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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

what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany 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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