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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

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
Widad Reply
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Ariel
?
Ariel
What is economics
Widad Reply
the study of how humans make choices under conditions of scarcity
AI-Robot
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn Reply
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn
what is ecnomics
Jan Reply
this is the study of how the society manages it's scarce resources
Belonwu
what is macroeconomic
John Reply
macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.
husaini
etc
husaini
difference between firm and industry
husaini Reply
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?
Toofiq Reply
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Jan
economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses
Jan
calculate the profit maximizing for demand and supply
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out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials
AI-Robot
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identify a demand and a supply curve
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i don't know
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there's a difference
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Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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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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