3.2 Polynomial function  (Page 2/3)

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3 : Roots having square root term occur in pairs 1+√3 and 1-√3.

4 : If a polynomial equation involves only even powers of x and all terms are positive, then all roots of polynomial equation are imaginary (complex). For example, roots of the quadratic equation given here are complex.

${x}^{4}+2{x}^{2}+4=0$

Descartes rules of signs

Descartes rules are :

(i) Maximum number of positive real roots of a polynomial equation f(x) is equal to number of sign changes in f(x).

(ii) Maximum number of negative real roots of a polynomial equation f(x) is equal to number of sign changes in f(-x).

The signs of the terms of polynomial equation $f\left(x\right)={x}^{3}+3{x}^{2}-12x+3=0$ are “+ + - +”. There are two sign changes as we move from left to right. Hence, this cubic polynomial can have at most 2 positive real roots. Further, corresponding $f\left(-x\right)=-{x}^{3}+3{x}^{2}+12x+3=0$ has signs of term given as “- + + +“. There is one sign change involved here. It means that polynomial equation can have at most one negative root.

Zero polynomial

The function is defined as :

$y=\mathrm{f\left(x\right)}=0$

The polynomial “0”, which has no term at all, is called zero polynomial. The graph of zero polynomial is x-axis itself. Clearly, domain is real number set R, whereas range is a singleton set {0}.

Constant function

It is a polynomial of degree 0. The value of constant function is constant irrespective of values of "x". The image of the constant function (y) is constant for all values of pre-images (x).

$y=\mathrm{f\left(x\right)}=c$

The graph of a constant function is a straight line parallel to x-axis. As “y = (f(x) = c” holds for real values of “x”, the domain of constant function is "R". On the other hand, the value of “y” is a single valued constant, hence range of constant function is singleton set {c}.We can treat constant function also as a linear function of the form f(x) = c with m=0. Its graph is a straight line like that of linear function.

There is an interesting aspect about periodicity of constant function. A polynomial function is not periodic in general. A periodic function repeats function values after regular intervals. It is defined as a fuction for which f(x+T) = f(x), where T is the period of the function. In the case of constant function, function value is constant whatever be the value of independent variable. It means that $\mathrm{f\left(x}+{a}_{1}\right)=\mathrm{f\left(x}+{a}_{2}\right)=..........\mathrm{f\left(x\right)}=c$ . Clearly, it meets the requirement with the difference that there is no definite or fixed period like "T". The relation of periodicity, however, holds for any change to x. We, therefore, summarize (it is also the accepted position) that constant function is a periodic function with no period.

Linear function

Linear function is a polynomial of order 1.

$f\left(x\right)={a}_{0}x+{a}_{1}$

It is also expressed as :

$f\left(x\right)=mx+c$

The graph of a linear function is a straight line. The coefficient of “x” i.e. m is slope of the line and c is y-intercept, which is obtained for x = 0 such that f(0) = c. It is clear from the graph that its domain and range both are real number set R.

Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
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?
China
Cied
types of nano material
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 nano technology
what is system testing?
how did you get the value of 2000N.What calculations are needed to arrive at it
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What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x