student workbook with scaffolded practice unit 2a · 2017. 8. 9. · program workbook pages...

Student Workbookwith Scaffolded Practice

Unit 2A

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1 2 3 4 5 6 7 8 9 10

ISBN 978-0-8251-7456-8 U2A

Copyright © 2014

J. Weston Walch, Publisher

Portland, ME 04103

www.walch.com

Printed in the United States of America

EDUCATIONWALCH

This book is licensed for a single student’s use only. The reproduction of any part, for any purpose, is strictly prohibited.

© Common Core State Standards. Copyright 2010. National Governor’s Association Center for Best Practices and

Council of Chief State School Officers. All rights reserved.

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Program pages

Workbook pages

Introduction 5

Unit 2A: Polynomial RelationshipsLesson 1: Polynomial Structures and Operating with Polynomials

Lesson 2A.1.1: Structures of Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2A-4–U2A-16 7–16

Lesson 2A.1.2: Adding and Subtracting Polynomials . . . . . . . . . . . . . . . . . . . U2A-17–U2A-27 17–26

Lesson 2A.1.3: Multiplying Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2A-28–U2A-39 27–36

Lesson 2: Proving IdentitiesLesson 2A.2.1: Polynomial Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2A-46–U2A-62 37–46

Lesson 2A.2.2: Complex Polynomial Identities . . . . . . . . . . . . . . . . . . . . . . . . . U2A-63–U2A-75 47–56

Lesson 2A.2.3: The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2A-76–U2A-91 57–66

Lesson 3: Graphing Polynomial FunctionsLesson 2A.3.1: Describing End Behavior and Turns . . . . . . . . . . . . . . . . . . .U2A-100–U2A-118 67–76

Lesson 2A.3.2: The Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .U2A-119–U2A-138 77–86

Lesson 2A.3.3: Finding Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U2A-139–U2A-156 87–96

Lesson 2A.3.4: The Rational Root Theorem. . . . . . . . . . . . . . . . . . . . . . . . . .U2A-157–U2A-172 97–106

Lesson 4: Solving Systems of Equations with PolynomialsLesson 2A.4.1: Solving Systems of Equations Graphically . . . . . . . . . . . . . .U2A-182–U2A-209 107–118

Lesson 5: Geometric SeriesLesson 2A.5.1: Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U2A-221–U2A-242 119–128

Lesson 2A.5.2: Sum of a Finite Geometric Series . . . . . . . . . . . . . . . . . . . . .U2A-243–U2A-270 129–138

Lesson 2A.5.3: Sum of an Infinite Geometric Series . . . . . . . . . . . . . . . . . . .U2A-271–U2A-290 139–148

Station ActivitiesSet 1: Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U2A-318–U2A-323 149–154

Set 2: Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .U2A-328–U2A-331 155–158

Coordinate Planes 159–186

Table of Contents

CCSS IP Math III Teacher Resource© Walch Educationiii

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The CCSS Mathematics III Student Workbook with Scaffolded Practice includes all of the student pages from the Teacher Resource necessary for your day-to-day classroom use. This includes:

• Warm-Ups

• Problem-Based Tasks

• Practice Problems

• Station Activity Worksheets

In addition, it provides Scaffolded Guided Practice examples that parallel the examples in the TRB and SRB. This supports:

• Taking notes during class

• Working problems for preview or additional practice

The workbook includes the first Guided Practice example with step-by-step prompts for solving, and the remaining Guided Practice examples without prompts. Sections for you to take notes are provided at the end of each sub-lesson. Additionally, blank coordinate planes are included at the end of the full unit, should you need to graph.

The workbook is printed on perforated paper so you can submit your assignments and three-hole punched to let you store it in a binder.

CCSS IP Math III Teacher Resource© Walch Educationv

Introduction

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UNIT 2A • POLYNOMIAL RELATIONSHIPSLesson 1: Polynomial Structures and Operating with Polynomials

U2A-4© Walch EducationCCSS IP Math III Teacher Resource

2A.1.1

Name: Date:

Warm-Up 2A.1.1A band is holding a small concert. The seating is being sold in sections, and different sections have different numbers of seats. A diagram showing the sections and the number of seats in each section is shown as follows.

