



Key equations
general form of a polynomial function 
$$f(x)={a}_{n}{x}^{n}+\mathrm{...}+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$$ 
Key concepts
 A power function is a variable base raised to a number power. See
[link] .
 The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
 The end behavior depends on whether the power is even or odd. See
[link] and
[link] .
 A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See
[link] .
 The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See
[link] .
 The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See
[link] and
[link] .
 A polynomial of degree
$\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ will have at most
$\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$
x intercepts and at most
$\text{\hspace{0.17em}}n1\text{\hspace{0.17em}}$ turning points. See
[link] ,
[link] ,
[link] ,
[link] , and
[link] .
Section exercises
Verbal
Explain the difference between the coefficient of a power function and its degree.
The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.
If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?
In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.
As
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ decreases without bound, so does
$\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$ As
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ increases without bound, so does
$\text{\hspace{0.17em}}f\left(x\right).$
What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?
What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As
$\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty \text{\hspace{0.17em}}$ and as
$\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty .\text{\hspace{0.17em}}$
The polynomial function is of even degree and leading coefficient is negative.
Algebraic
For the following exercises, identify the function as a power function, a polynomial function, or neither.
$f(x)={\left({x}^{2}\right)}^{3}$
$f(x)=\frac{{x}^{2}}{{x}^{2}1}$
$f(x)=2x\left(x+2\right){\left(x1\right)}^{2}$
For the following exercises, find the degree and leading coefficient for the given polynomial.
Degree = 2, Coefficient = –2
$x\left(4{x}^{2}\right)(2x+1)$
Degree =4, Coefficient = –2
${x}^{2}{\left(2x3\right)}^{2}$
For the following exercises, determine the end behavior of the functions.
$f\left(x\right)={x}^{4}$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty $
$f\left(x\right)={x}^{3}$
$f\left(x\right)={x}^{4}$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty $
$f\left(x\right)={x}^{9}$
$f(x)=2{x}^{4}3{x}^{2}+x1$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty $
$f(x)={x}^{2}(2{x}^{3}x+1)$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty $
For the following exercises, find the intercepts of the functions.
Questions & Answers
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is system testing?
AMJAD
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field .
1Electronicsmanufacturad IC ,RAM,MRAM,solar panel etc
2Helth and MedicalNanomedicine,Drug Dilivery for cancer treatment etc
3 Atomobile MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:
OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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