Hybridoma Technology ( Production , Purification , and Application )
AA Section 11-3 Day 1
1. Section 11-3
Factoring Special Cases
Tuesday, March 3, 2009
2. Factoring:
Rewriting a polynomial as a product of factors
Tuesday, March 3, 2009
3. Factoring:
Rewriting a polynomial as a product of factors
1. Greatest Common Factor
Tuesday, March 3, 2009
4. Factoring:
Rewriting a polynomial as a product of factors
1. Greatest Common Factor
2. Binomial Square Factoring
Tuesday, March 3, 2009
5. Factoring:
Rewriting a polynomial as a product of factors
1. Greatest Common Factor
2. Binomial Square Factoring
3. Difference of Squares Factoring
Tuesday, March 3, 2009
6. Factoring:
Rewriting a polynomial as a product of factors
1. Greatest Common Factor
2. Binomial Square Factoring
3. Difference of Squares Factoring
4. Other Methods of Factoring
Tuesday, March 3, 2009
7. Factoring:
Rewriting a polynomial as a product of factors
1. Greatest Common Factor
2. Binomial Square Factoring
3. Difference of Squares Factoring
4. Other Methods of Factoring
There’s trial-and-error, too, but that just takes too long.
Tuesday, March 3, 2009
8. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
Tuesday, March 3, 2009
9. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4
Tuesday, March 3, 2009
10. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x
Tuesday, March 3, 2009
11. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(
Tuesday, March 3, 2009
12. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3
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13. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x
Tuesday, March 3, 2009
14. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x -
Tuesday, March 3, 2009
15. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1
Tuesday, March 3, 2009
16. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1)
Tuesday, March 3, 2009
17. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5
Tuesday, March 3, 2009
18. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5x
Tuesday, March 3, 2009
19. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5xy
Tuesday, March 3, 2009
20. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5xy(
Tuesday, March 3, 2009
21. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5xy(3x2 + xy - 7y)
Tuesday, March 3, 2009
22. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5xy(3x2 + xy - 7y)
All we did here was go through the numbers first, then the
variables in alphabetical order, finding factors that the terms
have in common.
Tuesday, March 3, 2009
23. Example 1: Factor.
a. 12x2 - 4x b. 15x3y + 5x2y2 - 35xy2
4x(3x - 1) 5xy(3x2 + xy - 7y)
All we did here was go through the numbers first, then the
variables in alphabetical order, finding factors that the terms
have in common.
To check your answer, re-distribute the GCF and see if you
get what you started with.
Tuesday, March 3, 2009
34. (x + 4) 2
(x + 4)(x + 4)
2 + 4x + 4x + 16
x
2 + 8x + 16
x
Tuesday, March 3, 2009
35. NOTICE
(x + 4) 2
First term:
Tuesday, March 3, 2009
36. NOTICE
(x + 4) 2
First term: x2
Tuesday, March 3, 2009
37. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Tuesday, March 3, 2009
38. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Middle term:
Tuesday, March 3, 2009
39. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Middle term: 8x
Tuesday, March 3, 2009
40. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Middle term: 8x
How does this compare with what we started out with?
Tuesday, March 3, 2009
41. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Middle term: 8x
How does this compare with what we started out with?
Last term:
Tuesday, March 3, 2009
42. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Middle term: 8x
How does this compare with what we started out with?
Last term: 16
Tuesday, March 3, 2009
43. NOTICE
(x + 4) 2
First term: x2
What do you notice about it compared to what we
began with?
Middle term: 8x
How does this compare with what we started out with?
Last term: 16
What’s happening?
Tuesday, March 3, 2009
44. A pattern emerges...
A perfect square trinomial will have the following things
occur:
Tuesday, March 3, 2009
45. A pattern emerges...
A perfect square trinomial will have the following things
occur:
1. The first term will be a perfect square.
Tuesday, March 3, 2009
46. A pattern emerges...
A perfect square trinomial will have the following things
occur:
1. The first term will be a perfect square.
2.The last term will be a perfect square.
Tuesday, March 3, 2009
47. A pattern emerges...
A perfect square trinomial will have the following things
occur:
1. The first term will be a perfect square.
2.The last term will be a perfect square.
3.The middle term will be 2 times the product of the square
roots of the first and last terms.
Tuesday, March 3, 2009
49. Example 2: Factor.
a. 9x2 + 12x + 4
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009
50. Example 2: Factor.
a. 9x2 + 12x + 4
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009
51. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009
52. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x
Check to see if the first and last terms are perfect squares.
3x
Tuesday, March 3, 2009
53. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x
Check to see if the first and last terms are perfect squares.
3x
Tuesday, March 3, 2009
54. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x
Tuesday, March 3, 2009
55. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Tuesday, March 3, 2009
56. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
Tuesday, March 3, 2009
57. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2)
Tuesday, March 3, 2009
58. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x)
Tuesday, March 3, 2009
59. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x) = 12x
Tuesday, March 3, 2009
60. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x) = 12x
Final Answer:
Tuesday, March 3, 2009
61. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x) = 12x
Final Answer:
(3x
Tuesday, March 3, 2009
62. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x) = 12x
Final Answer:
(3x 2)
Tuesday, March 3, 2009
63. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x) = 12x
Final Answer:
(3x + 2)
Tuesday, March 3, 2009
64. Example 2: Factor.
a. 9x2 + 12x + 4
3x · 3x 2·2
Check to see if the first and last terms are perfect squares.