Section D:4 seats

Stage

Section A:6 seats

Section B:15 seats

Section C:6 seats

Section F:4 seats

Section E:12 seats

1. Mario bought all the seats in sections A and D. How many total seats did he buy?

2. Evan bought all the seats in sections B and C. How many total seats did he buy?

3. Tara is trying to buy enough seats for a total of 18 people. The only seats remaining are in sections E and F. Will she be able to buy enough seats for her group?

Lesson 2A.1.1: Structures of Expressions

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U2A-9CCSS IP Math III Teacher Resource

2A.1.1© Walch Education

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Scaffolded Practice 2A.1.1Example 1

Identify the terms in the expression 5a2 – a + 7. What is the highest power of the variable a?

1. Rewrite any subtraction using addition.

2. List the terms being added.

3. Identify the power of the variable a in each term.

4. Determine the highest power of a.

continued

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U2A-10CCSS IP Math III Teacher Resource 2A.1.1

© Walch Education

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Example 2

Identify the terms in the expression –2x8 + 3x2 – x + 11, and note the coefficient, variable, and power of each term.

Example 3

Write a polynomial function in descending order that contains the terms –x, 10x5, 4x3, and x2. Determine the degree of the polynomial function.

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2A.1.1

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Problem-Based Task 2A.1.1: Laying TileA contractor is creating a design using different-sized rectangular tiles. The area of each tile is shown in the diagram below. What is the total area of the shown strip of tile? (Diagram not shown to scale.)

36 in2 3x in2 x2 in2

What is the total area of the shown

strip of tile?

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© Walch EducationU2A-15

CCSS IP Math III Teacher Resource 2A.1.1

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Identify the terms in each expression, and note the coefficient, variable, and power of each term.

1. y4 + 13

2. 8c3 – c2 + 8c

3. 12z5 + 9z2 – z – 7

4. –5m10 + m8 + 5m6 – m4

Write a polynomial function using the given terms. Determine the degree of each polynomial function.

5. 30x, x8, –3x3

6. 14x2, –6x3, 10, –x6

7. –x5, 2x, 22, x7, 7x3

Practice 2A.1.1: Structures of Expressions

continued

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2A.1.1

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Each of the figures below is divided into separate parts with each area written within that part. Find the total area of each figure. All units are in square inches.

8.

9x2

12x

9.

3x25x

8

10.

3x3

8

6x2

2x

14

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Warm-Up 2A.1.2Alyssa is creating a blanket using two types of fabric. The dimensions of the blanket, and the two different fabrics, are shown in the diagram that follows. Alyssa is still deciding on some of the dimensions, and the variable x represents the undecided lengths.

w = 42 in

l = 72 in

x

2x

2x

x

1. The perimeter of a rectangle is the sum of its sides. What will be the perimeter of the completed blanket?

2. Write an expression to show the width of the inner piece of fabric.

3. Write an expression to show the length of the inner piece of fabric.

Lesson 2A.1.2: Adding and Subtracting Polynomials

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Simplify (2x2 + x + 10) + (7x2 + 14).

1. Rewrite the sum so that any like terms are together.

2. Find the sum of any constants.

3. Find the sum of any terms with the same variable raised to the same power by adding the coefficients of the terms.

continued

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Example 2

Simplify (6x4 – x3 – 3x2 + 20) + (10x3 – 4x2 + 9).

Example 3

Simplify (–x6 + 7x2 + 11) – (12x6 + 4x5 – 2x + 1).

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2A.1.2

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Problem-Based Task 2A.1.2: Garden PerimeterMarcus is planting a vegetable garden, and he has already determined what some of the sections of the garden will be. Below is a diagram of the garden, with all units in feet. (Diagram not shown to scale.) Marcus wants to place a fence around the corn. What is the perimeter of the section for corn?

