-10 -8 -6 -4 -2 2 4 6 8 10 12
y
25
20
15
10
5
-5
-10
-15
-20
-25
D)
x
-12 -10 -8 -6 -4 -2 2 4 6 8 10 12
y
25
20
15
10
5
-5
-10
-15
-20
-25
x
-12 -10 -8 -6 -4 -2 2 4 6 8 10 12
y
25
20
15
10
5
-5
-10
-15
-20
-25
Page53
46) f(x)=x2+x–56
x2–x–42
x
–10–8-6-4-2 2 4 6 810
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
x
–10–8-6-4-2 2 4 6 810
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
A)
x
–10–8-6-4-2 2 4 6 810
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
x
–10–8-6-4-2 2 4 6 810
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
B)
x
–10–8-6-4-2 2 4 6 810
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
x
–10–8-6-4-2 2 4 6 810
y
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
C)
x
–10–8–6–4–2 246810
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8–6–4–2 246810
y
10
8
6
4
2
-2
-4
-6
-8
-10
D)
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–10–8-6-4-2 2 4 6 810
y
10
8
6
4
2
-2
-4
-6
-8
-10
Page54
47) f(x)=x2+9x+20
(x–2)2
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
B)
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
C)
x
-12 -8 -4 4 8 12
y
72
48
24
-24
-48
-72
x
-12 -8 -4 4 8 12
y
72
48
24
-24
-48
-72
D)
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
Page55
48) f(x)=(x–3)(x–3)
x2–36
x
-16 -8 8 16
y
6
4
2
-2
-4
-6
x
-16 -8 8 16
y
6
4
2
-2
-4
-6
A)
x
-16 -8 8 16
y
12
8
4
-4
-8
-12
x
-16 -8 8 16
y
12
8
4
-4
-8
-12
B)
x
-16 -8 8 16
y
6
4
2
-2
-4
-6
x
-16 -8 8 16
y
6
4
2
-2
-4
-6
C)
x
-12 -8 -4 4 8 12
y
4
2
-2
-4
x
-12 -8 -4 4 8 12
y
4
2
-2
-4
D)
x
-12 -8 -4 4 8 12
y
4
2
-2
-4
x
-12 -8 -4 4 8 12
y
4
2
-2
-4
Page56
49) f(x)=x2+7x+10
x2–1
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
B)
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
C)
x
-12 -8 -4 4 8 12
y
24
16
8
-8
-16
-24
x
-12 -8 -4 4 8 12
y
24
16
8
-8
-16
-24
D)
x
-12 -8 -4 4 8 12
y
24
16
8
-8
-16
-24
x
-12 -8 -4 4 8 12
y
24
16
8
-8
-16
-24
Page57
50) f(x)=x2+5x
(x–5)2
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
A)
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
B)
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
C)
x
-12 -8 -4 4 8 12
y
30
20
10
-10
-20
-30
x
-12 -8 -4 4 8 12
y
30
20
10
-10
-20
-30
D)
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
x
-12 -8 -4 4 8 12
y
12
8
4
-4
-8
-12
Page58
Solvetheproblem.
51) Decidewhichoftherationalfunctionsmighthavethegivengraph.
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A) f(x)=1–1
xB) f(x)=1+1
xC) f(x)=1
x
–1 D) f(x)=1–x
52) Decidewhichoftherationalfunctionsmighthavethegivengraph.
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A) f(x)=x+2
xB) f(x)=x+2 C) f(x)=x+1
xD) f(x)=2x+1
x
53) Decidewhichoftherationalfunctionsmighthavethegivengraph.
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A) f(x)=1
x2B) f(x)=1
xC) f(x)=1
2x D) f(x)=x2
Page59
54) Decidewhichoftherationalfunctionsmighthavethegivengraph.
x
–5–4–3–2–1 12345
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
–5–4–3–2–1 12345
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) R(x)=x–2
(x+2)(x–3) B) R(x)=x+2
(x–2)(x+3)
C) R(x)=x–2
(x+2)2(x–3)2D) R(x)=2–x
(x+2)(x–3)
55) DeterminewhichrationalfunctionR(x)hasagraphthatcrossesthex–axisat–1,touchesthex–axisat–4,
hasverticalasymptotesatx=–2andx=3,andhasonehorizontalasymptoteaty=–2.
A) R(x)=–2(x+1)(x+4)2
(x+2)2(x–3)
,x≠–2,3 B) R(x)=–2(x–3)(x+2)2
(x+4)2(x+1)
,x≠–4,–1
C) R(x)=–(x+1)(x+4)2
2(x–2)2(x+3)
,x≠2,–3 D) R(x)=–2(x+1)(x+4)
(x+2)(x–3) ,x≠–2,3
2 SolveAppliedProblemsInvolvingRationalFunctions
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Solvetheproblem.
1) ArarespeciesofinsectwasdiscoveredintherainforestofCostaRica.Environmentaliststransplantthe
insectintoaprotectedarea.Thepopulationoftheinsecttmonthsafterbeingtransplantedis
P(t)=45(1+0.6t)
(3+0.02t) .
a) Whatwasthepopulationwhent=0?
b) Whatwillthepopulationbeafter10years?
c) Whatisthelargestvaluethepopulationcouldreach?
2) TheconcentrationCofacertaindruginapatientʹsbloodstreamisgivenby
30t
t2+49
.
a) FindthehorizontalasymptoteofC(t).
b) Usingagraphingutility,determinethetimeatwhichtheconcentrationishighest.
Page60
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
3) Acanintheshapeofarightcircularcylinderisrequiredtohaveavolumeof700cubiccentimeters.The
topandbottomaremadeupofamaterialthatcosts8¢persquarecentimeter,whilethesidesaremadeof
materialthatcosts5¢persquarecentimeter.Whichfunctionbelowdescribesthetotalcostofthematerial
asafunctionoftheradiusrofthecylinder?
A) C(r)=0.16πr2+70
rB) C(r)=0.16πr2+140
r
C) C(r)=0.08πr2+140
rD) C(r)=0.08πr2+70
r
4) Theconcentrationofadruginthebloodstream,measuredinmilligramsperliter,canbemodeledbythe
function,C(t)=12t+4
3t2+2
,wheretisthenumberofminutesafterinjectionofthedrug.Whenwillthedrugbe
atitshighestconcentration?Approximateyouranswerroundedtotwodecimalplaces.
A) t=0.55minutesaftertheinjectionisgiven B) t=3.65minutesaftertheinjectionisgiven
C) atthetimeofinjectio
n
D) t=4minutesaftertheinjectionisgiven
5) Aclosedboxwithasquarebasehastohaveavolumeof3000 cubicinches.Findafunctionforthesurface
areaofthebox.
A) S(x)=2x2+12,000
xB) S(x)=2x2+18,000
x
C) S(x)=x2+12,000
xD) S(x)=2x2+3000
x
6) EconomistsusewhatiscalledaLeffercurvetopredictthegovernmentrevenuefortaxratesfrom0%to
100%.Economistsagreethattheendpointsofthecurvegenerate0revenue,butdisagreeonthetaxrate
thatproducesthemaximumrevenue.Supposeaneconomistproducesthisrationalfunction
R(x)=10x(100–x)
50+x,whereRisrevenueinmillionsatataxrateofxpercent.Useagraphingcalculator
tographthefunction.Whattaxrateproducesthemaximumrevenue?Whatisthemaximumrevenue?
A) 36.6%;$268million B) 41.2%;$264million
C) 34.0%;$271million D) 35.8%;$276million
7) EconomistsusewhatiscalledaLeffercurvetopredictthegovernmentrevenuefortaxratesfrom0%to
100%.Economistsagreethattheendpointsofthecurvegenerate0revenue,butdisagreeonthetaxrate
thatproducesthemaximumrevenue.Supposeaneconomistproducesthisrationalfunction
R(x)=10x(100–x)
25+x,whereRisrevenueinmillionsatataxrateofxpercent.Useagraphingcalculator
tographthefunction.Whattaxrateproducesthemaximumrevenue?Whatisthemaximumrevenue?
A) 30.9%;$382million B) 27.0%;$379million
C) 28.8%;$272million D) 38.4%;$383million
8) EconomistsusewhatiscalledaLeffercurvetopredictthegovernmentrevenuefortaxratesfrom0%to
100%.Economistsagreethattheendpointsofthecurvegenerate0revenue,butdisagreeonthetaxrate
thatproducesthemaximumrevenue.Supposeaneconomistproducesthisrationalfunction
R(x)=10x(100–x)
75+x,whereRisrevenueinmillionsatataxrateofxpercent.Useagraphingcalculator
tographthefunction.Whattaxrateproducesthemaximumrevenue?Whatisthemaximumrevenue?
A) 39.6%;$209million B) 34.9%;$207million
C) 37.5%;$210million D) 35.8%;$209million
Page61
9) EconomistsusewhatiscalledaLeffercurvetopredictthegovernmentrevenuefortaxratesfrom0%to
100%.Economistsagreethattheendpointsofthecurvegenerate0revenue,butdisagreeonthetaxrate
thatproducesthemaximumrevenue.Supposeaneconomistproducesthisrationalfunction
R(x)=10x(100–x)
15+x,whereRisrevenueinmillionsatataxrateofxpercent.Useagraphingcalculator
tographthefunction.Whattaxrateproducesthemaximumrevenue?Whatisthemaximumrevenue?
A) 26.5%;$469million B) 29.7%;$467million
C) 31.4%;$464million D) 28.1%;$470million
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
10) Aboxhasabasewhoselengthistwiceitswidth.Thevolumeoftheboxis7000cubicinches.
a) Findafunctionforthesurfaceareaofthebox.
b) Whatarethedimensionsoftheboxthatminimizessurfacearea?
11) Theformulay=fx
x–f
modelstherelationshipsneededtofocusanimagewhereyisthedistancebetween
thefilmandprojectorlens,xisthedistancebetweenthemovescreenandtheprojectorlens,andfisthe
focallength.
a) Sketchthegraphofthisrationalfunctionforf=5centimeters.
x
–10 102030
y
30
20
10
-10
x
–10 102030
y
30
20
10
-10
b) BobandCarolareshowinghomemovies.Usethegraphtodescribethedesireddistancebetweenthe
filmandtheprojectorlensasCarolmovestheprojectorfurtherfromthescreen.
Page62
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
12) Whichofthefollowingfunctionscouldhavethisgraph?
A) y=(x+1)(x–4)2
(x–2)2(x–6) B) y=(x–2)(x–6)2
(x+1)2(x–4)
C) y=(x–2)2(x–6)
(x+1)(x–4)2D) y=2(x–2)2(x–6)
(x+1)(x–4)2
5.4 PolynomialandRationalInequalities
1 SolvePolynomialInequalities
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheinequality.
