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## Algebra 2

### Course: Algebra 2>Unit 3

Lesson 1: 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, squared. A polynomial is a sum of monomials, like 3, x, squared, plus, 6, x, minus, 1.
If A, equals, B, dot, 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, start superscript, 5, end superscript.
• 8, x, start superscript, 5, end superscript, equals, left parenthesis, 2, x, squared, right parenthesis, left parenthesis, 4, x, cubed, right parenthesis
• 8, x, start superscript, 5, end superscript, equals, left parenthesis, 8, x, right parenthesis, left parenthesis, x, start superscript, 4, end superscript, right parenthesis
• 8, x, start superscript, 5, end superscript, equals, left parenthesis, 2, x, right parenthesis, left parenthesis, 2, x, right parenthesis, left parenthesis, 2, x, right parenthesis, left parenthesis, x, squared, right parenthesis
Notice that when you multiply each expression on the right, you get 8, x, start superscript, 5, end superscript.

### Reflection question

Andrei, Amit and Andrew were each asked to factor the term 20, x, start superscript, 6, end superscript as the product of two monomials. Their responses are shown below.
AndreiAmitAndrew
20, x, start superscript, 6, end superscript, equals, left parenthesis, 2, x, right parenthesis, left parenthesis, 10, x, start superscript, 5, end superscript, right parenthesis20, x, start superscript, 6, end superscript, equals, left parenthesis, 4, x, cubed, right parenthesis, left parenthesis, 5, x, cubed, right parenthesis20, x, start superscript, 6, end superscript, equals, left parenthesis, 20, x, squared, right parenthesis, left parenthesis, x, cubed, right parenthesis
1) Which of the students factored 20, x, start superscript, 6, end superscript 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, equals, 2, dot, 3, dot, 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, cubed, we can write the prime factorization of 10 as 2, dot, 5 and write x, cubed as x, dot, x, dot, x. Therefore, this is the complete factorization of 10, x, cubed:
10, x, cubed, equals, 2, dot, 5, dot, x, dot, x, dot, x

2) Which of the following is the complete factorization of 6, x, squared?

3) Which of the following is the complete factorization of 14, x, start superscript, 4, end superscript?

## Finding missing factors of monomials

#### Review: integer factorization

Suppose we know that 56, equals, 8, b for some integer b. How can we find the other factor?
Well, we can solve the equation 56, equals, 8, b 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, start superscript, 5, end superscript, equals, left parenthesis, 4, x, cubed, right parenthesis, left parenthesis, C, right parenthesis for some monomial C. We can find C by dividing 8, x, start superscript, 5, end superscript by 4, x, cubed:
\begin{aligned}8x^5&=(4x^3)(C)\\ \\ \dfrac{8x^5}{4x^3}&=\dfrac{(4x^3)(C)}{4x^3}&&\small{\gray{\text{Divide both sides by }4x^3}}\\ \\\\\\ 2x^2&=C&&\small{\gray{\text{Simplify with properties of exponents}}} \end{aligned}
We can check our work by showing that the product of 4, x, cubed and 2, x, squared is indeed 8, x, start superscript, 5, end superscript.
\begin{aligned}(\purpleC{4}\tealD {x^3})(\purpleC{2}\tealD{x^2})&=\purpleC 4\cdot \purpleC{2}\cdot \tealD {x^3}\cdot \tealD{x^2}\\ \\ &=\purpleC{8}\tealD{x^5} \end{aligned}

4) Find the missing factor B that makes the following equality true.
28, x, start superscript, 5, end superscript, equals, left parenthesis, B, right parenthesis, left parenthesis, 7, x, right parenthesis

5) Find the missing factor C that makes the following equality true.
40, x, start superscript, 9, end superscript, equals, left parenthesis, C, right parenthesis, left parenthesis, 4, x, cubed, right parenthesis
C, equals

## A note about multiple factorizations

Consider the number 12. We can write four different factorizations of this number.
• 12, equals, 2, dot, 6
• 12, equals, 3, dot, 4
• 12, equals, 12, dot, 1
• 12, equals, 2, dot, 2, dot, 3
However, there is only one prime factorization of the number 12, i.e. 2, dot, 2, dot, 3.
The same idea holds with monomials. We can factor 18, x, cubed in many ways. Here are a few different factorizations.
• 18, x, cubed, equals, 2, dot, 9, dot, x, cubed
• 18, x, cubed, equals, 3, dot, 6, dot, x, dot, x, squared
• 18, x, cubed, equals, 2, dot, 3, dot, 3, dot, x, cubed
Yet there is only one complete factorization!
18, x, cubed, equals, 2, dot, 3, dot, 3, dot, x, dot, x, dot, x

## Challenge problems

6*) Write the complete factorization of 22, x, y, squared.
22, x, y, squared, equals

7*) The rectangle below has an area of 24, x, cubed square meters and a length of 4, x, squared meters.
A rectangle with the width labeled width and the length being four x squared. Inside the rectangle is twenty four x cubed.
What is the width of the rectangle?
start text, W, i, d, t, h, end text, equals
meters