Using Distribution with Polynomials Copyright Scott Storla 2015

Copyright Scott Storla 2015

Using Distribution with Polynomials


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


3 2x

3x 6

Simplify


2 1 5y y

2y 2 5y

7 2y

Simplify


3 11 5 2 1x x

Simplify

2 23 6q q

33 15 2 2x x

2 23 18q q

22 18q

31 17x


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


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


23(7 11 5)x x

33x 15221x

Simplify


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


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


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


Using Distribution with Polynomials


Polynomials and the Product Rule for Exponents


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


2 34 3y y

Simplify

512y

2 34 3 y y

2 34 3 y y


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


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


2 3 5 3 3( ) 3 ( 2 ) 2 ( )y y y y y y y

6 6 66 2y y y

65y

Simplify


Distributing a Monomial


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


2 3x x

2x x

22 6x x

2 3x

Simplify


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


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


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


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


Multiplying Two Linear Binomial Factors Using

FOIL

F irst

O utside

I nside

L ast


Simplify

Multiplying Two Linear Binomial Factors Using FOIL

3 1 4x x

First

23 11 4x x

23x

Outside

12x

Inside

Last

x 4


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


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.


2 2( 2)( 2)x x x

Simplify

4 3 22x x x 4 3 2 4x x x

22 2 4x x


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


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


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


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


An Introduction to Polynomials

Distribution


Some Common Polynomials Products


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


2 2 22m n m mn n

Perfect Square Trinomial of a Difference


, 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


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


2 2m n m n m n m n m n

Difference of Two Squares


, 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


2 2m n m n m n m n m n

Difference of Two Squares


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


Some Common Polynomials Products

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Using Distribution with Polynomials Copyright Scott Storla 2015