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## Algebra 1 (Eureka Math/EngageNY)

### Unit 4: Lesson 1

Topic A: Lessons 1-2: Factoring monomials- Intro to factors & divisibility
- Intro to factors & divisibility
- Factors & divisibility
- 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

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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

**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

## 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:

### Check your understanding

## 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:

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.

### Check your understanding

## 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!

## Challenge problems

## Want to join the conversation?

- i need help with this i have already tried a lot but i just don't get it.(10 votes)
- Let me see if I can help.

One way to look at factorizing monomials is pulling them apart into their basic parts. Like you had a peanut butter and jelly sandwich. Pulling that into its basic parts, you could get bread times bread times peanut butter times jelly. That would be its complete factorization.

Now for a monomial

Say you have a monomial (or one term) that is 24x^2. There are a few ways to factor this. One example would be: (4x)(6x) This is not a complete factorization, but it is still a valid way to factorize it. Another way to do it would be (3)(8x^2) Its complete factorization would be 2 times 2 times 2 times 3 times x times x.

When factoring, you just need to make sure that when you preform all the operations, they still come out to be the original number. Like I need to make sure (4x)(6x) = 24x^2. Let's check. 4 times 6 equals 24, and x times x equals x^2. Those two multiplied together equals 24x^2.

I hope that helped! :)(22 votes)

- I'm having a hard time understanding how to find the width of the rectangle,how should i go about finding the width of a rectangle?(5 votes)
- Length times width equals area. So to find width, you would divide the area by length.

hope that helps!(10 votes)

- I am still confused on how to tell if it is a monomial or not. Can you help me?(5 votes)
- A monomial is a single term polynomial, with a non-negative integer exponent.

Typically each term in a polynomial is separated by addition/subtraction.(5 votes)

- If it has 1 term, it is a monomial. If it has 2 terms, it is a binomial. 3 terms or more is a polynomial.(2 votes)
- Slight clarification: All of them are polynomials. The classification as monomial, binomial, or trinomial just gives more descriptive information about the polynomial. You omitted trinomial from you list. It has 3 terms.(5 votes)

- i hate khan acadeemy no ofence but i dont learn a thing(0 votes)
- you might be suffering from missing fundamentals. Before approaching a unit, try and see what its prerequisites are first.(7 votes)

- philosophically why do we simplify?(1 vote)
- I don't know about the philosophy but I do know why we do simplification. Basically, it makes the math waayyy less hairy. I believe that 932/1864 * 3/12 is harder to solve than 1/2 * 1/4. :o. it also makes sure everyone has the same answer! If you don't simplify, you might get totally different numbers that don't look at all even close, 345/690 and 17/34 don't look related. however, if you simplified, you would know they are indeed the same number, 1/2(3 votes)

- So a simple way of doing it is by dividing the total by the number given and then subtracting the exponent given from the total and then what you get is your answer?(2 votes)
- Technically, yes. A better way to think about it is to think that you are dividing the coefficients and the exponents separately.

For example, let's say we have to find A in (A)4x^2=8x^3. We would divide both sides by 4x^2. First, divide the coefficients: 8/4=2. Then, divide the exponents: x^3/x^2=x*x*x/x*x=x. Multiplying our two results together will give us our answer: 2x.

Hope this helps!

Jonathan Myung(0 votes)

- Isn't 3x^2+6x-1 a trinomial instead of a polynomial?(0 votes)
- A polynomial literally means many (poly) numbers (nomial). So a trinomial is a form of polynomial.(2 votes)

- A monomial is a multiplication with 1 number.(1 vote)
- A monomial is one term. It can be as simple as a number (like 3) or a variable (like "x"). But it can be more complicated than that. 2x(4xy) is a monomial. All numbers and variables are connected by multiplication.(1 vote)

- i don't get the last two(0 votes)
- 6) just asks for the prime factorization, so 22 = 2 • 11, so just add that to the variables to get 2 • 11 x • y • y. Prime factorization helps us find GCF and LCM

7) this is popular way to create problems, note that A = l w, so we can set an equation

24 x^3 = 4 x^2 w divide both sides by 4 x^2, we get 4 • 6 xxx/4 xx and canceling we get 6x.(3 votes)