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<para>This module is from<link document="col10614">Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr.</para><para>A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.</para><para>The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.</para><para>Objectives of this module: be able to divide a polynomial by a monomial, understand the process and be able to divide a polynomial by a polynomial.</para>


  • Dividing a Polynomial by a Monomial
  • The Process of Division
  • Review of Subtraction of Polynomials
  • Dividing a Polynomial by a Polynomial

Dividing a polynomial by a monomial

The following examples illustrate how to divide a polynomial by a monomial. The division process is quite simple and is based on addition of rational expressions.

a c + b c = a + b c

Turning this equation around we get

a + b c = a c + b c

Now we simply divide c into a , and c into b . This should suggest a rule.

Dividing a polynomial by a monomial

To divide a polynomial by a monomial, divide every term of the polynomial by the monomial.

Sample set a

3 x 2 + x 11 x . Divide every term of   3 x 2   +   x 11   by   x . 3 x 2 x + x x 11 x = 3 x + 1 11 x

8 a 3 + 4 a 2 16 a + 9 2 a 2 . Divide every term of   8 a 3   +   4 a 2 16 a   + 9   by   2 a 2 . 8 a 3 2 a 2 + 4 a 2 2 a 2 16 a 2 a 2 + 9 2 a 2 = 4 a + 2 8 a + 9 2 a 2

4 b 6 9 b 4 2 b + 5 4 b 2 . Divide every term of 4 b 6 9 b 4 2 b + 5 by 4 b 2 . 4 b 6 4 b 2 9 b 4 4 b 2 2 b 4 b 2 + 5 4 b 2 = b 4 + 9 4 b 2 + 1 2 b 5 4 b 2

Practice set a

Perform the following divisions.

2 x 2 + x 1 x

2 x + 1 1 x

3 x 3 + 4 x 2 + 10 x 4 x 2

3 x + 4 + 10 x 4 x 2

a 2 b + 3 a b 2 + 2 b a b

a + 3 b + 2 a

14 x 2 y 2 7 x y 7 x y

2 x y 1

10 m 3 n 2 + 15 m 2 n 3 20 m n 5 m

2 m 2 n 2 3 m n 3 + 4 n

The process of division

In Section [link] we studied the method of reducing rational expressions. For example, we observed how to reduce an expression such as

x 2 2 x 8 x 2 3 x 4

Our method was to factor both the numerator and denominator, then divide out common factors.

( x 4 ) ( x + 2 ) ( x 4 ) ( x + 1 )

( x 4 ) ( x + 2 ) ( x 4 ) ( x + 1 )

x + 2 x + 1

When the numerator and denominator have no factors in common, the division may still occur, but the process is a little more involved than merely factoring. The method of dividing one polynomial by another is much the same as that of dividing one number by another. First, we’ll review the steps in dividing numbers.

  1. 35 8 .  We are to divide 35 by 8.
  2. Long division showing eight dividing thirty five. This division is not performed completely.   We try 4, since 32 divided by 8 is 4.
  3. Long division showing eight dividing thirty five, with four at quotient's place. This division is not performed completely. Multiply 4 and 8.
  4. Long division showing eight dividing thirty five, with four at quotient's place. Thirty two is written under thirty five. This division is not performed completely Subtract 32 from 35.
  5. Long division showing eight dividing thirty five, with four at quotient's place. Thirty two is written under thirty five and three is written as the subtraction of thirty five and thirty two. Since the remainder 3 is less than the divisor 8, we are done with the 32 division.
  6. 4 3 8 .   The quotient is expressed as a mixed number.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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, Algebra ii for the community college. OpenStax CNX. Jul 03, 2014 Download for free at http://cnx.org/content/col11671/1.1
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