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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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