Algebra I Name: page 1 of 15

CLASS NOTES: Chapter 2 sections 5, 6, 8 and 9 Multiplication and Division §2 – 5: The Distributive Property PROPERTIES - These are the RULES of Algebra!! Commutative Property of… Addition:

!

a + b = b + a Multiplication:

!

a " b = b " a Associative Property of… Addition:

!

a + b( ) + c = a + b + c( ) Multiplication:

!

a " b( ) " c = a " b " c( ) Identity Property of … Addition:

!

a + 0 = a Multiplication:

!

a " 1 = a Inverse Property of… Addition:

!

a + "a( ) = 0

Multiplication:

!

a "1a

= 1

Distributive property of… Multiplication with respect to addition.

!

a b + c( ) = ab + ac

This means you can change the ORDER.

Order stays the same but you can change the GROUPING (change the parentheses).

• Any number plus zero is itself. • Any number times one is itself.

• Any number plus its opposite is zero.

• Any number times its

reciprocal is one.

You must have addition or subtraction inside the parentheses to distribute!!!!!!! For example,

!

a b • c( ) = abc , NOT

!

ab • ac

NEW!!!

Algebra I Name: page 2 of 15

Question: Where does the distributive property fall in the ORDER of OPERATIONS? Answer: Multiplication!!

Question: How could this possibly work???

Answer:

!

5 • 83 = or

!

5 • 83 = 5 80 + 3( ) = 5( ) 80( ) + 5( ) 3( ) = 400 + 15 = 415

EX 1 Use the distributive property to simplify each expression.

(a)

!

6 x + 7( ) =

(b)

!

8 4 " x( ) =

(c)

!

x " 2( )5 =

(d)

!

"2 x + 9( ) =

!

a b + c( ) = ab + ac

a b " c( ) = ab " ac

!

a b + c( )

This implies (means) MULTIPLY!

Remember… Use the commutative property to make this easier.

Algebra I Name: page 3 of 15

Definitions: Term - Any part of an expression that is separated by addition (+) or

subtraction (-). Factor - Any part of an expression that is separated by multiplication (

!

"). Like Term - (or Similar Term) Terms that contain the exact same variable(s)

and the exact same power(s) of those variables. (Ex: 2xy and -3xy are similar terms because the both have exactly the same variables with the same powers.)

Coefficient - The number in front of the variables in a term. (Ex:

!

5 is the coefficient of the term

!

5 x 2 y .) Combining - We can combine (ADD or SUBTRACT) only those terms that are

similar by adding or subtracting the coefficients of the similar terms.

EX 2 Simplify by using the distributive property and combining like terms. (a)

!

2 " 3 x + 1( ) =

(b)

!

3 4x " 9( ) + 2 x + 1 =

(c)

!

2 x + 3 + x( )4 =

(d)

!

2 14 + 16( ) =

Hint: Remember ORDER of OPERATIONS. Sometimes it is easier to simplify within parentheses first, than to use the distributive property.

Algebra I Name: page 4 of 15

EX 3 Simplify by using the distributive property and combining like terms. (a)

!

5x + 7 x " 3( ) =

(b)

!

4 2 x + y( ) + 8 x " 3y( ) =

(c)

!

4a + 6b " a + 3b = (d)

!

8n " 3 3n " 1( ) =

(e)

!

7 9k " 3( ) " 5 6k " 5( ) " 3k =

Algebra I Name: page 5 of 15

§2 – 6: Rules for Multiplication Question: How many different ways can you write a multiplied by b? Rules for MULTIPLICATION… Multiplicative Property of 0…

!

a • 0 = 0 (Any number times 0 is 0.) Identity Property of Multiplication…

!

a • 1 = a (Any number times 1 is ITSELF.) Multiplicative Property of -1…

!

a • "1( ) = "a (Any number times negative 1 is its opposite.)

POSITIVE or NEGATIVE???

If you multiply… Your answer is…

!

+( ) • +( ) +

!

"( ) • "( ) +

!

+( ) • "( ) or

!

"( ) • +( ) -

EVEN number of negatives multiplied +

ODD number of negatives multiplied -

Algebra I Name: page 6 of 15

EX 1 Simplify each expression. (Be careful! Some are multiplication examples; some are addition, and some will require distribution!)

(a)

!

2( ) 5( ) "20( ) =

(b)

!

"23( ) 0( ) 4( ) =

(c)

!

2 "3( ) 12( ) "1( ) =

(d)

!

6x( ) "3y( ) =

(e)

!

6x + "11x( ) = (f)

!

6 a " 5b( ) =

(g)

!

"6 a " 5b( ) =

(h)

!

5x " 3 2 x " 9( ) =

(i)

!

"2 x " 3y( ) =

Algebra I Name: page 7 of 15

EX 2 Simplify each expression. (a)

!

3x " 4 x " 2( ) =

(b)

!

"1( ) 3a " b + 8( ) =

(c)

!

" "4 x " y( )[ ] =

(d)

!

"7x( ) "4y( )4 =

(e)

!

"4 + 7x( ) "2( ) =

(f)

!

