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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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
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Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
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
7.5 and 37.5
Nando
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
the 28th term is 175
Nando
192
Kenneth
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
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write down the polynomial function with root 1/3,2,-3 with solution
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if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
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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
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Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
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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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