using distribution with polynomials copyright scott storla 2015
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Simplify Copyright Scott Storla 2015TRANSCRIPT
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Copyright Scott Storla 2015
Using Distribution with Polynomials
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Copyright Scott Storla 2015
The Distributive Property of Multiplication over Addition
Property – The Distributive Property
A product, where one or more of the factors contains terms, can be rewritten as the sum of products. ( )a b c ab ac
Example: 3( 2) 3 3(2)x x
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Copyright Scott Storla 2015
3 2x
3x 6
Simplify
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Copyright Scott Storla 2015
2 1 5y y
2y 2 5y
7 2y
Simplify
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3 11 5 2 1x x
Simplify
2 23 6q q
33 15 2 2x x
2 23 18q q
22 18q
31 17x
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4 1 3 4a a
2 2 1 4 2 8y y y
Simplify
4 4 3 4a a
7a
2 2 2 8 4 8y y y
4 14y
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2 24 7 9 45 5 5( ) x xx x
Simplify
2 29 45 5 4 20 7 xx x x
213 6 26x x
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23(7 11 5)x x
33x 15221x
Simplify
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2 25( 4 3) 8( 12 )y y y y
Simplify
2 25 20 15 8 96 8y y y y
23 12 111y y
2 22(8 7 ) 9( 4 2 2 )p p p p
2 216 2 14 36 18 18p p p p
216 22 2p p
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Simplify
3 2 3 27 2( ) 1x x x x
2 28 5( 2) ( 4)p p p p
3 2(3 ) 2(3 )x x
3 2 3 2
3 2
7 2 2 1
9 3 1
x x x x
x x
3 6 6 24 3x x
x
2 2
2
8 5 5 10 4
6 6 14
p p p p
p p
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Copyright Scott Storla 2015
2 3 3 2 316 2 14 36 18 18 2p p p p p
Simplify
2 2
2
20 20 4 12 12 3 4
9 16 12
x x x x
x x
2 2
2
6 8 40 2 3
5 39 4
a a a a
a a
2 3 3 2 32(8 7 ) 9( 4 2 2 ) 2p p p p p
2 26 8( 5 ) 2 ( 3 )a a a a
2 24( 5 ) 4( 5 ) 3(4 ) 3(4 ) 4x x x x
3 252 16 2p p
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Copyright Scott Storla 2015
Using Distribution with Polynomials
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Copyright Scott Storla 2015
Polynomials and the Product Rule for Exponents
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The Commutative Property of Multiplication The order of the factors doesn’t affect the product.
Example: 3 4 4 3
Note: Division is not commutative.
The Associative Property of Multiplication The grouping of the factors doesn’t affect the product. Example: 3 (4 5) (3 4) 5
Note: Division is not associative.
Property – The Product Rule for Exponents Products of powers with a common base can be written as the common base raised to the sum of the exponents.
Example: 4 2 4 2 63 3 3 3
4 2 4 2 63 3 3 3
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2 34 3y y
Simplify
512y
2 34 3 y y
2 34 3 y y
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4 2(7 )( 2 )( 3 )x x x
742x
2 3 43 ( ) (6 )y y yy
2 3 53 ( ) (6 )y y y
5 53 (6 )y y
53y
Simplify
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3 2 4(2 )(3 ) ( )(4 )a a a a5 56 4a a
510a
4 2 5 3 3( ) 3 ( 2 ) 2 ( )y y y y y y 6 6 66 2y y y
67y
2 3 4 2 25 ( 2 ) 2 (2 ) 5 ( )h h h h h h h 5 5 510 4 5h h h
5h
Simplify
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2 3 5 3 3( ) 3 ( 2 ) 2 ( )y y y y y y y
6 6 66 2y y y
65y
Simplify
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Copyright Scott Storla 2015
Distributing a Monomial
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Copyright Scott Storla 2015
The Distributive Property of Multiplication over Addition
Property – The Distributive Property
A product, where one or more of the factors contains terms, can be rewritten as the sum of products. ( )a b c ab ac
Example: 23 ( 2) 3 3(2)x x x
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Copyright Scott Storla 2015
2 3x x
2x x
22 6x x
2 3x
Simplify
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Copyright Scott Storla 2015
