<< Chapter < Page Chapter >> Page >

Sample set b

Factor 24 x 2 41 x + 12 .

Factor the first and last terms.

24 x 2 12
24 x , x 12 , 1
12 x , 2 x 6 , 2
8 x , 3 x 4 , 3
6 x , 4 x

Rather than starting with the 24 x , x and 12 , 1 , pick some intermediate values, 8 x and 3 x , the 6 x and 4 x , or the 6 and 2 , or the 4 and 3 .

24 x 2 41 x + 12 = ( 8 x 3 ) ( 3 x 4 )

Got questions? Get instant answers now!

Practice set b

Factor 48 x 2 + 22 x - 15 .

( 6 x + 5 ) ( 8 x 3 )

Got questions? Get instant answers now!

Factor 54 y 2 + 39 y w 28 w 2 .

( 9 y 4 w ) ( 6 y + 7 w )

Got questions? Get instant answers now!

The collect and discard method of factoring a x 2 + b x + c

Collect and discard method

Consider the polynomial 6 x 2 + x 12 . We begin by identifying a and c . In this case, a = 6 and c = 12 . We start out as we would with a = 1 .

6 x 2 + x 12 : ( 6 x ) ( 6 x )

Now, compute a c .

a c = ( 6 ) ( 12 ) = 72

Find the factors of 72 that add to 1, the coefficient of x , the linear term. The factors are 9 and 8 . Include these factors in the parentheses.

6 x 2 + x 12 : ( 6 x + 9 ) ( 6 x 8 )

But we have included too much. We must eliminate the surplus. Factor each parentheses.

6 x 2 + x 12 : 3 ( 2 x + 3 ) 2 ( 3 x 4 )

Discard the factors that multiply to a = 6 . In this case, 3 and 2. We are left with the proper factorization.

6 x 2 + x 12 = ( 2 x + 3 ) ( 3 x 4 )

Sample set c

Factor 10 x 2 + 23 x 5 .

Identify a = 10 and b = 5 .

10 x 2 + 23 x 5 ; ( 10 x ) ( 10 x )

Compute

a c = ( 10 ) ( 5 ) = 50

Find the factors of 50 that add to + 23 , the coefficient of x , the linear term. The factors are 25 and 2 . Place these numbers into the parentheses.

10 x 2 + 23 x 5 : ( 10 x + 25 ) ( 10 x 2 )

We have collected too much. Factor each set of parentheses and eliminate the surplus.

10 x 2 + 23 x 5 : ( 5 ) ( 2 x + 5 ) ( 2 ) ( 5 x 1 )

Discard the factors that multiply to a = 10 . In this case, 5 and 2.

10 x 2 + 23 x 5 = ( 2 x + 5 ) ( 5 x 1 )

Got questions? Get instant answers now!

Factor 8 x 2 30 x 27 .

Identify a = 8 and c = 27 .

8 x 2 30 x 27 : ( 8 x ) ( 8 x )

Compute

a c = ( 8 ) ( 27 ) = 216

Find the factors of 216 that add to 30 , the coefficient of x , the linear term. This requires some thought. The factors are 36 and 6. Place these numbers into the parentheses.

8 x 2 30 x 27 : ( 8 x 36 ) ( 8 x + 6 )

We have collected too much. Factor each set of parentheses and eliminate the surplus.

8 x 2 30 x 27 : ( 4 ) ( 2 x 9 ) ( 2 ) ( 4 x + 3 )

Discard the factors that multiply to a = 8 . In this case, 4 and 2.

8 x 2 30 x 27 = ( 2 x 9 ) ( 4 x + 3 )

Got questions? Get instant answers now!

Factor 18 x 2 5 x y 2 y 2 .

Identify a = 18 and c = 2 .

18 x 2 5 x y 2 y 2 : ( 18 x ) ( 18 x )

Compute

a c = ( 18 ) ( 2 ) = 36

Find the factors of 36 that add to 5 , the coefficient of x y . In this case, 9 and 4. Place these numbers into the parentheses, affixing y to each.

18 x 2 5 x y 2 y 2 : ( 18 x 9 y ) ( 18 x + 4 y )

We have collected too much. Factor each set of parentheses and eliminate the surplus.

18 x 2 5 x y 2 y 2 : ( 9 ) ( 2 x y ) ( 2 ) ( 9 x + 2 y )

Discard the factors that multiply to a = 18 . In this case, 9 and 4.

18 x 2 5 x y 2 y 2 = ( 2 x y ) ( 9 x + 2 y )

Got questions? Get instant answers now!

Practice set c

Factor 6 x 2 + 7 x 3 .

( 3 x 1 ) ( 2 x + 3 )

Got questions? Get instant answers now!

Factor 14 x 2 31 x 10 .

( 7 x + 2 ) ( 2 x 5 )

Got questions? Get instant answers now!

