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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Factoring is an essential skill for success in algebra and higher level mathematics courses. Therefore, we have taken great care in developing the student's understanding of the factorization process. The technique is consistently illustrated by displaying an empty set of parentheses and describing the thought process used to discover the terms that are to be placed inside the parentheses.The factoring scheme for special products is presented with both verbal and symbolic descriptions, since not all students can interpret symbolic descriptions alone. Two techniques, the standard "trial and error" method, and the "collect and discard" method (a method similar to the "ac" method), are presented for factoring trinomials with leading coefficients different from 1. Objectives of this module: be reminded of products of polynomials, be able to determine a second factor of a polynomial given a first factor.

Overview

  • Products of Polynomials
  • Factoring

Products of polynomials

Previously, we studied multiplication of polynomials (Section [link] ). We were given factors and asked to find their product , as shown below.

Given the factors 4and 8, find the product. 4 8 = 32 . The product is 32.

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Given the factors 6 x 2 and 2 x 7 , find the product.

6 x 2 ( 2 x 7 ) 12 x 3 42 x 2

The product is 12 x 3 42 x 2 .

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Given the factors x 2 y and 3 x + y , find the product.

( x 2 y ) ( 3 x + y ) = 3 x 2 + x y 6 x y 2 y 2 = 3 x 2 5 x y 2 y 2

The product is 3 x 2 5 x y 2 y 2 .

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Given the factors a 8 and a 8 , find the product.

( a + 8 ) 2 = a 2 + 16 a + 64

The product is a 2 + 16 a + 64 .

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Factoring

Now, let’s reverse the situation. We will be given the product, and we will try to find the factors. This process, which is the reverse of multiplication, is called factoring .

Factoring

Factoring is the process of determining the factors of a given product.

Sample set a

The number 24 is the product, and one factor is 6. What is the other factor?

We’re looking for a number ( ) such that 6 ( ) = 24 . We know from experience that ( ) = 4 . As problems become progressively more complex, our experience may not give us the solution directly. We need a method for finding factors. To develop this method we can use the relatively simple problem 6 ( ) = 24 as a guide.
To find the number ( ) , we would divide 24 by 6.

24 6 = 4

The other factor is 4.

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The product is 18 x 3 y 4 z 2 and one factor is 9 x y 2 z . What is the other factor?

We know that since 9 x y 2 z is a factor of 18 x 3 y 4 z 2 , there must be some quantity ( ) such that 9 x y 2 z ( ) = 18 x 3 y 4 z 2 . Dividing 18 x 3 y 4 z 2 by 9 x y 2 z , we get

18 x 3 y 4 z 2 9 x y 2 z = 2 x 2 y 2 z

Thus, the other factor is 2 x 2 y 2 z .

Checking will convince us that 2 x 2 y 2 z is indeed the proper factor.

( 2 x 2 y 2 z ) ( 9 x y 2 z ) = 18 x 2 + 1 y 2 + 2 z 1 + 1 = 18 x 3 y 4 z 2

We should try to find the quotient mentally and avoid actually writing the division problem.

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The product is 21 a 5 b n and 3 a b 4 is a factor. Find the other factor.

Mentally dividing 21 a 5 b n by 3 a b 4 , we get

21 a 5 b n 3 a b 4 = 7 a 5 1 b n 4 = 7 a 4 b n 4

Thus, the other factor is 7 a 4 b n 4 .

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Practice set a

The product is 84 and one factor is 6. What is the other factor?

14

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The product is 14 x 3 y 2 z 5 and one factor is 7 x y z . What is the other factor?

2 x 2 y z 4

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Exercises

In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor.

10 a , 5

2 a

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21 b , 7 b

3

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20 x 3 , 4

5 x 3

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8 x 4 , 4 x

2 x 3

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16 y 5 , 2 y

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6 x 2 y , 3 x

2 x y

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9 a 4 b 5 , 9 a 4

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15 x 2 b 4 c 7 , 5 x 2 b c 6

3 b 3 c

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25 a 3 b 2 c , 5 a c

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18 x 2 b 5 , 2 x b 4

9 x b

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22 b 8 c 6 d 3 , 11 b 8 c 4

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60 x 5 b 3 f 9 , 15 x 2 b 2 f 2

4 x 3 b f 7

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39 x 4 y 5 z 11 , 3 x y 3 z 10

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147 a 20 b 6 c 18 d 2 , 21 a 3 b d

7 a 17 b 5 c 18 d

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121 a 6 b 8 c 10 , 11 b 2 c 5

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1 8 x 4 y 3 , 1 2 x y 3

1 4 x 3

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7 x 2 y 3 z 2 , 7 x 2 y 3 z

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5 a 4 b 7 c 3 d 2 , 5 a 4 b 7 c 3 d

d

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14 x 4 y 3 z 7 , 14 x 4 y 3 z 7

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12 a 3 b 2 c 8 , 12 a 3 b 2 c 8

1

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6 ( a + 1 ) 2 ( a + 5 ) , 3 ( a + 1 ) 2

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8 ( x + y ) 3 ( x 2 y ) , 2 ( x 2 y )

4 ( x + y ) 3

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14 ( a 3 ) 6 ( a + 4 ) 2 , 2 ( a 3 ) 2 ( a + 4 )

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26 ( x 5 y ) 10 ( x 3 y ) 12 , 2 ( x 5 y ) 7 ( x 3 y ) 7

13 ( x 5 y ) 3 ( x 3 y ) 5

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34 ( 1 a ) 4 ( 1 + a ) 8 , 17 ( 1 a ) 4 ( 1 + a ) 2

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( x + y ) ( x y ) , x y

( x + y )

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( a + 3 ) ( a 3 ) , a 3

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48 x n + 3 y 2 n 1 , 8 x 3 y n + 5

6 x n y n 6

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0.0024 x 4 n y 3 n + 5 z 2 , 0.03 x 3 n y 5

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Exercises for review

( [link] ) Simplify ( x 4 y 0 z 2 ) 3 .

x 12 z 6

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( [link] ) Simplify { [ ( | 6 | ) ] } .

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( [link] ) Find the product. ( 2 x 4 ) 2 .

4 x 2 16 x + 16

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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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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