and

ZEROS=ROOTS=SOLUTIONSEquals x intercepts

Long Division

1. What do I multiply first term of divisor by to get first term of dividend?

2. Multiply entire divisor by answer to step 1.

3. Subtract result of step 2(distribute the negative)

4. Bring down next term.

5. Start with step 1.Repeat

#5 Dividing a polynomial by a polynomial (Long Division)

2

822

x

xx

822 2 xxxx

xx 22 x4 8

4

84 x0

Check

)2)(4( xx

8422 xxx

822 xx

x

x2

)2(xx

x

x4 )2(4 x

POLYNOMIALS – DIVIDINGEX – Long division

• (5x³ -13x² +10x -8) / (x-2)

5x³ - 13x² + 10x - 8x - 2

5x²

5x³ - 10x²- ( )

-3x² + 10x

- 3x

-3x² + 6x- ( )

4x - 8

4x - 8- ( )

+ 4

0

R 0

36x

#6 Dividing a polynomial by a polynomial (Long Division)

7

42133

36

x

xx

3x

36 7xx 42

6

426 3 x0

Check

)6)(7( 33 xx

4276 336 xxx

4213 36 xx

)7( 33 xx

)7(6 3x

42137 363 xxx

3

6

x

x

3

36

x

x

f(x) = x + 2LinearFunction

Degree = 1

Maximum Number of

Zeros: 1

Polynomial Functions

f(x) = x2 + 3x + 2QuadraticFunction

Degree = 2Maximum Number of

Zeros: 2

Polynomial Functions

f(x) = x3 + 4x2 + 2

Cubic Function

Degree = 3

Maximum Number of

Zeros: 3

Polynomial Functions

Quartic Function

Degree = 4

Maximum Number of

Zeros: 4

Polynomial Functions

EXAMPLE: ODD

A function is odd if the degree which is greatest is odd and even if the degree which is greatest is even

7x 3 x 2 x 2

Example: even

64c 4 48c 2 9

End Behavior

Behavior of the graph as x approaches positive infinity (+∞) or negative infinity (-∞)

The expression x→+∞ : as x approaches positive infinity

The expression x→-∞ : as x approaches negative infinity

End Behavior of Graphs of Linear Equations

f(x)→+∞ as x→+∞ f(x)→-∞ as x→-∞

f(x) = x

f(x)→-∞ as x→+∞ f(x)→+∞ as x→-∞

f(x) = -x

End Behavior of Graphs of Quadratic Equations

f(x)→+∞ as x→+∞ f(x)→+∞ as x→-∞

f(x) = x²

f(x)→-∞ as x→+∞ f(x)→-∞ as x→-∞

f(x) = -x²

End Behavior…

Four Possibilities Up on both ends Down on both ends Up on the right & Down on the left Up on the left & Down on the right

End Behavior…

Four Prototypes: Up on both ends… y = x2

Down on both ends… y = -x2

Up on the right & Down on the left… y = x3

Up on the left & Down on the right… y = -x3

End Behavior…

Notation: Up on both ends…

Down on both ends…

Up on the right & Down on the left…

Up on the left & Down on the right…

x y

x y

x y

x y

x y

x y

x y

x y

Investigating Graphs of Polynomial Functions

1. Use a Graphing Calculator to graph each function then analyze the functions end behavior by filling in this statement: f(x)→__∞ as x→+∞ and f(x)→__∞ as x→-∞

a. f(x) = x³ c. f(x) = x4 e. f(x) = x5 g. f(x) = x6

b. f(x) = -x³ d. f(x) = -x4 f. f(x) = -x5 h. f(x) = -x6

Investigating Graphs of Polynomial Functions

How does the sign of the leading coefficient affect the behavior of the polynomial function graph as x→+∞?

How is the behavior of a polynomial functions graph as x→+∞ related to its behavior as x→-∞ when the functions degree is odd? When it is even?

End Behavior for Polynomial Functions

For the graph of

If an>0 and n even, then f(x)→+∞ as x→+∞ and f(x)→+∞ as x→-∞

If an>0 and n odd, then f(x)→+∞ as x→+∞ and f(x)→-∞ as x→-∞

If an<0 and n even, then f(x)→-∞ as x→+∞ and f(x)→-∞ as x→-∞

If an<0 and n odd, then f(x)→-∞ as x→+∞ and f(x)→+∞ as x→-∞

11 1 0( ) .........n n

n nf x a x a x a x a

24

Using the Leading Coefficient to Describe End Behavior: Degree is EVEN

If the degree of the polynomial is even and the leading coefficient is positive, both ends ______________.

If the degree of the polynomial is even and the leading coefficient is negative, both ends ________________.

4( )f x x 4( )f x x

, ( ) _____

, ( ) _____

As x f x

As x f x

, ( ) _____

, ( ) _____

As x f x

As x f x

25

Using the Leading Coefficient to Describe End Behavior: Degree is ODD

If the degree of the polynomial is odd and the leading coefficient is positive, the graph falls to the __________ and rises to the ______________.

If the degree of the polynomial is odd and the leading coefficient is negative, the graph rises to the _________ and falls to the _______________.

5( )f x x5( )f x x

, ( ) _____

, ( ) _____

As x f x

As x f x

, ( ) _____

, ( ) _____

As x f x

As x f x

Graphing Polynomial Functions

f(x)= -x4 – 2x³ + 2x² + 4x

x -3 -2 -1 0 1 2 3

f(x)

Determining End Behavior

Match each function with its graph.

4 2

3 2

( ) 5 4

( ) 3 2 4

f x x x x

h x x x x

47)(

43)(7

26

xxxkxxxxg

A. B.

C. D.

29

For example, there are an infinite number of polynomials of degree 3 whose zeros are -4, -2, and 3. They can be expressed in the form: 4 2 3f x a x x x

4 2 3f x x x x

4 2 3f x x x x

2 4 2 3f x x x x

Many correct answers

POLYNOMIALS – DIVIDINGEX – Long division

(5x³ -13x² +10x -8) / (x-2)

5x³ - 13x² + 10x - 8x - 2

5x²

5x³ - 10x²- ( )

-3x² + 10x

- 3x

-3x² + 6x- ( )

4x - 8

4x - 8- ( )

+ 4

0

R 0

#7 Dividing a polynomial by a polynomial (Long Division)

1

124

x

xx

1001 234 xxxxx

3x

34 xx 3x 2x

2x

23 xx 22x x0

x2 x2

xx 22 2 x2 1

2

12 x22 x3

1

3

x