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M314: October/November 2013
Sun
Mon
Tue
20 21
22
Wed
Thu
23
24
Quarter 1
Retention
Unit 2
Practice ACT
Unit 3
Unit 2 WS
No Class
Unit 3 WS
Unit 1
Late Start Day
Fri
Sat
25
26
Quarter 1
Retention
Test
Unit 1 WS
27 28
29
30
Unit 4
Factor by GCF
Classifying/
Multiplying
Quadratics
WS 2
31
Factor
Trinomials with
a=1
Nov. 1
Factor Trinomials
with a=1
Factoring
Patterns
WS 4
WS 5
WS 3
2
WS 1
3
4
5
Factor
Trinomials with
a≠1
6
Review
7
Quiz
8
Factor Trinomials
with a≠1
WS 7
WS 6
10 11
Veterans’
Day
No School
(Thank any
veterans
you see!)
12
13
Factoring
Cubics
WS 8
14
Mixed Problems
Review
9
Teachers’
Institute
No School
16
15
Test
WS 9
2013-14
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M314—Algebra II
Unit 4: Polynomials
Notes for 4.1: Classifying Polynomials
Name: _______________________________
Date: ____________ Teacher: ___________
Section 4.1: Classifying/Multiplying Polynomials
Part 1: Warmup Puzzles
a) The following diamonds have a pattern. You can use the two side numbers from the diamond to
get the top and bottom numbers. Work with your group to try to identify the pattern:
5
3
2
7
-5
4
-2
-1
-3
-1
9
8
2
-20
14
15
b) Think you have the pattern? Try filling these in to see if you have it!
7
2
4
1
-3
2
-9
-4
Part 2: Classifying Polynomials
●A polynomial is an ___________ involving ___________ and ___________.
Examples:
●The degree of a polynomial in one variable is the ___________ ___________ in the
polynomial.
Examples:
●The term of a polynomial is any part of the polynomial that is joined to the
expression using ___________ or ___________.
Examples:
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Classifying Polynomial by Number of Terms:
Name
Number of Terms Definition
Examples
Classifying Polynomial by Degree:
Name
Degree
Definition
Examples
Example 1:
Polynomial
Degree Name by Degree
Number of
Terms
Name by Number of
Terms
x + 4
x3 + x2 + 2x
2x2 – (3x – 5)
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Part 3: Multiplying Polynomials Using the Area Model
Ex 1: (x + 7)(2x – 5)
Ex 2: 2(2x + 1)(3x – 2)
Ex 3: (x + 4)(x – 4)
Ex 4: (x + 3)2
Ex 5: (x + 2)(x2 + 3x – 1)
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Part 4: Multiplying Polynomials Using FOIL
(x + 1)(2x – 3)
Part 5: Writing Quadratics in Standard Form
●A quadratic equation is an equation of the form ________________.
Example:
●To put a quadratic equation into standard form, ________________ then
________________.
Example:
Part 6: Group Practice
Multiply 7x(2x + 3)(2x – 3):
Classify By Degree:
Classify by Number of Terms:
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Algebra 2 – M314
Unit 4 – Quadratic Equations
WS 1 – General Form of Quadratic Equations
Name _______________________
Teacher _____________________
Part I: Simplify each expression (if possible). Classify each by the numbers of terms and by
the degree.
Polynomial
Degree
Name by
Degree
Number of
Terms
Name by Number of
Terms
1) x + 4
2) x3 + x2 + 2x
3) 2x2 – (3x – 5)
4) 6(2 - x2) + 3(2x2 + 2x)
5) 3x2(x – 2) – 7
6)
½
(6x2 – 8x) + 8
7) 3x3 + x2 – 4x + 2x3
8) 7x(x – 2) + 5(3x2)
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Part II: Multiply the following binomials using the area model (a box).
9)  5x  4   2x  4 
10) 12x  5   7x  9 
9) _________
10) ________
11)
 x  3  x  3
12)
 3x  8   3x
 7
11) ________
12) ________
13)
 9x
 11  2x  7 
14)
 x  6
2
13) ________
14) ________
15)  5x  4 
2
16)
 x  4  x
2

 3x  1
15) ________
16) ________
M. Giblin
William Fremd High School
10/25/2013
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Part III: Write the following quadratic equations in standard form.
