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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.Objectives of this module: be able to multiply a polynomial by a monomial, be able to simplify +(a + b) and -(a - b), be able to multiply a polynomial by a polynomial.

Overview

  • Multiplying a Polynomial by a Monomial
  • Simplifying + ( a + b ) and ( a + b )
  • Multiplying a Polynomial by a Polynomial

Multiplying a polynomial by a monomial

Multiplying a polynomial by a monomial is a direct application of the distributive property.

Distributive property

The product of a monomial a and a binomial b plus c is equal to ab plus ac. This is the distributive property. In the expression, there are two arrows originating from the monomial, a, and pointing towards the terms b and c of the binomial.

The distributive property suggests the following rule.

Multiplying a polynomial by a monomial

To multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial and then add the resulting products together.

Sample set a

10 a b 2 c ( 125 a 2 ) = 1250 a 3 b 2 c

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Practice set a

Determine the following products.

( a 2 2 b + 6 ) 2 a

2 a 3 4 a b + 12 a

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8 a 2 b 3 ( 2 a + 7 b + 3 )

16 a 3 b 3 + 56 a 2 b 4 + 24 a 2 b 3

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4 x ( 2 x 5 + 6 x 4 8 x 3 x 2 + 9 x 11 )

8 x 6 + 24 x 5 32 x 4 4 x 3 + 36 x 2 44 x

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( 3 a 2 b ) ( 2 a b 2 + 4 b 3 )

6 a 3 b 3 + 12 a 2 b 4

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5 m n ( m 2 n 2 + m + n 0 ) , n 0

5 m 3 n 3 + 5 m 2 n + 5 m n

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Use a calculator. 6.03 ( 2.11 a 3 + 8.00 a 2 b )

12.7233 a 3 + 48.24 a 2 b

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Simplifying + ( a + b ) And - ( a + b )

+ ( a + b ) And - ( a + b )

Oftentimes, we will encounter multiplications of the form

+ 1 ( a + b ) or - 1 ( a + b )

These terms will actually appear as

+ ( a + b ) and - ( a + b )

Using the distributive property, we can remove the parentheses.

Removal of a set of parentheses preceded by a plus sign using the distributive property. See the longdesc for a full description.

The parentheses have been removed and the sign of each term has remained the same.

Removal of a set of parentheses preceded by a minus sign using the distributive property. See the longdesc for a full description.

The parentheses have been removed and the sign of each term has been changed to its opposite.

  1. To remove a set of parentheses preceded by a " + " sign, simply remove the parentheses and leave the sign of each term the same.
  2. To remove a set of parentheses preceded by a “ ” sign, remove the parentheses and change the sign of each term to its opposite sign.

Sample set b

Simplify the expressions.

( 6 x 1 ) .

This set of parentheses is preceded by a “ + ’’ sign (implied). We simply drop the parentheses.

( 6 x 1 ) = 6 x 1

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( 14 a 2 b 3 6 a 3 b 2 + a b 4 ) = 14 a 2 b 3 6 a 3 b 2 + a b 4

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( 21 a 2 + 7 a 18 ) .

This set of parentheses is preceded by a “ ” sign. We can drop the parentheses as long as we change the sign of every term inside the parentheses to its opposite sign.

( 21 a 2 + 7 a 18 ) = 21 a 2 7 a + 18

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( 7 y 3 2 y 2 + 9 y + 1 ) = 7 y 3 + 2 y 2 9 y 1

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Practice set b

Simplify by removing the parentheses.

( a 2 6 a + 10 )

a 2 6 a + 10

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( x + 2 y )

x 2 y

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( 5 m 2 n )

5 m + 2 n

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( 3 s 2 7 s + 9 )

3 s 2 + 7 s 9

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Multiplying a polynomial by a polynomial

Since we can consider an expression enclosed within parentheses as a single quantity, we have, by the distributive property,

Finding the product of the binomials 'a plus b' and 'c plus d', using the distributive property. See the longdesc for a full description.

For convenience we will use the commutative property of addition to write this expression so that the first two terms contain a and the second two contain b .

Questions & Answers

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
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.
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
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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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
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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
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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
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name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
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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
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Prasenjit 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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