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f ( x ) = 2 x 3 9 x 2 + 13 x 6 ;   x 1

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f ( x ) = 2 x 3 + x 2 5 x + 2 ;   x + 2

2 ,   1 ,   1 2

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f ( x ) = 3 x 3 + x 2 20 x + 12 ;   x + 3

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f ( x ) = 2 x 3 + 3 x 2 + x + 6 ; x + 2

2

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f ( x ) = 5 x 3 + 16 x 2 9 ; x 3

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x 3 + 3 x 2 + 4 x + 12 ; x + 3

3

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4 x 3 7 x + 3 ; x 1

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2 x 3 + 5 x 2 12 x 30 , 2 x + 5

5 2 ,   6 ,   6

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For the following exercises, use the Rational Zero Theorem to find all real zeros.

x 3 3 x 2 10 x + 24 = 0

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2 x 3 + 7 x 2 10 x 24 = 0

2 ,   4 ,   3 2

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x 3 + 2 x 2 9 x 18 = 0

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x 3 + 5 x 2 16 x 80 = 0

4 ,   4 ,   5

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x 3 3 x 2 25 x + 75 = 0

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2 x 3 3 x 2 32 x 15 = 0

5 ,   3 ,   1 2

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2 x 3 + x 2 7 x 6 = 0

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2 x 3 3 x 2 x + 1 = 0

1 2 ,   1 + 5 2 ,   1 5 2

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3 x 3 x 2 11 x 6 = 0

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2 x 3 5 x 2 + 9 x 9 = 0

3 2

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2 x 3 3 x 2 + 4 x + 3 = 0

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x 4 2 x 3 7 x 2 + 8 x + 12 = 0

2 ,   3 ,   1 ,   2

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x 4 + 2 x 3 9 x 2 2 x + 8 = 0

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4 x 4 + 4 x 3 25 x 2 x + 6 = 0

1 2 ,   1 2 ,   2 ,   3

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2 x 4 3 x 3 15 x 2 + 32 x 12 = 0

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x 4 + 2 x 3 4 x 2 10 x 5 = 0

1 ,   1 ,   5 ,   5

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8 x 4 + 26 x 3 + 39 x 2 + 26 x + 6

3 4 ,   1 2

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For the following exercises, find all complex solutions (real and non-real).

x 3 8 x 2 + 25 x 26 = 0

2 ,   3 + 2 i ,   3 2 i

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x 3 + 13 x 2 + 57 x + 85 = 0

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3 x 3 4 x 2 + 11 x + 10 = 0

2 3 ,   1 + 2 i ,   1 2 i

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x 4 + 2 x 3 + 22 x 2 + 50 x 75 = 0

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2 x 3 3 x 2 + 32 x + 17 = 0

1 2 ,   1 + 4 i ,   1 4 i

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Graphical

For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Confirm with the given graph.

f ( x ) = x 4 x 2 1

1 positive, 1 negative

Graph of f(x)=x^4-x^2-1.
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f ( x ) = x 3 2 x 2 5 x + 6

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f ( x ) = x 3 2 x 2 + x 1

3 or 1 positive, 0 negative

Graph of f(x)=x^3-2x^2+x-1.
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f ( x ) = x 4 + 2 x 3 12 x 2 + 14 x 5

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f ( x ) = 2 x 3 + 37 x 2 + 200 x + 300

0 positive, 3 or 1 negative

Graph of f(x)=2x^3+37x^2+200x+300.
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f ( x ) = x 3 2 x 2 16 x + 32

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f ( x ) = 2 x 4 5 x 3 5 x 2 + 5 x + 3

2 or 0 positive, 2 or 0 negative

Graph of f(x)=2x^4-5x^3-5x^2+5x+3.
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f ( x ) = 2 x 4 5 x 3 14 x 2 + 20 x + 8

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f ( x ) = 10 x 4 21 x 2 + 11

2 or 0 positive, 2 or 0 negative

Graph of f(x)=10x^4-21x^2+11.
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Numeric

For the following exercises, list all possible rational zeros for the functions.

f ( x ) = x 4 + 3 x 3 4 x + 4

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f ( x ) = 2 x 3 + 3 x 2 8 x + 5

± 5 ,   ± 1 ,   ± 5 2

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f ( x ) = 3 x 3 + 5 x 2 5 x + 4

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f ( x ) = 6 x 4 10 x 2 + 13 x + 1

± 1 ,   ± 1 2 ,   ± 1 3 ,   ± 1 6

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f ( x ) = 4 x 5 10 x 4 + 8 x 3 + x 2 8

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Technology

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

f ( x ) = 6 x 3 7 x 2 + 1

1 ,   1 2 ,   1 3

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f ( x ) = 4 x 3 4 x 2 13 x 5

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f ( x ) = 8 x 3 6 x 2 23 x + 6

2 ,   1 4 ,   3 2

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f ( x ) = 12 x 4 + 55 x 3 + 12 x 2 117 x + 54

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f ( x ) = 16 x 4 24 x 3 + x 2 15 x + 25

5 4

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Extensions

For the following exercises, construct a polynomial function of least degree possible using the given information.

Real roots: –1, 1, 3 and ( 2 , f ( 2 ) ) = ( 2 , 4 )

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Real roots: –1 (with multiplicity 2 and 1) and ( 2 , f ( 2 ) ) = ( 2 , 4 )

f ( x ) = 4 9 ( x 3 + x 2 x 1 )

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Real roots: –2, 1 2 (with multiplicity 2) and ( 3 , f ( 3 ) ) = ( 3 , 5 )

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Real roots: 1 2 , 0, 1 2 and ( 2 , f ( 2 ) ) = ( 2 , 6 )

f ( x ) = 1 5 ( 4 x 3 x )

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Real roots: –4, –1, 1, 4 and ( 2 , f ( 2 ) ) = ( 2 , 10 )

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Real-world applications

For the following exercises, find the dimensions of the box described.

The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.

8 by 4 by 6 inches

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The length, width, and height are consecutive whole numbers. The volume is 120 cubic inches.

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The length is one inch more than the width, which is one inch more than the height. The volume is 86.625 cubic inches.

5.5 by 4.5 by 3.5 inches

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The length is three times the height and the height is one inch less than the width. The volume is 108 cubic inches.

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The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.

8 by 5 by 3 inches

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For the following exercises, find the dimensions of the right circular cylinder described.

The radius is 3 inches more than the height. The volume is 16 π cubic meters.

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The height is one less than one half the radius. The volume is 72 π cubic meters.

Radius = 6 meters, Height = 2 meters

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The radius and height differ by one meter. The radius is larger and the volume is 48 π cubic meters.

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The radius and height differ by two meters. The height is greater and the volume is 28.125 π cubic meters.

Radius = 2.5 meters, Height = 4.5 meters

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80. The radius is 1 3 meter greater than the height. The volume is 98 9 π cubic meters.

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Questions & Answers

preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply
hil
Roy Reply
hi
Roy Reply
A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center
Abdulfatah Reply
Practice Key Terms 6

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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