17) y   3x  5   2x  7 
18) y  3  x  2   x  2 
17) ________
18) ________
19) y   2x  3   x  2 
20) y   x  2   x  4 
19) ________
20) ________
21) y   7  x 
22) y   x  3   5x  4 
2
21) ________
22) ________
23) y    x  5 
2

24) y   x  3  2x 2  x  4

23) ________
24) ________
M. Giblin
William Fremd High School
10/25/2013
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M314—Algebra II
Unit 4: Factoring
Notes for 4.2: Factor by GCF
Name: _______________________________
Date: ____________ Teacher: ___________
Section 4.2: Factor by GCF
Part 1: ACT Warmup
1.  3 x 2  2 x 3 y 0  is equivalent to:
A.
B.
C.
D.
E.
2. For all x,  x  4  x  5  ?
0
5x 6
6x 6
5x 5 y
6x 5
F.
G.
H.
I.
J.
x 2  20
x 2  x  20
2x 1
2 x2 1
2 x 2  x  20
Show Work:
Show Work:
Part 2: Warmup Puzzles
a) Do you remember the diamond pattern from yesterday? Then fill in the diamond!
9
2
3
3
-7
9
-5
-9
b) Part (a) isn’t too difficult. It’s harder to go the other way:
6
9
-15
20
5
10
-2
-9
The main question to ask yourself with diamond problems:
“What two numbers ______________ to the ______________ and
_______________ to the ______________?”
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Part 3: Review Distributing
Instructions: Distribute the following quantities.
1) y2 (y  9)
2) 5x2 y2 (y  3x)
3) x 2 (x 4  x 3  x  1)
Part 4: Factoring out the Greatest Common Factor (GCF)
The Greatest Common Factor of a polynomial is an expression that divides evenly with
every term of the polynomial.
When Factoring Out the GCF, think of “distributing in reverse.”
Example 1: 2x2  6
Example 2: 4xy2  12x 2 y  4xy
Example 3: y 3  9y 2
Example 4: x 6  x5  x 3  x2
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Part 4: Factoring by Grouping
t 2  6t  2t  12
Step 1
• Write the first two terms in the first row.
Step 2
• Write the second two terms in the second row.
Step 3
• Factor the GCF out of the columns and the rows.
1
Step 4
• Write your final answer (called a factorization)
Factorization:
Example 5: Factor x2  3x  4x  12 by grouping.
Don’t combine like terms
when factoring by grouping!
Factorization:
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Example 6: Factor x2  7x  x  7 by grouping.
Factor out a ______ from
the second row.
Factorization:
Example 7: Factor mp  mn  qp  qn by grouping.
Factorization:
Example 8: Factor 2d3  10d  d2  5 by grouping.
Factor out a ______ from
the second row.
Factorization:
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Algebra 2 – M314
Unit 4 – Quadratic Equations
WS 2 – Factoring GCF (Greatest Common Factor)
& By Grouping
Name _______________________
Teacher _____________________
Part I: Factor each expression by GCF.
1) y2  5y
2) 4a2  2a
3) y 3  9y2
4) x 3  8x2
5) 3y2  3y  9
6) 5x 3  10x 2  15x
7) 6x 2  3x 4
8) 4ab  6ac  12ad
9) 8xy  10xz  14xw
10) 4x2 y  12xy2
11) 5x 2 y 3  15x 3 y2
12) x 6  x 5  x 3  x 2
13) 24x 3  36x 2  72x
14) 10a 4  15a2  25a
15) 4ab2  6a2 b
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Part II: Factor by grouping.
16) ax  3a  bx  3b
17) ac  ad  bc  bd
18) b3  b2  2b  2
19) y2  8y  y  8
20) t 2  6t  2t  12
21) x 2  5x  4x  20
22) 4x 2  3x  20x  15
23) 2x 4  6x 2  5x2  15
24) x 2  3x  4x  12
25) Multiple Choice: The formula for the surface area of a right circular cone, including the
base, is A   rs   r ,
Where A is the surface area, r is the radius, and s is the length of the
vertex to the edge of the cone. Which of the following represents an
equivalent formula for A?
