I keep mixing up GCF and LCM, anyone has a good way to memorize the difference?

try the ladder method. It is very helpful to me.

What is the difference between binomials and monomials.

A binomial has 2 terms (2 items being added or subtracted). Examples: 3x^3y + 6xy; 7 - 5y A monomial has 1 term. Examples of monomials: 4; 5ab^2; 7x/8 Hope this helps.

Why is it so hard to understand GCF when its used in a lot of ways

IIt is hard to learn at first, because you are not used to looking at two different numbers and instead of t usual math operations, such as adding subtracting, your breaking them down. into smaller numbers and finding out what they have in common.(AKA greatest common factor)

I still don't understand the difference between monomials and polynomials. When you add 3x+3x it's 6x. That's not a polynomial. That's a monomial!

I'm not exactly sure what your asking but I know this much: A monomial is a one term polynomial. A bionomial is a two term polynomial. A trinomial is a three term polynomial.

Why do we have to show 3 factors multiplied together, for example, one of the example questions I got wrong was because I showed the factors of 12 by 6 by 2, but the correct way of factoring the 12 is by 2 by 2 by 3. Why?

You need to keep factoring until you get the prime factors of the numbers, so you can easily find the gcf of various numbers.

When factoring, are you able to have more than two terms? for example 24x3, you could write that as 12x x 2x2 but you could also write it as 3x x 4x x 2x. Does it still work in that case?

Notice that in your examples you can still go further: *12* x 2 becomes *6x2* x 2, which in turn becomes: *3x2* x 2 x 2 Which is the stopping point, because every number is prime. Prime numbers can't be factored out further since they aren't divisible by anything other than 1 and themselves. As for your second example: 3 x *4* x 2 => *4* is not a prime, we can factor it out further: 3 x *2x2* x 2 As you can see, we arrived at the same expression. Hope that helps!

what is the gcf 4(12)+4(8)

Multiply: 48 + 32 Now find the GCF. Or, factors the numbers down to prime factors and find all the common factors. The GCF = 16

When factoring the coefficient, do you use the prime factorization or the GCF of those two numbers?

You want to find the GCF. You can use prime factorization of each monomial to help you find the GCF. Hope this helps.

Is there a trick to finding GCF of large number such as 625? It takes me forever to do these big numbers.

When you see large numbers like those, it comes in handy to know basic tricks to figure out if that number has small factors, such as 4, 9, and ll. In the case of 625, you can already tell that it is a multiple of 5, so divide it by 5, and you get 125. 125 is also divisible by 5, so divide by 5 again. You get 25, which is also divisible by 5. Then you get 5, which cannot be broken down further. You got four 5's during that search. Now, you just need another number. Say we have 45. 45 factors into a 3, 3, and a 5. We can tell that both of these numbers have a 5, so 5 is the GCF of 625 and 45. Just note that in order to find the GCF and/or LCM, you need two integers at the very least. Hope this helps. Jonathan Myung

What is the difference between binomials and monomials

A monomial is a polynomial with one term (such as x or 3 or y^2). A binomial is a polynomial with two terms (such as 3x + 2 or x^2 + 3x). A trinomial is next with 3 terms (x^2+4x+5).

Main content

Course: Integrated math 3 > Unit 2

Lesson 2: Greatest common factor

Greatest common factor of monomials

Google Classroom

Learn how to find the GCF (greatest common factor) of two monomials or more.

What you should be familiar with before this lesson

A monomial is an expression that is the product of constants and nonnegative integer powers of

x

, like

3 x^{2}

. A polynomial is a sum of monomials.

You can write the complete factorization of a monomial by writing the prime factorization of the coefficient and expanding the variable part. Check out our Factoring monomials article if this is new to you.

What you will learn in this lesson

In this lesson, you will learn about the greatest common factor (GCF) and how to find this for monomials.

Review: Greatest common factors in integers

The greatest common factor of two numbers is the greatest integer that is a factor of both numbers. For example, the GCF of

12

and

18

6

We can find the GCF for any two numbers by examining their prime factorizations:

$12 = 2 \cdot 2 \cdot 3$ ‍
$18 = 2 \cdot 3 \cdot 3$ ‍

Notice that

12

and

18

have a factor of

2

and a factor of

3

in common, and so the greatest common factor of

12

and

18

2 \cdot 3 = 6

Greatest common factors in monomials

The process is similar when you are asked to find the greatest common factor of two or more monomials.

Simply write the complete factorization of each monomial and find the common factors. The product of all the common factors will be the GCF.