2xTomatoes Cucumbers

Pumpkins

Carrots

Corn

Lettuce x

x

l = 40 ft

w = 35 ft

What is the perimeter of the

section for corn?

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Simplify each expression.

1. (–x6 + 10x3 + 5x + 6) + (12x3 – 8x + 7)

2. (14y3 + 8y2 – 8y – 19) + (18y2 + 5y – 14)

3. (20x4 + x3 + 18) – (3x4 + 14x3 + 11x2 + 2)

4. (11x5 – 4x4 + 19x) – (15x4 + 13x3 – 6x + 10)

5. (–10z4 + 2z3 + 14z2 + 15) + (5z3 – 17z2 – 13z + 8)

6. (–4x6 + 3x4 + 20) – (–17x6 + 12x5 – 6x4)

7. (12h2 – 9h – 15) – (3h3 + 7h2 + 8h + 10)

The perimeter of a rectangle is the sum of its sides. Find the perimeter of a rectangle with each given length and width. All measurements are given in centimeters.

8. length: x – 5; width: x + 10

9. length: x2 + 1; width: 4x

10. length: –8x + 24; width: 2x + 3

Practice 2A.1.2: Adding and Subtracting Polynomials

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2A.1.3

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Warm-Up 2A.1.3Charlie is fencing in a rectangular area of his backyard for a garden, but he hasn’t yet decided on the specific dimensions. The diagram below shows the general dimensions of the area Charlie is planning to fence. He knows the hardware store from which he will buy the fencing charges by the foot.

x feet

2x feet

x feet

2x feet

1. Write an expression, using x, to show the perimeter of the fence.

2. The style of fence Charlie wants costs $5 per foot. Use the expression you wrote for the perimeter to find an expression for the total cost of the fence.

3. If Charlie decides to use 15 feet for the length of x, how much will the total amount of fence cost?

Lesson 2A.1.3: Multiplying Polynomials

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Simplify the expression (x2 + 3)(x + 6).

1. Rewrite the product using the Distributive Property.

2. Use properties of exponents to simplify the expression.

3. Simplify any remaining products.

continued

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Example 2

Simplify the expression (–5x + 2)(3x2 – x + 4).

Example 3

Simplify the expression (3x4 + 10x2 – 4x)(x3 – 8x2 + x).

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2A.1.3

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Problem-Based Task 2A.1.3: Not Quite Set in StoneA landscaper is drafting a plan for a stone walkway. Some of the dimensions of the walkway have not yet been determined. The design is below, with all dimensions shown in feet. (Diagram not shown to scale.) What expression can be used to represent the area of the walkway?

x

25 x

x

x

25

1010

x

x

What expression can be used to

represent the area of the walkway?

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Simplify each expression.

1. (11x + 3)(–x2 + 7)

2. (6x3 + 5x2 – 1)(2x3 + 4)

3. (–y2 + 10)(8y3 + 2y)

4. (10x5 + 4x2)(2x2 – 6x + 3)

5. (–3x4 + 5x3 – x2)(x3 – 6)

6. (7y6 – 9y4 + 2)(4y3 + y2 – 1)

7. (5x2 + 4x + 3)(7x3 – 5x – 2)

The area of a rectangle is found using the formula length • width. Find the area of a rectangle with the given length and width. All measurements are given in meters.

8. length: x + 8; width: 3x – 2

9. length: x2 + 1; width: 4x + 10

10. length: 6x + 5; width: 2x2 – 3

Practice 2A.1.3: Multiplying Polynomials

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UNIT 2A • POLYNOMIAL RELATIONSHIPSLesson 2: Proving Identities


2A.2.1

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Warm-Up 2A.2.1A new playground is being built at a school. The rubberized ground under the playground will have the design shown below.

a

b

c

d

A planning committee is considering a few different sizes for the playground. The committee would like to calculate the area of the rubberized ground for the different sizes. What is the area of the ground for a playground with each of the following dimensions for a, b, c, and d ? The area of a rectangle is the product of the length and width, or A = lw.

1. a = 20 feet, b = 10 feet, c = 15 feet, d = 10 feet



Lesson 2A.2.1: Polynomial Identities

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Use a polynomial identity to expand the expression (x – 14)2.

1. Determine which identity is written in the same form as the given expression.

2. Replace a and b in the identity with the terms in the given expression.

3. Simplify the expression by finding any products and evaluating any exponents.

continued

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© Walch Education

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Example 2

Use a polynomial identity to factor the expression x3 + 125.

Example 3

Use a polynomial identity to expand the expression (3x2 – 2x + 8)2.

Example 4

Show how 362 can be evaluated using polynomial identities.

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Problem-Based Task 2A.2.1: How Big Is the Dog Park?City planners are designing a new dog park. The park will be a square, with the dimensions shown in the diagram below. The area of the park is the square of the side length: area = 2802 feet2. Without using a calculator, what is the area of the dog park? Use polynomial identities to support your answer.

280 feet

280 feet

Without using a calculator, what is the area of the

dog park?

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2A.2.1

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Use polynomial identities to expand or factor each expression.

1. (x – 16)2

2. (4x + 21)2

3. 100x2 – 4

4. x2 – 81

5. 27x3 + 343

6. 512x3 – 8

7. (x2 + x + 3)2

Find the area of a square with the given side length, without using a calculator. The area of a square is the square of the side length, or area = side2.

8. side length = 530 feet

9. side length = 470 meters

10. side length = 310 centimeters

Practice 2A.2.1: Polynomial Identities

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Warm-Up 2A.2.2Isaac is comparing the sizes of three different storage boxes by calculating the volume of each box. The volume of each box is found using the formula volume = (length)(width)(height), or V = lwh. Use this formula and the given dimensions to calculate the volume of each box.

1. length: 20 inches; width: 20 inches; height: 16 inches



Lesson 2A.2.2: Complex Polynomial Identities

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© Walch Education

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Find the result of (10 + 7i)(10 – 7i).

1. Determine whether an identity can be used to rewrite the expression.

2. Identify a and b in the sum of squares.

3. Simplify the equation as needed.

continued

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Example 2

Factor the expression 9x2 + 169.

Example 3

Find the result of (8x + 15i)(8x – 15i).

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Problem-Based Task 2A.2.2: Measuring Electrical Impedance

Impedance measures the total opposition that a circuit presents to an electric current. The impedance

of an element can be represented using a complex number V + Ii, where V is the element’s voltage

and I is the element’s current. If the impedance of Element 1 is Z1 = 12 + i, and the impedance of

Element 2 is Z2 = 14 + i, the total impedance of the two elements in parallel is

1 1

1 2

+Z Z

. What is the

total impedance, in fractional form, for the two elements in parallel?

What is the total impedance,

in fractional form, for the

two elements in parallel?

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Use properties of complex numbers to expand or factor each expression.

1. (20 + i)(20 – i)

2. (4 + 3i)(4 – 3i)

3. (7x + i)(7x – i)

4. x2 + 64

5. 25x2 + 484

6. (9x + 15i)(9x – 15i)

7. 36x2 + 144

Use the following information to solve problems 8–10.

Impedance measures the total opposition that a circuit presents to an electric current.

The impedance of an element can be represented using a complex number V + Ii,

where V is the element’s voltage and I is the element’s current. If the impedance of

Element 1 is Z1 and the impedance of Element 2 is Z

2, the total impedance of the

two elements in parallel is 1 1

1 2

+Z Z

. Find the total impedance for each given set of

elements.

8. Element 1: 10 + i; Element 2: 8 + 2i

9. Element 1: 13 + 2i; Element 2: 11 + i

10. Element 1: 11 + i; Element 2: 12 + 3i

Practice 2A.2.2: Complex Polynomial Identities

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2A.2.3

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Warm-Up 2A.2.3Isabella is on a road trip with her family. She records the time her family has been traveling, in hours, and the total distance they have traveled, in miles, in the table below.

Hours Miles2 1303 1954 260

1. How many miles is her family traveling each hour?

2. If her family travels at the same rate for the next hour, what will be the total distance traveled after 5 hours?

3. If her family continues to travel at the same rate, what distance will they have traveled after 6.5 hours?

Lesson 2A.2.3: The Binomial Theorem

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Use the Binomial Theorem to expand (6x + 2y)3.

1. Create Pascal’s Triangle to the appropriate row.

2. Identify the row of Pascal’s Triangle with the coefficients of the expanded expression.

3. Write the expanded expression with the coefficients and powers of each term.

4. Evaluate each term.

continued

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Example 2

Find term 4 from row 6 of Pascal’s Triangle.

Example 3

Find term 4 in the expanded expression of (4x – 9y)7.

Example 4

Kamali has 7 unique bracelets, and is trying to decide which 3 to wear on a date. Use Pascal’s Triangle to find the number of different combinations of 3 bracelets that Kamali could choose from the 7 she owns.

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2A.2.3

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Problem-Based Task 2A.2.3: Combinations of CandidatesThere are 9 students running for 4 openings on the student council. Only students who are running for election can be elected; no write-in candidates are allowed. An elected student can only hold one position at a time. Use Pascal’s Triangle to find the number of different ways 4 students can be elected from the group of 9 candidates.

Use Pascal’s Triangle to find the number of different ways 4 students

can be elected from the group of 9

candidates.

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Practice 2A.2.3: The Binomial TheoremFind the coefficient of the given term in the specified row of Pascal’s Triangle.

1. row 4, term 1

2. row 10, term 3

Expand each binomial using the Binomial Theorem.

3. (–8x + 4)3

4. (2x + y)5

Find the given term in the expanded form of each binomial.

5. (5x + 1)9, term 7

6. (–x – 2y)12, term 3

7. (4x + y)14, term 13

Tamara is choosing cities to visit on her train trip across Europe. For problems 8–10, use Pascal’s Triangle and the given numbers to determine the number of different combinations of unique cities Tamara can visit, if there are 15 total cities along the route. The train moves in one direction, so she can only stop in each city once.

8. number of cities to visit: 3



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UNIT 2A • POLYNOMIAL RELATIONSHIPSLesson 3: Graphing Polynomial Functions


2A.3.1

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Warm-Up 2A.3.1The following graphed function represents the concentration, in parts per million, of a particular medication in the bloodstream after t hours. Use the graph to answer the questions that follow.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42

123456789

101112131415161718192021222324

C(t)

Time (hours)

Conc

entr

atio

n (p

arts

per

mill

ion)

1. When is the concentration increasing? Decreasing?

2. After how many hours is the concentration of medicine at its highest? What is the concentration at this time?

3. What is the x-intercept and what does it represent?

4. What is the y-intercept and what does it represent?

Lesson 2A.3.1: Describing End Behavior and Turns

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Determine the end behavior, maximum number of turning points, and maximum number of real roots of the function f(x) = 6x5 – 3x4 + 2x + 7.

1. Identify the leading coefficient and degree of the polynomial function.

2. Determine the end behavior of the function.

3. Determine the maximum number of turning points of the polynomial function.

4. Determine the maximum number of real roots of the polynomial function.

continued

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Example 2

Describe the end behavior of the given graph of g(x). Determine whether the graph represents an even-degree or odd-degree function, and determine the number of real roots.

Example 3

Use a graphing calculator to graph the function p(x) = –x4 + 3x2 + 4. Summarize the end behavior and turning points of the function.

Example 4

Create a rough sketch of the graph of a sixth-degree polynomial function with a positive leading coefficient.

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Problem-Based Task 2A.3.1: It’s Electric!The growth of demand for electricity in the United States changes from year to year. Below is a sketch of a polynomial function that represents that approximate growth since 1950 and includes projections to 2040. Consider a possible polynomial function that could represent this information. Using end behavior, turning points, and roots, do you expect the growth of electricity demand to increase or decrease from 2040 to 2050? Explain your reasoning.

1950 1960 1970 1980 1990 2000 2010 2020 2030 2040

2

4

6

8

10

12

D(t)

Years

App

roxi

mat

e gr

owth

of e

lect

ricity

dem

and

(per

cent

)

Source: U.S. Energy Information Administration: Annual Energy Outlook 2013

Do you expect the growth of electricity demand to increase or decrease from 2040 to 2050?

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For problems 1 and 2, determine the end behavior, the maximum number of turning points, and the maximum number of real roots of each function.

1. ( ) 2 6 54 3f x x x x=− + +

2. ( ) 4 7 83 2g x x x= − +

For problems 3 and 4, describe the end behavior of each graph. Determine whether the graph represents an even-degree or odd-degree function, and determine the number of real roots.

3. f(x)

4.

f(x)

For problems 5 and 6, create a rough sketch of a possible graph of the function described.

5. a fifth-degree polynomial function with a negative leading coefficient

6. a fourth-degree polynomial function with a positive leading coefficient

Practice 2A.3.1: Describing End Behavior and Turns

continued

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2A.3.1

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The following graph models the price of a particular stock over a period of time. Use this graph to complete problems 7–10.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

255075

100125150175200225250275300325350375400425450475500525550575600

P(t)

Time (days)

Pric

e of

sto

ck (d

olla

rs)

7. Estimate the turning points of the graph of this function.

8. What do the turning points mean in terms of the price of the stock?

9. Describe the end behavior of this graph.

10. If this graph were modeled by a polynomial function, what is the least degree the equation could have? Explain your answer.

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Warm-Up 2A.3.2A factory produces sweatshirts for a sports team, and packages the sweatshirts in boxes of 20. Every hour, the factory produces 9,284 sweatshirts.

1. How many full boxes of sweatshirts are packaged in 1 hour?

2. How many sweatshirts are produced but not packaged in 1 hour?

3. If the remaining sweatshirts were set aside, what fraction of a box would they make up?

Lesson 2A.3.2: The Remainder Theorem

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Find the quotient of (x2 – 5x – 20) ÷ (x – 4) using polynomial long division.

1. Set up the division.

2. Divide the leading term of the dividend by the leading term of the divisor.

continued

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Example 2

Find the quotient of (x2 – 5x – 20) ÷ (x – 4) using synthetic division.

Example 3

Find the quotient of (3x3 + 16x2 + 18x + 8) ÷ (x + 4) using synthetic division.

Example 4

Use synthetic substitution to evaluate p(x) = x2 – 32 for x = –7.

Example 5

If the remainder of (x2 + kx + 34) ÷ (x – 5) is –21, what is the value of k?

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Problem-Based Task 2A.3.2: When Is the Next Dose Due?The amount of a certain medication remaining in the bloodstream t hours after taking the medicine is modeled by the equation M(t) = –x3 + 5x2 + 3x + 18. Package directions recommend taking a second dose 4–6 hours after the initial dose. Use synthetic substitution to show that these directions are accurate.

Use synthetic substitution to show that these directions are

accurate.

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2A.3.2

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For problems 1 and 2, use synthetic division to find each quotient.

1. (x3 – 7x2 + 36) ÷ (x – 3)

2. (2x4 – 6x2 + 8x + 2) ÷ (x + 2)

For problems 3 and 4, use synthetic substitution to evaluate each function.

3. p(x) = 3x2 + 6x + 10 for x = –4

4. p(x) = x5 + 4x3 + 2x – 16 for x = 2

For problems 5 and 6, find the value of k.

5. (x2 + kx + 10) ÷ (x – 1) has a remainder of 4.

6. (x2 + 5x + k) ÷ (x + 6) has a remainder of 9.

