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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
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
Do you know which machine is used to that process?
s.
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
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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