# 3.1 Rational expressions: dividing polynomials

 Page 1 / 2
<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>

## Overview

• 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.

$\frac{a}{c}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{b}{c}=\frac{a+b}{c}$

Turning this equation around we get

$\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$

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.

## Practice set a

Perform the following divisions.

$\frac{2{x}^{2}+x-1}{x}$

$2x+1-\frac{1}{x}$

$\frac{3{x}^{3}+4{x}^{2}+10x-4}{{x}^{2}}$

$3x+4+\frac{10}{x}-\frac{4}{{x}^{2}}$

$\frac{{a}^{2}b+3a{b}^{2}+2b}{ab}$

$a+3b+\frac{2}{a}$

$\frac{14{x}^{2}{y}^{2}-7xy}{7xy}$

$2xy-1$

$\frac{10{m}^{3}{n}^{2}+15{m}^{2}{n}^{3}-20mn}{-5m}$

$-2{m}^{2}{n}^{2}-3m{n}^{3}+4n$

## 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

$\frac{{x}^{2}-2x-8}{{x}^{2}-3x-4}$

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

$\frac{\left(x-4\right)\left(x+2\right)}{\left(x-4\right)\left(x+1\right)}$

$\frac{\overline{)\left(x-4\right)}\left(x+2\right)}{\overline{)\left(x-4\right)}\left(x+1\right)}$

$\frac{x+2}{x+1}$

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. $\frac{35}{8}.$  We are to divide 35 by 8.
2.   We try 4, since 32 divided by 8 is 4.
3. Multiply 4 and 8.
4. Subtract 32 from 35.
5. Since the remainder 3 is less than the divisor 8, we are done with the 32 division.
6. $4\frac{3}{8}.$   The quotient is expressed as a mixed number.

I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
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
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!