3x 2
Check to see if the middle term is 2 times the product of the
square roots of the first and last terms.
2(3x · 2) = 2(6x) = 12x
Final Answer:
(3x + 2)2
Tuesday, March 3, 2009
65. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
d. x2 + 7x + 14
Tuesday, March 3, 2009
66. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x
d. x2 + 7x + 14
Tuesday, March 3, 2009
67. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x 3)
d. x2 + 7x + 14
Tuesday, March 3, 2009
68. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3)
d. x2 + 7x + 14
Tuesday, March 3, 2009
69. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2
d. x2 + 7x + 14
Tuesday, March 3, 2009
70. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2 (y
d. x2 + 7x + 14
Tuesday, March 3, 2009
71. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2 (y -
d. x2 + 7x + 14
Tuesday, March 3, 2009
72. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2 (y - 10)
d. x2 + 7x + 14
Tuesday, March 3, 2009
73. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2 (y - 10)2
d. x2 + 7x + 14
Tuesday, March 3, 2009
74. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2 (y - 10)2
d. x2 + 7x + 14
(x
Tuesday, March 3, 2009
75. Example 2: Factor.
b. x2 - 6x + 9 c. y2 - 20y + 100
(x - 3) 2 (y - 10)2
d. x2 + 7x + 14
(x
14 is not a perfect square!
Cannot factor with this method.
Tuesday, March 3, 2009
79. Difference of Squares
Factoring
For all a and b,
a2 - b2 =
(a + b)(a - b)
Tuesday, March 3, 2009
80. Difference of two squares
This only works for the following conditions:
Tuesday, March 3, 2009
81. Difference of two squares
This only works for the following conditions:
1. You must have a binomial.
Tuesday, March 3, 2009
82. Difference of two squares
This only works for the following conditions:
1. You must have a binomial.
2.Both terms must be perfect squares.
Tuesday, March 3, 2009
83. Difference of two squares
This only works for the following conditions:
1. You must have a binomial.
2.Both terms must be perfect squares.
3.There must be subtraction!
Tuesday, March 3, 2009
91. Example 3: Factor.
a. 64x2 - 81
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009
92. Example 3: Factor.
a. 64x2 - 81
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009
93. Example 3: Factor.
a. 64x2 - 81
8x · 8x
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009
94. Example 3: Factor.
a. 64x2 - 81
8x · 8x
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009
95. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Tuesday, March 3, 2009
96. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Tuesday, March 3, 2009
97. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
Tuesday, March 3, 2009
98. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
(8x
Tuesday, March 3, 2009
99. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
(8x (8x
Tuesday, March 3, 2009
100. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
(8x 9)(8x
Tuesday, March 3, 2009
101. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
(8x 9)(8x 9)
Tuesday, March 3, 2009
102. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
(8x + 9)(8x 9)
Tuesday, March 3, 2009
103. Example 3: Factor.
a. 64x2 - 81
8x · 8x 9 · 9
Check to see if the first and last terms are perfect squares.
Is it a subtraction problem?
Answer:
(8x + 9)(8x - 9)
Tuesday, March 3, 2009
104. Example 3: Factor.
b. r2 - 121 c. y2 + 100
e. x4 - 16
d. 25x4y6 - 36z8
Tuesday, March 3, 2009
105. Example 3: Factor.
b. r2 - 121 c. y2 + 100
( )( )
e. x4 - 16
d. 25x4y6 - 36z8
Tuesday, March 3, 2009
106. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r )( r )
e. x4 - 16
d. 25x4y6 - 36z8
Tuesday, March 3, 2009
107. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r 11)( r 11)
e. x4 - 16
d. 25x4y6 - 36z8
Tuesday, March 3, 2009
108. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11)
e. x4 - 16
d. 25x4y6 - 36z8
Tuesday, March 3, 2009
109. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
e. x4 - 16
d. 25x4y6 - 36z8
Tuesday, March 3, 2009
110. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6 - 36z8
Tuesday, March 3, 2009
111. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6 - 36z8
( )( )
Tuesday, March 3, 2009
112. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6 -36z8
(5x2y3 )(5x2y3 )
Tuesday, March 3, 2009
113. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3 6z4)(5x2y3 6z4 )
Tuesday, March 3, 2009
114. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3+ 6z4)(5x2y3 - 6z4 )
Tuesday, March 3, 2009
115. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3+ 6z4)(5x2y3 - 6z4 ) ( )( )
Tuesday, March 3, 2009
116. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3+ 6z4)(5x2y3 - 6z4 ) (x2 )( x2 )
Tuesday, March 3, 2009
117. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3+ 6z4)(5x2y3 - 6z4 ) (x2 4)( x2 4 )
Tuesday, March 3, 2009
118. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3+ 6z4)(5x2y3 - 6z4 ) (x2+ 4)( x2 - 4 )
Tuesday, March 3, 2009
119. Example 3: Factor.
b. r2 - 121 c. y2 + 100
(r +11)( r - 11) Cannot be factored
Not a difference
e. x4 - 16
d. 25x4y6
- 36z8
(5x2y3+ 6z4)(5x2y3 - 6z4 ) (x2+ 4)( x2 - 4 )
(x2 + 4)(x + 2)(x - 2)
Tuesday, March 3, 2009