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Course: Algebra 2 > Unit 3
Lesson 2: Greatest common factorGreatest 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 , like . 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 and is .
We can find the GCF for any two numbers by examining their prime factorizations:
Notice that and have a factor of and a factor of in common, and so the greatest common factor of and is .
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 and :
Notice that and have one factor of and one factor of in common. Therefore, their greatest common factor is or .
Check your understanding
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 .
For example, consider the monomials and :
- Since the lowest power of
is , that will be the variable part of the GCF. - You could then find the GCF of
and , which is , and multiply this by to obtain , the GCF of the monomials!
This is especially helpful to understand when finding the GCF of monomials with very large powers of . For example, it would be very tedious to completely factor monomials like and !
Challenge Problems
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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