2
a) A  2rs
d) A  r 2 (L  s)
b) A  2r 2 s
e) A  r(r  s)
c) A  r(L  s)
M. Giblin
William Fremd High School
10/25/2013
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M314—Algebra II
Unit 4: Factoring
Notes for 4.3: Factoring Trinomials
Name: _______________________________
Date: ____________ Teacher: ___________
Section 4.3: Factoring Trinomials where a=1
Part 1: ACT Opener
1. The operation x  y stands for
A. 2
x y
. Which of the following is equal to 7  3?
x y
1
2
B. 4
C. 5
1
4
D. 6
E. 10
Part 2: Warmup Puzzles
a) Here are some more diamonds to work out:
12
-30
-20
2
7
1
-8
-3
-9
-45
-24
16
-8
-12
-12
-10
b) One of these will not work:
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Part 3: Factoring Trinomials when constant term is positive
Attention!
When the constant term is ___________, the binomials will have the same ______.
Example 1: Factor the following trinomial.
x 2  7x  12
12
(x + ___ )(x + ___ )
7
Example 2: Factor the following trinomial
x 2  10x  21
21
-10
You Try 1:
You Try 2:
x 2  13x  36
x 2  24x  63
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Part 4: Factoring Trinomials when constant term is negative
Attention!
When the constant term is ___________, the binomials will have different ________.
Example 3: Factor the following trinomial.
x 2  2x  15
Example 4: Factor the following trinomial
x 2  x  30
You Try 3:
You Try 4:
x 2  x  72
x 2  3x  4
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Group Practice: Factor these polynomials correctly and show
your work, and you can start your homework!
1)
x2  27x  90
2)
x2  22x  72
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Algebra 2 – M314
Unit 4 – Factoring
WS 3 – Factoring Trinomials w/a=1
Name _______________________
Teacher _____________________
Part I: Factor the following trinomials. If it is not possible to factor, then write can’t factor.
1) x 2  3x  2
2) x 2  5x  6
3) x 2  x  6
4) x 2  x  6
5) x 2  7x  10
6) x 2  10x  24
7) x 2  5x  14
8) x 2  3x  40
9) x 2  2x  63
10) x 2  13x  36
11) x 2  10x  75
12) x 2  7x  44
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13) x 2  x  12
14) x 2  8x  12
15) x2  7x  12
16) x 2  x  12
17) x2  4x  12
18) x2  8x  9
Part II: Mixed Problems – Factor the following expressions. Factor out the GCF first.
19) 2x2  4x  2
21) 3x 3 y2  9y
GCF First! GCF First! 20) 2x 3  2x  24
22) 5x2 y 3  15x 3 y2
GCF First! GCF First! Part III: Solve each of the problems below.
23) Manufacturing: A machine will cut a small square of plastic from a larger square.
a) Write an expression for the remaining area. b) Factor the expression.
4
M. Giblin
William Fremd High School
10/25/2013
x
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M314—Algebra II
Unit 4: Factoring
Notes for 4.4: Factoring Trinomials
Name: _______________________________
Date: ____________ Teacher: ___________
Section 4.4: More Factoring Trinomials Where a=1
Part 1: ACT Opener
1.
3  2  1 4  ?
a. -4
b. -2
c. 2
d. 4
e. 10
Part 2: Warmup Puzzles
a) Here are some more diamonds to work out:
-99
60
36
-28
-2
19
-15
12
b) What is the pattern to the following four diamonds? ___________________________________
25
81
1
4
10
-18
2
4
c) What is the pattern to the following four diamonds? ___________________________________
-9
-16
-25
-1
0
0
0
0
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Part 2: Factoring Out the GCF First
Example 1: Factor the following trinomial:
2x 2  14x  24
Step 1
Step 2
Step 3
• Factor out the GCF first
• Use a diamond to factor the trinomial
• Include the GCF with your final answer
Factorization:
Example 2: Factor the following trinomial:
Always factor the
GCF out first!
2x 3  20x 2  42x
Factorization:
Don’t forget to include the
GCF in your final answer!
