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Math 3 Unit 3 Part 2 Day 1 – Graphing Polynomial Functions

Expression Type of Function

9

x-2

3x2 – x + 2

4x3 + x2 + x + 1

5x4 + x2 + 10

X5 + 2x2 + 5

3c2 + 4c – 2/c

Graphs of Polynomial Functions

A.

Type of Function___________

Degree (even or odd)________

Number of Zeros___________

Left Behavior______________

Right Behavior_____________

B.


Degree (even or odd)_________

Number of Zeros____________


Right Behavior______________

C.



Number of Zeros__________

Left Behavior_____________

Right Behavior____________

D.

Type of Function____________

Degree (even or odd)__________


Left Behavior_______________


E.

Type of Function____________

Degree (even or odd)_________




F.

Type of Function__________


Number of Zeros__________

Left Behavior_____________

Right Behavior_____________

Left Behavior: Right Behavior:

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Summary of Behavior:

Odd Degree Even Degree

Positive vs. Negative

Positive vs. Negative

End Behavior Examples:

Is the degree even or odd? Is the leading coefficient greater than or less than 0 (positive or negative?

Determine the end behaviors.

Degree?

Leading Coefficient?

Left Behavior?

Right Behavior?

Degree?


Left Behavior?

Right Behavior?

Degree?


Left Behavior?

Right Behavior?

Degree?


Left Behavior?

Right Behavior?

Multiplicity

Double Root Triple Root 2

2

2

( 2)

( 3)

( 1)

y x

y x

y x

Double Root means:

What does the graph look like?

3

3

3

( 2)

( 3)

( 1)

y x

y x

y x

Triple Root means:

What does the graph look like?

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Graphing Polynomial Functions

Sketch the graphs of each equation.

Step 1: Plot the x intercepts and indicate if any roots have multiplicity.

Step 2: Determine the degree and whether the function is an even or an odd degree function.

Step 3: Determine the end behaviors.

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

X Intercepts: X Intercepts X Intercepts

Degree Degree Degree

Left Behavior Left Behavior Left Behavior

Right Behavior Right Behavior Right Behavior

4. 2( ) (2 )( 4) (1 )f x x x x 5. 3( ) ( 2)( 4) ( 1)( 2)f x x x x x 6. 3 3( ) ( 2) ( 1)f x x x

X Intercepts: X Intercepts X Intercepts

Degree Degree Degree

Left Behavior Left Behavior Left Behavior

Right Behavior Right Behavior Right Behavior

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Math 3 Unit 3 Part 2 Day 2– Even and Odd FUNCTIONS

Even and Odd Functions (Not the same as even and odd degree)

Determine graphically and algebraically whether a function has symmetry and whether it is even, odd, or neither.

The characterization of a function as “even”, “odd”, or “neither” can be made either by using the equation of the function or by

looking at its graph.

Even Functions Odd Functions

Equation Example: 2( ) 1f x x

X F(x)

-2

-1

0

1

2

Equation Example: ( ) 2f x x

X F(x)

-2

-1

0

1

2

Graph the function above:

***Very Important Trick*** EVEN functions should look the same when you take your

paper and ____________________________________.

Algebraically:

Replace x with –x and compare the result to f(x).

If ( ) ( )f x f x the function is EVEN.

Example: 2( ) 1f x x

Steps:

1. Substitute x with – x

2. Simplify and compare result with given function

(they should be the same)

Graph the function above:

***Very Important Trick*** ODD functions should look the same when you take your

paper and ____________________________________.

Algebraically:

Replace x with –x and compare the result to f(x).

If ( ) ( )f x f x the function is ODD.

Example: ( ) 2f x x

Steps:

1. Substitute x with – x

2. Simplify and compare result with given function

(they should be opposites)

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Practice:

For Questions 1-3, classify the functions as “even”, “odd”, or “neither” based on the table of values or the list of ordered pairs.

1. 2. 3.

For question 4, classify as even, odd, or neither based on the given graph.

x ( )r x

-2 -10

-1 -4

1 -4

2 -10

x -6 -2 2 6

( )s x 1 -2 -5 -8

{(1,3), ( 1, 3), (4,11), ( 4, 11)}

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For questions 5-8, determine algebraically whether the function is even, odd, or neither.