Factor 48 x 2 + 22 x 15 .

( 6 x + 5 ) ( 8 x 3 )

Got questions? Get instant answers now!

Factor 10 x 2 23 x w + 12 w 2 .

( 5 x 4 w ) ( 2 x 3 w )

Got questions? Get instant answers now!

Exercises

Factor the following problems, if possible.

x 2 + 3 x + 2

( x + 2 ) ( x + 1 )

Got questions? Get instant answers now!

2 x 2 + 7 x + 5

( 2 x + 5 ) ( x + 1 )

Got questions? Get instant answers now!

2 x 2 + 11 x + 12

( 2 x + 3 ) ( x + 4 )

Got questions? Get instant answers now!

3 x 2 x 4

( 3 x 4 ) ( x + 1 )

Got questions? Get instant answers now!

4 x 2 + 8 x 21

( 2 x 3 ) ( 2 x + 7 )

Got questions? Get instant answers now!

9 a 2 7 a + 2

not factorable

Got questions? Get instant answers now!

16 y 2 - 26 y + 3

( 8 y 1 ) ( 2 y 3 )

Got questions? Get instant answers now!

10 x 2 + 29 x + 10

( 5 x + 2 ) ( 2 x + 5 )

Got questions? Get instant answers now!

81 a 2 + 19 a + 2

not factorable

Got questions? Get instant answers now!

24 x 2 34 x + 5

( 6 x 1 ) ( 4 x 5 )

Got questions? Get instant answers now!

24 x 2 + 26 x 5

( 6 x 1 ) ( 4 x + 5 )

Got questions? Get instant answers now!

6 x 2 + 5 x y + y 2

( 3 x + y ) ( 2 x + y )

Got questions? Get instant answers now!

For the following problems, the given trinomial occurs when solving the corresponding applied problem. Factor each trinomial. You do not need to solve the problem.

5 r 2 24 r 5 .

It takes 5 hours to paddle a boat 12 miles downstream and then back. The current flows at the rate of 1 mile per hour. At what rate was the boat paddled?

( 5 r + 1 ) ( r 5 )

Got questions? Get instant answers now!

x 2 + 5 x 84 .

The length of a rectangle is 5 inches more than the width of the rectangle. If the area of the rectangle is 84 square inches, what are the length and width of the rectangle?

Got questions? Get instant answers now!

x 2 + 24 x 145 .

A square measures 12 inches on each side. Another square is to be drawn around this square in such a way that the total area is 289 square inches. What is the distance from the edge of the smaller square to the edge of the larger square? (The two squares have the same center.)

( x + 29 ) ( x 5 )

Got questions? Get instant answers now!

x 2 + 8 x 20 .

A woman wishes to construct a rectangular box that is open at the top. She wishes it to be 4 inches high and have a rectangular base whose length is three times the width. The material used for the base costs $2 per square inch, and the material used for the sides costs $1.50 per square inch. The woman will spend exactly $120 for materials. Find the dimension of the box (length of the base, width of the base, and height).

Got questions? Get instant answers now!

For the following problems, factor the trinomials if possible.

16 x 2 8 x y 3 y 2

( 4 x + y ) ( 4 x 3 y )

Got questions? Get instant answers now!

12 a 2 + 7 a b + 12 b 2

not factorable

Got questions? Get instant answers now!

8 a 2 + 10 a b 6 b 2

2 ( 4 a 2 + 5 a b 3 b 2 )

Got questions? Get instant answers now!

12 b 4 + 30 b 2 a + 12 a 2

6 ( 2 b 2 + a ) ( b 2 + 2 a )

Got questions? Get instant answers now!

30 a 4 b 4 3 a 2 b 2 6 c 2

Got questions? Get instant answers now!

3 a 6 3 a 3 b 2 18 b 4

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

Got questions? Get instant answers now!

20 a 2 b 2 + 2 a b c 2 6 a 2 c 4

Got questions? Get instant answers now!

14 a 2 z 2 40 a 3 z 2 46 a 4 z 2

2 a 2 z 2 ( 7 20 a 23 a 2 )  or  2 a 2 z 2 ( 23 a 2 + 20 a 7 )

Got questions? Get instant answers now!

Exercises for review

( [link] ) Simplify ( a 3 b 6 ) 4 .

Got questions? Get instant answers now!

( [link] ) Find the product. x 2 ( x 3 ) ( x + 4 ) .

x 4 + x 3 12 x 2

Got questions? Get instant answers now!

( [link] ) Find the product. ( 5 m 3 n ) 2 .

Got questions? Get instant answers now!

( [link] ) Solve the equation 5 ( 2 x 1 ) 4 ( x + 7 ) = 0 .

x = 11 2

Got questions? Get instant answers now!

( [link] ) Factor x 5 8 x 4 + 7 x 3 .

Got questions? Get instant answers now!

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
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





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