Cache County School District 2013-2014
Secondary II Math
Utah Integrated
Mathematics Core
Student Edition - Honors
Unit 4:
Factoring
Secondary II Unit 4 – Factoring and Solving Quadratics by Factoring:
Table of Contents
Homework Help (QR Codes and links to videos, tutorials, examples)………………….
Section 4.1 – Greatest Common Factor, Teacher Notes ......................................................
Notes, Assignment ......................................................................................................
Section 4.2 – Factoring by Grouping, Teacher Notes ..........................................................
Notes, Assignment .......................................................................................................
Section 4.3 – Difference of Square and Perfect Square Trinomials Task
Teacher Notes, Notes, Assignment ..............................................................................
Section 4.4 – Factoring when a=1 Task, Teacher Notes .....................................................
Notes, Assignment .......................................................................................................
Factoring Matching Activity……………………………………………………………
Section 4.1-4.4 Review Worksheet ……………………………………………………….
Section 4.5 – Factoring when a does not equal 1, Teacher Notes .......................................
Notes, Assignment .......................................................................................................
Section 4.6 –Additional Factoring, Teacher Notes ..............................................................
Notes, Assignment .......................................................................................................
Section 4.7 – Factoring Review Activity, Review Assignment ……………………………....
Section 4.8 – Factoring Simple Quadratic Expressions over the Complex Number System
Task, Teacher Notes, Notes, Assignment………………………….………………………
Factoring Review Puzzle …………………..…………………………………
Review Worksheets






#1 – G.C.F
#2 – Difference of Squares
#3 – Factoring Trinomials 1
#4 – Factoring Binomials and Trinomials
#5 - Reviewing Factoring Skills
#6 – Factoring Polynomials Completely
Secondary II Unit 4: Factoring and Solving Quadratics by Factoring
Homework Help
Section 4.1
http://goo.gl/Y6azf
video
http://goo.gl/UAFAV
http://goo.gl/I9fJn
Section 4.2
http://goo.gl/FBr3H
http://goo.gl/f2MMb
http://goo.gl/A8QMy
Section 4.3
video
Difference of squares
http://goo.gl/sogMf
http://goo.gl/HTmYP
http://goo.gl/qEaWs
http://goo.gl/IWngu
http://goo.gl/pjUoT
Section 4.4
http://goo.gl/GdDfw
Section 4.5
http://goo.gl/Xn4mI
http://goo.gl/Pa8z0
http://goo.gl/Ma7Y0
Section 4.6
No additional resources for this section. Use resources above. And www.cachemath2.wordpress.com
Section 4.7
No additional resources for this section. This section is a review. Use resources above.
And www.cachemath2.wordpress.com
Section 4.7
No additional resources for this section. This section is a review. Use resources above.
And www.cachemath2.wordpress.com
Unit 4 Lesson 1 – Greatest Common Factor
Task 4.1
Name_________________________________
_
Date_________ Hour_______
Complete the task following these simple rules:
1. You can only use multiplication.
2. You can only use the numbers 2, 3, 5, and 7 and they may be repeated.
40 = _________∙_________∙ __________∙_________
36 = _________∙_________∙ __________∙_________
48 =
72 =
128 =
135 =
675 =
112 =
210 =
189000 =
What is special about the numbers I had you use?
What does it mean to be a “multiple” of a number?
What does it mean to be a “factor” of a number?
What other methods can I use to find these prime factorizations without guessing?
Let’s try a few examples using these methods:
Unit 4 Lesson 1 – Greatest Common Factor
Notes 4.1
Note that:
21𝑥 4 = 3 ∙ 7 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 - and - 18𝑥 2 = 2 ∙ 3 ∙ 3 ∙ 𝑥 ∙ 𝑥 and so on.
Using prime factorizations, find the GCF between the following sets of numbers.
Example 1:
a.
40 𝑎𝑛𝑑 36
b.
56 𝑎𝑛𝑑 70
c.
21𝑥 4 𝑎𝑛𝑑 18𝑥 2
d.
70𝑥 3 𝑎𝑛𝑑 42𝑥 2
e.
60𝑎𝑏 𝑎𝑛𝑑 126𝑎2 𝑏 2
f.
42𝑥 3 𝑎𝑛𝑑 90𝑥
g.
6𝑥 3 𝑦 𝑎𝑛𝑑 20𝑥𝑦 2
h.
4𝑥 3 𝑦 2 𝑎𝑛𝑑 18𝑥 2 𝑦
Example 2:
a.
12, 28, 36
b.
63, 81, 18
21, 35, 63
d.
25, 50, 60
c.
Example 3: Find the GCF of the following terms. Then, factor out the GCF and rewrite an equivalent
polynomial expression.
a.
6𝑥 2 + 9𝑥
b.
12𝑥 4 + 21𝑥 2 − 15𝑥
c.
2𝑥 2 + 6𝑥 + 8
d.
8𝑤 4 − 3𝑤 3 + 5𝑤 2
e.
9𝑧 3 − 3𝑧 2 + 15𝑧
f.
4𝑥 2 − 12𝑥 − 16
g.
5𝑥 5 + 10𝑥 4 + 15𝑥 3
h.
−8𝑥 4 − 32𝑥 3 + 16𝑥 2
Unit 4 Lesson 1 – Greatest Common Factors
Ready, Set, Go! - Assignment 4.1
Name______________________________
Date_________ Hour_______
http://goo.gl/Y6azf
Ready
1. True or False. If the statement is false, then give the correct statement.
a.
There are only nine prime numbers. _____________________
b.
The prime factorization of 32 𝑖𝑠 23 ∙ 3 _______________________
c.
The integer 51 is a prime number. ________________________
d.
The GCF for the integers 12 and 16 is 4. ______________________
e.
The GCF for the integers 10 and 21 is 1. ______________________
f.
The GCF for the polynomial 2𝑥 2 − 6𝑥𝑦 2 𝑖𝑠 𝑥 4 𝑦 3 ____________________________
g.
For the polynomial 2𝑥 2 𝑦 − 6𝑥𝑦 2 you could factor out either 2𝑥𝑦 𝑜𝑟 − 2𝑥𝑦. _______________
h.
The greatest common factor for the polynomial 8𝑎3 𝑏 − 12𝑎2 𝑏 𝑖𝑠 4𝑎𝑏. ____________________
i.
