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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 x = x 3 + 3 x 2 12 x + 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 - x = - x 3 + 3 x 2 + 12 x + 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.

Polynomials

Zero polynomial

The function is defined as :

y = f(x) = 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 = f(x) = c

Constant function

Constant function is a polynomial of degree 0.

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 f(x + a 1 ) = f(x + a 2 ) = .......... f(x) = 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 x = a 0 x + a 1

It is also expressed as :

f x = m x + c

Linear function

Linear function is a polynomial of degree 1.

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.

Questions & Answers

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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