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Example 3: Factor the following trinomial:
3x 3 y 2  36x 2 y 2  84xy 2
Factorization:
Example 4: Factor the following trinomial:
x 2  xy  30y 2
Factorization:
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Factor These on Your Own:
Practice 1:
2x 2 y  48xy  126y
Practice 2:
2x 3  98x
Practice 3:
4x 3  32x 2  64x
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Algebra 2 – M314
Unit 4 – Factoring
WS 4 – More Factoring Trinomials w/a=1
Name _______________________
Teacher _____________________
Part I: Factor the following trinomials. If it is not possible to factor, then write can’t factor.
1) 2x2  2x  12
2) 3x 2  21x  30
3) 5x 2  50x  120
4) x 3  3x 2  2x
5) x 2 y  5xy  6y
6) x 3 y2  x 2 y2  6y2 x
7) x 2  13xy  36y2
8) x 2  10xy  75y2
9) x 2  7xy  44y2
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10) 2x 3  10x 2  28x
11) 10x 2  30x  400
12) x 5  2x 4  63x 3
Part II: Mixed Problems – Factor the following expressions. Factor out the GCF first if possible.
13) x2  16
14) x2  11x  18
15) 2x 2  10x  3x  15
16) 9y2  6y 4
17) x 3  3x 2  x  3
18) 2x 2  8
19) ab  fa  bc  fc
20) 6y 3  y2  12y  2
M. Giblin
William Fremd High School
10/25/2013
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M314—Algebra II
Unit 4: Factoring
Notes for 4.5: Factoring Binomials and Trinomials
Name: _______________________________
Date: ____________ Teacher: ___________
Section 4.5: Factoring Binomials and Trinomials
Part 1: Warmup Puzzles
a) All of the diamonds in this row have something in common.
9
25
16
25
6
10
-8
-10
What do the side numbers have in common?
___________________________
b) These diamond puzzles also have something in common. What did you find?
-4
-9
-1
-16
0
0
0
0
What do the side numbers have in common?
___________________________
c) What’s wrong with these ones?
1
25
36
4
0
0
0
0
What do these puzzles have in common?
___________________________
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Part 2: Perfect Square Trinomials
Example 1: There is a connection between our diamond puzzles and factoring Perfect Square Trinomials.
See if you can figure it out what it is!
x 2  10x  25  (x  5)(x  5)  (x  5)2
25
5
5
10
How are they related?
Example 2: Factor x2  8x  16
x 2  8x  16  (x 
)(x 
)
Use FOIL
to check
your work.
Summary: Use the diamond method to factor x2  bx  c
x 2  bx  c  (x 
)(x 
)
These numbers
are from the
__________
Perfect Square Trinomial:
a 2  2ab  b2 
a2  2ab  b2 
Example 3: Factor 4x2  20x  25 using the formula above.
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Part 3: The Difference of Two Squares
Example 4: There is a connection between our diamond puzzles and factoring the Difference of Two
Squares. See if you can figure it out what it is!
-36
0
x 2  36  x 2  0x  36  (x 
)(x 
)
How are they related?
Example 5: Factor x2  4
x 2  4  (x 
)(x 
)
Use FOIL
to check
your work.
Difference of Two Squares:
a 2  b2 
Example 6: Factor x2  64 using the formula above.
Example 7: Factor 4x2  25 using the formula above.
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Part 4: The Sum of Two Squares
Example 8: There is a connection between our diamond puzzles and factoring the Sum of Two Squares.
See if you can figure it out what it is!
9
0
x 2  9  x 2  0x  9 
Does this factor? Why not?
Sum of Two Squares:
a 2  b2
Group Practice: Factor these polynomials correctly and show
your work, and you can start your homework!
1)
x2  14x  49
2)
9x2  25
3)
36  49x2
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Algebra 2 – M314
Unit 4 – Factoring
WS 5 – Factoring Trinomials (a=1) Continued
Name _______________________
Teacher _____________________
Part I: Factor the following binomials. If it is not possible to factor, then write can’t factor.
You can check your answer by using FOIL.
1) x 2  9
2) x 2  16
3) x 2  16
4) 9x2  25
5) 100x 2  81
6) 4x 2  49
7) x 2  64
8) 9x 2  1
9) x 2  y2
10) 4x2  25
11) 36  49x2
12) 9x2  121y2
Part II: Factor the following perfect square trinomials.
