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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

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
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Dominic
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Lukman Reply
_3_2_1
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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
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Ifeanyi
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x*x=2
felecia
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×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
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4
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x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
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(61/11,41/11,−4/11)
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Need help solving this problem (2/7)^-2
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x+2y-z=7
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-1
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NTQ Reply
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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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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