𝑥 − 7 = 7 − 𝑥 for any real number x. ____________________
j.
−3𝑥 2 + 6𝑥 = −3𝑥(𝑥 − 2) for any real number x. ________________________
2.
Find the greatest common factor (GCF) for each group of integers or monomials.
a.
40, 48, 88
b.
76, 84, 100
c.
66𝑎3 , 72𝑎2 𝑏, 120𝑎4 𝑏 3
d.
81𝑥 2 𝑦 3 𝑧, 200𝑥 3 𝑦 2 𝑧, 539𝑥 4 𝑦 4 𝑧
3.
Complete the factorization of each monomial. (These are not prime factorizations)
a.
27𝑥 = 9(
c.
24𝑡 2 = 8𝑡(
e.
36𝑦 5 = 4𝑦 2 (
g.
−14𝑚4 𝑛3 = 2𝑚4 (
)
)
)
)
b.
51𝑦 = 3𝑦 (
)
d.
18𝑢2 = 3𝑢 (
f.
42𝑧 4 = 3𝑧 2 (
h.
−96𝑎3 𝑏 4 𝑐 5 = −12𝑎𝑏 3 𝑐 3 (
)
)
)
Set
Factor out the GCF in each expression. Then, factor out the GCF and use it to write the polynomial in an
equivalent form.
4.
2𝑤 + 4𝑡
5.
12𝑥 − 18𝑦
6.
24𝑎 − 36𝑏
7.
𝑥 3 − 6𝑥
8.
5𝑎𝑥 + 5𝑎𝑦
9.
ℎ5 + ℎ3
10.
−6ℎ5 𝑦 2 + 3ℎ3 𝑦 6
11.
2𝑥 3 − 6𝑥 2 + 8𝑥
12.
6𝑥 3 + 18𝑥 2 + 24𝑥
14.
15𝑥 2 − 9𝑥𝑦 2 + 6𝑥 2 𝑦
13.
12𝑥 4 𝑡 + 30𝑥 3 𝑡 − 24𝑥 2 𝑡 2
Go!
First factor out the GCF and rewrite the expression in an equivalent form; and then factor out the
opposite of the GCF and rewrite the expression in an equivalent form..
15.
8𝑥 − 8𝑦
16.
−5𝑥 2 + 10𝑥
17.
𝑎−6
18.
4 − 7𝑎
19.
−30𝑏 4 + 75𝑏 3
20.
−2𝑥 3 + 6𝑥 2 − 2𝑥
21.
12𝑢5 𝑣 6 + 18𝑢2 𝑣 3 − 15𝑢4 𝑣 5
22.
−𝑥 + 5
Unit 4 Lesson 2 – Factor by Grouping
Notes 4.2
Factor out the GCF in each expression and then write an equivalent form of the equation.
(𝑥 − 3)𝑎 + (𝑥 − 3)𝑏
(𝑦 + 4)3 + (𝑦 + 4)𝑧
1.
2.
3.
𝑥(𝑥 − 1) − 5(𝑥 − 1)
4.
𝑎(𝑎 + 1) − 3(𝑎 + 1)
5.
𝑚(𝑚 + 9) + (𝑚 + 9)
6.
𝑤(𝑤 + 2)2 + 8(𝑤 + 2)2
Use grouping to write the polynomials in an equivalent form by factor each polynomial completely.
Recall that you must factor out the GCF first if possible.
7.
𝑏𝑥 + 𝑏𝑦 + 𝑐𝑥 + 𝑐𝑦
8.
𝑥3 + 𝑥2 − 𝑥 − 1
9.
12𝑥 3 + 2𝑥 2 − 30𝑥 − 5
10.
21𝑘 3 − 84𝑘 2 + 15𝑘 − 60
11.
𝑥𝑎 + 𝑎𝑦 + 3𝑦 + 3𝑥
12.
𝑎𝑏𝑐 − 3 + 𝑐 − 3𝑎𝑏
Additional Notes/Examples:
Unit 4 Lesson 2 – Factor by Grouping
Ready, Set, Go! - Assignment 4.2
Name______________________________
Date_________ Hour_______
http://goo.gl/FBr3H
Ready
Use grouping to write the polynomials in an equivalent form by factor each polynomial completely.
1.
𝑥𝑦 + 2𝑦 + 3𝑥 + 6
2.
𝑎𝑥 + 3𝑦 − 3𝑥 − 𝑎𝑦
3.
x 3  x 2  2x  6
4.
x 3  3x 2  4x  12
5.
x 3  3x 2  x  3
6.
x 3  2x 2  5x  10
7.
x3  x 2  x  1
8.
1  x  x 2  x3
9.
x 3  2x 2  14x  7x 2
10.
x 3  x 2  2  2x
Set
Factor each expression completely, (by grouping).
11.
12𝑎3 − 9𝑎2 + 4𝑎 − 3
12.
2𝑝3 + 5𝑝2 + 6𝑝 + 15
13.
12𝑛3 + 4𝑛2 + 3𝑛 + 1
14.
5𝑛3 − 10𝑛2 + 3𝑛 − 6
15.
3𝑛3 − 4𝑛2 + 9𝑛 − 12
16.
𝑚3 − 𝑚2 + 2𝑚 − 2
Go!
Factor each expression completely
17.
40𝑥𝑦 + 30𝑥 − 100𝑦 − 75
19.
90𝑎𝑢 − 36𝑎𝑣 − 150𝑦𝑢 + 60𝑦𝑣
18.
140𝑎𝑏 − 60𝑎2 + 168𝑏 − 72𝑎
20.
16𝑥 2 𝑐 + 8𝑥𝑦𝑑 − 16𝑥 2 𝑑 − 8𝑥𝑦𝑐
Unit 4 Lesson 3 – Difference of Squares and Perfect Square Trinomials
Task 4.3
Name______________________________
Date_________ Hour_______
Work with your partner to complete the following table and answer the questions bellow:
Factoring Difference of Squares
Factors
Find the product – show work
Final Product
(x + 2)(x – 2)
(x + 3)(x – 3)
(2x + 1)(2x – 1)
(3x – 2 )(3x + 2)
Compare the factors in the first column and write at least two things they have in common (look for
patterns):
*
*
Compare the products in the third column and write at least two things they have in common (look for
patterns):
*
*
Using your observations, write a formula for the Difference of Two Squares
𝑎2 − 𝑏 2 =
Using what you have learned about difference of two squares, factor the following.