For problems 7–10, use the Remainder Theorem to solve each problem.

7. The area in square feet of a rectangular garden can be expressed as the product of the garden’s length and width, or A(x) = 3x2 + 13x + 14. If the width of the garden is (x + 2) feet, what is the length of the garden?

8. The area in square meters of a rectangular patio can be expressed as the product of the patio’s length and width, or A(x) = 7x2 – 34x – 24. If the length of the patio is (x – 4) meters, what is the width of the patio?

9. A generator produces voltage using levels of current modeled by I(t) = t + 4, where t > 0 represents the time in seconds. The power of the generator can be modeled by P(t) = 0.5t3 + 8t2 + 24t. If voltage is calculated by dividing P(t) by I(t), what expression represents the voltage of the generator?

10. A second generator produces voltage using levels of current modeled by I(t) = t + 4, where t > 0 represents the time in seconds. The power of the generator can be modeled by P(t) = 0.2t3 + 8.8t2 + 32t. What expression represents the voltage of this generator?

Practice 2A.3.2: The Remainder Theorem

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Warm-Up 2A.3.3Gregory is the proud new owner of a sailboat, the Cutty Snark. It’s the most common type of sailboat, a Bermuda sloop, and has two sails: a headsail in front, and a mainsail in back. The mainsail is in the shape of a right triangle. On the Cutty Snark, one leg of the triangular mainsail is 31 feet shorter than the other. The longest side of the mainsail is 41 feet. In order to make the proper adjustments, Gregory must know the area of the sail.

1. What is the length of the short leg in terms of x?

2. Using the Pythagorean Theorem, what quadratic equation relates the three sides of the mainsail?

3. What are the factors of this equation?

4. What is the area of the mainsail?

Lesson 2A.3.3: Finding Zeros

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Given the equation x3 + 4x2 – 3x – 18 = 0, state the number and type of roots of the equation if one root is –3.

1. Use synthetic division to find the depressed polynomial.

2. Factor the depressed polynomial to find the remaining factors.

3. State the number of roots and type of roots of the equation.

continued

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Example 2

Find all the zeros of the function f(x) = x3 + x2 + 11x + 51. Verify the zeros by creating a graph.

Example 3

Write the simplest polynomial function with integral coefficients that has the zeros 5 and 3 – i.

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Problem-Based Task 2A.3.3: In Hot WaterA household hot-water tank is designed to store enough hot water to meet the demands of all the members of the household. If the tank is too small, the household will continuously run out of hot water. If the tank is too large, more than enough hot water will be stored, causing a waste of resources. The design of hot-water tanks varies depending on location, use, and resources.

Luz wants to buy a new hot-water tank that is shaped like a cylinder with a hemisphere on the

top. Using the formula 2

33 2V r r hπ π= + , where r is the radius of the tank and h is the height of the

cylinder, she found that this tank has an approximate volume of 99,000p cm3. The height of the

cylindrical portion of the tank is 90 cm. The spot in Luz’s basement for the hot-water tank can only

accommodate a tank that is not wider than 55 cm. Does Luz have enough space for this particular

hot-water tank? Explain your reasoning.

Does Luz have enough space for this particular hot-water tank?

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2A.3.3

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For problems 1–3, determine the number and type of roots for each equation using one of the given roots. Then find each root.

1. x3 – 7x + 6 = 0; 1

2. x3 – 3x2 + 25x + 29 = 0; –1

3. x3 – 4x2 – 3x + 18 = 0; 3

For problems 4–6, find all the zeros of each function. Then graph each function to verify your answers.

4. f(x) = x2 + 4x – 12

5. f(x) = x3 – 3x2 + x + 5

6. f(x) = x3 – 4x2 – 7x + 10

For problems 7 and 8, write the simplest polynomial function with integral coefficients that has the given zeros.

7. –5, –1, 3, 7

8. 4, 2 + 3i

Use the following information to solve problems 9 and 10.

