Unit 5 – Polynomial Functions Mr. Rives

NAME: _____________________________ PERIOD:________DAY TOPIC ASSIGNMENT1 -Vocabulary for Polynomials

-Add/Subtract Polynomials-Identifying Number of Real Zeros for a graph from calculator

6.1 # 1-18

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

3 Long Division of Polynomials (begin synthetic division) 6.3 # 3, 4, 13, 15, 164 -Synthetic Division and Synthetic Substitution

-Remainder Theorem6.3 # 20-22, 24-26, 31, 32, 49

5 -Synthetic Division and Synthetic Substitution-Remainder Theorem

Worksheet (p.11 in packet)

6 REVIEW TO BE ANNOUNCED7 QUIZ (50 points) ENJOY THE BREAK8 Factor Theorem

Factoring Higher Degree PolynomialsSum/Diff of Two CubesGrouping

6.4 # 17-23, 34, 35, 50

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

10 Rational Roots TheoremSolving Polynomial Equations by FactoringMultiplicity of Roots

6.5 # 2-4, 11-13

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-Fundamental Theorem of Algebra-Irrational and Complex Conjugate Roots Theorems

6.6 # 1, 2, 7, 8, 15, 16, 20-21

13 More on 6.6 TBA14 REVIEW P 474 # 2-54 (even –

this might change) 15 TEST-entire unit

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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

Leading coefficient cubic term quadratic term linear term

Facts about polynomials:1. classify by the number of terms it contains2. A polynomial of more than three terms does not usually have a special name3. 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 Trinomial/

2

binomial

3

4 Quartic

5 quintic

Practice

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

a. b.

c. d.

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

a.) b.)

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.

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a.) b.)

Description:

c.) d.)

Description: Description:

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

U5 Day 2 Multiplying Polynomials (Section 6.2)

WARM UP1-2 Evaluate 3-4 Simplify

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XMIN = -5XMAX = 5YMIN = -5YMAX = 5

Description: from left to right the graph increases, decreases slightly, and increases

XMIN = -5XMAX = 5YMIN = -15YMAX = 10

XMIN = -5XMAX = 5YMIN = -5YMAX = 5

1. 2. 3.) x – 2(3x-1) 4.)

5.) 6.)

WARM UP Part 2Multiply

Multiplying Polynomials

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

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

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a

-3

2 -5a 2a

U5 Day 3 Long Division Polynomials (Section 6.3)Review Days 1 and 2

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Holt Algebra 2

6-2 Multiplying Polynomials

(y2 – 7y + 5)(y2 – y – 3) Find the product.

Example 2B: Multiplying Polynomials

Multiply each term of one polynomial by each term of the other. Use a table to organize the products.

–15–5y5y2

21y7y2–7y3

–3y2–y3y4

y2 –y –3y2

–7y

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The top left corner is the first term in the product. Combine terms along diagonals to get the middle terms. The bottom right corner is the last term in the product.

y4 + (–7y3 – y3 ) + (5y2 + 7y2 – 3y2) + (–5y + 21y) – 15 y4 – 8y3 + 9y2 + 16y – 15

Alternate Method – Table

Classify the each polynomial by degree and number of terms.1. 2.

Perform the indicated operation.3. 4.

5. 6. (x – 1) (x – 2) (x + 3)

Just for fun try the following long division without your calculator

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

3169/15 =

Let’s do one together:/(y-3)

The Setup:Write the dividend (the part on the inside) in standard form,

including any terms with a coefficient of 0.

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

Practice

y – 3 2y3 – y2 + 0y + 25

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5.

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 ____________.

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Let’s Try These Together

Synthetic Substitution – using synthetic division to evaluate polynomials. Usethe Remainder Theorem.

Example:P(x) =

Try These

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U5 Day 5 (Section 6.3 cont.)

Use this time to complete any skipped problems for days 1-4.Ready to Go On?

6-3 Lesson Practice Quiz

1. Divide by using long division. ( ) ÷ (x + 2)

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2. Divide by using synthetic division. ÷ (x + 2)

3. Use synthetic substitution to evaluate P(x) = for x = 5 and x = –1.

If time allows start on homeworkU5 Day 5 Homework Worksheet – show all work

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4. Find an expression for the height of a parallelogram whose area is represented by and whose base is represented by (x + 3).

U5 Day 6 Quiz ReviewShow all work-be organized-write answers on the lines provided.

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

1. ___________________________________

2. ___________________________________

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

Multiply:3. 4.

______________________________ ____________________________5. 6. Expand

_______________________________ __________________________________

III. Divide using LONG division: Write the quotient, with the remainder, if there is one, as a fraction, on the answer line.7. 8.

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____________________________ __________________________9. Divide using SYNTHETIC division: . Write the quotient, with the remainder, if there is one, as a fraction.

______________________________

10. If , find using synthetic division.

________

11. Is a factor of ? 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

Complete each statement below.

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A polynomial with 2 terms is called a ________________The degree of is____________.U5 Day 8 Factoring (Section 6.4)

Warm Up

Factor each expressiona.) 3x – 18y b.) c.)

Use the distributive propertya.) (x – 10) (2x + 7) b.)

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) = . Example 2: Is (x + 4) a factor of P(x) =

You Trya.) Is (x+2) a factor of P(x) = . b.) Is (3x - 6) a factor of P(x) = .

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

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Factor by GroupingCommon binomial factor – Write as two binomials in simplified form

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

Exampe1:

You Try

a.) b.) c.)

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

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Use the Table feature on your calculator to check problems a and b from above (You Try Section)a.) 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 Up1.) 2.)

Example 1: (Identify a and b) Example 2:

a = ________ a = ________

b = ________ b = ________

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Example 3: a = ________ b = ________

You Trya.) b.) c.)

d.) e.) Challenge

Application

Closure

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

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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.

FactorExample 1: (Factor out the GCF)

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

You Trya.) b.)

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Let’s look at the graph.

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 = ?

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

Compare this to y = M = ______

SKETCH THE FUNCTIONS

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

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

Compare this to y = M = ______

SKETCH THE FUNCTIONS

3. What is the multiplicity in the following: y =

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

SKETCH THE FUNCTION

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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) = b.)

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:

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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 work1. Let .

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.

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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 . 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.

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