1.
𝑥 2 − 49 =
3.
4𝑛2 − 100 =
2.
4.
𝑚2 − 16 =
25𝑦 2 − 64 =
Factoring a Perfect Square Trinomial
Factors
Find the product – show work
Final Product
(x + 2)2
(x + 3) 2
(2x + 1) 2
(3x + 2) 2
Compare the products in the first column and list at least two things they have in common (look for
patterns):
*
*
Compare the factors in the third column and list as least two things they have in common (look for
patterns):
*
*
Using your observations, write a formula for the square of a binomial or a perfect square with a sum
(𝑎 + 𝑏)2 =
Using what you have learned about perfect squares or squared binomials, find the following products
without using the distributive property and writing down your work.
1. (𝑥 + 2)2 =
2. (𝑥 + 5)2 =
3. (2𝑥 + 3)2 =
4. (3𝑥 + 1)2 =
Multiply one squared binomial and find the product to complete the table
Factors (Squared
Binomial)
Find the product – show work
Final Product
(𝑥 − 2)2
(𝑥 − 6)2
(2𝑥 − 3)2
(3𝑥 − 2𝑦)2
Write 2 observation or patterns that you see regarding the factors and/or their product.
*
*
Using your observations, write a formula for the square of a binomial or a perfect square with a
difference
(𝑥 − 𝑦)2 =
Using what you have learned about perfect squares or squared binomials, find the following products
without using the distributive property:
1. (𝑥 − 2)2 =
2. (𝑥 − 5)2 =
3. (2𝑥 − 3)2 =
A trinomial is a perfect square trinomial if…
1.
the first and last terms are of the form 𝑎2 𝑎𝑛𝑑 𝑏 2 (𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒𝑠)
2. the middle term is 2𝑎𝑏 𝑜𝑟 − 2𝑎𝑏. (2 ∙ 𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚 ∙ 𝑙𝑎𝑠𝑡 𝑡𝑒𝑟𝑚)
Unit 4 Lesson 3 – Difference of Squares and Perfect Square
Trinomials
Ready, Set, Go! - Assignment 4.3
Name______________________________
Date_________ Hour_______
http://goo.gl/sogMf
Ready
1. True or False. If false, explain why.
a.
The polynomial 𝑥 2 + 16 is a difference of two squares.
b.
The polynomial 𝑥 2 − 8𝑥 + 16 is a perfect square trinomial.
c.
The polynomial 9𝑥 2 + 21𝑥 + 49 is a perfect square trinomial.
d.
(4𝑥 2 + 4) = (2𝑥 + 2)2 for any real number x.
e.
The polynomial 16𝑦 + 1 is a prime polynomial.
f.
The polynomial 𝑥 2 + 9 can be factored as (𝑥 + 3)(𝑥 + 3).
g.
The polynomial 4𝑥 2 − 4 is factored COMPLETELY as 4(𝑥 2 − 1).
h.
(𝑦 2 − 2𝑦 + 1) = (𝑦 − 1)2 for any real number y.
i.
2𝑥 2 − 18 = 2(𝑥 − 3)(𝑥 + 3) for any real number x.
Set
Determine whether each polynomial can be written as a difference of two squares, a perfect square
trinomial, or neither of these.
2.
𝑥 2 − 20𝑥 + 100
3.
𝑥 2 − 10𝑥 − 25
4.
𝑦 2 − 40
5.
𝑎2 − 49
6.
4𝑦 2 + 12𝑦 + 9
7.
9𝑎2 − 30𝑎 − 25
8.
𝑥 2 − 8𝑥 + 64
9.
𝑥 2 + 4𝑥 + 4
10.
9𝑦 2 − 25𝑐 2
11.
9𝑥 2 + 4
12.
9𝑎2 + 6𝑎𝑏 + 𝑏 2
13.
4𝑥 2 − 4𝑥𝑦 + 𝑦 2
Go!
Write an equivalent form of each polynomial by factoring each polynomial completely.
14.
𝑎2 − 144
15.
4𝑥 2 − 9
16.
1 − 49𝑐 2
17.
100𝑘 2 − 49
18.
𝑓 2 − 36
20.
2𝑥 2 − 8
21.
𝑥 2 + 2𝑥 + 1
22.
𝑦 2 + 4𝑦 + 4
23.
𝑤 2 + 10𝑤 + 25
24.
𝑏 2 − 6𝑏 + 9
25.
25𝑦 2 − 10𝑦 + 1
26.
9𝑦 2 − 12𝑦 + 4
19.
20𝑞 2 − 5𝑟 2
27.
144𝑥 2 + 24𝑥 + 1
28.
𝑥 2 − 2𝑥𝑦 + 𝑦 2
29.
9𝑤 2 + 42𝑤 + 49
30.
4𝑡 2 + 20𝑡 + 25
31.
5𝑥 2 − 125
32.
−2𝑥 2 + 18
33.
𝑎3 − 𝑎𝑏 2
34.
𝑥2𝑦 − 𝑦
35.
12𝑎2 + 36𝑎 + 27
36.
−5𝑦 2 + 50𝑦 − 125
37.
𝑥 3 − 2𝑥 2 𝑦 + 𝑥𝑦 2
38.
𝑥 3 𝑦 + 2𝑥 2 𝑦 2 + 𝑥𝑦 3
Unit 4 Lesson 4 – Factoring 𝑎𝑥 2 + 𝑏 + 𝑐 𝑤𝑖𝑡ℎ 𝑎 = 1
Task 4.4
Name______________________________
Date_________ Hour_______
Tic-Tac-But-No-Toe
Part 1: In the following tic tac’s there are four numbers. Find the relationship that the two numbers on
the right have with the two numbers on the left.
-90 10
36 -6
-36 -6
-30 -6
1
-9
-12 -6
0
6
-1
5
-49
7
120 30
-81
9
24
-6
0
-7
34 4
0
-9
-10 -4
-72
24
16 4
-6 -3
49 -7
8
-1
-14
21 -3
4
Observations:
1.
What did you find?
2.
Did it follow the pattern every time?
2
-7
Part 2: Use your discoveries from Part 1 to complete the following Tic Tac’s.
9
16
6
-35
10
-10
7
2
4
45
6
-3
-5
14
-5
-2
-15
-6
-72
72
2
-5
-1
-38
-36
-22
5
9
Did your discovery work in every case?