13) x 2  2x  1
14) x 2  14x  49
15) x 2  14x  49
16) x2  2x  1
17) x2  4x  4
18) x2  6x  9
19) x 2  16x  64
20) x 2  8x  16
21) x2  20x  100
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Part III: Mixed Practice. Factor the following trinomials and binomials, if possible. Otherwise,
write “can’’t factor.”
22) x 2  9
23) x2  18x  81
24) x2  24x  144
25) x2  100
26) x 2  4
27) x2  1
28) x 2  12x  36
29) x 2  6x  9
30) x 2  10x  25
31) x2  22x  121
32) x2  1
33) 3x 3  x 2  12x  4
Part IV: Here some challenging ones!
34) 25x2  10x  1
35) 4x2  20x  25
36) 4x2  12x  9
37) x 4  1 (a difference of 2 squares!)
M. Giblin
William Fremd High School
10/25/2013
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M314—Algebra II
Unit 4: Factoring
Notes for 4.6: Factoring Trinomials
Name: _______________________________
Date: ____________ Teacher: ___________
Section 4.6: Factoring Trinomials where a≠1
Part 1: Opener
1) A farmer purchases a rectangular plot of land that is (2x – 7) meters by (3x + 5) meters. Find
the area of his plot of land in terms of x.
(2x – 7)
(3x + 5)
2) Multiply (5x  2)(3x  8) using the box method.
3) Factor x2  7x  10 using the diamond method and check your work with the box.
Making Connections:
Where do these
numbers come from?
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Part 2: Factoring Trinomials where
a≠1
Factor 2x2  9x  4
Put 2●4 into the top of the diamond and 9 into the bottom. (Put the first coefficient times the last coefficient in the box.)
Fill in the diamond.
Factor using the area model (a box).
Check your work using FOIL. Summary:
ax  bx  c
2
ac
b
ax2  bx  c
ax2
c
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Example 4: Factor 2x2  11x  12 .
Example 5 (You Try): Factor 3x2  2x  1 .
Example 6 (You Try): Factor 4x2  5x  6 .
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Group Practice: Factor these polynomials correctly and show
your work, and you can start your homework. You need to
draw your diamonds and boxes yourselves on these!
1)
2x2  13x  15
2)
3x2  7x  6
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Algebra 2 – M314
Unit 4 – Factoring
WS 6 – Factoring Trinomials w/ a  1
Name _______________________
Teacher _____________________
Factor the following trinomials.
1) 2x2  9x  4
2) 2x2  5x  3
1) __________
2) __________
3) 5x 2  4x  12
4) 2x 2  5x  7
3) __________
4) __________
5) 2x2  13x  15
6) 3x 2  7x  6
5) __________
6) __________
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7) 3x2  5x  2
8) 4x2  5x  6
7) __________
8) __________
9) 4x2  8x  3
10) 2x2  7x  6
9) __________
10) _________
11) 2x2  9x  5
12) 5x2  12x  4
11) _________
12) _________
You’ll have to make your own boxes for the last two
13) 4x  11x  6
2

14) 5x  8x  4
2
13) _________
14) _________
M. Giblin
William Fremd High School
10/25/2013
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Algebra 2 – M314
Unit 4 – Factoring
Review for Quiz
Name _______________________
Date
Teacher _____________________
Part I: Name the following expressions by the degree and by the number of terms.
1) 5x  13
2) 7x 2  3x  1
1) ______________
______________
2) ______________
______________
3) 9
4) 10x 2  3x  8x 5  1
3) ______________
______________
4) ______________
______________
Part II: a) Simplify each expression.
and the number of terms.

5) 5x 2  2x  7  5 9x  x 2  3

b) Name the polynomial according to the degree
5) ______________ ___________
__________
6) 3x 5  x 2 (2x 4  x  1)
6) ______________ ___________
__________
7) (x  1)(x 3  x2  x  1)
7) ______________ ___________
__________
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Part IV: Multiply the following binomials.
8)  2x  1  5x  2 
9)
 4x  1  4x  1
8) ______________
9) ______________
10)
 x  8  x  2
11)  2x  3 
2
10) ______________
11) ______________
12) (4  x)(2x  3)
13) 3  3x  1  x  2 
12) ______________
13) ______________
Part III: Factor each of the following.