1) (x+5)(x+1)>0
A) (–∞
,
–5)or(–1
,
∞)B)(
–5
,
–1) C) (–∞
,
–5) D) (–1
,
∞)
2) (x+2)(x–7)≤0
A) [–2
,
7] B) (–∞
,
–2] C) [7
,
∞)D)(
–∞
,
–2]or[7
,
∞)
3) x2–7x≥0
A) (–∞
,
0]or[7
,
∞) B) [0,7] C) (–∞
,
–7]or[0,∞)D)[
–7
,
0]
4) x2+7x≥0
A) (–∞
,
–7]or[0,∞) B) [0,7] C) (–∞
,
0]or[7
,
∞)D)[
–7
,
0]
5) x2+7x≤0
A) [–7
,
0] B) [0,7] C) (–∞
,
–7]or[0,∞)D)(
–∞
,
0]or[7
,
∞)
6) x2–5x≤0
A) [0,5] B) [–5
,
0] C) (–∞
,
–5]or[0,∞)D)(
–∞
,
0]or[5
,
∞)
7) x2–81>0
A) (–∞
,
–9)or(9
,
∞)B)(
–9
,
9)
C) (–∞
,
–81)or(81
,
∞)D)(
–81
,
81)
8) x2–9≤0
A) [–3
,
3] B) (–∞
,
–3]or[3
,
∞)C)(
–∞
,
–9]or[9
,
∞)D)[
–9
,
9]
9) 24(x2–1)>55x
A) –∞,–3
8
or8
3,∞B) –3
8,8
3C) –∞,–8
3
or3
8,∞D) –8
3,3
8
10) x2–6x≥–8
A) (–∞
,
2]or[4
,
∞)B)[2
,
4] C) (–∞
,
2] D) [4
,
∞)
11) x(x–3)≥4
A) (–∞
,
–1]or[4
,
∞)B)[
–1
,
4] C) (–∞
,
–1] D) [4
,
∞)
12) 3x2+13x<10
A) –5,2
3B) (–∞,–5)or2
3,∞C) –∞,2
3D) (–5
,
∞)
13) (a+3)(a+2)(a–1)>0
A) (–3
,
–2)or(1
,
∞)B)(
–∞
,
–3)or(–2
,
1) C) (1
,
∞)D)(
–∞
,
–2)
14) (b+5)(b–2)(b–7)
<
0
A) (–∞
,
–5)or(2
,
7) B) (7
,
∞)C)(
–∞
,
2) D) (–5
,
2)or(7
,
∞)
15) (x+1)(x2+x+1)>
A) (–1
,
∞)B)(
–∞
,
–1) C) (–1
,
1) D) (–∞
,
–1)or(1,∞)
16) x2–4x+3>0
A) (–∞
,
1)or(3
,
∞)B)(1
,
3) C) (–∞
,
1) D) (3
,
∞)
17) x2–4x–5≤0
A) [–1
,
5] B) (–∞
,
–1] C) [5
,
∞)D)(
–∞
,
–1]or[5
,
∞)
18) x3+5x2–24x>0
A) (–8
,
0)or(3
,
∞)B)(
–∞
,
–8)or(0,3) C) (–3
,
0)or(8
,
∞)D)(
–8
,
∞)
19) x(x+3)(5–x)≥0
A) (–∞
,
–3]or[0,5] B) [–3,0]or[5,∞)C)[
–3,5] D) [0,5]
20) x4<9x2
A) (–3
,
0)or(0,3) B) (–∞
,
–3)or(3
,
∞)C)(
–3
,
0)or(3
,
∞)D)(
–∞
,
–3)or(0,3)
21) x3>4x2
A) (4
,
∞) B) (0,4) C) (–∞
,
0)or(4
,
∞)D)(
–∞
,
4)
22) x4–5x2–36>0
A) (–∞
,
–3)or(3
,
∞)B)(
–3
,
3)
C) (–∞
,
–3)or(–2,2)or(3
,
∞)D)(
–3
,
–2)or(2,3)
23) x3≥27
A) [3
,
∞)B)(
–∞
,
3] C) (–∞
,
–3]or[3
,
∞)D)[
–3
,
3]
Page64
Solvetheproblem.
24) Forwhatpositivenumberswillthecubeofanumberexceed9 timesitssquare?
A) {x|x>9} B) x|0
<
x
<
9} C) x|x>81} D) x|0
<
x
<
81}
25) Aballisthrownverticallyupwardwithaninitialvelocityof192 feetpersecond.Thedistanceinfeetofthe
ballfromthegroundaftertsecondsiss=192t–16t2.Forwhatintervaloftimeistheballmorethan560
abovetheground?
A) {x∣5sec
<
x
<
7sec} B) {x∣4.5 sec
<
x
<
7.5sec}
C) {x∣11sec
<
x
<
13 sec} D) {x∣5.5 sec
<
x
<
6.5sec}
26) Aballisthrownverticallyupwardwithaninitialvelocityof128 feetpersecond.Thedistanceinfeetofthe
ballfromthegroundaftertsecondsiss=128t–16t2.Forwhatintervalsoftimeistheballlessthan112
abovetheground(afteritistosseduntilitreturnstotheground)?
