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Find the greatest common factor: 16 x 2 , 24 x 3 .

8 x 2

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Find the greatest common factor: 27 y 3 , 18 y 4 .

9 y 3

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Find the greatest common factor of 14 x 3 , 8 x 2 , 10 x .

Solution

Factor each coefficient into primes and write
the variables with exponents in expanded form.
Circle the common factors in each column.
Bring down the common factors.
Multiply the factors.
.
The GCF of 14 x 3 and 8 x 2 , and 10 x is 2 x
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Find the greatest common factor: 21 x 3 , 9 x 2 , 15 x .

3 x

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Find the greatest common factor: 25 m 4 , 35 m 3 , 20 m 2 .

5 m 2

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Factor the greatest common factor from a polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, 12 as 2 · 6 or 3 · 4 ), in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:

2 ( x + 7 ) factors 2 · x + 2 · 7 2 x + 14 product

Here, we will start with a product, like 2 x + 14 , and end with its factors, 2 ( x + 7 ) . To do this we apply the Distributive Property “in reverse”.

Distributive property

If a , b , c are real numbers, then

a ( b + c ) = a b + a c and a b + a c = a ( b + c )

The form on the left is used to multiply. The form on the right is used to factor.

So how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!

Factor: 2 x + 14 .

Solution

Step 1: Find the GCF of all the terms of the polynomial. Find the GCF of 2x and 14. .
Step 2: Rewrite each term as a product using the GCF. Rewrite 2x and 14 as products of their GCF, 2.
2 x = 2 x
14 = 2 7
.
Step 3: Use the Distributive Property 'in reverse' to factor the expression. 2 ( x + 7 )
Step 4: Check by multiplying the factors. Check:
.
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Factor: 4 x + 12 .

4( x + 3)

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Factor: 6 a + 24 .

6( a + 4)

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Notice that in [link] , we used the word factor as both a noun and a verb:

Noun 7 is a factor of 14 Verb factor 2 from 2 x + 14

Factor the greatest common factor from a polynomial.

  1. Find the GCF of all the terms of the polynomial.
  2. Rewrite each term as a product using the GCF.
  3. Use the Distributive Property ‘in reverse’ to factor the expression.
  4. Check by multiplying the factors.

Factor: 3 a + 3 .

Solution

.
.
Rewrite each term as a product using the GCF. .
Use the Distributive Property 'in reverse' to factor the GCF. .
Check by multiplying the factors to get the original polynomial.
.
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Factor: 9 a + 9 .

9( a + 1)

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Factor: 11 x + 11 .

11( x + 1)

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The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

Factor: 12 x 60 .

Solution

.
.
Rewrite each term as a product using the GCF. .
Factor the GCF. .
Check by multiplying the factors.
.
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Factor: 11 x 44 .

11( x − 4)

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Factor: 13 y 52 .

13( y − 4)

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Now we’ll factor the greatest common factor    from a trinomial    . We start by finding the GCF of all three terms.

Factor: 3 y 2 + 6 y + 9 .

Solution

.
.
Rewrite each term as a product using the GCF. .
Factor the GCF. .
Check by multiplying.
.
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Factor: 4 y 2 + 8 y + 12 .

4( y 2 + 2 y + 3)

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Factor: 6 x 2 + 42 x 12 .

6( x 2 + 7 x − 2)

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In the next example, we factor a variable from a binomial    .

Factor: 6 x 2 + 5 x .

Solution

6 x 2 + 5 x
Rewrite each term as a product. .
Factor the GCF. x ( 6 x + 5 )
Check by multiplying.
x ( 6 x + 5 )
x 6 x + x 5
6 x 2 + 5 x
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Questions & Answers

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.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Practice Key Terms 1

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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