14) ab2  3b2
15) 3x  6
14)
15)
C.Grattoni
William Fremd High School
10/25/2013
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16) 9a2 b  27a 3b2
17) 7xy 5  14y 3
16)
17)
18) ac  ad  bc  bd
19) x 3  2x 2  4x  8
18)
19)
20) x 3  2x2  x  2
21) 2w 6  2w 5  w  1
20)
21)
22) x 2  9x  36
23) x 2  x  12
22)
23)
24) x 2  6x  40
25) x 2  10x  24
24)
25)
C.Grattoni
William Fremd High School
10/25/2013
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26) x 2  36
27) 25x 2  16
26)
27)
28) x 2  100
29) 2x2  18
28)
29)
30) x 2  8x  16
31) x 2  18x  81
30)
31)
32) 3x 2  6x  3
33) x 2  10x  1
32)
33)
C.Grattoni
William Fremd High School
10/25/2013
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M314—Algebra II
Unit 4: Factoring
Notes for 4.7: Factoring Trinomials
Name: _______________________________
Date: ____________ Teacher: ___________
Section 4.7: Strategies for Factoring
Part 1: Opener
An 8” by 10” photograph is framed with a white, rectangular border around it. The border is
x inches all around the photograph. Write an expression that accurately describes the area
of the framed area:
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2 terms
•Look for the difference of 2 squares
•If the equation is the sum of 2 squares then can't factor!
x 2  36
3 terms
• Use the diamond method
x 2  8x  20
9x 2  121
a=1
3 terms a≠1
4 terms
Try these:
•Use the diamond and box
•GCF vertically and horizontally in box
3x 2  10x  8
•Group first and last 2 terms
•GCF from first 2 terms •GCF from last 2 terms
x 2  6x  3x  18
a) 4x 2  32x  80
b) 3x 3  3x 2  12x  12
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Algebra 2 – M314
Unit 4 – Factoring
WS 7 – Factoring Trinomials w/ a  1
Name _______________________
Teacher _____________________
Part 1: Factor the following trinomials.
1) 4x2  11x  6
2) 6x2  x  2
1) __________
2) __________
3) 3x 2  16x  21
4) 10x 2  17x  3
3) __________
4) __________
5) 3x2  14x  16
6) 3x2  10x  8
5) __________
6) __________
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7) 30x2  x  1
8) 3x 2  8x  5
7) __________
8) __________
9) 5x 2  8x  3
10) 3x 2  13x  10
9) __________
10) _________
11) 3x2  13x  10
12) 3x2  5x  8
11) _________
12) _________
13) 3x2  11x  4
14) 2x2  3x  20
13) _________
14) _________
M. Giblin
William Fremd High School
10/25/2013
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Part 2: Factor the following polynomials completely. Always remember to take the GCF out
first!! You must show all of your work to receive credit.
15) Factor x 2  16y2 completely.
a) (x  4)(x  4)
b) (x  4y)2
d) (x  4y)(x  4y)
e) Does not factor
15) _________
c) (x  4y)2
16) Factor 3x 3  x 2  3x  1 completely.
a) x 2 (3x  1)(3x  1)
b) (3x  1)(x 2  1)
d) (3x  1)(x  1)(x  1)
e) Does not factor
16) _________
c) (3x  1)(x  1)2
17) Factor 3x 2  12 completely.
