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## Integrated math 3

### Course: Integrated math 3>Unit 2

Lesson 2: Greatest common factor

# Greatest common factor of monomials

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, squared. 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 is 6.
We can find the GCF for any two numbers by examining their prime factorizations:
• 12, equals, start color #11accd, 2, end color #11accd, dot, 2, dot, start color #e07d10, 3, end color #e07d10
• 18, equals, start color #11accd, 2, end color #11accd, dot, start color #e07d10, 3, end color #e07d10, dot, 3
Notice that 12 and 18 have a factor of start color #11accd, 2, end color #11accd and a factor of start color #e07d10, 3, end color #e07d10 in common, and so the greatest common factor of 12 and 18 is start color #11accd, 2, end color #11accd, dot, start color #e07d10, 3, end color #e07d10, equals, 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, cubed and 4, x:
• 10, x, cubed, equals, start color #11accd, 2, end color #11accd, dot, 5, dot, start color #e07d10, x, end color #e07d10, dot, x, dot, x
• 4, x, equals, start color #11accd, 2, end color #11accd, dot, 2, dot, start color #e07d10, x, end color #e07d10
Notice that 10, x, cubed and 4, x have one factor of start color #11accd, 2, end color #11accd and one factor of start color #e07d10, x, end color #e07d10 in common. Therefore, their greatest common factor is start color #11accd, 2, end color #11accd, dot, start color #e07d10, x, end color #e07d10 or 2, x.

1) What is the greatest common factor of 9, x, squared and 6, x?

2) What is the greatest common factor of 12, x, start superscript, 5, end superscript and 8, x, cubed?

3) What is the greatest common factor of 5, x, start superscript, 7, end superscript, 30, x, start superscript, 4, end superscript, and 10, x, cubed?

## 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 start color #11accd, 6, end color #11accd, start color #e07d10, x, start superscript, 5, end superscript, end color #e07d10 and start color #11accd, 4, end color #11accd, start color #e07d10, x, squared, end color #e07d10:
• Since the lowest power of x is start color #e07d10, x, squared, end color #e07d10, that will be the variable part of the GCF.
• You could then find the GCF of start color #11accd, 6, end color #11accd and start color #11accd, 4, end color #11accd, which is start color #11accd, 2, end color #11accd, and multiply this by start color #e07d10, x, squared, end color #e07d10 to obtain start color #11accd, 2, end color #11accd, start color #e07d10, x, squared, end color #e07d10, the GCF of the monomials!
\begin{aligned} \text{The GCF of }\blueD 6\text{ and }\blueD 4\text{ is }\blueD 2\qquad&\quad\text{The lowest power of }\goldD{x^5}\text{ and }\goldD{x^2}\text{ is }\goldD{x^2} \\ \searrow\quad&\quad\swarrow \\ \LARGE\text{GCF}(\blueD 6\goldD{x^5},\blueD 4\goldD{x^2})=\blueD 2&\LARGE \goldD{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, start superscript, 100, end superscript and 16, x, start superscript, 88, end superscript!

## Challenge Problems

4*)What is the greatest common factor of 20, x, start superscript, 76, end superscript and 8, x, start superscript, 92, end superscript?

5*) What is the greatest common factor of 40, x, start superscript, 5, end superscript, y, squared and 32, x, squared, y, cubed?

## 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!

## Want to join the conversation?

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

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!