<< Chapter < Page Chapter >> Page >

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

0 with multiplicity 2, –2 with multiplicity 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

2 3 with multiplicity 5 , 5 with multiplicity 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

3 2 with multiplicity 2, 0 with multiplicity 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

0 with multiplicity 6 , 2 3 with multiplicity 2

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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).
Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!
Graph of a positive odd-degree polynomial with zeros at x=-2, and 3.

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!
Graph of a positive even-degree polynomial with zeros at x=-2,, and 3.

–2, 3 each with multiplicity 2

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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 ) .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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 ) .

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

f ( x ) = 2 x 3 3 x 1

Got questions? Get instant answers now!

f ( x ) = x 4 + x

global min ( .63, – .47 )

Got questions? Get instant answers now!

f ( x ) = x 4 + 3 x 2

Got questions? Get instant answers now!

f ( x ) = x 4 x 3 + 1

global min ( .75,  .89)

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

Questions & Answers

the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
x^20+x^15+x^10+x^5/x^2+1
Urmila Reply
evaluate each algebraic expression. 2x+×_2 if ×=5
Sarch Reply
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
Awe Reply
By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
jaks
Ryan
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
Shahid Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask