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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.

Check your understanding

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

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!
The GCF of 6 and 4 is 2The lowest power of x5 and x2 is x2GCF(6x5,4x2)=2x2\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!

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