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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: know how to factor a polynomial using the grouping method and when to try the grouping method.

Overview

  • Using Grouping to Factor a Polynomial
  • Knowing when to Try the Grouping Method

Using grouping to factor a polynomial

Sometimes a polynomial will not have a particular factor common to every term. However, we may still be able to produce a factored form for the polynomial.

The polynomial x 3 3 x 2 6 x 18 has no single factor that is common to every term. However, we notice that if we group together the first two terms and the second two terms, we see that each resulting binomial has a particular factor common to both terms.

The polynomial 'x cubed plus three x squared minus six x minus eighteen'. The first two terms of the polynomial have x square in common, and the last two terms of the polynomial have negative six in common.

Factor x 2 out of the first two terms, and factor 6 out of the second two terms.

x 2 ( x + 3 ) 6 ( x + 3 )

Now look closely at this binomial. Each of the two terms contains the factor ( x + 3 ) .

Factor out ( x + 3 ) .
( x + 3 ) ( x 2 6 ) is the final factorization.

x 3 + 3 x 2 6 x 18 = ( x + 3 ) ( x 2 6 )

Knowing when to try the grouping method

We are alerted to the idea of grouping when the polynomial we are considering has either of these qualities:

  1. no factor common to all terms
  2. an even number of terms

When factoring by grouping, the sign ( + or ) of the factor we are taking out will usually (but not always) be the same as the sign of the first term in that group.

Sample set a

Factor 8 a 2 b 4 4 b 4 + 14 a 2 7 .

  1. We notice there is no factor common to all terms.
  2. We see there are four terms, an even number.
  3. We see that terms 1 and 2 have + 4 b 4 in common (since the 1st term in the group is + 8 a 2 b 4 ) .
  4. We notice that the 3rd and 4th terms have + 7 in common (since the 1st term in the group is + 14 a 2 ).

    The equation eight a squared b to the fourth power minus four b to the fourth power plus fourteen a squared minus seven equals the sum of the product of four b to the fourth power and two a square minus one, and the product of seven and two a square minus 1. The two terms on the right side have two a square minus one in common. 8 a 2 b 4 4 b 4 + 14 a 2 7 = (2a 2 -1)(4b 4 +7)

Practice set a

Use the grouping method to factor the following polynomials.

a x a y b x b y

( a + b ) ( x + y )

2 a m + 8 m + 5 a n + 20 n

( 2 m + 5 n ) ( a + 4 )

a 2 x 3 + 4 a 2 y 3 + 3 b x 3 + 12 b y 3

( a 2 + 3 b ) ( x 3 + 4 y 3 )

15 m x + 10 n x 6 m y 4 n y

( 5 x 2 y ) ( 3 m + 2 n )

40 a b x 24 a b x y 35 c 2 x + 21 c 2 x y

x ( 8 a b 7 c 2 ) ( 5 3 y )

When factoring the polynomial 8 a 2 b 4 4 b 4 14 a 2 7 in Sample Set A, we grouped together terms1 and 2 and 3 and 4. Could we have grouped together terms1 and 3 and 2 and 4? Try this.
8 a 2 b 4 4 b 4 + 14 a 2 7 =

yes

Do we get the same result? If the results do not look precisely the same, recall the commutative property of multiplication.

Exercises

For the following problems, use the grouping method to factor the polynomials. Some polynomials may not be factorable using the grouping method.

2 a b + 3 a + 18 b + 27

( 2 b + 3 ) ( a + 9 )

x y 7 x + 4 y 28

x y + x + 3 y + 3

( y + 1 ) ( x + 3 )

m p + 3 m q + n p + 3 n q

a r + 4 a s + 5 b r + 20 b s

( a + 5 b ) ( r + 4 s )

14 a x 6 b x + 21 a y 9 b y

12 m x 6 b x + 21 a y 9 b y

3 ( 4 m x 2 b x + 7 a y 3 b y )  Not factorable by grouping

36 a k 8 a h 27 b k + 6 b h

a 2 b 2 + 2 a 2 + 3 b 2 + 6

( a 2 + 3 ) ( b 2 + 2 )

3 n 2 + 6 n + 9 m 3 + 12 m

8 y 4 5 y 3 + 12 z 2 10 z

Not factorable by grouping

x 2 + 4 x 3 y 2 + y

x 2 3 x + x y 3 y

( x + y ) ( x 3 )

2 n 2 + 12 n 5 m n 30 m

4 p q 7 p + 3 q 2 21

Not factorable by grouping

8 x 2 + 16 x y 5 x 10 y

12 s 2 27 s 8 s t + 18 t

( 4 s 9 ) ( 3 s 2 t )

15 x 2 12 x 10 x y + 8 y

a 4 b 4 + 3 a 5 b 5 + 2 a 2 b 2 + 6 a 3 b 3

a 2 b 2 ( a 2 b 2 + 2 ) ( 1 + 3 a b )

4 a 3 b c 14 a 2 b c 3 + 10 a b c 2 35 b c 4

5 x 2 y 3 z + 3 x 3 y w 10 y 3 z 2 6 w x y z

y ( 5 y 2 z + 3 x w ) ( x 2 2 z )

a 3 b 2 c d + a b c 2 d x a 2 b x y c x 2 y

5 m 10 n 17 p 3 m 6 n 7 p 4 40 m 4 n 10 q t 2 + 8 p q t 2

( m 6 n 7 p 3 8 q t 2 ) ( 5 m 4 n 10 p )

Exercises for review

( [link] ) Simplify ( x 5 y 3 ) ( x 2 y ) .

( [link] ) Use scientific notation to find the product of ( 3 × 10 5 ) ( 2 × 10 2 ) .

6 × 10 3

( [link] ) Find the domain of the equation y = 6 x + 5 .

( [link] ) Construct the graph of the inequality y 2 .

A horizontal line with arrows on both ends.

A number line with arrows on each end, labeled from negative three to three in increments of one. There is a closed circle at negative two. A dark arrow is originating from this circle, and heading towrads the right of negative two.

( [link] ) Factor 8 a 4 b 4 + 12 a 3 b 5 8 a 2 b 3 .

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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Source:  OpenStax, Algebra ii for the community college. OpenStax CNX. Jul 03, 2014 Download for free at http://cnx.org/content/col11671/1.1
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