Download Lecture notes 6

Section 6.1
FACTORING OUT THE GCF YouTube Video
3 x  4   3 x  12
3 x  12  3 x  4 
EX 1/
Factor
Distributi ng
Factoring
24 + 36x
24 + 36x
What is the greatest common factor?___________
EX 2/ Factor 4x 2 + 6x 3
4x 2 + 6x 3
What is the greatest common factor?___________
Do the two terms have an x in common, how many?__________
EX 2/ Factor 4x 2 y3 + 6x 3yc
4x 2 y3 + 6x 3yc
What is the greatest common factor?___________
Do the two terms have an x in common?__________
Do the two terms have a y in common?__________
3
2
1. Factor 6 x  9 x  12 x
2
3. Factor 8c  4c  32
2. Factor 12 x
4. Factor
3
 6x 2
40a 3b 2  36a 2 c  12ab 3
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Section 6.2
Factoring - 4 terms YouTube Video
Box Method
mx  3qx  my  3qy
 16m 3  4m 2  4m  1
Factor by Grouping
mx  3qx  my  3qy
 16m 3  4m 2  4m  1
`
8x 2  4 x  2 x  1
8x 2  4 x  2 x  1
`
xy 2  3xy  5 y  15
xy 2  3xy  5 y  15
`
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Section 6.3
Factoring: 3-terms, a=1:
𝟐
𝒙 + 𝒃𝒙 + 𝒄
YouTube Video
 X  5 X  2
_______X 2
________X
________
The last two numbers multiply to give you the third term.
The last two numbers add to give you the middle term
x2  8x  12
X
_______ X
_______
1) What two numbers when multiplied give you the last number (find the factors of 12)
Test numbers :
____·____
____·____
____·____
2) What same two numbers when added give you the middle number (add the factors of 12)
____+____
3)
x2  8x  12
____+____
____+____
X
_______ X
_______
4) Check it
 x  6 X  2
x 2 - bx - c or x 2 + bx - c , then one of the numbers is negative,
x 2 - bx + c , then both of the numbers are negative.
x 2  bx  c , then both of the numbers are positive.
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1. Fill in the blanks.
2
(a) To factor x + 6x + 8 , you need two numbers that add up to_______ and have a product of_______.
2
Then x + 6x + 8 = (
)(
)
2
(b) To factor x - 6x + 8 , you need two numbers that add up to_______ and have a product of_______.
2
Then x - 6x + 8 = (
)(
)
2
(c) To factor x - 2x - 3 , you need two numbers that add up to_______ and have a product of________.
2
Then x - 2x - 3 = (
)(
)
2
d) To factor x + 2x - 3 , you need two numbers that add up to_______ and have a product of________.
2
Then x + 2x - 3 =
2. Factor each trinomial.
In each case, check by multiplying your answer. This allows you to check to see if your answer is correct.
example: 𝒃𝟐 + 𝒃 − 𝟑𝟎 = (𝑏 − 5)(𝑏 + 6)
check: (b - 5)(b + 6) = b2 + b - 30 , so the answer is correct.
2
2
(a) x + 7x + 10
(b) x 2 − 8𝑥 + 15
(c) x  20 x  64
(d) x 2  4 xy  4 y 2
(e) x 2  16 xy  17 y 2
(f) 2 x 3 y  12 x 2 y  10 xy
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Section 6.4
𝟐
Factoring: 3-terms, a not equal to 1: 𝒂𝒙 + 𝒃𝒙 + 𝒄
YouTube Video
6 x 2  11x  30
Something new, so a little different
What two numbers when multiplied give you 6(-30)=____
6 x 2  11x  30
What same two numbers when added give you the middle number____
Setting the problem up
1) What two numbers when multiplied
give you the number -180
2) What same two numbers when added you
give you the middle number -11
Test numbers
____·____
____·____
____·____
____·____
____·____
____·____
____and ____
____and ____
____and ____
____and ____
____and ____
____and ____
3) So we change -11x into the two factors
____+____
____+____
____+____
____+____
____+____
____+____
6 x 2  11x  30
6 x 2  ______  ______  30
4)
The Box Method
or
Factor by Grouping
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3x 2 8x  5
2x2 1x  6
6x 2  x 15
5x 2  7 x  6
35 y2  34 y  8
9x4 18x2  8
9x 2 15x  4
4x 2 15x  9
8x2  8x  6
Hint: common factor
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Section 6.5
𝟐
𝟐
Factoring: 2-terms: 𝒙 − 𝒄
YouTube Video
a 2  b 2  a  b a  b 
2
a 2  2ab  b 2  a  b 
x2  9
4  y2
2 x 2  18 x 4
y 4  16
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2-TERMS (Difference and Sum of Cubes)
YouTube Video


a 3  b3  a  b  a 2  ab  b 2
a 3  b3  a  b  a 2  ab  b 2


x 3  27
y3  8
8 y 3  27 z 6
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Section 6.6
YouTube Video
Look for common FACTORS.
Factor 4x 2 + 6x 3
Factor - 6x 3 - 4x 2  2x
2-TERMS
a 2  b 2  a  ba  b
16  x 2 


64  x 3 


27 x 3  y 3 
a3  b3  a  b a 2  ab  b 2
a3  b3  a  b a 2  ab  b2
3-TERMS
x  3x  4
2
X
_______X
What two numbers when multiplied give you the last number (c)
and when added gives you the middle number (b).
_______
2 x 2  11x  12
4- TERMS
Box Method
mx  3qx  my  3qy
Factor by Grouping
mx  3qx  my  3qy
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Section 6.7
𝟐
Solving a𝒙 + 𝒃𝒙 + 𝒄 = 𝟎
x  32x  1  0
YouTube Video
What do you think the answers are?____________
Solving using factoring
x 2 6x  5
9h  1  10h 2
 2a 2  8  0
24 x  2 x 2
6 x3  13x 2  5x  0
xx  12  27
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Applications of solving a𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 YouTube Video
1) An envelope's length is 4 cm more than its width. The area is 96 cm 2 . Find the length and the width.
Define Length=
Width=
Equation 1:
Equation 2:
2) The base of a cardboard box has a width that is 4 feet less than the length. The area is 117 ft 2 . Find the
length and the width.
Define Length=
Width=
Equation 1:
Equation 2:
3) An object is dropped from a height of 64 feet, how long will it take to hit the ground? It’s position
equation is 𝑠 = −16𝑡 2 + 64
𝐻𝑖𝑛𝑡: 𝑇ℎ𝑒 𝑔𝑟𝑜𝑢𝑛𝑑 ℎ𝑎𝑠 𝑎 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 0, 𝑠𝑜 𝑠 = 0
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