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4 x 3 y 5 2 x y 3 is a binomial of degree 8. The degree of the first term is 8.

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3 x + 10 is a binomial of degree 1.

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Linear quadratic cubic

Polynomials of the first degree are called linear polynomials.
Polynomials of the second degree are called quadratic polynomials.
Polynomials of the third degree are called cubic polynomials.
Polynomials of the fourth degree are called fourth degree polynomials.
Polynomials of the n th degree are called n th degree polynomials.
Nonzero constants are polynomials of the 0th degree.

Some examples of these polynomials follow:

4 x 9 is a linear polynomial.

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3 x 2 + 5 x 7 is a quadratic polynomial.

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8 y 2 x 3 is a cubic polynomial.

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16 a 2 32 a 5 64 is a 5th degree polynomial.

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x 12 y 12 is a 12th degree polynomial.

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7 x 5 y 7 z 3 2 x 4 y 7 z + x 3 y 7 is a 15th degree polynomial. The first term is of degree 5 + 7 + 3 = 15 .

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43 is a 0th degree polynomial.

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Classification of polynomial equations

As we know, an equation is composed of two algebraic expressions separated by an equal sign. If the two expressions happen to be polynomial expressions, then we can classify the equation according to its degree. Classification of equations by degree is useful since equations of the same degree have the same type of graph. (We will study graphs of equations in Chapter 6.)

The degree of an equation is the degree of the highest degree expression.

Sample set a

x + 7 = 15 .

This is a linear equation since it is of degree 1, the degree of the expression on the left of the " = " sign.

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5 x 2 + 2 x 7 = 4 is a quadratic equation since it is of degree 2.

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9 x 3 8 = 5 x 2 + 1 is a cubic equation since it is of degree 3. The expression on the left of the " = " sign is of degree 3.

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y 4 x 4 = 0 is a 4th degree equation.

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a 5 3 a 4 = a 3 + 6 a 4 7 is a 5th degree equation.

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y = 2 3 x + 3 is a linear equation.

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y = 3 x 2 1 is a quadratic equation.

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x 2 y 2 4 = 0 is a 4th degree equation. The degree of x 2 y 2 4 is 2 + 2 = 4 .

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Practice set a

Classify the following equations in terms of their degree.

3 x + 6 = 0

first, or linear

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9 x 2 + 5 x 6 = 3

quadratic

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25 y 3 + y = 9 y 2 17 y + 4

cubic

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5 x 7 = 3 x 5 2 x 8 + 11 x 9

eighth degree

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Exercises

For the following problems, classify each polynomial as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term.

5 x + 7

binomial; first (linear); 5 , 7

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4 x 2 + 9

binomial; second (quadratic); 4 , 9

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a 4 + 1

binomial; fourth; 1 , 1

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5 x

monomial; first (linear); 5

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5 x 3 + 2 x + 3

trinomial; third (cubic); 5 , 2 , 3

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41 a 3 + 22 a 2 + a

trinomial; third (cubic); 41 , 22 , 1

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2 c 6 + 0

monomial; sixth; 2

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9 g

monomial; first (linear); 9

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3 y z 6 y + 11

trinomial; second (quadratic); 3 , 6 , 11

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7 a b 2 c 2 + 2 a 2 b 3 c 5 + a 14

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x 4 y 3 z 2 + 9 z

binomial; ninth; 1 , 9

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6 + 3 x 2 y 5 b

binomial; eighth; 6 , 3

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9 + 3 x 2 + 2 x y 6 z 2

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3 x 2 y 0 z 4 + 12 z 3 , y 0

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4 x y 3 z 5 w 0 , w 0

monomial; ninth; 4

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Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it.

9 z = 12 x 18

linear

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y 5 + y 3 = 3 y 2 + 2

cubic

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x 2 + x 4 = 7 x 2 2 x + 9

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2 y + 5 x 3 + 4 x y = 5 x y + 2 y

quadratic

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8 a + 2 b = 4 b 8

linear

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2 x 5 8 x 2 + 9 x + 4 = 12 x 4 + 3 x 3 + 4 x 2 + 1

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x + 3 x 5 = x + 2 x 5

fifth degree

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3 x 2 y 4 + 2 x 8 y = 14

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10 a 2 b 3 c 6 d 0 e 4 + 27 a 3 b 2 b 4 b 3 b 2 c 5 = 1 , d 0

19th degree

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The expression 4 x 3 9 x 7 is not a polynomial because

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The expression a 4 7 a is not a polynomial because

. . . there is a variable in the denominator

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Is every algebraic expression a polynomial expression? If not, give an example of an algebraic expression that is not a polynomial expression.

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Is every polynomial expression an algebraic expression? If not, give an example of a polynomial expression that is not an algebraic expression.

yes

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How do we find the degree of a term that contains more than one variable?

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Exercises for review

( [link] ) Use algebraic notation to write “eleven minus three times a number is five.”

11 3 x = 5

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( [link] ) Simplify ( x 4 y 2 z 3 ) 5 .

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( [link] ) Find the value of z if z = x u s and x = 55 , u = 49 , and s = 3 .

z = 2

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( [link] ) List, if any should appear, the common factors in the expression 3 x 4 + 6 x 3 18 x 2 .

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( [link] ) State (by writing it) the relationship being expressed by the equation y = 3 x + 5 .

The value of y is 5 more then three times the value of  x .

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Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
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.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
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 . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,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
Hello
Uday
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
Prasenjit Reply
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.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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