Polynomial Functions and End Behavior



Unit 5 – Polynomial Functions

Mr. Rives

NAME: _____________________________ PERIOD:________

|DAY |TOPIC |ASSIGNMENT |

|1 |-Vocabulary for Polynomials |6.1 # 1-18 |

| |-Add/Subtract Polynomials | |

| |-Identifying Number of Real Zeros for a graph from calculator | |

|2 |Multiplying Polynomials |6.2 # 1-8, 10, 18-25 |

|3 |Long Division of Polynomials (begin synthetic division) |6.3 # 3, 4, 13, 15, 16 |

|4 |-Synthetic Division and Synthetic Substitution |6.3 # 20-22, 24-26, 31, 32, 49 |

| |-Remainder Theorem | |

|5 |-Synthetic Division and Synthetic Substitution |Worksheet |

| |-Remainder Theorem |(p.11 in packet) |

|6 |REVIEW |TO BE ANNOUNCED |

|7 |QUIZ (50 points) |ENJOY THE BREAK |

|8 |Factor Theorem |6.4 # 17-23, 34, 35, 50 |

| |Factoring Higher Degree Polynomials | |

| |Sum/Diff of Two Cubes | |

| |Grouping | |

|9 |More on 6.4 |6.4 26 – 30(skip 27), 33, 34, 36 |

|10 |Rational Roots Theorem |6.5 # 2-4, 11-13 |

| |Solving Polynomial Equations by Factoring | |

| |Multiplicity of Roots | |

|11 |Rational Roots Theorem and Solving Polynomial Equations with the help of a calculator |6.5 # 24-26 (Use RRT), 27-29 |

|12 |-Writing Functions Given Zeros |6.6 # 1, 2, 7, 8, 15, 16, 20-21 |

| |-Fundamental Theorem of Algebra | |

| |-Irrational and Complex Conjugate Roots Theorems | |

|13 |More on 6.6 |TBA |

|14 |REVIEW |P 474 # 2-54 (even – this might |

| | |change) |

|15 |TEST-entire unit | |

U5 Day 1 Polynomial Functions (Section 6.1)

An expression that is a real number, a variable, or a product of a real number and a variable with whole-number exponents _______________________________

A _______________________ is a monomial or the sum of monomials. Standard form is written in descending order of exponents.

The exponent of the variable in a term is the ______________________

constant

[pic]

Leading coefficient cubic term quadratic term linear term

Facts about polynomials:

1. classify by the number of terms it contains

2. A polynomial of more than three terms does not usually have a special name

3. Polynomials can also be classified by degree.

4. the degree of a polynomial is: ____________________________________ ____________________________________________________________

|Degree |Name using degree |Polynomial example |Alternate Example |Number of Terms |

|0 | |-9 |11 |Monomial/monomial |

|1 | |x-4 |4x | |

|2 | |[pic] |[pic] |Trinomial/binomial |

|3 | |[pic] |[pic] | |

|4 |Quartic | | | |

|5 |quintic | | | |

Practice

1. Write each polynomial in standard form. Then classify it by degree and by the number of terms.

a. [pic] b. [pic]

c. [pic] d. [pic]

2. ADDING and SUBTRACTING Polynomials. Write your answer in standard form.

a.) [pic] b.) [pic]

3. Graph each polynomial function on a calculator.

Read the graph from left to right and describe when it increases or decreases.

Determine the number of x-intercepts. Sketch the graph.

a.) [pic] b.) [pic]

Description:

c.) [pic] d.) [pic]

Description: Description:

Closure: Describe in words how to determine the degree of a polynomial.

U5 Day 2 Multiplying Polynomials (Section 6.2)

WARM UP

1-2 Evaluate 3-4 Simplify

1. [pic] 2. [pic] 3.) x – 2(3x-1) 4.) [pic]

5.) [pic] 6.)[pic]

WARM UP Part 2

Multiply

Multiplying Polynomials

Distribute the x and then distribute the 2. Combine like terms and simplify.

