Factoring with Common Monomial Factor

Factoring with Common Monomial Factor

Module 1 - Lesson 1

Multiple Choice

Pretest

What is the greatest common factor (GCF) of 24 and 54?

• 1

• 2

• 4

• 6

Multiple Choice

What is the GCF of 20, 24, and 40?

• 1

• 2

• 3

• 4

Multiple Choice

What is the GCF of

 $x^2$  and  $x^9$  ?

•  $x^2$  

•  $x^7$  

•  $x^9$  

•  $x^{11}$  

Multiple Choice

What are the factors of 7x - 7?

• 7(x - 1)

• 7(x + 1)

• 7(1 - x)

• 7x - 1

Multiple Choice

What is the product of x - 6 and 2?

• 
  

  $x^2-12$
  

• $2x-12$

• $x\ -12$

• $2x\ -6$

What is your score?

If you're perfect, then well done! But if you're below 5, well you will learn more about this module.

Recap

Let's have a review on the definition of greatest common factor.

Definition of Greatest Common Factor

The greatest common factor of a number refers to the largest positive integer that divides each of the integers. For integers x and y, the greatest common factor of x and y is denoted as:

 $\gcd\left(x,y\right)$  .

Example Problem of GCF

Find the GCF of 12 and 18

We must find the GCF of 12 and 18 using the listing method:

12 = 2 * 2 * 3

18 = 2 * 3 * 3

As we can see, there is a common factor between these two numbers - 6. So, the GCF of 12 and 18 is 6.

Finding the Greatest Common Monomial Factor

Before we go into the main topic, let's have a recap on what is a common monomial factor.

Common Monomial Factor

Common monomial factor (CMF) refers to a number, variable, or combination which can be found in a given polynomial.

Finding the Greatest Common Monomial Factor

Suppose we have the expression 2x - 4. As we notice, the equation can be written as:

 $2\cdot x-2\cdot2$  
Notice that 2 is the common factor to both terms. Therefore, we rewrite this expression as:
 $2\left(x-2\right)$  
Since 2 is the common factor between 2x and -4 and there are no other factors aside from 1, we call 2 as the greatest common monomial factor. 

Example Problem # 1: Find the GCF of 4x³ and 8x².

Step 1. Factor each monomial.

 $4x^3=2\cdot2\cdot x\cdot x\cdot x$  
 $8x^3=2\cdot2\cdot2\cdot x\cdot x$  
Step 2. Find the common factors.
We can notice that we can eliminate easily 2 * 2 and x * x. These are the common factors of the expressions.
Step 3. Rewrite the factors as products.
 $2\cdot2\cdot x\cdot x=4x^2$  
Therefore the GCMF of the given pair of expressions is  $4x^2$  .

Example Problem # 2: Find the GCF of 15y⁶ and 9z.

Step 1. 

 $15y^6=3\cdot5\cdot y\cdot y\cdot y\cdot y\cdot y\cdot y$  
 $9z=3\cdot3\cdot z$  
Step 2.
Common factor: 3
No need to go into step 3 since 3 is the common factor of the given expressions.
Therefore the GCMF of  the given pair of expression is 3.
 

Factoring Polynomials

In this lesson, we will see how GCF is used in factoring polynomials.

Example Problem # 1: Factor 6x + 3x².

Step 1. Determine the number of terms.

There are two terms - 6x and 3x².

Step 2. Determine the GCF of each monomial.

6x = 3*2*x

3x² = 3*x*x

Step 3. Determine the factors and find their products.

3*x = 3x

Example Problem # 1: Factor 6x + 3x².

Step 4. Find the other factor by dividing each monomial by the GCF.
 $\frac{6x}{3x}+\frac{3x^2}{3x}=2+x$  

Step 5. Write in factored form.
 $6x+3x^2=3x\left(2+x\right)$  

Example Problem # 2: Factor 12x⁴y⁵z³ - 15xy³z⁴.

Step 1. There are two terms, 12x⁴y⁵z³ and 15xy³z⁴.

Step 2.

12x⁴y⁵z³ = 3*2*2*x*x*x*x*y*y*y*y*y*z*z*z

15xy³z⁴ = 3*5*x*y*y*y*z*z*z*z

Step 3. 3*x*y*y*y*z*z*z = 3xy³z³

Example Problem # 2: Factor 12x⁴y⁵z³ - 15xy³z⁴.

Step 4.
 $\frac{12x^4y^5z^3}{3xy^3z^3}-\frac{15xy^3z^4}{3xy^3z^3}=4x^3y^2-5z$  

Step 5.
 $12x^4y^5z^3-15xy^3z^4=3xy^3z^3\left(4x^3y^2-5z\right)$  

WE'RE DONE!!!

Have you learned something today? Well, I hope you learned something in our lesson. Prepare for a short quiz here also in Quizizz. I'll send the link later. Goodbye and thank you for listening!