A) {x∣0sec
<
x
<
1secand7sec>x>8 sec} B) {x∣1 sec
<
x
<
7 sec}
C) {x∣0sec
<
x
<
0.5secand7.5sec>x>8 sec} D) {x∣0sec
<
x
<
3.5secand4.5sec>x>8 sec}
27) Therevenueachievedbysellingxgraphingcalculatorsisfiguredtobex(42 –0.2x)dollars.Thecostof
eachcalculatoris$14.Howmanygraphingcalculatorsmustbesoldtomakeaprofit(revenue–cost)ofat
least$946.20?
A) {x∣57
<
x
<
83} B) {x∣22
<
x
<
48} C) {x∣58
<
x
<
56} D) {x∣59
<
x
<
81}
28) Therevenueachievedbysellingxgraphingcalculatorsisfiguredtobex(49 –0.5x)dollars.Thecostof
eachcalculatoris$37.Howmanygraphingcalculatorsmustbesoldtomakeaprofit(revenue–cost)ofat
least$59.50?
A) {x∣7
<
x
<
17} B) {x∣10
<
x
<
20} C) {x∣8
<
x
<
16} D) {x∣9
<
x
<
15}
2 SolveRationalInequalities
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheinequality.
1) x–1
x+2
<0
A) (–2
,
1) B) (–∞
,
–2)or(1
,
∞)C)(1
,
∞)D)(
–∞
,
–2)
2) x–1
x+2
>0
A) (–∞
,
–2)or(1
,
∞)B)(
–2
,
1) C) (1
,
∞)D)(
–∞
,
–2)
3) x–2
x+7
<1
A) (–7
,
∞)B)(
–∞
,
–7)or(2
,
∞)C)(
–7
,
2) D) (–∞
,
–7)
4) x+15
x+9
<3
A) –∞
,
–9or–6
,
∞B) –9
,
–6 C) –∞
,
–6or 9
,
∞D) –∞
,
–9 or9
,
∞
5) x+40
x
<14
A) (–∞
,
0)or(4
,
10) B) (0,4)or(10
,
∞) C) (0,4)or(4
,
10) D) (–∞
,
0) or(10
,
∞)
Page65
6) (x–6)(x+6)
x
≤0
A) (–∞
,
–6]or(0,6] B) [–6
,
0)or(0,6] C) (–∞
,
–6]or[6
,
∞)D)[
–6
,
0)or[6
,
∞)
7) (x+11)(x–8)
x–1
≥0
A) [–11
,
1)or[8
,
∞)B)(
–∞
,
–11]or(1,8] C) (–∞
,
–11]or[8
,
∞)D)[
–11
,
1)or(1,8]
8) (x–2)2
x2–9
>0
A) (–∞
,
–3)or(3
,
∞)B)(
–3
,
2)or(2
,
3) C) (–∞
,
–3)or(2
,
3) D) (–3
,
2)or(3
,
∞)
9) (x–1)(3–x)
(x–2)2
≤0
A) (–∞
,
1]or[3,∞)B)(
–∞
,
–3]or(–2,–1)or[1,∞)
C) (–∞
,
–3)or(–1,∞)D)(
–∞
,
1)or(3,∞)
10) 3x
7–x
<x
A) (0,4)or(7
,
∞)B)(4
,
7) C) (7
,
∞)D)(
–∞
,
4)or(7
,
∞)
11) 2x
7–x
≥2x
A) (–∞
,
0]or[6
,
7) B) [0,6]or[7
,
∞)C)[7
,
∞)D)(
–∞
,
6]or[7
,
∞)
12) 14
x–6
>12
x–1
A) (–29
,
1)or(6
,
∞)B)(
–∞
,
–29)or(1
,
6) C) (–∞
,
–29)or(6
,
∞)D)(
–29
,
1)or(1
,
6)
13) x2(x–10)(x+1)
(x–6)(x+9)
≥0
A) (–∞
,
–9)or[–1
,
6)or[10
,
∞)B)(
–9
,
–1]or(6
,
10]
C) (–∞
,
–9)or[10
,
∞)D)(
–∞
,
–9)or[–1
,
0)or(0,6)or[10
,
∞)
14) 5x2–3x–8
x+5
≤0
A) (–∞,–5)or–1,8
5B) (–∞,–5]or–1,8
5C) (–5,–1]or8
5,∞D) (–∞,–1]or8
5,∞
5.5 TheRealZerosofaPolynomialFunction
1 UsetheRemainderandFactorTheorems
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
UsetheRemainderTheoremtofindtheremainderwhenf(x)isdividedbyx–c.
1) f(x)=x4+8x3+12x2;x+1
A) R=5B)R=–5C)R=21 D) R= –21
Page66
2) f(x)=5x6–3x3+8;x+1
A) R=16 B) R=10 C) R=8D)R=6
UsetheFactorTheoremtodeterminewhetherx–cisafactoroff(x).
3) f(x)=x3+3x2–8x+10;x+5
A) Yes B) No
4) f(x)=x3+5x2–12x+14;x–7
A) Yes B) No
5) f(x)=x4–5x2–36;x–3
A) Yes B) No
6) f(x)=x4–32x2–144;x–12
A) Yes B) No
7) f(x)=x4+7x3+4x2+26x–14;x+7
A) Yes B) No
8) f(x)=x4+9x3+7x2+55x–72;x–9
A) Yes B) No
9) f(x)=24x3+82x2–37x–55;x+11
3
A) Yes B) No
10) f(x)=42x3+65x2–34x–22;x–11
6
A) Yes B) No
11) f(x)=9x4+26x3–3x2+x–3;x+3
A) Yes B) No
12) f(x)=7x4+20x3–3x2+x+3;x+3
A) Yes B) No
13) f(x)=3x3+11x2–19x+5;x+5
A) Yes B) No
14) f(x)=7x3+17x2–11x–3;x+3
A) Yes B) No
Page67
2 UsetheRationalZerosTheoremtoListthePotentialRationalZerosofaPolynomialFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Listthepotentialrationalzerosofthepolynomialfunction.Donotfindthezeros.
