Factoring GCF and Grouping-- Unit 4 Quiz#1

Intro to Factoring Unit 4 Quiz: GCF and Grouping

Copy each problem, show work for each problem, and describe what you did on each problem.

Example for #1

The Greatest Common Factor, the GCFF, is the biggest factor in common, including the variables. So, the GCF of 24x² and 36x is:

24x² factors to 12⋅2xx or 3⋅8xx or 6⋅4xx

36x factors to 2⋅18x or 9⋅4x or 12⋅3x

We can see that the largest or Greatest Common Factor is 12x.

Multiple Choice

#1 What is the Greatest Common Factor of 12x² and 16x?

• 6x

• 6x²

• 4x

• 2x

Example for #2

Factor: 28x²y - 21xy

28x²y factors to: 2⋅14 xxy or 4⋅7xxy

21xy factors to: 21⋅1xy or 3⋅7xy

We can see the GCF is 7xy

So, write the GCF first, then the parenthesis. In the parenthesis write the parts the terms don't have in common.

7xy(4x - 3)

You can use Distributive property to make sure this factoring works. So, if you multiply you would go back to 28x²y - 21xy.

Multiple Choice

#2. Factor 49x²y - 7xy

• 7y(7x - x)

• 7xy(7x + 1)

• 7xy(7x -1)

• 1x(49xy + 1)

Example for #3--- Factor: 18x³y³ - 36x²y² + 60x³y²

18x³y³ factors to: 6⋅3 xxxyyy

36x²y² factors to: 6⋅6xxyy

60x³y² factors to: 6⋅10xxxyy

We can see the GCF is 6xxyy

So, write the GCF first, then the parenthesis. In the parenthesis write the parts the terms don't have in common.

6x²y²(3xy - 6 +10x)

You can use Distributive property to make sure this factoring works. So, if you multiply you would go back to 28x²y - 21xy.

Multiple Choice

#3. Factor: 8xy² - 20x²y² - 28x³y²

• 2(4x -10y2 -14y)

• 2x²y²(4x -10y² -14y)

• 4xy²(x -5y -7x³y)

• 4xy²(2 -5x -7x²)

Example for #4-5 Factor by Grouping

Factor: 3x(8x - 1) +1(8x - 1)

Answer: (3x + 1)(8x -1) Factored! :)

Notice that this problem shows you the second step in the distributive property. If you were to multiply (3x + 1)(8x -1), you would separate the (3x + 1) to show the double distribution. You would write as 3x(8x -1) + 1(8x - 1)

Multiple Choice

#4 Factor 7x(9x -2)-1(9x -2)

• (7x - 1)(9x - 2)

• (7x + 1)(9x - 2)

• (7x - 1)(9x + 1)

• (7x - 2)(9x - 1)

Multiple Choice

#5 Factor 10x(x -2) + 6(x -2)

• (10x - 2)(6x - 12)

• (10x + 6)(x - 2)

• (10x - 6)(x - 2)

• (10x + 6)(x + 2)

Example for #6-11 Factor by Grouping

Factor: 21x² -3x -7x + 1 Answer: (3x -1)(7x -1)

step 1) Make 2 groups: 21x² -3x and -7x + 1

step 2) Factor out the GCF from each group:

The GCF of 21x² -3x is 3x because 21x² factors to 7⋅3xx, and 3x factors to 3⋅1x

The GCF of -7x + 1 is -1 because 7x factors to 7⋅1x and 1 factors to 1⋅1, and you use the negative sign that's with the -7x term.

So, factor out the GCF from each group to get: 3x(7x - 1) -1(7x - 1)

step 3) Write binomial factors

(3x -1)(7x -1)

Multiple Choice

#6 Factor:15x² + 5x - 6x + 2

The first step when you factor by grouping is to....

• Make a 1 group with 15x² + 5x - 6x

• Make 3 groups with 5x - 6x + 2

• Make 2 groups; one with 5x -6x and another group with 15x² + 5x

• Make 2 groups; one with 15x² + 5x and another group with - 6x + 2

Multiple Select

#7 Factor: 15x² + 5x - 6x + 2

The second step when you factor by grouping is to....

*Choose 2 that apply

• Find and factor out the GCF of 15x² + 5x which is 5x

• Find and factor out the GCF of 15x² + 5x which is x

• Find and factor out the GCF of - 6x + 2 which is -2

• Find and factor out the GCF of - 6x + 2 which is +1x

Multiple Choice

#8 Factor: 15x² + 5x - 6x - 2

Also, in the second step when you factor by grouping you....

• write with the GCF factored out 5x(3x+ 1) -2(3x + 1)

• write with the GCF factored out 5x(3x - 1) -2(3x + 1)

• write with the GCF factored out 5x(3x + 2) -2(3x + 5)

• write with the GCF factored out 5x(3x + 1) +6(3x + 1)

Multiple Select

#9 Factor: 15x² + 5x - 6x - 2

The third and final step when you factor by grouping is to....

*Choose 3 that apply

• write as two parenthesis being multiplied (5x - 2)(3x + 1)

• write as two parenthesis being multiplied called binomial factors (5x - 2)(3x + 1)

• write (5x - 2)(3x + 1)

• eat a fried stick of butter, yum

Multiple Choice

#10 Factor by Grouping: 25x² + 10x - 20x - 8

*Show all work on your paper! Try your best to describe your work :)

• (5x + 4)(5x + 2)

• (5x - 4)(5x + 2)

• (5x - 2)(5x + 6)

• (5x - 4)(5x - 2)

Multiple Select

#11 What expressions are equivalent to 6x² -42x + 9x -21 as you factor by grouping?

*Choose 2 that apply.

• 6x(x - 7) - 3x(3x - 7)

• 6x(x - 7) + 3(3x - 7)

• (6x + 3)(x - 7)

• (6x - 3)(9x - 21)

Multiple Choice

#12 Charlie is factoring this problem. She would like you to fill in the blanks in the factoring process.

9x² - 12x + 18x -24

__( __ - __) +6(3x - __)

• 4x; 5x; 3; 3

• 3x; 3x; 4; 4

• 3x²; 3x; 4; 4

• 3x; 4x; 4; 3

Multiple Select

#13 How do you find the GCF of a polynomial?

Show how to find the GCF of 45x² + 9x and then, the GCF of x - 5 on your paper and show your work. **Choose 2 that apply.

• You factor each term into its multiples, including the variables, and find the smallest common factor.

• You factor 45x² to 9⋅5xx; and factor 9x to 9⋅1x;

  and then factor x to 1⋅1x;

  and factor 5 to 1⋅5

• You factor each term into its multiples, including the variables, and find the biggest or greatest common factor.

• You factor 45x² to 9⋅5xx; and factor 9x to 3⋅3xx;

  and then factor x to 1⋅1;

  and factor 5 to 1⋅5xxx.