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Chapter 6: Polynomials and Polynomial Functions

Recall

A xA is called the

_____________x is called the _____________

The operation involved is ___________

6.1: Using Properties of Exponents• Properties of Exponents:• 1. Product of powers• 2. Power of a power • 3. power of a product• 4. negative exponent• 5. zero exponent• 6. quotient of powers• 7. power of a quotient

nmnm aaa • When you multiply same bases,

you add exponents!!!!

2 7

42

3

3 2 4

1.) 4 4

2.)

3.) (6 )

4.) ( )

x

a

w v w

2 74 94 262,1442 4x 8x

1 3 1 36 a

3 8 4( )w v w 3 4 8w v 7 8w v

3 2 4 1 4( )w v w

3 36 a 3216a

Zero exponent:

Evaluate:01.) 5 1

02.) 0 !!!undefined

03.) ( 2) 1

014.) ( ) 1

9

ANYTHINGTO THE ZERO POWERIS ONE!!!!!!

is the reciprocal of :n na a

1, where 0n

na a

a

Negative exponents:

EXAMPLES:

21. 22

1

2 1

4

22. ( 9) 2

1

9

1

81

2

13.

3

23 9

3

0

4

2

You try!

1. -4

2. 8

13.

21

4.( 5)

Evaluating Exponential Expressions:

4 41. 6 6 4 46 06 1 232. 2

3 22 62 64

23. 3 2

2( 6) 2

1

6

1

36

2 3

1 2

2

YOU TRY: 1. 4 4

2. (3 )

3. (2 5)

SIMPLIFYING EXPONENTIALEXPRESSIONS:

***Basically, when we simplify, we make sure that our final answer has NO negative exponents in it!

2 31. 2x y What are my negative exponents???Yes, the x and the y bases so those are theONLY one’s that I will move to the denominator!

2

32.

c

d

So, both are negative so the c goes down to the denominator and the d goes up to the numerator!3

2

d

c

2 3

2

x y

33. (5 )a 3

1

(5 )a

1 3 1 3

1

5 a 3

1

125a

YOU TRY!

33 32

31. 2 2. 3. 5x y b

x

Simplify each expression completely.

853

92

34

336

.4

48 .3

45 .2

.1

xxx

aab

xx

yx

918 yx7000,40 x

211256 ba16x

Rewrite the expression with positive exponents.

3

2

33

5 .10

3 .9

2 .8

b

x

yx3

32

x

y

23x

3125

1

b

3

5

2

21

97

9 .5

2 .4

4 .3

6 .2

55 .1

y

x

2555 297

36

1

6

16 2

2

22222

16

1

4

14

xxx

32

1

2

15

39y

634 y 324 y6

34

y

6

64

y

Example 6

8

25

y

2

4

5

y 2

8

5

y

25

8y

Simplify the expression, only using positiveexponents:

3 4

3 7

3

2 4

0

2

2 4

3

4

1.

2. ( )

3. (2 )

4. ( )

5. 9

16.

5

7. 8

98.

x x

a

d

mn n

x y

p

q

Division Properties of Exponents

** *Quotient of Powers:To divide powers that have the same base, you subtract the exponents.This is called the quotient of powers property.

EXAMPLES:5

4

61.

65 46 6

4

6

82.

84 68 28 2

1

8 1

64

3

53.

y

y3 5y 2y 2

1

y

3

2

( 3)4.

3

3 23

13 3

4

45.

x

x4 4x 0x 1YOU TRY!!!

4 11 8 8

1 18 3 8

5 ( 2)1. 2. 3. 4.

5 ( 2)

m s

m s

***Power of a Quotient: Remember, that when you have a base to a power raised to a power, that youmultiply exponents!

EXAMPLES:22

1. ( )3

2

2

2

3

4

9

Remember, that you only SUBTRACT exponents,the bases HAVE TO BE THE SAME!!!! SO, that’s why we just multiply 2 by itself and 3 byitself

33

2.y

3

3

( 3)

y

3

27

y

37

3.4

3

3

7

4

3

3

4

7

64

343

Simplify Expressions using MultipleProperties:

2 2

4

2 91.

3

x y xy

x y

3 3

4

18

3

x y

xy 3 1 3 46x y 2 16x y

26x

y

4

2

22.

x

y

4

2 4

(2 )

( )

x

y

4 4

2 4

2 x

y 4

8

16x

y

YOU TRY:34

3 3 3

3 51. 2.

xy y x

x xy y

Simplify Expressions with NegativeExponents:

32

11.

x x

y y

2 3

1 1 3

x x

y y

6

1 3

x x

y y

1 ( 6)

1 ( 3)

x

y

5

4

x

y

4

5

y

x

12 4

22.

y x

x y

2 4 1

2 1 1

y x

x y

2 4

2 1

y x

x y

1

2 2 4

1 y

x y x

6 2

y

x y

1 2

6

y

x

1

6

y

x

6

1

x y

Simplify each expression completely.

?2or 2 larger, is Which .5

22 .4

.3

32 .2

3 .1

22

25342

47

1220

42

--

cbbc

y y

yx

xy

886561 yx

3

22 yx

11

1

y

181064 cb 22

Quotient of Powers

3

42

8

88

3

6

8

8 38

Simplify top 1st.

33

43

xy

y

x

xy

Simplify with multiple properties

3

23

x

y

14

2

2

y

x

x

y

yx6

1