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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

how to convert general to standard form with not perfect trinomial
Camalia Reply
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
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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