5. 4 2f x 12x 3x 6. 8 7 5g x 6x x 4x

7. 2h x 12x 9x 8. 7 5k x x x

Operations with Polynomials

A. Adding B. Subtracting

1. 2 2 2 2(2 ) ( 3 4 3 )x xy y x xy y 1. 2 2 2 2(5 2 6 ) ( 3 5 )m mp p m mp p

C. Distributive Property D. Multiplying

1. 2 3 46 ( )a w a w aw 1. (3 2)(5 4)y y 2. 2( 1)( 2 3)x x x

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Math 3 Unit 3 Part 2 Day 3 – Operations with Polynomials Continued

E. Dividing

Placeholders:

1. 3 2x x 2. 4 2 1a a 3. 5 1x 4. 5 32 5x x

Method 1 - Long Division

Review Example: 25 579 Steps:

1.

2.

3.

4.

Steps to Dividing a Polynomial using Long Division:

1. Divide - Divide the first term of the outside into first term of the inside

2. Multiply - Multiply “that term” by the entire outside

3. Subtract - Change all the signs

4. Bring Down - Bring down enough terms so that the number of terms at the bottom is the same as

the number of terms on the outside

Write your answer as a fraction IF there is a remainder.

Ex 1:

24 2 24x x x

Ex 2:

27 15 58x x x

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Ex 3:

4 22 1 4 5 2 4y y y y

Ex 4:

4 32 3 5 6x x x x

Method 2: Synthetic Division

Synthetic division is used when the divisor is linear and has a leading coefficient of 1.

What is meant by “linear”?_______________________________________________

Ex 1: 3 1(2 4 6)( 3)x x x Ex 2:

31 6x x

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Ex 3: 5(3 1) ( 1)m m m Ex 4:

2( 28) ( 5)a a

Find the value of K so that the remainder is 5.

Ex 1: 2( 8 ) ( 5)x x k x Ex 2:

2( 44) ( 4)x kx x Ex 3: 2( 12 40) ( )x x x k

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Math 3 Unit 3 Part 2 Day 4 – The Remainder and Factor Theorems

I. Remainder Theorem

A. Evaluate 3 2( ) 3 8 5 7f x x x x at 2.x

B. Evaluate 4 3 2( ) 3 18 20 24f x x x x x at 5x .

In Summary the answer is _____________________________________________________.

II. Factor Theorem

A. Definition

A polynomial function )(xP has a factor of cx if and only if 0)( cP .

That is, cx is a factor if and only if c is a zero of the function P.

Ex 1: If (4) 0f this means….

_____________ is a zero

_____________ is a factor

Ex 2: If ( 7) 0f this means…

_____________ is a zero

_____________ is a factor

Ex 3: If 2

03

f

this means…

_____________ is a zero

_____________ is a factor

Ex 4: If (2) 5f this means…

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Practice: Decide whether the given x value is a zero of the function and if so determine the factor of

the polynomial.

1. 4 3 2( ) 2 5 8 4; 1f x x x x x x 2. (3 2( ) 2 5 6f x x x x ; x = -3

B. The Factor Theorem to factor polynomials.

A. Given 3 2( ) 2 11 18 9f x x x x and 3x is a zero (which means _______________), factor f(x).

B. Given that 2x is a factor of 3 2( ) 2 13 6h x x x x , use the factor theorem to factor.

C. Find the missing factors of 3 22 10 4 48 2( 4)(?)(?)x x x x .

D. Find the missing factors of 3 23 9 102 144 3( 3)(?)(?)x x x x

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Math 3 Unit 3 Part 2 Day 5 – Finding Zeros

I. Name Three other Names for Solutions:

_________________ , _________________ , __________________

Types of Zeros:

A. ___________________ which _______________________________________.

1._______________________

2._______________________

B. ____________________ which ______________________________________.

Total Number of Zeros is ______________________________

Double Root is________________________________________________________________.

Graph the following polynomials on a graphing calculator. Then identify how many zeros each has, how

many are real, and how many are imaginary, and if there are any double roots.

1. y = x5 + 4x4 -3x3 - 7x2 + 2

2. y = x4 + 2x2 + 8

3. y = x6 + 5x5 - x4 +5x2 - 5

4. y = x3 - 5x2 +8x - 4

What are the possible solutions for polynomials with a higher degree?

Degree of 3 Degree of 4 Degree of 5

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II. Find all Rational Zeros.

A. B. 3 2( ) 3 6 8f x x x x

III. Find all Zeros.

A. 4 3 2( ) 5 3 6f x x x x x B. 4 3( ) 2 3 18f x x x x

IV. Find all Solutions of the Equation.

A. 4 3 22 5 4 6x x x x B.

3 1x

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V. Open Ended Questions

A. Two of the zeros of 3 2( ) 2 4 2 4f x x x x are 1 and -1. What type of zero is the third zero?

Explain.

B. How many solutions does the function 5 3( ) 2 5f x x x x have? Explain.

C. Use a calculator to graph 4 2( ) 5 36f x x x . How many real and/or complex solutions does the

function have?