Can you give any explanation for this?
Unit 4 Lesson 4 – Factoring 𝑎𝑥 2 + 𝑏 + 𝑐 𝑤𝑖𝑡ℎ 𝑎 = 1
Notes 4.4
Notes
Factor each trinomial completely, if possible.
1.
𝑏 2 + 8𝑏 + 7
2.
𝑚2 + 𝑚 − 90
3.
𝑛2 − 10𝑛 + 9
4.
𝑚2 + 2𝑚 − 24
5.
𝑘 2 − 13𝑘 + 40
6.
𝑧 2 − 4𝑧 + 24
7.
2𝑛2 + 6𝑛 − 108
8.
5𝑛2 + 10𝑛 + 20
9.
4𝑣 2 − 4𝑣 − 8
11.
If the only factors of a polynomial are 1 and itself, then the polynomials is _______________.
Additional Notes/Examples:
10.
2𝑝2 + 2𝑝 − 4
Unit 4 Lesson 4 – Factoring 𝑎𝑥 2 + 𝑏 + 𝑐 𝑤𝑖𝑡ℎ 𝑎 = 1
Ready, Set, Go! - Assignment 4.4
Name______________________________
Date_________ Hour_______
http://goo.gl/GdDfw
Ready
1. State whether each of the following statements is true or false.
a.
𝑥 2 − 6𝑥 + 9 = (𝑥 − 3)2 ________________________
b.
𝑥 2 − 8𝑥 − 9 = (𝑥 − 8)(𝑥 − 9) ___________________________
c.
𝑥 2 − 10𝑥𝑦 + 9𝑦 2 = (𝑥 − 𝑦)(𝑥 − 9𝑦) __________________________
d.
𝑥 2 + 𝑥 + 1 = (𝑥 + 1)(𝑥 + 1) ________________________
e.
𝑥 2 + 𝑥𝑦 + 20𝑦 2 = (𝑥 + 5𝑦)(𝑥 − 4𝑦) _________________________
f.
𝑥 2 + 1 = (𝑥 + 1)2
2.
How can you check if you have factored a trinomial correctly?
3.
What should you always look for first when attempting to factor a polynomial completely?
__________________________
Set
Factor each polynomial completely. If the polynomial is prime, say so.
4.
𝑦 2 + 7𝑦 + 10
5.
𝑎2 − 6𝑎 + 8
6.
𝑚2 − 10𝑚 + 16
7.
𝑚2 − 17𝑚 + 16
8.
𝑚2 + 6𝑚 − 16
9.
𝑤 2 − 8 − 2𝑤
10.
−16 + 𝑚2 − 6𝑚
11.
𝑎2 − 2𝑎 − 12
12.
𝑥 2 + 3𝑥 + 3
13.
3𝑦 + 𝑦 2 − 10
14.
𝑚2 + 12𝑚 + 20
15.
𝑡 2 + 30𝑡 + 200
Go!
Factor each polynomial completely. If the polynomial is prime, say so.
16.
2𝑘 2 + 22𝑘 + 60
17.
4𝑣 2 − 30𝑣 + 40
18.
6𝑣 2 + 66𝑣 + 60
19.
2𝑝2 + 2𝑝 − 4
20.
4𝑣 2 − 4𝑣 − 8
21.
5𝑣 2 − 30𝑣 + 40
22.
5𝑛2 + 10𝑛 + 20
23.
2𝑛2 + 6𝑛 − 108
Factoring Matching Activity
Name______________________________
Date_________ Hour_______
Cut out each pair of quadratic equations and match each equation to its equivalent form.
Standard Form
Factored Form
1.
𝑦 = 𝑥 2 + 3𝑥 + 2
a.
𝑦 = (𝑥 − 1)(𝑥 + 3)
2.
𝑦 = 𝑥 2 + 2𝑥 − 3
b.
𝑦 = (𝑥 − 3)(𝑥 + 1)
3.
𝑦 = 𝑥 2 − 1𝑥 − 6
c.
𝑦 = (𝑥 + 1)(𝑥 − 5)
4.
𝑦 = 𝑥2 + 𝑥 − 6
d.
𝑦 = (𝑥 − 3)(𝑥 + 2)
5.
𝑦 = 𝑥 2 + 5𝑥 + 4
e.
𝑦 = (𝑥 − 2)(𝑥 − 5)
6.
𝑦 = 𝑥 2 + 2𝑥 − 3
f.
𝑦 = (𝑥 + 2)(𝑥 + 4)
7.
𝑦 = 𝑥2 − 𝑥 − 6
g.
𝑦 = (𝑥 + 1)(𝑥 + 4)
8.
𝑦 = 𝑥 2 + 6𝑥 + 5
h.
𝑦 = (𝑥 + 1)(𝑥 + 2)
9.
𝑦 = 𝑥 2 − 4𝑥 − 5
i.
𝑦 = (𝑥 + 2)(𝑥 − 6)
10.
𝑦 = 𝑥 2 − 4𝑥 − 12
j.
𝑦 = (𝑥 + 1)(𝑥 + 5)
11.
𝑦 = 𝑥 2 + 6𝑥 + 8
k.
𝑦 = (𝑥 + 2)(𝑥 − 3)
12.
𝑦 = 𝑥 2 − 5𝑥 + 6
l.
𝑦 = (𝑥 + 2)(𝑥 + 5)
13.
𝑦 = 𝑥 2 − 2𝑥 − 3
m.
𝑦 = (𝑥 − 2)(𝑥 + 3)
14.
𝑦 = 𝑥 2 + 7𝑥 + 12
n.
𝑦 = (𝑥 + 3)(𝑥 − 1)
15.
𝑦 = 𝑥 2 + 7𝑥 + 10
o.
𝑦 = (𝑥 − 3)(𝑥 − 2)
16.
𝑦 = 𝑥 2 − 7𝑥 + 10
p.
𝑦 = (𝑥 + 3)(𝑥 + 4)
Factoring Review Assignment Lesson 4.1-4.4
Name______________________________
Date_________ Hour_______
Review Assignment Covering Sections 4.1-4.4
Write an equivalent form of each expression by factoring each polynomial completely.
1.
𝑥4 − 𝑥3
2.
2𝑤 2 − 162
3.
4.
−𝑎3 − 100𝑎
5.
𝑥 3 − 2𝑥 2
6.
7.
4𝑟 2 + 9
8.
10.
𝑤 2 − 18𝑤 + 81
11.
13.
𝑎𝑥 + 𝑎𝑦 + 𝑐𝑥 + 𝑐𝑦
14.
𝑡 2 + 4𝑧 2
𝑤 2 + 30𝑤 + 81
𝑦 3 + 𝑦 2 − 4𝑦 − 4
9.
12.
15.
6𝑤 4 − 54𝑤 2
𝑥 3 + 7𝑥 2
𝑥 2 𝑤 2 + 9𝑥 2
6𝑤 2 − 12𝑤 − 18
−2𝑥 2 − 10𝑥 − 12
16.
−𝑎3 − 2𝑎2 − 𝑎
17.
32𝑥 2 − 2𝑥 4
18.
19.
𝑤 3 − 3𝑤 2 − 18𝑤
20.
18𝑤 2 + 𝑤 3 + 36𝑤
21.
9𝑦 2 + 1 + 6𝑦
22.
2𝑎2 + 1 + 3𝑎
23.
3ℎ2 𝑡 + 6ℎ𝑡 + 3𝑡
24.
6𝑥 3 𝑦 + 30𝑥 2 𝑦 2 + 36𝑥𝑦 3
26.
5 + 8𝑤 + 3𝑤 2
27.
𝑎𝑐 + 𝑥𝑐 + 𝑎𝑤 2 + 𝑥𝑤 2
29.
− 4𝑤 3 − 16𝑤 2 + 20𝑤
30.
25.
3𝑥 3 𝑦 2 − 3𝑥 2 𝑦 2 + 3𝑥𝑦 2
28. 𝑎3 + 𝑎𝑏 + 3𝑏 + 3𝑎2
20𝑤 2 + 100𝑤 + 40
− 3𝑦 3 + 6𝑦 2 − 3𝑦
Unit 4 Lesson 5 – Factoring 𝑎𝑥 2 + 𝑏 + 𝑐 𝑤𝑖𝑡ℎ 𝑎 ≠ 1
Notes 4.5
Notes on factoring trinomials when 𝒂 ≠ 𝟏.
Rewrite each of the following polynomials in an equivalent form by factoring each completely.
1.
3𝑥 2 + 7𝑥 + 2
2.
3𝑝2 − 2𝑝 − 5
3.
3𝑛2 − 8𝑛 + 4
4.
2𝑣 2 + 11𝑣 + 5
5.
2𝑛2 + 3𝑛 − 9
6.
2𝑛2 + 5𝑛 + 2
7.
36x2 + 12x + 1
8.
6x2 + 26x + 24
9.
9𝑘 2 + 66𝑘 + 21
Unit 4 Lesson 5 - Factoring 𝑎𝑥 2 + 𝑏 + 𝑐 𝑤𝑖𝑡ℎ 𝑎 ≠ 1
Ready, Set, Go! - Assignment 4.5
Name______________________________
Date_________ Hour_______
http://goo.gl/Ma7Y0
Ready
1. True or False.
a.
3𝑥 2 + 4𝑥 − 15 = (3𝑥 + 5)(𝑥 − 3) ____________________
b.
4𝑥 2 + 4𝑥 − 3 = (4𝑥 − 1)(𝑥 + 3) _____________________
c.
4𝑥 2 − 4𝑥 − 3 = (2𝑥 + 1)(2𝑥 − 3) _____________________
d.
4𝑥 2 + 8𝑥 + 3 = (2𝑥 + 1)(2𝑥 + 3) _____________________
2.
Explain trial-and-error factoring.
3.
What should you always first look for when factoring a polynomial?
Set
Factor each polynomial completely. If prime, say so.
4.
6𝑤 2 + 5𝑤 + 1
5.
4𝑥 2 + 11𝑥 + 6
6.
2𝑥 2 − 5𝑥 − 3
7.
2𝑎2 + 3𝑎 − 2
8.
4𝑥 2 + 16𝑥 + 15
9.
6𝑚2 − 𝑚 − 12
10.
12𝑥 2 + 5𝑥 − 2
11.
30𝑏 2 − 𝑏 − 3
12.
6𝑎2 + 𝑎 − 5
13.
2𝑥 2 + 15𝑥 − 8
14.
3𝑎2 + 20𝑎 + 12
15.
4𝑥 2 − 5𝑥 + 1
16.
4𝑥 2 + 7𝑥 + 3
17.
7𝑢2 + 11𝑢 − 6
18.
6𝑦 2 − 7𝑦 − 20
19.
5𝑚2 + 13𝑚 − 6
Unit 4 Lesson 6 – Additional Factoring
Notes 4.6
Factor each polynomial completely, if possible. If a polynomial is prime, say so.
1.
𝑥4 − 9
2.
𝑦 8 − 14𝑦 4 + 49
3.
𝑥 2𝑚 − 𝑦 2
4.
𝑥10 − 9
5.
𝑦8 − 4
6.
𝑎6 + 10𝑎3 + 25
7.
𝑧12 − 6𝑧 6 + 9
8.
𝑥6 − 8
9.
𝑎2𝑛 − 1
10.
𝑏 4𝑛 − 9
Unit 4 Lesson 6 – Additional Factoring
Task 4.6
Name______________________________
Date_________ Hour_______
1. Which of the following are not perfect square trinomials? Explain.
𝑎) 4𝑎6 − 6𝑎3 𝑏 4 + 9𝑏 8
𝑐) 900𝑦 4 − 60𝑦 2 + 1
𝑏) 1000𝑥 2 + 200𝑎𝑥 + 𝑎2
𝑑) 36 − 36𝑧 7 + 9𝑧14
2. Which of the following is not a difference of two squares? Explain.
𝑎) 16𝑎8 𝑦 4 − 25𝑐12
𝑐) 𝑡 90 − 1
3.
𝑏) 𝑎9 − 𝑏 4
𝑑) 𝑥 2 − 196
Factor each polynomial and explain how you decided which method to use.
𝑎) 𝑥 2 + 10𝑥 + 25
𝑏) 𝑥 2 − 10𝑥 + 25
𝑐) 𝑥 2 + 26𝑥 + 25
𝑑) 𝑥 2 − 25
𝑒) 𝑥 2 + 25
Unit 4 Lesson 6 – Additional Factoring
Ready, Set, Go! - Assignment 4.6
Name______________________________
Date_________ Hour_______
Factor each completely. If prime, say so.
1.
𝑦 6 − 27
2.
𝑎2𝑟 + 6𝑎𝑟 + 9
3.
𝑢6𝑛 − 4𝑢3𝑛 + 4
4.
𝑥 6 − 2𝑥 3 − 35
5.
𝑥 4 + 7𝑥 2 − 30
6.
𝑎20 − 20𝑎10 + 100
7.
𝑏16 + 22𝑏 8 + 121
8.
𝑥10 − 100
9.
𝑦8 − 9
10.
𝑦6 − 8
Answer Sheet for Factoring Around the Room
1.
13.
25.
2.
14.
26.
3.
15.
27.
4.
16.
28.
5.
17.
29.
6.
18.
30.
7.
19.
31.
8.
20.
32.
9.
21.
33.
10.
22.
34.
11.
23.
35.
12.
24.
Unit 4 Lesson 7 – Factoring Review
Assignment 4.7
Name______________________________
Date_________ Hour_______
Factor completely. If not factorable, write prime.
1. 3 x 2  15 x
2. x 3 y 4  x 2 y 3
1._____________________
2._____________________
3. 3x  4 xy  6 y
4. x 3  2 x 2  4 x  8
3._____________________
4._____________________
5. 2 x 3  8 x 2  3x  12
6. 2 x 3  x 2  x  1
5._____________________
6._____________________
7. x 2  4 x  3
8. x 2  5 x  24
7._____________________
8._____________________
9. x 2  x  30
10. 2 x 2  x  1
9._____________________
10.____________________
11. 3x 2  10 x  8
12. 4 x 2  15 x  4
11.____________________
12.____________________
13. x 2  9
14. 2 x 2  50
13.____________________
14.____________________
15. x 2  16
16. 𝑥 2 − 144
15.____________________
16.____________________
17. 3 x 3  3
18. 8 x 3  64
17.____________________
18.____________________
19. 𝑥 2 − 81𝑦 2
20. (x – 4)3(x – 2)2 – 3(x – 4)2(x – 2)2
19.____________________
20.____________________
Circle your answer.
21. Mrs. Rich is trying to carpet her room. If her room is a square and has an area of 4 x 2  25 y 2 , write
expressions that represent the length and width of her room.
Unit 4 Lesson 8 – Factoring Simple Quadratic Expressions over
the Complex Number System
Task/Notes 4.8
Name________________________________________
Date_________ Hour________
1. Consider the polynomial 𝑥 2 − 1. How would you factor this polynomial?
2. Consider the polynomial 𝑥 2 + 1. How would you factor this polynomial?
3. Can you solve the polynomial 𝑥 2 + 1 = 0?
4. How could you factor the polynomial 𝑥 2 + 1?
(
)(
)
Be sure to distribute your factorization below to be sure it is in fact an equivalent form.
To be able to factor 𝑥 2 + 1 you need to use imaginary numbers. To factor such examples you are
factoring over the set of Complex Numbers.
5.
Factor 𝑥 2 + 25 over the set of complex numbers.
6.
Factor 49𝑥 2 + 144𝑦 2 over the set of complex numbers.
In the next Unit we will discuss how to write more complicated quadratic expressions in factored form,
problems with complex roots. Today’s lesson will focus on more simple examples.
7.
Factor 𝑥 4 − 1 over the set of complex numbers.
8.
How would the factorization be different if you were to factor 𝑥 4 − 1 over the set of real
numbers? Explain.
9.
Factor 9𝑥 2 + 100 over the set of the complex numbers.
10.
Factor 64𝑦 2 + 121𝑥 2 over the set of complex numbers.
Unit 4 Lesson 8– Factoring Simple Quadratic Expressions
over the Complex Number System
Ready, Set, Go! Assignment - 4.8
Name__________________________________
Date_______ Hour________
Ready
Factor each over the complex number system.
1. 𝑥 2 + 1
2. 𝑥 4 − 1
3. 4𝑥 2 + 1
4. 16𝑦 2 + 9
5. 25𝑚2 + 16𝑛2
6. 36𝑎2 + 49
7. −100 + 𝑦 2
8. 121𝑥 2 + 4
9. 144𝑦 2 + 169𝑧 2
10. 225𝑎4 − 4
Factor each over a) The real number system:
b) The complex number system:
Set
The real number system.
The complex number system.
11a. 𝑥 4 − 1
11b. 𝑥 4 − 1
12a. 𝑦 4 − 4
12b. 𝑦 4 − 4
13a. 𝑧 4 − 9
13b. 𝑧 4 − 9
14a. 𝑎4 − 16
14b. 𝑎4 − 16
15a. 𝑏 4 − 25
15b. 𝑏 4 − 25
16a. 𝑐 4 − 36
16b. 𝑐 4 − 36
Go!
17a. 𝑥 4 − 𝑦 4
17b. 𝑥 4 − 𝑦 4
18a. 16𝑎2 − 25𝑏 2
18b. 16𝑎2 − 25𝑏 2
19a. 49 − 100𝑎4
19b. 49 − 100𝑎4
20a. 𝑥 8 − 1
20b. 𝑥 8 − 1
Factoring Cut-outs – Cut out each puzzle piece and reassemble so that the expressions and their factored
forms match up.
x2+4x-21
x2+3x-4
x2+6x+9
(x)(3x)
x2-4
(x-2)(x+2)
x2+8x+7
x2+7x+10
x2+6x
(x)(x)
(x+4)(x-1)
(x+5)(x+2)
(x+10)(x+2)
(x-2)(x+4)
(x+2)(x+10)
x2+9x+20
x2+20x+100
(x+5)(x+4)
x2-7x-18
(x+3) 2 x2+10x+25
(3x)(x) x2-1
(5x)(3x) x2+4x+4
x2+3x-4
(x+10)(x+2)
(x+4)(x-1)
(x+4)(x+3)
(x+7)(x-3)
x2+2x-8
x2+7x+10
x2+12x+20
x2-4x-5
3(x+2) 3x+6
x2-5x
15x2
x2
x2-64
x(x+6) x2+6x
(x+2) 2
(x+5)(x+2)
(x+6)(x+10)
(x+10) 2
(x-10)(x-4)
x2+12x+20
x2+9x+20
x2-14x+40
x2+3x-4
(x+9)(x-6)
x2+20x+100
(x+2)(x-9) 3x2
x2+12x+20
x(x+1) x2+x
(x-8)(x+8)
(x+5) 2
(5x)(3x) 3x+6
(x+1)(x-5)
(x+7)(x+1)
Factoring Review Worksheet # 1
The G.C.F.