Try These

If you are interested in using the Alternate Method (see example below), I set up the first one for you.

a.)[pic] b.)[pic]

| | | |

| | | |

[pic]

U5 Day 3 Long Division Polynomials (Section 6.3)

Review Days 1 and 2

Classify the each polynomial by degree and number of terms.

1. [pic] 2. [pic]

Perform the indicated operation.

3. [pic] 4. [pic]

5. [pic] 6. (x – 1) (x – 2) (x + 3)

Just for fun try the following long division without your calculator

(OH NOOOO!! Please don’t make me think – it’s almost winter break).

3169/15 =

Let’s do one together:

[pic]/(y-3)

The Setup:

Write the dividend (the part on the inside) in standard form, including any terms with a coefficient of 0.

[pic]

Setup a long division problem the same way you would when dividing numbers.

[pic]

Practice

[pic] 5. [pic]

6.

U5 Day 4 Synthetic Division (Section 6.3 cont.)

Synthetic division is a shorthand method of dividing a polynomial by a linear binomial by using only the _______________. For synthetic division to work, the polynomial must be written in standard form, using 0 and a coefficient for any missing terms, and the divisor must be in the form

(x – a).

In long division we divide and

subtract, in synthetic division we

____________ and ____________.

Let’s Try These Together

[pic]

Synthetic Substitution – using synthetic

division to evaluate polynomials. Use

the Remainder Theorem.

Example:

P(x) = [pic]

Try These

U5 Day 5 (Section 6.3 cont.)

Use this time to complete any skipped problems for days 1-4.

Ready to Go On?

[pic]

6-3 Lesson Practice Quiz

1. Divide by using long division.

([pic]) ÷ (x + 2)

2. Divide by using synthetic division.

[pic]÷ (x + 2)

3. Use synthetic substitution to evaluate

P(x) = [pic]for x = 5 and x = –1.

If time allows start on homework

U5 Day 5 Homework Worksheet – show all work

U5 Day 6 Quiz Review

Show all work-be organized-write answers on the lines provided.

I. Perform the indicated operation. Write the answer in standard form.

1. [pic]___________________________________

2. [pic]___________________________________

2a) The degree of your answer to #2 is_________ 2b) The leading coefficient in your answer is_______

Multiply:

3. [pic] 4. [pic]

______________________________ ____________________________

5. [pic] 6. Expand [pic]

_______________________________ __________________________________

III. Divide using LONG division: Write the quotient, with the remainder, if there is one, as a fraction, on the answer line.

7. [pic] 8. [pic]

____________________________ __________________________

9. Divide using SYNTHETIC division: [pic]. Write the quotient, with the remainder, if there is one, as a fraction.

______________________________

10. If [pic], find [pic] using synthetic division.

[pic]________

11. Is [pic] a factor of [pic]? Explain how you know. Show work.

Fill in the blanks for the chart below.

|Example of a function |Degree of the function |Name/type of function |

|[pic] | | |

| | | |

|[pic] | | |

|[pic] | | |

|[pic] | | |

|[pic] | | |

|[pic] | | |

Complete each statement below.

A polynomial with 2 terms is called a ________________The degree of [pic] is____________.

U5 Day 8 Factoring (Section 6.4)

Warm Up

Factor each expression

a.) 3x – 18y b.) [pic] c.) [pic]

Use the distributive property

a.) (x – 10) (2x + 7) b.) [pic]

The Remainder Theorem: if a polynomial is divided by (x – a), the remainder is the value of the function at a. So, if (x – a) is a factor of P(x), then P(a) = 0.

Determine Whether a Linear Binomial is a Factor

Example1: Is (x-3) a factor of P(x) = [pic]. Example 2: Is (x + 4) a factor of P(x) =

[pic]

You Try

a.) Is (x+2) a factor of P(x) = [pic]. b.) Is (3x - 6) a factor of P(x) = [pic].

Note: the binomial is not in the form (x – a)

Factor by Grouping

Common binomial factor – Write as two binomials in simplified form

a.) 2y(5x + 12) + 7(5x + 12) b.[pic] c.) [pic]

Exampe1:

You Try

a.) [pic] b.) c.) [pic]

Graphing Calculator Table Feature (compare original equation and factored form)

Use the Table feature on your calculator to check problems a and b from above (You Try Section)

a.) [pic] Which values of Y1 and Y2 are 0?____________

b.)

