The Foundation Copyright Scott Storla 2015. Wolfram Alpha Simplify Prime Factor the Polynomial...

Copyright Scott Storla 2015

The Foundation


Wolfram Alpha

12 4 3 12 4 3

12 12

Simplify

3 5 1

9 6 36 3

k k k

10 5 2 2 5w w w

Prime Factor the Polynomial

4 3 29 18 16 32k k k k

Solve

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Properties


A property allows us to use a general idea in specific situations.

For instance a property of fire is that it needs oxygen to burn.

We use this property when we blow on a struggling campfire or extinguish a frying pan fire with a cover.

Numbers and operations have properties too.


The Commutative Properties

The Commutative Property of Addition The Commutative Property of Multiplication

The order of the terms doesn’t affect the sum. The order of the factors doesn’t affect the product.

Example: 3 4 4 3 Example: 3 4 4 3

Note: Subtraction is not commutative. Note: Division is not commutative.

The Associative Properties

The Associative Property of Addition The Associative Property of Multiplication

The grouping of the terms doesn’t affect the sum. The grouping of the factors doesn’t affect the product.

Example: 3 4 5 3 4 5 Example: 3 4 5 3 4 5

Note: Subtraction is not associative. Note: Division is not associative.

The Distributive Property of Multiplication Over Addition

A sum with a common factor can be rewritten as the product of the common factor and the sum of the remaining factors.

Example: 3(4) 3(2) 3 4 2 and 3 4 2 3(4) 3(2)

The Identity Properties

The Additive Identity The Multiplicative Identity

0 is the additive identity. Adding 0 to an expression doesn’t change the value of the expression.

1 is the multiplicative identity. Multiplying an expression by 1 doesn’t change the value of the expression.

Example: 0 3 is equivalent to 3. Example: 1 5 is equivalent to 5.

The Inverse Properties

The Additive Inverse The Multiplicative Inverse

The expression which when added to the original expression gives a sum of 0.

The expression which when multiplied to the original expression gives a product of 1.

Example: The additive inverse of 8 is 8 . Example: The multiplicative inverse of 2 is 1/2.

The Commutative Properties

The Commutative Property of Addition The Commutative Property of Multiplication

The order of the terms doesn’t affect the sum. The order of the factors doesn’t affect the product.

Example: 3 4 4 3 Example: 3 4 4 3

Note: Subtraction is not commutative. Note: Division is not commutative.

The Associative Properties

The Associative Property of Addition The Associative Property of Multiplication

The grouping of the terms doesn’t affect the sum. The grouping of the factors doesn’t affect the product.

Example: 3 4 5 3 4 5 Example: 3 4 5 3 4 5

Note: Subtraction is not associative. Note: Division is not associative.

The Distributive Property of Multiplication Over Addition

A sum with a common factor can be rewritten as the product of the common factor and the sum of the remaining factors.

Example: 3(4) 3(2) 3 4 2 and 3 4 2 3(4) 3(2)


The Additive Identity The Multiplicative Identity

0 is the additive identity. Adding 0 to an expression doesn’t change the value of the expression.

1 is the multiplicative identity. Multiplying an expression by 1 doesn’t change the value of the expression.

Example: 0 3 is equivalent to 3. Example: 1 5 is equivalent to 5.

The Inverse Properties

The Additive Inverse The Multiplicative Inverse

The expression which when added to the original expression gives a sum of 0.

The expression which when multiplied to the original expression gives a product of 1.

Example: The additive inverse of 8 is 8 . Example: The multiplicative inverse of 2 is 1/2.


____________________________________

____________________________________

15 14 12 2

15 14 12 2

15 12 14 2

15 12 14 2

27 12

12 27

12 27

k j k j

k j k j

k k j j

k j

k j

j k

j k

Where it’s appropriate use a property to justify the step.

The commutative property of addition.

____________________________________

____________________________________

The distributive property.

Added.

___________________________________

____________________________________


Wrote adding an opposite as subtraction.

Wrote subtraction as adding an opposite.

Simplify


The Foundation


Property – The Commutative Property of Addition

English: The order of the terms doesn’t affect the sum.

Example: 3 4 4 3

Note: Subtraction is not commutative.

The Commutative properties are about order.

Property – The Commutative Property of Multiplication

English: The order of the factors doesn’t affect the product.

Example: 3 4 4 3

Note: Division is not commutative.


Property – The Associative Property of Addition

English: The grouping of the terms doesn’t affect the sum.

Example: 3 4 5 3 4 5

Note: Subtraction is not associative.

The Associative properties are about grouping.

Property – The Associative Property of Multiplication

English: The grouping of the factors doesn’t affect the product.

Example: 3 4 5 3 4 5

Note: Division is not associative.


2 3 2

2 2 3

Describe which property is being used to transform the upper expression to the lower expression. Don’t simplify the expression.


2 2 3

2 2 3

The associative property of addition.

8

8

t

t

The commutative property of multiplication.

4 1 3 1

1 1 4 3


1 1 4 3

1 1 4 3

The associative property of multiplication.


4 5 6

6 4 5

y y

y y



3 5 2 1

2 1 3 5

x x

x x


2(1) 2( 1) 5( 1)

2(1) 2( 1) 5( 1)

The associative property of addition.

2 3

2 3

k

kThe associative property of multiplication.


The Distributive property of multiplication over addition

Property – The Distributive Property of Multiplication over Addition

English: A sum of terms, each with a common factor, can be rewritten as the product of the common factor and the sum of the remaining factors.

Example: 3(2) 3(5) 3(2 5)

When we write 3(2) 3(5) as 3(2 5) we saywe have "factored out" the 3.


3 5

Using the distributive property to add or subtract natural numbers

3 1 5 1

1 3 5

1 8

8

Simplify using the distributive property.

4 6 3

4(1) 6(1) 3(1)

(1)(4 6 3)

(1)(13)

(13)(1) 8 1

13

5 4 1

5(1) 4(1) 1(1)

(1)(5 4 1)

(1)(10)

(10)(1)

10


Property – The Additive Identity

English: 0 is the additive identity. Adding a term of 0 to an expression doesn’t change the value of the expression.

Example: 3 0 is equivalent to 3


Property – The Multiplicative Identity

English: 1 is the multiplicative identity. Multiplying an expression by 1 doesn’t change the value of the expression.

Example: 1(5) is equivalent to 5.


Property – The Additive Inverse

English: The expression which when added to the original gives a sum of 0.

Example: The additive inverse of 8 is 8 .

The Inverse properties

Property – The Multiplicative Inverse

English: The expression which when multiplied to the original expression gives a product of 1.

Example: The multiplicative inverse of 2 is 1/2.

Note: 0 does not have a multiplicative inverse.


2 2

0


The additive inverse.

31

8

3

8

The multiplicative identity.

1 2

2

x

x

The multiplicative inverse.

The multiplicative identity.

15 2

5

1 2

x

x


____________________________________

Fill in the property which allows each step.

The associative property of multiplication

____________________________________

____________________________________

The multiplicative inverse

The multiplicative identity

2 1

1 2

2 1

1 2

1

k

k

k

k


___________________________________

____________________________________

2

22

2 3

2 3 2 3

2 3 2 3

(2 2 3) 3

(2 2) 3 3

2 3

Justifying the Product to a Power Property

The definition of an exponent

____________________________________

____________________________________

The associative property of multiplication

The commutative property of multiplication

____________________________________The associative property of multiplication

The definition of an exponent


Properties

The Foundation Copyright Scott Storla 2015. Wolfram Alpha Simplify Prime Factor the Polynomial...

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