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### Course: Algebra (all content) > Unit 10

Lesson 11: Factoring monomials- Which monomial factorization is correct?
- Factoring monomials
- Worked example: finding the missing monomial factor
- Worked example: finding missing monomial side in area model
- Factor monomials
- Greatest common factor of monomials
- Greatest common factor of monomials
- Greatest common factor of monomials

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# Factoring monomials

Learn how to completely factor monomial expressions, or find the missing factor in a monomial factorization.

#### What you should be familiar with before this lesson

A $x$ , like $3{x}^{2}$ . A $3{x}^{2}+6x-1$ .

**monomial**is an expression that is the product of constants and nonnegative integer powers of**polynomial**is a sum of monomials, likeIf $A=B\cdot C$ , then $B$ and $C$ are $A$ , and $A$ is $B$ and $C$ . To review this material, check out our article on Factoring and divisibility.

**factors**of**divisible**by#### What you will learn in this lesson

In this lesson, you will learn how to factor monomials. You will use what you already know about factoring integers to help you in this quest.

## Introduction: What is monomial factorization?

To

**factor**a monomial means to express it as a product of two or more monomials.For example, below are several possible factorizations of $8{x}^{5}$ .

$8{x}^{5}=(2{x}^{2})(4{x}^{3})$ $8{x}^{5}=(8x)({x}^{4})$ $8{x}^{5}=(2x)(2x)(2x)({x}^{2})$

Notice that when you multiply each expression on the right, you get $8{x}^{5}$ .

### Reflection question

## Completely factoring monomials

#### Review: integer factorization

To factor an integer completely, we write it as a product of primes.

For example, we know that $30=2\cdot 3\cdot 5$ .

#### And now to monomials...

To factor a monomial completely, we write the coefficient as a product of primes

*and*expand the variable part.For example, to completely factor $10{x}^{3}$ , we can write the prime factorization of $10$ as $2\cdot 5$ and write ${x}^{3}$ as $x\cdot x\cdot x$ . Therefore, this is the complete factorization of $10{x}^{3}$ :

### Check your understanding

## Finding missing factors of monomials

#### Review: integer factorization

Suppose we know that $56=8b$ for some integer $b$ . How can we find the other factor?

Well, we can solve the equation $56=8b$ for $b$ by dividing both sides of the equation by $8$ . The missing factor is $7$ .

#### And now to monomials...

We can extend these ideas to monomials. For example, suppose $8{x}^{5}=(4{x}^{3})(C)$ for some monomial $C$ . We can find $C$ by dividing $8{x}^{5}$ by $4{x}^{3}$ :

We can check our work by showing that the product of $4{x}^{3}$ and $2{x}^{2}$ is indeed $8{x}^{5}$ .

### Check your understanding

## A note about multiple factorizations

Consider the number $12$ . We can write four different factorizations of this number.

$12=2\cdot 6$ $12=3\cdot 4$ $12=12\cdot 1$ $12=2\cdot 2\cdot 3$

However, there is only $12$ , i.e. $2\cdot 2\cdot 3$ .

*one*prime factorization of the numberThe same idea holds with monomials. We can factor $18{x}^{3}$ in many ways. Here are a few different factorizations.

$18{x}^{3}=2\cdot 9\cdot {x}^{3}$ $18{x}^{3}=3\cdot 6\cdot x\cdot {x}^{2}$ $18{x}^{3}=2\cdot 3\cdot 3\cdot {x}^{3}$

Yet there is only one complete factorization!

## Challenge problems

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