For example, let's find the greatest common factor of

10 x^{3}

and

4 x

$10 x^{3} = 2 \cdot 5 \cdot x \cdot x \cdot x$ ‍
$4 x = 2 \cdot 2 \cdot x$ ‍

Notice that

10 x^{3}

and

4 x

have one factor of

2

and one factor of

x

in common. Therefore, their greatest common factor is

2 \cdot x

2 x

Check your understanding

1) What is the greatest common factor of $9 x^{2}$ ‍ and $6 x$ ‍?

2) What is the greatest common factor of $12 x^{5}$ ‍ and $8 x^{3}$ ‍?

3) What is the greatest common factor of $5 x^{7}$ ‍, $30 x^{4}$ ‍, and $10 x^{3}$ ‍?

A note on the variable part of the GCF

In general, the variable part of the GCF for any two or more monomials will be equal to the variable part of the monomial with the lowest power of

x

For example, consider the monomials

6 x^{5}

and

4 x^{2}

Since the lowest power of $x$ ‍ is $x^{2}$ ‍, that will be the variable part of the GCF.
You could then find the GCF of $6$ ‍ and $4$ ‍, which is $2$ ‍, and multiply this by $x^{2}$ ‍ to obtain $2 x^{2}$ ‍, the GCF of the monomials!

\begin{aligned} The GCF of 6 and 4 is 2 & The lowest power of x^{5} and x^{2} is x^{2} \\ ↘ & ↙ \\ GCF (6 x^{5}, 4 x^{2}) = 2 & x^{2} \end{aligned}

This is especially helpful to understand when finding the GCF of monomials with very large powers of

x

. For example, it would be very tedious to completely factor monomials like

32 x^{100}

and

16 x^{88}

Challenge Problems

4*)What is the greatest common factor of $20 x^{76}$ ‍ and $8 x^{92}$ ‍?

5*) What is the greatest common factor of $40 x^{5} y^{2}$ ‍ and $32 x^{2} y^{3}$ ‍?

To find the greatest common factor of

40 x^{5} y^{2}

and

32 x^{2} y^{3}

, let's factor each monomial completely, and see what is common.

$40 x^{5} y^{2} = 2 \cdot 2 \cdot 2 \cdot 5 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y$ ‍
$32 x^{2} y^{3} = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot x \cdot x \cdot y \cdot y \cdot y$ ‍

The monomials each have three factors of

2

, two factors of

x

, and two factors of

y

in common. Therefore, the greatest common factor of the polynomial is

2 \cdot 2 \cdot 2 \cdot x \cdot x \cdot y \cdot y

8 x^{2} y^{2}

Alternatively, notice that the lowest power of

x

x^{2}

and the lowest power of

y

y^{2}

. So the variable part of the GCF will be

x^{2} y^{2}

. The GCF of

40

and

32

8

, and so the GCF of the monomials is

8 x^{2} y^{2}

What's next?

To see how we can use these skills to factor polynomials, check out our next article on factoring out the greatest common factor!

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Brian
Posted 8 years ago. Direct link to Brian's post “I keep mixing up GCF and ...”
I keep mixing up GCF and LCM, anyone has a good way to memorize the difference?
Button navigates to signup pageComment on Brian's post “I keep mixing up GCF and ...”
(45 votes)
Answer
- Danica
  Posted 5 years ago. Direct link to Danica's post “try the ladder method. It...”
  try the ladder method. It is very helpful to me.
  Comment on Danica's post “try the ladder method. It...”
  (5 votes)
emily detering
Posted 7 years ago. Direct link to emily detering's post “What is the difference be...”
What is the difference between binomials and monomials.
Button navigates to signup pageComment on emily detering's post “What is the difference be...”
(21 votes)
Answer
- Kim Seidel
  Posted 7 years ago. Direct link to Kim Seidel's post “A binomial has 2 terms (2...”
  A binomial has 2 terms (2 items being added or subtracted).
  Examples: 3x^3y + 6xy; 7 - 5y
  