Name_____________________________________
Date___________ Hour_________
Directions: Find the missing factor of each expression below. Write the factor in the blank in the term. Then
find your answer in the Answer Bank and write its corresponding letter in the blank before the problem.
When you have finished, write the letters in order, starting with the first problem, to complete the statement
at the end of the activity.
1. _______ 98𝑎 = _________ (7𝑎)
2. _______ 15𝑎 = _________ (5)
3. _______ 12𝑎2 = _________ (6𝑎)
4. _______ 3𝑎2 𝑏 = _________ (𝑎)
5. _______ 18𝑎𝑏 = __________ (9𝑎)
6. _______ 27𝑎2 𝑏2 = _________ (3𝑎𝑏)
7. _______ 6𝑎 + 6𝑏 = __________ (𝑎 + 𝑏)
8. _______ 21𝑎 + 28 = ___________ (3𝑎 + 4)
9. _______ 42𝑎 + 54𝑏 = ___________ (7𝑎 + 9𝑏)
10. _______ 12𝑎 + 3𝑎2 = ___________ (4 + 𝑎)
11. _______ 15𝑎2 + 12𝑎 + 30 = _________ (5𝑎2 + 4𝑎 + 10)
12. _______ 2𝑏 2 − 2𝑏 = ___________ (𝑏 − 1)
Finding the Missing Factor
13. _______ 12𝑎2 𝑏 + 18𝑎2 𝑏 + 6𝑎 = _________ (2𝑎𝑏 + 3𝑎𝑏 + 1)
14. _______ 3𝑎3 𝑏2 − 3𝑎2 𝑏 3 + 3𝑎𝑏 4 = _________ (𝑎2 − 𝑎𝑏 + 𝑏 2 )
15. _______ 𝑎3 𝑏 + 2𝑎2 𝑏 2 + 4𝑎𝑏 3 = __________ (𝑎2 + 2𝑎𝑏 + 4𝑏 2 )
16. _______ −10𝑎𝑏 + 4𝑎𝑏 3 + 14𝑏 4 = _________ (−5𝑎 + 2𝑎𝑏 2 + 7𝑏 3 )
17. _______ 𝑎2 𝑏 7 − 2𝑎2 𝑏 6 + 𝑎5 𝑏 3 = __________ (𝑏 4 − 2𝑏 3 + 𝑎3 )
18. _______ −7𝑎 + 49𝑎2 − 14 = ___________ (−𝑎 + 7𝑎2 − 2)
19. _______ 15𝑎2 − 27𝑎 = __________ (5𝑎 − 9)
20. _______ 12𝑎3 + 30𝑎2 − 12𝑎 = ___________ (2𝑎2 + 5𝑎 − 2)
Answer Bank
A. 3a
Y. 3𝑎𝑏
2
O. 2b
S. 6
L. 6a
M. 𝑎2 𝑏 3
F. 14
P. 3
C. 2a
T. 3ab
R.9ab
N. ab
You can check your answers for this activity by multiplying each factor.
If one of the ___ ___ ___ ___ ___ ___ ___
___ ___
___
___ ___ ___ ___ ___ ___ ___ ___ ___ ___, you must use the Distributive Property.
I. 7
Factoring Review Worksheet # 2
Difference of Squares
Name_____________________________________
Date___________ Hour_________
Directions: Factor each polynomial, if possible, and write the factors in the space after the polynomial.
1. 𝑥 2 − 36 =
_________________________________________
2. 𝑥 2 − 64 =
_________________________________________
3. 𝑥 2 − 1 =
_________________________________________
4. 𝑥 2 + 16 =
_________________________________________
5. 4𝑥 2 − 121 =
_________________________________________
6. 2𝑥 2 − 25 =
__________________________________________
7. 𝑥 4 − 4 =
__________________________________________
8. 16𝑥 2 − 49 =
__________________________________________
9. 𝑥 2 𝑦 2 − 9 =
__________________________________________
10. 𝑥 3 − 144 =
__________________________________________
11. 25𝑥 2 𝑦 2 − 36 =
_________________________________________
12. 10𝑥 4 − 81 =
_________________________________________
13. 𝑥 6 𝑦 4 − 169 =
__________________________________________
14. 𝑥 6 𝑦 8 − 100 =
__________________________________________
15. 8𝑥 4 𝑦 2 − 25 =
__________________________________________
Factoring Review Worksheet # 3
Factoring Trinomials 1
Name_____________________________________
Date___________ Hour_________
Directions: Factor each trinomial, if possible, and write your answer in the space provided.
1. 𝑥 2 + 5𝑥 + 6 = _____________________________________________________________________
2. 𝑥 2 − 6𝑥 − 7 = _____________________________________________________________________
3. 𝑥 2 − 12𝑥 + 32 = ____________________________________________________________________
4. 𝑥 2 − 4𝑥 + 4 = _____________________________________________________________________
5. 𝑥 2 − 9𝑥 + 8 = _____________________________________________________________________
6. 𝑥 2 + 𝑥 − 20 = _____________________________________________________________________
7. 𝑥 2 − 𝑥 − 30 = _____________________________________________________________________
8. 𝑥 2 − 16𝑥 + 60 = ____________________________________________________________________
9. 𝑥 2 − 3𝑥 − 28 = _____________________________________________________________________
10. 𝑥 2 − 2𝑥 − 15 = ___________________________________________________________________
11. 𝑥 2 + 3𝑥 + 2 = _____________________________________________________________________
12. 𝑥 2 − 15𝑥 + 36 = ___________________________________________________________________
13. 𝑥 2 − 6𝑥 + 5 = _____________________________________________________________________
14. 𝑥 2 − 12𝑥 + 27 = ___________________________________________________________________
15. 𝑥 2 − 6𝑥 − 40 = ____________________________________________________________________
16. 𝑥 2 − 21𝑥 + 108 = _________________________________________________________________
Factoring Review Worksheet # 4
Factoring Binomials and Trinomials
Name_____________________________________
Date___________ Hour_________
Directions: Factor each polynomial and write the factors in the space after the polynomial.