a) 3(x 2  4)
b) 3(x  4)(x  4)
d) 3(x  2)
e) Does not factor
2
17) _________
c) 3(x  2)(x  2)
18)  x2  x  12
18) _________
a)  (x  3)(x  4)
b) (  x  3)(x  4)
d) (x  3)(x  4)
e) Does not factor
M. Giblin
William Fremd High School
c) (x  3)(x  4)
10/25/2013
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19) 2x 3  4x 2  2x
19) _________
a) 2x(x  1)2
b) 2x(x  1)(x  1)
d) 2x(x  2x  1)
e) Does not factor
2
c) x(2x  2)(x  1)
20) 7ac  ad  7bc  bd
20) _________
a) (a  b)(7c  d)
b) a(7c  d)  b(7c  d)
d) 7(a  b)(c  d)
e) Does not factor
c) (a  b)(7c  d)
21) 11x(x 2  1)  (x 2  1)
a) (11x  1)(x 2  1)
b) 11x(x 2  1)
d) (11x  1)(x  1)
e) Does not factor
2
21) _________
c) (11x  1)(x  1)(x  1)
22) x 4 y2  x 4 y  5x 3 y  2x 3 y2
a) (x 3 y  1)(5  2xy)
b) (x 3 y  1)(5y  2x 2 )
d) (x 4 y  1)(5  2y2 )
e) Does not factor
M. Giblin
William Fremd High School
22) _________
c) x 3 y(xy  x  5  2y)
10/25/2013
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M314—Algebra II
Unit 4: Factoring
Notes for 4.8: Sum/Difference of Cubes
Name: _______________________________
Date: ____________ Teacher: ___________
Section 4.8: Sum/Difference of Cubes
Part 1: Opener
Factor the following polynomial completely. Hint: Don’t forget to factor out the GCF first.
4x 3  14x2  60x
a) 2(2x 3  7x 2  30x)
b) (2x  3)(2x 2  10x)
c) 4x(x 2  7x  15)
d) 2x(2x  5)(x  6)
e) 2x(2x  7x  30)
2
Show your work:
Answer:
Part 2: Warm Up By Reviewing Cube Roots
When taking the cube root, ask yourself “What number do I have to cube
to get what is under the radical?”
Warmup #1:
3
8
Warmup #2:
3
27
Warmup #3:
3
x3
Warmup #4:
3
8x 3
“What number do I have to cube to get 8?”
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Part 3: The Sum of Two Cubes
We can factor a “cubic trinomial” when it is a sum of two cubes:
a 3  b3  (a  b)(a2  ab  b2 )
Factor 27x 3  8
Decide if you have a sum of two cubes. Look for: two terms, addition, and cubes. Identify "a" and "b" by taking the cube roots of the two terms.
Substitute into the formula from above.
Simplify.
Example 1: x 3  64
Example 2: 8x 3  1
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Part 4: The Difference of Two Cubes
We can factor a “cubic trinomial” when it is a difference of two cubes:
a 3  b3  (a  b)(a2  ab  b2 )
Note: This is just like the sum of two cubes, the formula is just a tiny bit different!
Example 3: x 3  8
Example 4: 8x 3  27y 3
Using SOP SOPS to memorize the pattern:
Signs: S O P
Trinomial: S O P S
Sum of Two Cubes:
a 3  b3  (a  b)(a2  ab  b2 )
Difference of Cubes: a 3  b3  (a  b)(a2  ab  b2 )
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Part 5: Practice Factoring When You Have Only Two Terms
Don’t forget to factor out the GCF first!
1) x 3  27
2) x 3  64
3) 8x 3  64
4) 125x 3  1
5) x 2  16
6) 24x 3  81
7) 36x 2  100
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Algebra 2 – M314
Unit 4 – Factoring
WS 8 – Factoring the Difference and Sum
Of Two Cubes
Name _______________________
Teacher _____________________
Part I: Factor the following expressions.
1) x 3  8
2) x 3  27
3) 8x 3  27
4) a 3  b3
5) 8x 3  1
6) x 3  64y 3
7) x 3  27
8) x 3  125
9) 8x 3  y 3
10) x 3  125
11) x 3  64
12) 27x 3  1
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Part II: Review - Factor the following expressions. Always factor out the GCF first!
13) 3x2  300
14) 2x2  5x  2
15) x 2  3x  10
16) 2x 2  3x  1
17) 4x 2  8x  4
18) 4x 2  400
19) 8x 3  64y 3 (Take out the GCF first!!!)
20) 9x 2  4
21) x 3  16x
22) x 2  8x  12
Part III: Extra Credit - Factor
M. Giblin
William Fremd High School
x6  8 .
10/25/2013
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Algebra 2 – M314
Unit 4 – Factoring
Pop Quiz
Name _______________________
Teacher _____________________
Polynomial Pop Quiz
1)
Which of the following is a quadratic monomial?
a. x2 + 3x – 4
c. 14x2
2)
3)
b. 3x2 + 9
d. 5(x2 + 2x)
Multiply the binomials.