"9 "8 x " 5( ) =

(g)

!

6a + 7b " 4a " 3 " 5b =

Algebra I Name: page 8 of 15

EX 3 Simplify each expression. (a)

!

"8 "19( ) " 7 "19( ) " 5 "15( ) =

(b)

!

"7 3a + b( ) " 2 5a " 2b( ) =

(c)

!

4 2 "5a + b( ) " b[ ] " 10 b " 4a( ) =

Remember… ORDER of OPERATIONS tells us what to do 1st!!!

Algebra I Name: page 9 of 15

§2 – 8: The Reciprocal of a Real Number Definitions: Reciprocals - Any two numbers whose product is one. EX 1 The reciprocal of an integer is one over that integer.

(a) The reciprocal of

!

3 is

!

13

(b) The reciprocal of

!

"5 is

!

"15

EX 2 The reciprocal of a fraction can be found by flipping that

fraction.

(a) The reciprocal of

!

27

is

!

72

(b) The reciprocal of

!

"14

is

!

"41

or just

!

"4

EX 3 What is the reciprocal of…

(a)

!

5 (b)

!

1

8

(c)

!

"9 (d)

!

7

6

(e)

!

"22

7 (f)

!

a

b

(g)

!

1 (h)

!

x (i)

!

"1.6 (j)

!

0

Note: Reciprocals have the same sign. Both are positive or both are negative!

By the way… did you know that the position of the negative in a fraction doesn’t matter???

!

"a

b =

"a

b =

a

"b

Algebra I Name: page 10 of 15

Rules for MULTIPLICATION… Inverse Property of Multiplication (a.k.a. Property of Reciprocals)…

!

a •1a

= 1 (Any number times its reciprocal is

!

1.)

EX 4 Fill in the blank with the number that completes each equation. (a)

!

x • _____ = 1

(b)

!

1

8• _____ = 1

(c)

!

_____ • "2

3= 1

(d)

!

1

k• k = _____

(e)

!

_____ •1

5= 1

REMEMBER: Multiplying Fractions…

⇒ Multiply across the numerator ⇒ Multiply across the denominator

Ex:

!

1

4•

1

5=

1 • 1

4 • 5=

1

20

Ex:

!

9m •1

3=

9m

1•

1

3=

9m • 1

1 • 3=

9m

3= 3m

Algebra I Name: page 11 of 15

EX 5 Simplify each expression.

(a)

!

6x •1

6=

(b)

!

1

6"12( ) =

(c)

!

112 "1

4

#

$ % %

&

' ( ( "

1

7

#

$ % %

&

' ( ( =

(d)

!

12ab "1

3

#

$ % %

&

' ( ( =

(e)

!

1

x42 xy( ) =

(f)

!

1

39x( ) =

(g)

!

"1

2

#

$ % %

&

' ( ( "4xy( ) =

(h)

!

"64ab "1

8

#

$ % %

&

' ( ( =

Algebra I Name: page 12 of 15

EX 6 Simplify each expression.

(a)

!

"1

6"24x + 6y( ) =

(b)

!

1

324a " 3b( ) =

(c)

!

121

4x +

1

3

"

# $ $

%

& ' ' (

1

212 x ( 6( ) =

Algebra I Name: page 13 of 15

§2 – 9: Dividing Real Numbers Definitions: Division - Multiplication by the reciprocal.

POSITIVE or NEGATIVE??? The rules are the same for division as they are for multiplication. Division with Fractions: To divide by a fraction, we multiply by its reciprocal.

MULTIPLY and FLIP!!

a divided by b can be written in the following ways…

!

a

b = a ÷ b = a •

1

b

the

reciprocal of b

To divide by b, we multiply by the reciprocal of b.

This is what DIVISION means!!

This means…

!

15 ÷ "3

2

#

$ % %

&

' ( ( = 15 ) "

2

3

#

$ % %

&

' ( (

Algebra I Name: page 14 of 15

EX 1 Simplify each expression.

(a)

!

8a ÷ "2( ) =

(b)

!

"12 ÷ "5

6

#

$ % %

&

' ( ( =

(c)

!

"16

"1

2

=

(d)

!

"6 ÷6

x= (e)

!

16 ÷ "1

4

#

$ % %

&

' ( ( =

(f)

!

9

"1

7

= (g)

!

"30x

15

#

$ % %

&

' ( ( =

(h)

!

a ÷1

b=

NEVER… Cancel when dividing!!! Multiply and flip FIRST!

!

a

b = a ÷ b = a •

1

b

Algebra I Name: page 15 of 15

Remember some of the RULES…

EX 1 Any number divided by itself is .

!

" x ÷ x = x •1

x=

x

x= 1

" x ÷ x = 1

"x

x= 1

EX 2 Zero divided by any number is .

!

" 0 ÷ x = 0 •1

x= 0

" 0 ÷ x = 0

"0

x= 0

EX 3 Any number divided by one is . EX 4 Any number divided by negative one is . EX 5 One divided by any integer is the of that

integer. EX 6 Any number divided by zero is .

!

" x ÷ 0 = no way! !

"x

0= error! !