D. Using the factor theorem, prove whether 2 is a solution to the equation 3 22 25 50 0.x x x

E. Name all the combinations of possible solutions the polynomial can have. 5 3( ) 2 5 2f x x x x

F. Prove whether x + 3 is a factor of the polynomial 3 23 2 6x x x

G. What are the two types of zeros?

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Math 3 Unit 3 Part 2 Day 6 – Writing Polynomial Equations

Find a polynomial function )(xP that has the indicated zeros. Simplify when directed to do so.

A. 1, 2, and -3 as zeros and Simplify.

What is the degree?________

B. zeros 2i and -3 and Simplify.

What is the degree?_________

C. Real coefficients and a zero s 3 – 7i and

Simplify.

What is the degree?___________

D. zeros of 3, -3, and 2 i

What is the degree?___________

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E. one zero is 4-5i and Simplify.


F. real coefficients and two zeros of 2i and 7i


G. zeros are 5, ½, and -3

What is the degree?______

H. real coefficients and a zeros of 8-i and Simpify.

What is the degree?______

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Write the polynomial function as the product of factors and list the zeros.

A. 3 2( ) 1f x x x x B. 3 2( ) 2 3 11 6f x x x x

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Math 3 Unit 3 Part 2 Day 7 - Polynomial Word Problems and the Closure Property

I. The Closure Property

Closure Property How do you know which operations polynomials are closed under?

Whole Numbers - 0,1,2,3…. Does not include fractions or decimals or negative numbers.

CLOSED – Means that when you start with a whole number and perform the operation, you end with a

whole number.

Are whole numbers

CLOSED under

addition?

Are whole numbers

CLOSED under

subtraction?

Are whole numbers

CLOSED under

multiplication?

Are whole numbers

CLOSED under division?

Counting Numbers - 1,2,3… Does not include fractions or decimals or negative numbers

CLOSED – Means that when you start with a counting number and perform the operation, you end with

a counting number.

Are counting numbers

CLOSED under

addition?


CLOSED under

subtraction?


CLOSED under

multiplication?



Integers - …-3,-2,-1,0,1,2,3… Does not include fractions or decimals

CLOSED – Means that when you start with integers and perform the operation, you end with integers.

Are integers CLOSED

under addition?

Are integers CLOSED

under subtraction?

Are integers CLOSED

under multiplication?

Are integers CLOSED

under division?

Complex Numbers/Imaginary

CLOSED – Means that when you start with an imaginary number and perform the operation, you end

with an imaginary number.

Are imaginary

numbers CLOSED

under addition?

Are imaginary numbers

CLOSED under

subtraction?

Are imaginary

numbers CLOSED

under multiplication?

Are imaginary numbers


Polynomials

Are polynomials CLOSED under addition?

Are polynomials CLOSED under subtraction?

Are polynomials CLOSED under multiplication? Are polynomials CLOSED under division?

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Word Problems

1. The function 2( ) 2.75 36.9 104.9f x x x describes the United States trade balance with Mexico in

billions of dollars for the period 1994-1998 (1994 corresponds to x = 4).

A. According to the function when was the U.S. deficit greatest (largest negative value)?

B. When was trade balanced (y = 0) for the period 1994-1998?

C. If the function continues to model U.S.-Mexican trade, will trade be balanced again? When?

2. The function 4 3 2( ) 0.004656 0.0875 0.5177 0.959 1.247G x x x x x describes the annual average

price of a gallon of gasoline during the period 1990-2000 where x = 0 represents 1990.

A. Describe the price of gasoline according to this function including highs, lows, recent changes in the

price, increasing, ,decreasing etc…

B. Within the domain given, how is the price of gasoline best characterized?

C. If this function continues to model the price of gasoline for the next three years, what kind of

change in prices should we expect?

D. According to this function will gasoline reach a price of $2.00 per gallon? If so, when? If not, why?

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3. Stephen has a set of plans to build a wooden box. He wants to reduce the volume of the box to 105

cubic meters. He would like to reduce the length of each dimension by the same amount. The plans call

for the box to be 10 inches by 8 inches by 6 inches. Write and solve a polynomial to find out how much

Stephen should take from each dimension?

4. The height of a box that Joan is shipping is 3 inches less than the width of the box. The length is 2

more than twice the width. The volume of the box is 1540 cubic inches. What are the dimensions of

the box?

5. From 1985 through 1995, the actual and projected number, C (in millions), of home computers sold in

the United Stated cam be modeled by 3 20.0092( 8 40 400)C t t t where t = 0 represents 1990.

During which year are 8.51 million computers projected to be sold?

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