<< Chapter < Page Chapter >> Page >
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.

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

Practice set a

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

14

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

21 b , 7 b

3

Got questions? Get instant answers now!

20 x 3 , 4

5 x 3

Got questions? Get instant answers now!

8 x 4 , 4 x

2 x 3

Got questions? Get instant answers now!

16 y 5 , 2 y

Got questions? Get instant answers now!

6 x 2 y , 3 x

2 x y

Got questions? Get instant answers now!

9 a 4 b 5 , 9 a 4

Got questions? Get instant answers now!

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

3 b 3 c

Got questions? Get instant answers now!

25 a 3 b 2 c , 5 a c

Got questions? Get instant answers now!

18 x 2 b 5 , 2 x b 4

9 x b

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

4 x 3 b f 7

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

7 a 17 b 5 c 18 d

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

1 4 x 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

d

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

4 ( x + y ) 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

( x + y )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

6 x n y n 6

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Exercises for review

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

x 12 z 6

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

4 x 2 16 x + 16

Got questions? Get instant answers now!

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask