2.32.3 polynomial functions of higher degree. quick review

2.32.32.32.3

Polynomial Functions of Higher Polynomial Functions of Higher DegreeDegree

Quick Review

2

3 2

2

3

Factor the polynomial into linear factors.

1. 3 11 4

2. 4 10 24

Solve the equation mentally.

3. ( 2) 0

4. 2( 2) ( 1) 0

5. ( 3)( 5) 0

x x

x x x

x x

x x

x x x

Quick Review Solutions

2

3 2

2

3

Factor the polynomial into linear factors.

1. 3 11 4

2. 4 10 24

Solve the equation mentally.

3. ( 2) 0

4. 2( 2) ( 1) 0

3 1 4

2 2 3 4

0,

5. ( 3)(

2

2, 1

5) 0

x x

x x x

x x

x x

x x x

x x

x x x

x x

x x

0, 3, 5x x x

What you’ll learn about• How to use transformations to sketch graphs of

Polynomial Functions• How to use the “Leading Coefficient Test” to

determine End Behavior of Polynomial Functions• How to find and use Zeros of Polynomial

Functions• How to use the Intermediate Value Theorem to

locate zeros• … and whyThese topics are important in modeling various

aspects of nature and can be used to provide approximations to more complicated functions.

Example Graphing Transformations of Monomial

Functions4

Describe how to transform the graph of an appropriate monomial function

( ) into the graph of ( ) ( 2) 5. Sketch ( ) and

compute the -intercept.

n

nf x a x h x x h x

y

Example Graphing Transformations of Monomial Functions

4

Describe how to transform the graph of an appropriate monomial function

( ) into the graph of ( ) ( 2) 5. Sketch ( ) and

compute the -intercept.

n

nf x a x h x x h x

y

4

4

4

You can obtain the graph of ( ) ( 2) 5 by shifting the graph of

( ) two units to the left and five units up. The -intercept of ( )

is (0) 2 5 11.

h x x

f x x y h x

h

Cubic Functions

Quartic Function

Local Extrema and Zeros of Polynomial

FunctionsA polynomial function of degree n

has at most n – 1 local extrema and at most n zeros.

Example Applying Polynomial Theory

4 3Describe the end behavior of ( ) 2 3 1 using limits.g x x x x

Example Applying Polynomial Theory

4 3Describe the end behavior of ( ) 2 3 1 using limits.g x x x x

lim ( )xg x

Example Finding the Zeros of a Polynomial

Function3 2Find the zeros of ( ) 2 4 6 .f x x x x

Example Finding the Zeros of a Polynomial

Function

3 2Find the zeros of ( ) 2 4 6 .f x x x x

3 2

Solve ( ) 0

2 4 6 0

2 1 3 0

0, 1, 3

f x

x x x

x x x

x x x

Example Sketching the Graph of a Factored

Polynomial3 2Sketch the graph of ( ) ( 2) ( 1) .f x x x

Example Sketching the Graph of a Factored

Polynomial

3 2Sketch the graph of ( ) ( 2) ( 1) .f x x x

The zeros are 2 and 1. The graph crosses the -axis at 2 because

the multiplicity 3 is odd. The graph does not cross the -axis at 1 because

the multiplicity 2 is even.

x x x x

x x

Intermediate Value Theorem

If a and b are real numbers with a < b and if f is continuous on the interval [a,b], then f takes on every value between f(a) and f(b). In other

words, if y0 is between f(a) and f(b), then y0=f(c) for some number c in [a,b].