quartic polynomials look at the two graphs and discuss the questions given below. 1. how can you...

Quartic Polynomials

Look at the two graphs and discuss the questions given below.

1. How can you check to see if both graphs are functions?

3. What is the end behavior for each graph?

4. Which graph do you think has a positive leading coeffient? Why?

5. Which graph do you think has a negative leading coefficient? Why?

2. How many x-intercepts do graphs A & B have?

Graph BGraph A-5 -4 -3 -2 -1 1 2 3 4 5

-14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

12

14

AAT-A IB - HR Date: 3/3/2014 ID Check•Obj: SWBAT evaluate functions using synthetic division and determine whether a binomial is a factor of a polynomial by synthetic substitution.Bell Ringer: pg 363 #18-28 evens (Show table for Casios)

HW Requests: Pg 357 #13-37 odds, use calculator; pg 363 #11-29 oddsStudy- Quiz Sect7.1-7.3 Read Sect7.4HW: Pg 368 #13, 17, 21, 25,29, 31-33 , 37-40 Study -Quiz Section 7.1-7.3 Read Section 7.4Announcements:Quiz this week“There is something in every one of you that waits and listens for the sound of the genuine in yourself. It is the only true guide you will ever have. And if you cannot hear it, you will all of your life spend your days on the ends of strings that somebody else pulls.” ― Howard Thurman

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3Glencoe – Algebra 2Chapter 7: Polynomial Functions

Example 3Solve each equation.

x 6 x 7

x 6 x 7 0

x 2 6 x 7 0

x 7 x 1 0

x 7 0 or x 10

x 7

x 49

x 1

x 3 x 10

x 2 3 x 10

a 1 b 3 c 1

x 3 32 4 1 1

2 1

x

3 13

2 or x

3 13

2

x

3 13 24

9 6 13 13

4

11 3 13

2

The Remainder Theorem

If a polynomial f(x) is divided by x-a, the remainder is the constant f(a), and

Dividend=quotient *divisor +remainder

Where is a polynomial with degree one less than the degree of f(x)

( ) ( ) ( - ) ( )f x q x x a f a

( )q x


4 3 2

4 3 2

Let ( ) 6 8 5 13

find (4)

(4) (4) 6(4) 8(4) 5(4) 13

(4) 2

(4) 33

therefor

#1 Direct Substitution

e when ( ) is divid

56 38

ed b

4 128 20

y (

# 2 Synthe

4) the r

tic

1

emainder = 33

3

f

f x

M

f x x x x x

f

f

ethod

Method

f

x

Use synthetic division

4 1 6 8 5

Sub

13

33

4 8 0 20

1 2 0

stitutio

5

n

Remember Lesson 5-3

for Synthetic Division


5 3Let ( ) 3 5 57.

Find the remainder when divided by ( 2)

____________________________________

2 3 0 5 0 0 57

6 12 14 28 56

3 14 16 7 28

f x x x

x

Remainder = 1

The Factor Theorem

The binomial - is a factor of the polynomial ( ) if and only if ( ) 0x a f x f a

3 2Let ( ) 5 12 36

Is the binomial 3 a factor of the polynomial ( ) ?

3 1 5 -12 -36

3 24 36

1 8 2 01

f x x x x

x f x

Since the remainder is 0, x-3 is a factor of the

polynomial.

When you divide the polynomial by one of the binomial factors , the quotient is called a depressed equation.

The Factor Theorem

3

2

2

2

The polynomial 5 12 36 can be factored as

(x-3)

The polynomial is the depressed polynomial,

which

(x 8 12).

also ma

x

y be factorable

12

.

8

x

x x

x

x

The Factor Theorem

4 3Is a factor - 2 2 2of ?x x x x

2 1 1 0 2 2

2 2 4 12

1 101 2 6

(x-2)

Is NOT a factor

Remainder = 10, therefore

The Factor Theorem

3 21. 8 42x x x

3 22. 2 15 2 120x x x

4 3 23. 6x 13 36 43 30x x x

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials.

( 7)x

(2 5)x

( 2)x

1.( 3)( 2)x x

2.( 6)( 4)x x

23.(3 8 5)(2 3)x x x

Comprehensive Graphs

The most important features of the graph of a polynomial function are:

1. intercepts,2. extrema,3. end behavior.

A comprehensive graph of a polynomial function will exhibit the following features:

1. all x-intercepts (if any),2. the y-intercept,3. all extreme points (if any),4. enough of the graph to exhibit end

behavior.

13

Solving Using Quadratic Form

1. Rewrite in standard form.2. Check if first term can be

written as the square of the middle term.