1) f(x)=11x4–x2+2
A) ±1
11,±2
11,±1,±2B)
±1
2,±11
2,±1,±11
C) ±1
11,±2
11,±1,±2,±11 D) ±1
11,±1
2,±1,±2,±11
2) f(x)=6x4+3x3–4x2+2
A) ±1
6,±1
3,±1
2,±2
3,±1,±2B)
±1
6,±1
3,±1
2,±2
3,±1,±2,±3
C) ±1
6,±1
3,±1
2,±1,±2D)
±1
2,±3
2,±1,±2,±3,±6
3) f(x)=–2x3+4x2–3x+8
A) ±1
2,±1,±2,±4,±8B)±1
4,±1
2,±1,±2,±4,±8
C) ±1
8,±1
4,±1
2,±1,±2,±4,±8D)±1
2,±1,±2,±4
4) f(x)=–4x4+3x2–4x+6
A) ±1
4,±1
2,±3
4,±3
2,±1,±2,±3,±6B)±1
6,±1
2,±1
3,±2
3,±4
3,±1,±2,±4
C) ±1
4,±1
2,±3
4,±3
2,±1,±2,±3,±4,±6D)±1
4,±1
2,±2
3,±3
4,±3
2,±1,±2,±3,±6
5) f(x)=3x5–4x2+3x–1
A) ±1,±1
3B) ±1,±3C)
±1,±3,±1
3D) ±3,±1
3
6) f(x)=x5–4x2+4x+2
A) ±1,±2B)
±1,±1
2C) ±1
4,±1
2,±2D)
±2,±1
2
7) f(x)=x5–4x2+5x+21
A) ±1,±7
,
±3
,
±21 B) ±1,±1
7,±1
3,±1
21
C) ±1,±1
7,±1
3,±1
21,±7,±3,±21 D) ±1,±7
,
±3
Page68
3 FindtheRealZerosofaPolynomialFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
UsetheRationalZerosTheoremtofindalltherealzerosofthepolynomialfunction.Usethezerostofactorfover
therealnumbers.
1) f(x)=x4+7x2–144
A) –3,3;f(x)=(x–3)(x+3)(x2+16) B) –4,4;f(x)=(x–4)(x+4)(x2+9)
C) 3;f(x)=(x–3)2(x2+16) D) –3
,
–4
,
3
,
4;f(x)=(x–3)(x+3)(x–4)(x+4)
2) f(x)=x3+3x2–4x–12
A) –3
,
–2
,
2;f(x)=(x+3)(x+2)(x–2) B) –2
,
2
,
3;f(x)=(x+2)(x–2)(x–3)
C) –3;f(x)=(x+3)(x2–x–4) D) –2;f(x)=(x+2)(x2+x–6)
3) f(x)=5x3–9x2–6x+8
A) –1,4
5,2;f(x)=(5x–4)(x–2)(x+1) B) –2,4
5,1;f(x)=(5x–4)(x–1)(x+2)
C) –1,5
4,–2;f(x)=(5x–4)(x–2)(x+1) D) 1,5
4,–2;f(x)=(5x–4)(x–1)(x+2)
4) f(x)=5x3–4x2+25x–20
A) 4
5;f(x)=(5x–4)(x2+5) B) 5,4
5,1;f(x)=(5x–4)(x–1)(x–5)
C) –5,–1,4
5;f(x)=(5x–4)(x+1)(x+5) D) 20;f(x)=(x–20)(5x2+1)
5) f(x)=4x4–7x3+11x2–14x+6
A) 1,3
4;f(x)=(x–1)(4x–3)(x2+2)
B) 2,3
4;f(x)=(x–2)(4x–3)(x2+1)
C) –2,–1,1,3
4;f(x)=(x–1)(4x–3)(x+1)(x+2)
D) –2,–1,1,–3
4;f(x)=(x–1)(4x+3)(x+1)(x+2)
6) f(x)=4x4–8x3+5x2–2x+1
A) 1,multiplicity2;f(x)=(x–1)2(4x2+1) B) –1,1;f(x)=(x–1)(x+1)(4x2+1)
C) norealroots;f(x)=(x2+1)(4x2+1) D) –1,multiplicity2;f(x)=(x+1)2(4x2+1)
Findtheinterceptsofthefunctionf(x).