1. 𝑥2 − 𝑥 − 6 = _____________________________________________________________________
2. 𝑥2 − 9𝑥 + 20 = _____________________________________________________________________
3. 𝑥2 + 9𝑥 + 14 = _____________________________________________________________________
4. 𝑥2 − 9𝑥 + 8 = _____________________________________________________________________
5. 3𝑥 2 + 16𝑥 + 16 = __________________________________________________________________
6. 𝑥2 − 13𝑥 + 40 = ____________________________________________________________________
7. 𝑥2 − 6𝑥 + 8 = _____________________________________________________________________
8. 𝑥2 + 2𝑥 − 3 = _____________________________________________________________________
9. 𝑥2 − 𝑥 − 2 = _____________________________________________________________________
10. 𝑥2 + 8𝑥 + 16 = ___________________________________________________________________
11. 2𝑥 3 − 16𝑥 2 = _____________________________________________________________________
12. 4𝑥 + 28 = _____________________________________________________________________
13. 2𝑥 2 − 𝑥 − 6 = _____________________________________________________________________
14. 2𝑥 3 + 2𝑥 2 = _____________________________________________________________________
15. 𝑥2 − 11𝑥 + 24 = ___________________________________________________________________
16. 6𝑥 2 + 42𝑥 = _____________________________________________________________________
Directions: Factor each trinomial. Hint: one factor of each polynomial is a factor of the polynomial in the
next problem. Always check your work.
17. 3𝑥 2 − 11𝑥 − 4 = _______________________________________________________________
18. 6𝑥 2 − 𝑥 − 1 =
_______________________________________________________________
19. 2𝑥 2 + 13𝑥 − 7 = _______________________________________________________________
20. 𝑥 2 − 𝑥 − 56 =
_______________________________________________________________
21. 𝑥 2 − 5𝑥 − 24 = _______________________________________________________________
22. 4𝑥 2 + 11𝑥 − 3 = _______________________________________________________________
23. 2𝑥 2 + 11𝑥 + 15 = ______________________________________________________________
24. 6𝑥 2 + 11𝑥 − 10 = ______________________________________________________________
25. 12𝑥 2 + 𝑥 − 6 =
_______________________________________________________________
26. 4𝑥 2 + 15𝑥 + 9 = _______________________________________________________________
27. 𝑥 2 + 12𝑥 + 27 = _______________________________________________________________
28. 5𝑥 2 + 52𝑥 + 63 = ______________________________________________________________
29. 10𝑥 2 − 𝑥 − 21 = _______________________________________________________________
30. 12𝑥 2 − 28𝑥 + 15 = _____________________________________________________________
31. 18𝑥 2 − 3𝑥 − 10 = ______________________________________________________________
Factoring Review Worksheet # 5
Reviewing Factoring Skills
Name_____________________________________
Date___________ Hour_________
Directions: Read each statement and decide whether it is true or false. If it is true, write “true”. If it is false, write
“false” and provide an example or an explanation that will make the statement true.
1. 10 is the GCF of 20 and 40. ________________________________________________________
______________________________________________________________________________
2. 36 is a square number. ________________________________________________________
______________________________________________________________________________
3. 𝑥(𝑥 2 + 1) = 𝑥 3 + 𝑥. ________________________________________________________
______________________________________________________________________________
4. 𝑥 2 − 16 cannot be factored. _______________________________________________________
______________________________________________________________________________
5. 2𝑥 4 is the GCF of 10𝑥 4 + 12𝑥 2 + 2. ________________________________________________
______________________________________________________________________________
6. (𝑥 + 3) is a factor of 𝑥 2 + 𝑥 − 12. __________________________________________________
______________________________________________________________________________
7. (2𝑥 2 + 1)(𝑥 2 − 1) = 2𝑥 4 − 𝑥 2 − 1. ________________________________________________
______________________________________________________________________________
8. 𝑥 + 1 is a factor of 𝑥 2 + 1. ________________________________________________________
______________________________________________________________________________
9. The product of two binomials is always a trinomial. ____________________________________
10. 51 is a prime number. ___________________________________________________________
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11. 2𝑥 + 1 is a factor of 6𝑥 2 − 5𝑥 − 4. _________________________________________________
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12. 16𝑥 3 is a perfect square. ________________________________________________________
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13. 1 may be a GCF. _______________________________________________________________
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14. 𝑥 + 3 is a factor of 𝑥 4 + 3𝑥 3 + 𝑥 2 + 3𝑥. _____________________________________________
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15. 2𝑥 2 − 5𝑥 − 12 cannot be factored. _________________________________________________
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16. 2𝑥 3 + 4𝑥 2 − 16𝑥 is factored completely as 𝑥(2𝑥 2 + 4𝑥 − 16). __________________________
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17. 𝑥 3 + 27 is the sum of two cubes. ___________________________________________________
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18. 𝑥 2 + 1𝑥 + 3 cannot be factored. ___________________________________________________
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19. (𝑥 3 − 8) = (𝑥 − 2)(𝑥 2 − 2𝑥 + 4). _________________________________________________
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20. The number of false statements in this exercise is a factor of 20. __________________________
Factoring Review Worksheet # 6
Factoring Polynomials Completely
Name_____________________________________
Date___________ Hour_________
Directions: Factor each polynomial completely. See “flow chart” in textbook to help you.
1. 2𝑥 2 − 6𝑥 + 4 = _______________________________________________________________
2. 𝑥 3 + 3𝑥 2 − 4𝑥 =_______________________________________________________________
3. 3𝑥 2 − 12𝑥 − 36 =_______________________________________________________________
4. 16𝑥 2 + 16𝑥 + 4 =_______________________________________________________________
5. 3𝑥 2 − 27 =____________________________________________________________________
6. −𝑥 + 4𝑥 3 =_______________________________________________________________
7. 25𝑥 4 − 100𝑥 2 =_______________________________________________________________
8. 𝑥 4 − 1 =_______________________________________________________________
9. 15𝑥 2 − 9𝑥 − 6 =_______________________________________________________________
10. 12𝑥 2 + 38𝑥 + 16 =______________________________________________________________
11. 30𝑥 3 + 21𝑥 2 + 3𝑥 =_____________________________________________________________
12. 2𝑥 6 − 8 =_______________________________________________________________
13. 𝑥 4 − 𝑦 4 = _______________________________________________________________
14. 𝑥 2 𝑦 − 9𝑦 + 3𝑥 2 − 27 =__________________________________________________________