# Greatest common factor of monomials

CCSS Math: HSA.SSE.B.3, HSF.IF.C.8

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 $3x^2$. 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 $18$ is $6$.We can find the GCF for any two numbers by examining their prime factorizations:

- $12=\blueD{2}\cdot 2\cdot \goldD{3}$
- $18=\blueD{2}\cdot \goldD3\cdot 3$

Notice that $12$ and $18$ have a factor of $\blueD{2}$ and a factor of $\goldD{3}$ in common, and so the greatest common factor of $12$ and $18$ is $\blueD{2}\cdot \goldD{3}=6$.

## 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 $10x^3$ and $4x$:

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

Notice that $10x^3$ and $4x$ have one factor of $\blueD{2}$ and one factor of $\goldD{x}$ in common. Therefore, their greatest common factor is $\blueD2\cdot \goldD{x}$ or $2x$.

### 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 $x$.

For example, consider the monomials $\blueD{6}\goldD{x^5}$ and $\blueD{4}\goldD{x^2}$:

- Since the lowest power of $x$ is $\goldD{x^2}$, that will be the variable part of the GCF.
- You could then find the GCF of $\blueD6$ and $\blueD4$, which is $\blueD2$, and multiply this by $\goldD{x^2}$ to obtain $\blueD2\goldD{x^2}$, the GCF of the monomials!

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 $32x^{100}$ and $16x^{88}$!

## 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!