<< 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!

Questions & Answers

how does the planets on our solar system orbit
cheten Reply
how many Messier objects are there in space
satish Reply
did you g8ve certificate
Richard Reply
what are astronomy
Issan Reply
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
Roaul
I want to be an astronomer. That's my dream
Astrit
Who named the the whole galaxy?
Shola Reply
solar Univers
GPOWER
what is space
Richard
what is the dark matter
Richard
what are the factors upon which the atmosphere is stratified
Nicholas Reply
is the big bang the sun
Folakemi Reply
no
Sokak
bigbang is the beginning of the universe
Sokak
but thats just a theory
Sokak
nothing will happen, don't worry brother.
Vansh
what does comet means
GANGAIN Reply
these are Rocky substances between mars and jupiter
GANGAIN
Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.
Aarya
hllo
John
hi
John
qt rrt
John
r u there
John
hey can anyone guide me abt international astronomy olympiad
sahil
how can we learn right and true ?
Govinda Reply
why the moon is always appear in an elliptical shape
Gatjuol Reply
Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.
Hemen
what's see level?
lidiya Reply
Did you mean eye sight or sea level
Minal
oh sorry it's sea level
lidiya
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
w astronomy and cosmology!
Michele
another theory of universe except big ban
Albash Reply
how was universe born
Asmit Reply
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
Karuna Reply
I think it couldn't possible on earth
Nagalakshmi
in this time i don't Know
Michele
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.
Christopher
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

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