7) f(x)=x3+2x2–5x–6
A) x–intercepts:–3
,
–1
,
2;y–intercept:–6B)x
–intercepts:–2
,
1
,
3;y–intercept:–6
C) x–intercept:–3;y–intercept:–6D)x
–intercept:–1;y–intercept:–6
Page69
8) f(x)=4x3–19x2+19x+6
A) x–intercepts:–1
4,2,3;y–intercept:6B)x
–intercepts:1
4,2,–3;y–intercept:6
C) x–intercepts:3
4,–1,2;y–intercept:6D)x
–intercepts:–3
4,–1,–2;y–intercept:6
9) f(x)=x3–4x2–x+4
A) x–intercepts:1,–1,4;y–intercept:4B)x
–intercepts:–1,1
,
–4;y–intercept:4
C) x–intercepts:1,1
,
4;y–intercept:4D)x
–intercepts:1,–1,–4;y–intercept:4
10) f(x)=2x3–x2–14x+7
A) x–intercepts:1
2,7,–7;y–intercept:7B)x
–intercepts:–1
2,7,–7;y–intercept:7
C) x–intercepts:2,7,–7;y–intercept:7D)x
–intercepts:–2,7,–7;y–intercept:7
11) f(x)=3x4–5x3+11x2–15x+6
A) x–intercepts:1,2
3;y–intercept:6B)x
–intercepts:3,2
3;y–intercept:6
C) x–intercepts:–3,–1,1,2
3;y–intercept:6D)x
–intercepts:–3,–1,1,–2
3;y–intercept:6
12) f(x)=2x4–4x3+3x2–2x+1
A) x–intercept:1;y–intercept:1B)x
–intercepts:–1
,
1;y–intercept:1
C) x–intercepts:none;y–intercept:1D)x
–intercept:–1;y–intercept:1
13) f(x)=4x2(x–2)3
A) x–intercepts:0,2;y–intercept:0B)x
–intercepts:0,2;y–intercept:4
C) x–intercepts:0,–2;y–intercept:4D)x
–intercepts:0,–2;y–intercept:0
14) f(x)=(x+5)(x–3)(x+3)
A) x–intercepts:–5
,
–3
,
3;y–intercept:–45 B) x–intercepts:–3
,
3
,
5;y–intercept:45
C) x–intercepts:–5
,
–3
,
3;y–intercept:45 D) x–intercepts:–3
,
3
,
5;y–intercept:–45
15) f(x)=6x–x3
A) x–intercepts:0,6,–6;y–intercept:0B)x
–intercepts:0,6,–6;y–intercept:6
C) x–intercepts:0,–6;y–intercept:0D)x
–intercepts:0,–6;y–intercept:6
16) f(x)=(x+1)(x–5)(x–1)2
A) x–intercepts:–1,1,5;y–intercept:–5B)x
–intercepts:–1,1,5;y–intercept:5
C) x–intercepts:–1,1,–5;y–intercept:–5D)x
–intercepts:–1,1,–5;y–intercept:5
17) f(x)=–x2(x+8)(x2–1)
A) x–intercepts:–8
,
–1,0,1;y–intercept:0B)x
–intercepts:–1,0,1,8;y–intercept:0
C) x–intercepts:–8
,
0,1;y–intercept:–8D)x
–intercepts:–8
,
–1,0,1;y–intercept:–8
18) f(x)=–x2(x+7)(x2+1)
A) x–intercepts:–7
,
0;y–intercept:0B)x
–intercepts:–7
,
–1,0,1;y–intercept:0
C) x–intercepts:–7
,
–1,0;y–intercept:7D)x
–intercepts:–7
,
–1,0;y–intercept:–7
Page70
19) f(x)=x2(x–6)(x–4)
A) x–intercepts:0,6
,
4;y–intercept:0B)x
–intercepts:0,–6
,
–4;y–intercept:0
C) x–intercepts:0,6
,
4;y–intercept:24 D) x–intercepts:0,–6
,
–4;y–intercept:24
20) f(x)=(x–4)2(x2–25)
A) x–intercepts:–5
,
4
,
5;y–intercept:–400 B) x–intercepts:–5
,
4
,
5;y–intercept:400
C) x–intercepts:4
,
25;y–intercept:100 D) x–intercepts:–4
,
–25;y–intercept:100
4 SolvePolynomialEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheequationintherealnumbersystem.
1) x3+2x2–5x–6=0
A) {–3
,
–1
,
2} B) {–2
,
1
,
3} C) {–3
,
–1} D) {1
,
3}
2) x3+6x2–14x+16=0
A) {–8} B) {8} C) {–8
,
8} D) {1}
3) 2x3–x2–14x+7=0
A) 1
2,7,–7 B) –1
2,7,–7 C) {2,7,–7}D){–2,7,–7}
4) 2x3–9x2+7x+6=0
A) –1
2,2,3 B) 1
2,2,–3 C) 3
2,–1,2 D) –3
2,–1,–2
5) 2x3–x2+2x–1=0
A) 1
2B) 1
2,–1 C) –2,1
2,–1 D) –2,–1
2,–1
6) x4–3x3+5x2–x–10=0
A) {–1,2} B) {–2,1} C) {–1,–2} D) {1,2}
7) x4–12x2–64=0
A) {–4
,
4} B) {–2,2} C) {–8
,
8} D) {–4
,
–2,2,4}
8) x4–8x3+16x2+8x–17=0
A) {–1,1} B) {–4,4} C) {–1,4} D) {–4,1}
9) 2x4–2x3+x2–5x–10=0
A) {–1,2} B) {1,–2} C) –10
2,10
2D) –5
2,5
2
10) 2x4–15x3+57x2–103x+39=0
A) 3,1
2B) –3,–1
2C) 3,–1
2D) –3,1
2
Page71
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Theequationhasasolutionrintheintervalindicated.Approximatethissolutioncorrecttotwodecimalplaces.
11) x3–8x–3=0;–3≤r≤–2
12) x3–8x–3=0;–1≤r≤0
13) x4–x3–7x2+5x+10=0;–3≤r≤–2
14) x4–x3–7x2+5x+10=0;2<r≤3
5 UsetheTheoremforBoundsonZeros
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findaboundontherealzerosofthepolynomialfunction.
1) f(x)=x4–7x2–18
A) –19and19 B) –18and18 C) –25 and25 D) –26 and26
2) f(x)=x5+4x4+3x3+2x2–3x+4
A) –5and5B)
–4and4C)
–11 and11 D) –16 and16
3) f(x)=6x3–x2+0.3x–0.06
A) –1and1B)
–1.36and1.36 C) –6 and6D)
–2and2
6 UsetheIntermediateValueTheorem
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
UsetheIntermediateValueTheoremtodeterminewhetherthepolynomialfunctionhasazerointhegiven
interval.
1) f(x)=6x3+8x2–8x–7;[1,2]
A) f(1)=–1andf(2)=57;yes B) f(1)=1 andf(2)=57;no
C) f(1)=–1andf(2)=–57;no D) f(1)=1 andf(2)=–57;yes
2) f(x)=3x5–4x3+3x2–5;[1,2]
A) f(1)=–3andf(2)=71;yes B) f(1)=3 andf(2)=71;no
C) f(1)=–3andf(2)=–71;no D) f(1)=3 andf(2)=–71;yes
3) f(x)=4x4–7x2–1;[1,2]
A) f(1)=–4andf(2)=35;yes B) f(1)=4 andf(2)=36;no
C) f(1)=–4andf(2)=–35;no D) f(1)=4 andf(2)=–35;yes
4) f(x)=9x4–7x3–4x–2;[–1,0]
A) f(–1)=18andf(0)=–2;yes B) f(–1)=18 andf(0)=2;no
C) f(–1)=–18andf(0)=–2;no D) f(–1)= –18 andf(0)=2;yes
5) f(x)=7x3+8x+8;[–1,0]
A) f(–1)=–7andf(0)=8;yes B) f(–1)= –7 andf(0)=–8;no
C) f(–1)=7andf(0)=8;no D) f(–1)=7 andf(0)=–8;yes
Page72
Solvetheproblem.
6) Thepolyniomialfunctionf(x)=6x3+25x2+12x–7hasexactlyonepositivezero.
UsetheIntermediateValueTheoremtoapproximatethezerocorrectto2decimalplaces.
A) 0.33 B) 0.50 C) 0.66 D) 0.10
5.6 ComplexZeros;FundamentalTheoremofAlgebra
1 UsetheConjugatePairsTheorem
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Informationisgivenaboutapolynomialf(x)whosecoefficientsarerealnumbers.Findtheremainingzerosoff.
1) Degree3;zeros:5
,
4–i
A) 4+iB)
–5C)
–4+iD)nootherzeros
2) Degree4;zeros:i,2+i
A) –i,2–iB)2–iC)
–2+i,2–iD)
–i,–2+i
3) Degree4;zeros:2–5i,8i
A) 2+5i,–8i B) –2+5i,–8i C) –2–5i,–8i D) 2+5i,8–i
4) Degree3;zeros:–5
,
5–5i
A) 5+5i B) –5+5i C) 5
,
5+5i D) 5
,
–5+5i
5) Degree5;zeros:8
,
4+5i,–8i
A) 4–5i,8i B) –4–5i,8i C) –4+5i,8i D) –8
,
4–5i,8i
6) Degree5;zeros:2
,
i,–2i
A) –i,2i B) –2
,
–iC)
–2
,
2i D) –2
,
–i,2i
7) Degree6;zeros:2
,
2+i,–1–i,0
A) 2–i,–1+iB)
–2+i,1–iC)
–2
,
2–i,–1+iD)
–2–i,1+i
8) Degree6;zeros:–5
,
2
,
5–5i,–2+i
A) 5+5i,–2–iB)
–5+5i,2–iC)5
,
5+5i D) 5
,
5+5i,–2–i
2 FindaPolynomialFunctionwithSpecifiedZeros
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Formapolynomialf(x)withrealcoefficientshavingthegivendegreeandzeros.
1) Degree3:zeros:1+iand–8
A) f(x)=x3+6x2–14x+16 B) f(x)=x3+6x2+16x–14
C) f(x)=x3+x2–14x+16 D) f(x)=x3–8x2–14x–12
2) Degree:3;zeros:–4and3–2i
A) f(x)=x3–2x2–11x+52 B) f(x)=x3–x2–11x+52
C) f(x)=x3–x2+11x+52 D) f(x)=x3–2x2+5x–52
3) Degree:3;zeros:–2and3+i.
A) f(x)=x3–4x2–2x+20 B) f(x)=x3–8x2+2x+20
C) f(x)=x3–6x2–10x+20 D) f(x)=x3–4x2–10x+20
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4) Degree:4;zeros:–1,2,and1–2i.
A) f(x)=x4–3x3+5x2–x–10 B) f(x)=x4–x3+x2+9x–10
C) f(x)=x4–x3+3x2–5x–10 D) f(x)=x4–3x3–3x2+7x+6
5) Degree:4;zeros:2iand–4i
A) f(x)=x4+20x2+64 B) f(x)=x4+20x2–4x+64
C) f(x)=x4–4x2+64 D) f(x)=x4–2x3+20x2+64
6) Degree:4;zeros:1,–1,and4–2i
A) f(x)=x4–8x3+16x2+8x–17 B) f(x)=x4–8x3+16x2+8x+17
C) f(x)=x4+8x3+16x2–8x–17 D) f(x)=x4+8x3+16x2–8x+17
7) Degree:5;zeros:2,–3i,and4–i
A) f(x)=x5–10x4+42x3–124x2+297x–306 B) f(x)=x5–10x4–42x3–124x2+297x+306
C) f(x)=x5–10x4+26x3–124x2–72x–306 D) f(x)=x5–10x4+26x3–124x2+72x+306
3 FindtheComplexZerosofaPolynomialFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Usethegivenzerotofindtheremainingzerosofthefunction.
