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## Algebra 1 (Eureka Math/EngageNY)

### Course: Algebra 1 (Eureka Math/EngageNY)>Unit 4

Lesson 1: Topic A: Lessons 1-2: Factoring monomials

# 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 monomial is an expression that is the product of constants and nonnegative integer powers of $x$, like $3{x}^{2}$. A polynomial is a sum of monomials, like $3{x}^{2}+6x-1$.
If $A=B\cdot C$, then $B$ and $C$ are factors of $A$, and $A$ is divisible by $B$ and $C$. To review this material, check out our article on Factoring and divisibility.

#### 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 $8{x}^{5}$.
• $8{x}^{5}=\left(2{x}^{2}\right)\left(4{x}^{3}\right)$
• $8{x}^{5}=\left(8x\right)\left({x}^{4}\right)$
• $8{x}^{5}=\left(2x\right)\left(2x\right)\left(2x\right)\left({x}^{2}\right)$
Notice that when you multiply each expression on the right, you get $8{x}^{5}$.

### Reflection question

Andrei, Amit and Andrew were each asked to factor the term $20{x}^{6}$ as the product of two monomials. Their responses are shown below.
AndreiAmitAndrew
$20{x}^{6}=\left(2x\right)\left(10{x}^{5}\right)$$20{x}^{6}=\left(4{x}^{3}\right)\left(5{x}^{3}\right)$$20{x}^{6}=\left(20{x}^{2}\right)\left({x}^{3}\right)$
1) Which of the students factored $20{x}^{6}$ correctly?

## 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 $10{x}^{3}$:
$10{x}^{3}=2\cdot 5\cdot x\cdot x\cdot x$

2) Which of the following is the complete factorization of $6{x}^{2}$?

3) Which of the following is the complete factorization of $14{x}^{4}$?

## 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 $8{x}^{5}=\left(4{x}^{3}\right)\left(C\right)$ 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 $8{x}^{5}$.
$\begin{array}{rl}\left(4{x}^{3}\right)\left(2{x}^{2}\right)& =4\cdot 2\cdot {x}^{3}\cdot {x}^{2}\\ \\ & =8{x}^{5}\end{array}$

4) Find the missing factor $B$ that makes the following equality true.
$\phantom{\rule{2em}{0ex}}28{x}^{5}=\left(B\right)\left(7x\right)$

5) Find the missing factor $C$ that makes the following equality true.
$\phantom{\rule{2em}{0ex}}40{x}^{9}=\left(C\right)\left(4{x}^{3}\right)$
$C=$

## 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 one prime factorization of the number $12$, i.e. $2\cdot 2\cdot 3$.
The 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!
$18{x}^{3}=2\cdot 3\cdot 3\cdot x\cdot x\cdot x$

## Challenge problems

6*) Write the complete factorization of $22x{y}^{2}$.
$22x{y}^{2}=$

7*) The rectangle below has an area of $24{x}^{3}$ square meters and a length of $4{x}^{2}$ meters.
What is the width of the rectangle?
$\text{Width}=$
meters