  A monomial has 1 term.
  Examples of monomials: 4; 5ab^2; 7x/8
  
  Hope this helps.
  Comment on Kim Seidel's post “A binomial has 2 terms (2...”
  (26 votes)
priyanka :D
Posted 6 years ago. Direct link to priyanka :D's post “I still don't understand ...”
I still don't understand the difference between monomials and polynomials. When you add 3x+3x it's 6x. That's not a polynomial. That's a monomial!
Button navigates to signup pageComment on priyanka :D's post “I still don't understand ...”
(6 votes)
Answer
- Cochran
  Posted 6 years ago. Direct link to Cochran's post “I'm not exactly sure what...”
  I'm not exactly sure what your asking but I know this much:
  A monomial is a one term polynomial.
  A bionomial is a two term polynomial.
  A trinomial is a three term polynomial.
  Comment on Cochran's post “I'm not exactly sure what...”
  (17 votes)
Bector, Ajay
Posted 8 years ago. Direct link to Bector, Ajay's post “Why is it so hard to unde...”
Why is it so hard to understand GCF when its used in a lot of ways
Button navigates to signup pageComment on Bector, Ajay's post “Why is it so hard to unde...”
(9 votes)
Answer
- Mike G
  Posted 8 years ago. Direct link to Mike G's post “IIt is hard to learn at f...”
  IIt is hard to learn at first, because you are not used to looking at two different numbers and instead of t usual math operations, such as adding subtracting, your breaking them down. into smaller numbers and finding out what they have in common.(AKA greatest common factor)
  Comment on Mike G's post “IIt is hard to learn at f...”
  (7 votes)
Raindrop9
Posted 4 years ago. Direct link to Raindrop9's post “Why do we have to show 3 ...”
Why do we have to show 3 factors multiplied together, for example, one of the example questions I got wrong was because I showed the factors of 12 by 6 by 2, but the correct way of factoring the 12 is by 2 by 2 by 3. Why?
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(5 votes)
Answer
- QueenCobra
  Posted 4 years ago. Direct link to QueenCobra's post “You need to keep factorin...”
  You need to keep factoring until you get the prime factors of the numbers, so you can easily find the gcf of various numbers.
  Button navigates to signup page
  (7 votes)
trevit0435
Posted a year ago. Direct link to trevit0435's post “When factoring, are you a...”
When factoring, are you able to have more than two terms? for example 24x3, you could write that as 12x x 2x2 but you could also write it as 3x x 4x x 2x. Does it still work in that case?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Dowlgrett
  Posted a year ago. Direct link to Dowlgrett's post “Notice that in your examp...”
  Notice that in your examples you can still go further:
  12 x 2 becomes 6x2 x 2, which in turn becomes:
  
  3x2 x 2 x 2
  
  Which is the stopping point, because every number is prime. Prime numbers can't be factored out further since they aren't divisible by anything other than 1 and themselves.
  
  As for your second example:
  3 x 4 x 2 => 4 is not a prime, we can factor it out further:
  
  3 x 2x2 x 2
  
  As you can see, we arrived at the same expression. Hope that helps!
  Button navigates to signup page
  (7 votes)
alvarezdamaris340
Posted 3 years ago. Direct link to alvarezdamaris340's post “What is the difference be...”
What is the difference between binomials and monomials
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- David Severin
  Posted 3 years ago. Direct link to David Severin's post “A monomial is a polynomia...”
  A monomial is a polynomial with one term (such as x or 3 or y^2). A binomial is a polynomial with two terms (such as 3x + 2 or x^2 + 3x). A trinomial is next with 3 terms (x^2+4x+5).
  Button navigates to signup page
  (9 votes)
😊
Posted 7 years ago. Direct link to 😊's post “what is the gcf 4(12)+4(8...”
what is the gcf 4(12)+4(8)
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(4 votes)
Answer
- Kim Seidel
  Posted 7 years ago. Direct link to Kim Seidel's post “Multiply: 48 + 32 Now fi...”
  Multiply: 48 + 32
  Now find the GCF.
  Or, factors the numbers down to prime factors and find all the common factors.
  The GCF = 16
  Button navigates to signup page
  (6 votes)
BruinWarrior
Posted 3 years ago. Direct link to BruinWarrior's post “When factoring the coeffi...”
When factoring the coefficient, do you use the prime factorization or the GCF of those two numbers?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Kim Seidel
  Posted 3 years ago. Direct link to Kim Seidel's post “You want to find the GCF....”
  You want to find the GCF. You can use prime factorization of each monomial to help you find the GCF.
  Hope this helps.
  Comment on Kim Seidel's post “You want to find the GCF....”
  (4 votes)
😊
Posted 6 years ago. Direct link to 😊's post “Is there a trick to findi...”
Is there a trick to finding GCF of large number such as 625? It takes me forever to do these big numbers.
Button navigates to signup pageComment on 😊's post “Is there a trick to findi...”
(4 votes)
Answer
- Jonathan
  Posted 3 years ago. Direct link to Jonathan's post “When you see large number...”
  When you see large numbers like those, it comes in handy to know basic tricks to figure out if that number has small factors, such as 4, 9, and ll. In the case of 625, you can already tell that it is a multiple of 5, so divide it by 5, and you get 125. 125 is also divisible by 5, so divide by 5 again. You get 25, which is also divisible by 5. Then you get 5, which cannot be broken down further. You got four 5's during that search. Now, you just need another number. Say we have 45. 45 factors into a 3, 3, and a 5. We can tell that both of these numbers have a 5, so 5 is the GCF of 625 and 45.
  
  Just note that in order to find the GCF and/or LCM, you need two integers at the very least.
  
  Hope this helps.
  
  Jonathan Myung
  Comment on Jonathan's post “When you see large number...”
  (3 votes)