2 35 ( 4) 8 12y y y y
3 213 20 12y y y
3 2 35 20 8 12y y y y
3 2 24 2 ( 4 ) 1x x x x x
3 26 7 1x x
3 3 2 24 2 8 1x x x x
2 37 ( 2) (4 )p p p p 3 37 14 4p p p p
36 10p p
Simplify
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23 (2 4 7)y y y
3 26 12 21y y y
23 (2 )y y 3 (4 )y y 3 (7)y
Simplify
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2 24 (3 2 )p p p 2 4 312 4 8p p p 4 3 24 8 12p p p
2 3 2 24 ( 12 ) ( 6 5)x x x x x x
3 4 4 3 24 48 6 5x x x x x 4 3 247 10 5x x x
2 2( 4 2 ) ( 8 4 )k k k k k k 3 2 2 34 2 8 4k k k k k k
3 25 3 12k k k
Simplify
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3 3 2 3 22 ( ) 2( )( )( )y y y y y y y
4 6 5 3 22 2 2 2( )( )( )y y y y y y
4 6 5 62 2 2 ( 2 )y y y y
4 6 5 62 2 2 2y y y y
6 5 44 2 2y y y
Simplify
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Multiplying Two Linear Binomial Factors Using
FOIL
F irst
O utside
I nside
L ast
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Simplify
Multiplying Two Linear Binomial Factors Using FOIL
3 1 4x x
First
23 11 4x x
23x
Outside
12x
Inside
Last
x 4
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3 4m m
2 7 12m m
Simplify
2 4 3 12m m m
4 2 7p p
28 28 2 7p p p
22 28p p
3 2( 1)( 4)s s
5 3 24 4s s s
2 2 4(6 )(2 )a a a
2 4 4 612 6 2a a a a 6 4 28 12a a a
6 4 4 22 6 12k k k k
2 4 2( 6)( 2 )k k k
6 4 24 12k k k
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Copyright Scott Storla 2015
A General Procedure for Multiplying Polynomials
When multiplying two polynomial factors multiply each term in the first factor with every term in the second factor.
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2 2( 2)( 2)x x x
Simplify
4 3 22x x x 4 3 2 4x x x
22 2 4x x
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2 2( 1)( 1)x x x x
Simplify
4 3 2x x x 3 2x x x 2 1x x
4 2 2 1x x x
2 2( 2)( 2)x x x
4 3 22x x x 22 2 4x x
4 3 2 4x x x
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2 (3 1)( 4)x x x
Simplify
2(6 2 )( 4)x x x
3 2 26 24 2 8x x x x
3 26 22 8x x x
2 2( 2)( 2)x x x
4 3 22x x x 22 2 4x x
4 3 2 4x x x
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2( 1)( 1)r r r r
Simplify
2 2( )( 1)r r r r
4 3 2 3 2r r r r r r
4 32r r r
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Simplify
2( 2)( 4)( 2)k k k
3 2( 4 2 8)( 2)k k k k
4 3 2 3 22 4 8 2 4 8 16k k k k k k k
4 16k
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An Introduction to Polynomials
Distribution
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Some Common Polynomials Products
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2 2 22m n m mn n
Perfect Square Trinomial of a Sum
Identify “m” and “n”Then find the product using the special form.
, 6m r n
2 12 36r r
26r
2 2
2 2
2
2 6 6
m mn n
r r
3 , 4m k n
29 24 16k k
23 4k
2 22
2 23 2 3 4 4
m mn n
k k
1, 4m n n
216 8 1n n
21 4n
2 22
221 2 1 4 4
m mn n
n n
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2 2 22m n m mn n
Perfect Square Trinomial of a Difference
Identify “m” and “n”Then find the product using the special form.
, 5m t n
2 10 25t t
25t
2 22
2 22 5 5
m mn n
t t
3,m n h
2 6 9h h
23 h
2 22
2 23 2 3
m mn n
h h
7 , 4m m n
249 56 16m m
27 4m
2 22
2 27 2 7 4 4
m mn n
m m
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The Difference of Two Squares
2 2m n
2 2
m n m n
m n
m n m n
2 2m mn nm n
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2 2m n m n m n m n m n
Difference of Two Squares
Identify “m” and “n”Then find the product using the special form.
, 5m y n
2 25y
5 5y y
2 2
2 25
m n
y
, 7m k n
2 49k
7 7k k
2 2
2 27
m n
k
2 , 1m x n
24 1x
2 1 2 1x x
2 2
2 2
2 1
m n
x
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2 2m n m n m n m n m n
Difference of Two Squares
Identify “m” and “n”Then find the product using the special form.
3 , 4m y n
29 16y
3 4 3 4y y
2 23 4y
5 , 1m a n
225 1a
5 1 5 1a a
2 25 1a
9, 2m n v
24 81v
9 2 9 2v v
2 29 2v
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Some Common Polynomials Products