Which values of Y1 and Y2 are 0?____________

Closure

1. If (x – 3) is a factor of some polynomial P(x) what does that tell you about the remainder?

2. If you divide 5 into 80 what is your remainder? What does this tell you about the number 5 with regard to the number 20?

U5 Day 9 Factoring continued…

Warm Up

1.) [pic] 2.) [pic]

Example 1: [pic] (Identify a and b) Example 2: [pic]

a = ________ a = ________

b = ________ b = ________

Example 3: [pic] a = ________ b = ________

You Try

a.) [pic] b.) [pic] c.) [pic]

d.) [pic] e.) Challenge [pic]

Application

Closure

1. Describe one key difference between factoring the SUM of perfect squares VS the DIFFERENCE of perfect squares.

U5 Day 10 Real Roots in Polynomial Equations (Section 6.5)

From section 5-3 the Zero _________ Property defines how we can find the roots (or solutions) of the polynomial equation P(x) = 0 by setting each __________ equal to 0.

Factor

Example 1: (Factor out the GCF)

Example 2: Use a simple substitution here. I’ll show you.

You Try

a.) [pic] b.) [pic]

Multiplicity Calculator Exploration

Multiplicity simply means that a factor is repeated in a polynomial function.

By Definition: The multiplicity of root r is the number of times that x – r is a ___________ of P(x).

1. What is the multiplicity in the following: y = [pic]?

M = _____ What does the graph do if M is EVEN?

Compare this to y = [pic] M = ______

SKETCH THE FUNCTIONS

2. . What is the multiplicity in the following: y = [pic]?

M = _____ What does the graph do if M is ODD?

Compare this to y = [pic] M = ______

SKETCH THE FUNCTIONS

3. What is the multiplicity in the following: y =[pic]

There are two values for M. Let’s see what happens. Do you have a prediction?

SKETCH THE FUNCTION

4. Find the roots and the multiplicity of each root for y = (2x - 10)(x – 7)(x + 1)(x+1)

5. Identify the roots and state the multiplicity for each root: (Use your calculator.)

a.) f(x) = [pic] b.) [pic]

Closure: How is a real root with odd multiplicity different from a real root with even multiplicity? Explain (yes in words).

U5 Day 11 Rational Root Theorem (Section 6.5 cont.)

Warm Up

Example:

Step 3 Test on the possible rational roots. Look at the graph, which one seems possible.

Use Division and the Remainder Theorem to test.

Step 4 List all factors.

Step 5 Find all roots. Set each factor = 0. Sometimes you’ll need the quadratic formula.

(Ignore the numbering.)

Follow the directions. Just practice listing the possible roots.

Show all work

1. Let [pic].

a. List all the possible rational roots. (p/q’s)

b. Use a calculator to help determine which values are the roots and perform synthetic division with those roots.

c. Write the polynomial in factored form and determine the zeros of the function. List the multiplicity of each zero. (You will need to use the quadratic formula.)

2. Let [pic].

a. List all the possible rational roots. (p/q’s)

b. Use a calculator to help determine which values are the roots and perform synthetic division with those roots.

c. Write the polynomial in factored form and determine the zeros of the function. List the multiplicity of each zero.

-----------------------

XMIN = -5

XMAX = 5

YMIN = -5

YMAX = 5

[pic]

Description: from left to right the graph

increases, decreases slightly, and increases

again. There are 3 x-intercepts = 3 REAL ZEROS.

XMIN = -5

XMAX = 5

YMIN = -15

YMAX = 10

XMIN = -5

XMAX = 5

YMIN = -5

YMAX = 5

Alternate Method – Table

a

-3

2

-5a

[pic]

y – 3 2y3 – y2 + 0y + 25

Let’s look at the graph.

[pic]

[pic]

[pic]

4. Find an expression for the height of a parallelogram whose area is represented by [pic] and whose base is

represented by (x + 3).

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