1) f(x)=x4–32x2–144;zero:–2i
A) 2i,6
,
–6 B) 2i,6i,–6i C) 2i,12
,
–12 D) 2i,12i,–12i
2) f(x)=x3+7x2–16x+18;zero:1+i
A) 1–i,–9B)1–i,9C)
–9
,
9D)1–i,9i
3) f(x)=x3–2x2–11x+52;zero:–4
A) 3+2i,3–2i B) 6+4i,6–4i
C) 1+213
i,1–213iD)1+2i,1–2i
4) f(x)=x3–8x2+25x–26;zero:3+2i
A) 3–2i,2B)2–3i,2C)2
–3i,–2D)3–2i,–2
5) f(x)=2x4–19x3+71x2–109x+39;zero:3+2i
A) 3–2i,3,1
2B) 2–3i,–3,–1
2C) 2–3i,3,–1
2D) 3–2i,–3,1
2
6) f(x)=x5–10x4+42x3–124x2+297x–306;zero:3i
A) 2,–3i,4–i,4+iB)2,–3i,–4–i,–4+i
C) –2,–3i,4–i,4+iD)
–2,–3i,–4–i,–4+i
Findallzerosofthefunctionandwritethepolynomialasaproductoflinearfactors.
7) f(x)=x3–x2+9x–9
A) f(x)=(x–1)(x+3i)(x–3i) B) f(x)=(x–1)(x+3)(x–3)
C) f(x)=(x–1)(x+1)(x+9) D) f(x)=(x–25)(x+i)(x–i)
8) f(x)=x3+6x2+13x+10
A) f(x)=(x+2)(x+2+i)(x+2–i) B) f(x)=(x+2)(x+2+i)(x–2–i)
C) f(x)=(x+1)(x+2+i3
)(x–2–i3) D) f(x)=(x–1)(x+2+i3)(x+2–i3)
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9) f(x)=x3+4x2+2x–28
A) f(x)=(x–2)(x+3+i5
)(x+3–i5) B) f(x)=(x–2)(x+3+i5)(x–3–i5)
C) f(x)=(x+2)(x+5+3i)(x+5–3i) D) f(x)=(x+2)(x+5+3i)(x–5–3i)
10) f(x)=x4+29x2+100
A) f(x)=(x+2i)(x–2i)(x+5i)(x–5i) B) f(x)=(x+2i)2(x+5i)2
C) f(x)=(x+i)(x–i)(x+10i)(x–10i) D) f(x)=(x+2+5i)2(x+2–5i)2
11) f(x)=x4+2x3–4x2+8x–32
A) f(x)=(x+4)(x–2)(x–2i)(x+2i) B) f(x)=(x–1)(x+8)(x–2i)(x+2i)
C) f(x)=(x–i8
)(x+i8)(x–2)(x+2) D) f(x)=(x–4)(x–2)(x–2)(x+2)
12) f(x)=3x4–7x3+29x2–63x+18
A) f(x)=(3x–1)(x–2)(x+3i)(x–3i) B) f(x)=(3x+1)(x+2)(x+3i)(x–3i)
C) f(x)=(3x–1)(x–2)(x+3)(x–3) D) f(x)=(3x+1)(x+2)(x+3)(x–3)
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Ch.5 PolynomialandRationalFunctions
AnswerKey
5.1 PolynomialFunctionsandModels
1 IdentifyPolynomialFunctionsandTheirDegree
2 GraphPolynomialFunctionsUsingTransformations
3 IdentifytheRealZerosofaPolynomialFunctionandTheirMultiplicity
4 AnalyzetheGraphofaPolynomialFunction
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5 BuildCubicModelsfromData
5.2 PropertiesofRationalFunctions
1 FindtheDomainofaRationalFunction
2 FindtheVerticalAsymptotesofaRationalFunction
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3 FindtheHorizontalorObliqueAsymptotesofaRationalFunction
4 DemonstrateAdditionalUnderstandingandSkills
5.3 TheGraphofaRationalFunction
1 AnalyzetheGraphofaRationalFunction
2 SolveAppliedProblemsInvolvingRationalFunctions
5.4 PolynomialandRationalInequalities
1 SolvePolynomialInequalities
2 SolveRationalInequalities
5.5 TheRealZerosofaPolynomialFunction
1 UsetheRemainderandFactorTheorems
2 UsetheRationalZerosTheoremtoListthePotentialRationalZerosofaPolynomialFunction
3 FindtheRealZerosofaPolynomialFunction
4 SolvePolynomialEquations
5 UsetheTheoremforBoundsonZeros
6 UsetheIntermediateValueTheorem
5.6 ComplexZeros;FundamentalTheoremofAlgebra
1 UsetheConjugatePairsTheorem
2 FindaPolynomialFunctionwithSpecifiedZeros
3 FindtheComplexZerosofaPolynomialFunction
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