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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
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
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Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at 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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