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f ( x ) = x 5 2 x , between x = 1 and x = 2.

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f ( x ) = x 4 + 4 , between x = 1 and x = 3 .

f ( 1 ) = 3 and f ( 3 ) = 77. Sign change confirms.

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f ( x ) = −2 x 3 x , between x = –1 and x = 1.

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f ( x ) = x 3 100 x + 2 , between x = 0.01 and x = 0.1

f ( 0.01 ) = 1.000001 and f ( 0.1 ) = 7.999. Sign change confirms.

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For the following exercises, find the zeros and give the multiplicity of each.

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

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

0 with multiplicity 2, 3 2 with multiplicity 5, 4 with multiplicity 2

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

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

0 with multiplicity 2, –2 with multiplicity 2

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

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f ( x ) = ( 3 x + 2 ) 5 ( x 2 10 x + 25 )

2 3 with multiplicity 5 , 5 with multiplicity 2

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f ( x ) = x ( 4 x 2 12 x + 9 ) ( x 2 + 8 x + 16 )

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f ( x ) = x 6 x 5 2 x 4

0 with multiplicity 4 , 2 with multiplicity 1 , 1 with multiplicity 1

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f ( x ) = 3 x 4 + 6 x 3 + 3 x 2

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f ( x ) = 4 x 5 12 x 4 + 9 x 3

3 2 with multiplicity 2, 0 with multiplicity 3

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

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f ( x ) = 4 x 4 ( 9 x 4 12 x 3 + 4 x 2 )

0 with multiplicity 6 , 2 3 with multiplicity 2

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Graphical

For the following exercises, graph the polynomial functions. Note x - and y - intercepts, multiplicity, and end behavior.

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

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g ( x ) = ( x + 4 ) ( x 1 ) 2

x -intercepts, ( 1, 0 ) with multiplicity 2, ( 4 ,   0 ) with multiplicity 1, y - intercept ( 0 ,   4 ). As x , f ( x ) , as x , f ( x ) .

Graph of g(x)=(x+4)(x-1)^2.
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h ( x ) = ( x 1 ) 3 ( x + 3 ) 2

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k ( x ) = ( x 3 ) 3 ( x 2 ) 2

x -intercepts ( 3 , 0 ) with multiplicity 3, ( 2 , 0 ) with multiplicity 2, y - intercept ( 0 , 108 ) . As x , f ( x ) , as x , f ( x ) .

Graph of k(x)=(x-3)^3(x-2)^2.
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m ( x ) = 2 x ( x 1 ) ( x + 3 )

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n ( x ) = 3 x ( x + 2 ) ( x 4 )

x -intercepts ( 0 ,   0 ) ,   ( 2 ,   0 ) ,   ( 4 , 0 ) with multiplicity 1, y - intercept ( 0 ,   0 ) . As x , f ( x ) , as x , f ( x ) .

Graph of n(x)=-3x(x+2)(x-4).
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For the following exercises, use the graphs to write the formula for a polynomial function of least degree.

Graph of a negative odd-degree polynomial with zeros at x=-3, 1, and 3.

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

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Graph of a positive odd-degree polynomial with zeros at x=-2, and 3.

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

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For the following exercises, use the graph to identify zeros and multiplicity.

Graph of a negative even-degree polynomial with zeros at x=-4, -2, 1, and 3.

–4, –2, 1, 3 with multiplicity 1

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Graph of a positive even-degree polynomial with zeros at x=-2,, and 3.

–2, 3 each with multiplicity 2

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For the following exercises, use the given information about the polynomial graph to write the equation.

Degree 3. Zeros at x = –2, x = 1, and x = 3. y -intercept at ( 0 , –4 ) .

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

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Degree 3. Zeros at x = –5, x = –2 , and x = 1. y -intercept at ( 0 , 6 )

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Degree 5. Roots of multiplicity 2 at x = 3 and x = 1 , and a root of multiplicity 1 at x = –3. y -intercept at ( 0 , 9 )

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

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Degree 4. Root of multiplicity 2 at x = 4, and a roots of multiplicity 1 at x = 1 and x = –2. y -intercept at ( 0 , 3 ) .

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Degree 5. Double zero at x = 1 , and triple zero at x = 3. Passes through the point ( 2 , 15 ) .

f ( x ) = −15 ( x 1 ) 2 ( x 3 ) 3

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Degree 3. Zeros at x = 4 , x = 3 , and x = 2. y -intercept at ( 0 , −24 ) .

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Degree 3. Zeros at x = −3 , x = −2 and x = 1. y -intercept at ( 0 , 12 ) .

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

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Degree 5. Roots of multiplicity 2 at x = −3 and x = 2 and a root of multiplicity 1 at x = −2.

y -intercept at ( 0 ,   4 ) .

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Degree 4. Roots of multiplicity 2 at x = 1 2 and roots of multiplicity 1 at x = 6 and x = −2.

y -intercept at ( 0, 18 ) .

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

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Double zero at x = −3 and triple zero at x = 0. Passes through the point ( 1 , 32 ) .

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Technology

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.

f ( x ) = x 3 x 1

local max ( .58, – .62 ) , local min ( .58, –1 .38 )

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f ( x ) = 2 x 3 3 x 1

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f ( x ) = x 4 + x

global min ( .63, – .47 )

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f ( x ) = x 4 + 3 x 2

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f ( x ) = x 4 x 3 + 1

global min ( .75,  .89)

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Extensions

For the following exercises, use the graphs to write a polynomial function of least degree.

Graph of a positive odd-degree polynomial with zeros at x=--200, and 500 and y=50000000.

f ( x ) = ( x 500 ) 2 ( x + 200 )

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Real-world applications

For the following exercises, write the polynomial function that models the given situation.

A rectangle has a length of 10 units and a width of 8 units. Squares of x by x units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of x .

f ( x ) = 4 x 3 36 x 2 + 80 x

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Consider the same rectangle of the preceding problem. Squares of 2 x by 2 x units are cut out of each corner. Express the volume of the box as a polynomial in terms of x .

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A square has sides of 12 units. Squares x   + 1 by x   + 1 units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of x .

f ( x ) = 4 x 3 36 x 2 + 60 x + 100

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A cylinder has a radius of x + 2 units and a height of 3 units greater. Express the volume of the cylinder as a polynomial function.

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A right circular cone has a radius of 3 x + 6 and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is V = 1 3 π r 2 h for radius r and height h .

f ( x ) = π ( 9 x 3 + 45 x 2 + 72 x + 36 )

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Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
Practice Key Terms 4

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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