3. If possible, rewrite in quadratic form.

4. Factor, complete the square, or use the quadratic formula.

5. Solve for the variable.

Glencoe – Algebra 2Chapter 7: Polynomial Functions

Example 1Write each expression in

quadratic form, if possible.

a. x4 13x2 36

x2 2 13 x2 36

b. 625x4 196

25x2 2 14 2

c. 100x8 13x2 81

Not possible. x8 x2 2

d. x 6 x 8

x 2 6 x 8

14Glencoe – Algebra 2Chapter 7: Polynomial Functions

Example 2Solve each equation.

a. x4 13x2 36 0

x2 2 13 x2 36 0

x2 4 x2 9 0

QuadForm

Factor

x2 4 0 or x2 9 0

Zero Product Property

x2 4

x 2

x2 9

x 3

b. x23 6x

13 5 0

x

13 2 6 x

13 5 0

x

13 1 x

13 5 0

x13 10 or x

13 5 0

x13 1

x

13 3 13

x 1

x13 5

x

13 3 53

x 125

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.

Higher Degree Polynomial Functions and Graphs

an is called the leading coefficient n is the degree of the polynomial a0 is called the constant term

Polynomial Function

A polynomial function of degree n in the variable x is a function defined by

where each ai is real, an 0, and n is a whole number.01

11)( axaxaxaxP n

nn

n

x-Intercepts (Real Zeros)

Number Of x-Intercepts of a Polynomial Function

A polynomial function of degree n will have a maximum of n x- intercepts (real zeros).

Find all zeros of f (x) = -x4 + 4x3 - 4x2.x4 4x3 4x2 We now have a polynomial equation.

x4 4x3 4x2 0 Multiply both sides by 1. (optional step)

x2(x2 4x 4) 0 Factor out x2.

x2(x 2)2 0 Factor completely.

x2 or (x 2)2 0 Set each factor equal to zero.

x 0 x 2 Solve for x.

(0,0) (2,0)

Extrema Turning points – where the graph of a function changes from

increasing to decreasing or vice versa. The number of turning points of the graph of a polynomial function of degree n 1 is at most n – 1.

Local maximum point – highest point or “peak” in an interval function values at these points are called local maxima

Local minimum point – lowest point or “valley” in an interval function values at these points are called local minima

Extrema – plural of extremum, includes all local maxima and local minima

Extrema

Polynomial Functions

The largest exponent within the polynomial determines the degree of the polynomial.

Polynomial Function in

General FormDegree

Name of Function

1 Linear

2 Quadratic

3 Cubic

4 Quarticedxcxbxaxy 234

dcxbxaxy 23

cbxaxy 2

baxy


f(x) = 3

ConstantFunction

Degree = 0

Maximum Number of

Zeros: 0

f(x) = x + 2LinearFunction

Degree = 1

Maximum Number of

Zeros: 1


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

Degree = 2Maximum Number of

Zeros: 2


f(x) = x3 + 4x2 + 2

Cubic Function

Degree = 3

Maximum Number of

Zeros: 3


Quartic Function

Degree = 4

Maximum Number of

Zeros: 4


Leading Coefficient

The leading coefficient is the coefficient of the first term in a polynomial when the terms are written in descending order by degrees.

For example, the quartic function f(x) = -2x4 + x3 – 5x2 – 10 has a leading

coefficient of -2.

29

Leading Coefficient Test for Polynomial End Behavior1 0

For any polynomial function ( ) ... , the limits lim ( ) and

lim ( ) are determined by the degree of the polynomial and its leading

coefficient :

n

n x

x

n

f x a x a x a f x

f x n

a

Leading Coefficient Test

leadingcoefficien

t

degree ofpolynomial

Left Right

+ even ↑ + ↑ +

+ odd ↓ - ↑+

- even ↓- ↓ -

- odd ↑ + ↓-

Example

Use the Leading Coefficient Test to determine the end behavior of the graph of f (x) x3 3x2 x 3.

Falls left

yRises right

x

Number of Local Extrema

A linear function has degree 1 and no local extrema.

A quadratic function has degree 2 with one extreme point.

A cubic function has degree 3 with at most two local extrema.

A quartic function has degree 4 with at most three local extrema.

How does this relate to the number of turning points?

The Leading Coefficient Test

As x increases or decreases without bound, the graph of the polynomial function

f (x) anxn an-1x

n-1 an-2xn-2… a1x a0 (an 0)

eventually rises or falls. In particular,

For n odd: an 0 an 0


f (x) anxn an-1x

n-1 an-2xn-2… a1x a0 (an 0)


For n odd: an 0 an 0

If the leading coefficient is positive, the graph falls to the left and rises to the right.

If the leading coefficient is negative, the graph rises to the left and falls to the right.

Rises right

Falls left

Falls right

Rises left


f (x) anxn an-1x

n-1 an-2xn-2… a1x a0 (an 0)


For n even: an 0 an 0


f (x) anxn an-1x

n-1 an-2xn-2… a1x a0 (an 0)


For n even: an 0 an 0

If the leading coefficient is positive, the graph rises to the left and to the right.

If the leading coefficient is negative, the graph falls to the left and to the right.

Rises right

Rises left

Falls left

Falls right

The Leading Coefficient Test

Multiplicity and x-Intercepts

If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.

Example

Find the x-intercepts and

multiplicity of f(x) =2(x+2)2(x-3) Zeros are at

(-2,0)(3,0)

quartic polynomials look at the two graphs and discuss the questions given below. 1. how can you...

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