(2x + 6) (6x – 3)
a. 8x2 +30x +18
c. 12x2 +30x – 18
2) ____________
b. 12x2 + 42x – 18
d. 38x2 -18
Factor x3 + 4x2 + 5x + 20 by grouping
a. (x - 5) (x - 4)
c. (x2 + 5) (x + 4)
1) ____________
3) ____________
b. (x – 10) (x - 2)
d. (x + 5) (x2 + 4)
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4)
Factor 16x2 – 49
a. (4x – 49) (4x – 49)
c. (4x + 7)
(4x +7)
5)
5) ____________
b. 4( 7x2y4 – 8x5y)
d. 4xy(7xy – 8xy)
Factor 8x3 – 27
a. (2x + 3)(4x2 + 6x + 9)
c. (2x - 3)(4x2 + 6x + 9)
7)
b. (8x + 7) (8x – 7)
d. (4x – 7) (4x + 7)
Factor 28x2y4 – 32x5y
a. 4x2y(7y3 – 8x3)
c. 2xy3(14xy – 16x4)
6)
4) ____________
6) ____________
b. (2x - 3)(4x2 - 6x + 9)
d. (2x - 3)3
Factor 8x2 – 2x - 3
a. (8x – 3)(x + 1)
c. (2x + 1)(4x – 3)
7) ____________
b. (x - 6)(x + 4)
d. (2x – 1)(4x – 3)
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Algebra 2 – M314
Unit 4 – Factoring
WS 9 – Mixed Practice Factoring Polynomials
Name _______________________
Teacher _____________________
Factor the following polynomials completely. You will need all methods you have learned
previously: perfect square trinomials, difference of two squares, trinomials, factor by GCF,
sum/difference of two cubes, and factor by grouping. One polynomial will not factor at all!
1) x 2  9
2) x 2  2x  15
1) __________
2) __________
3) x 2  14x  49
4) 2x 2  9x  4
3) __________
4) __________
5) x 2  16
6) x 3  x2  4x  4
5) __________
6) __________
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7) 2x2 y  2xy  40y
8) 3x 3  36x
7) __________
8) __________
9) x 2  9x  36
10) 6x 2  13x  5
9) __________
10) _________
11) 8x 3  1
12) 2x2  20x  42
11) _________
12) _________
13) 8ab  12b  6a  9
14) 6x 2 z  9xz  60z
13) _________
14) _________
M. Giblin
William Fremd High School
10/25/2013
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Algebra 2 – M314
Unit 4 Quadratic Equations
Review for Test
Name _________________
Teacher _______________
Part 1: Simplify the following expressions. Then classify them by their name and their number
of terms.
1) a) Simplify 3x3 + x2 – 4x + 2x3
b) Classify your result by degree.
c) Classify your result by number of terms.
1) a)__________________
b)___________________
c)___________________
2) a) Simplify 7x(x – 2) + 5(3x2)
b) Classify your result by degree.
c) Classify your result by number of terms.
2) a)__________________
b)___________________
c)___________________
3) a) Simplify 3x2(x – 2) – 7
b) Classify your result by degree.
c) Classify your result by number of terms.
3) a)__________________
b)___________________
c)___________________
Part 1: Multiply the following.
4)  2x  1  x  7 
5)  5x  4   2x  1
6)  2x  5   2x  5 
7)
 3x  5y 
2
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8)
 x  3  3x
2

 2x  1
9)
 3x  2   3x  1
Part 3: Factor the following expressions completely. Always take out the GCF first.
10) 15x 3 y 4  10x 2 y2  5xy 3
11) 4x 4  8x 3  6x 2
12) 5x  5y  mx  my
13) 5m 4 n  10m 3 n 3
14) 12p 3  3q2 p
15) x 4  x 3  7x  7
16) x 2  12x  36
17) x 2  5x  36
18) x 2  20x  100
19) x 2  14x  48
20) x 2  14x  45
21) x 2  7x  18
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22) 9x 2  25y2
23) 64y 3  81y
24) 36x 2  49y2
25) 16x 2  48x  36
26) 2x2  162
27) 400x 4  25
28) 2x 4  2x 2
29) 7x 2  22x  3
30) 2x2  15x  18
31) x 3  64
32) x 3  125
33) 8x 3  1
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