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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

June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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