A portable fuel tank used to transport gasoline is in the shape of a rectangular prism. The tank is 5 feet longer than it is wide and 29 feet deeper than it is wide. The volume of the tank is 33,488 cubic feet.

9. What equation represents the volume of the portable fuel tank?

10. Use synthetic division to find all the roots of the polynomial equation. What are the dimensions of the fuel tank?

Practice 2A.3.3: Finding Zeros

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Warm-Up 2A.3.4The manufacturer of an MP3 player sends its product to various retailers. Each shipping box contains the same number of MP3 players. The table below shows the number of MP3 players each retailer received.

Retailer MP3 players received

Electronic Envy 210

Sounds Swell 240

Listen-Up 120

Mostly Music 150

1. What is the greatest possible number of MP3 players in each shipping box?

2. How many shipping boxes, in total, were sent?

3. Once the MP3 players were received, Mostly Music requested two additional boxes of MP3 players. How many additional MP3 players did Mostly Music request?

Lesson 2A.3.4: The Rational Root Theorem

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Use the Rational Root Theorem to list all the possible rational roots of the polynomial equation 2x4 – 3x3 + 4x2 – 9x + 6 = 0.

1. Identify the constant term and the leading coefficient of the polynomial equation.

2. Identify the factors of the constant term and the leading coefficient.

3. Identify the possible rational roots of the polynomial equation.

continued

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Example 2

Find all of the roots to the polynomial equation 4x3 – x2 + 36x – 9 = 0.

Example 3

Find a third-degree polynomial with rational coefficients that has the roots 6 and 3 – i.

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Problem-Based Task 2A.3.4: Fan-tastic!Many residential bathrooms have overhead fans to remove the humidity caused by running a shower or filling a bathtub. If a bathroom has too much moisture, or humidity, the walls, floor, and ceiling can rot. The best size for the overhead fan depends on the volume of the bathroom.

Homeowners who want to install a new fan can determine the appropriate size of fan to purchase

by first calculating the volume of the bathroom and then multiplying the product by 2

15. The

resulting value gives the minimum rate at which a fan should circulate the air in that bathroom, in

cubic feet per minute. For an overhead fan to be effective, the amount of air it circulates in cubic feet

per minute must be greater than the homeowner’s calculated air circulation rate based on the volume

of the bathroom.

Nolan is remodeling his bathroom. The volume of Nolan’s bathroom is represented by the equation f(x) = x3 – 30x2 + 296x – 960, with the room’s dimensions (length, width, and height) measured in feet. Nolan has selected a fan that moves the air at 130 cubic feet per minute. Is this fan appropriate for his bathroom? Explain your answer.

Is this fan appropriate for his bathroom?

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For problems 1–4, use the Rational Root Theorem to list all of the possible rational roots for each polynomial equation.

1. x3 + 3x2 + 3x + 5 = 0

2. x3 + 2x2 + 5x – 9 = 0

3. 4x3 + x2 + 2x – 1 = 0

4. 2x3 – 6x2 + 3x + 12 = 0

For problems 5–8, find all of the solutions of each polynomial equation.

5. 2x3 + 11x2 + 8x – 21 = 0

6. x3 + 3x2 + 3x + 9 = 0

7. x3 – 9x2 + 22x – 10 = 0

8. 9x4 – 24x3 – 16x2 + 20x + 3 = 0

For problems 9 and 10, find a third-degree polynomial with rational coefficients that has the given numbers as roots.

9. –3 and 2i

10. 4 and 3 2−

Practice 2A.3.4: The Rational Root Theorem

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UNIT 2A • POLYNOMIAL RELATIONSHIPSLesson 4: Solving Systems of Equations with Polynomials


2A.4.1

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Warm-Up 2A.4.1

The height of a ball thrown into the air can be modeled by the equation f(x) = –16x2 + 18x + 6, where

f(x) is the height of the ball in feet x seconds after the ball is thrown. A paintball launched into the air

is modeled by the equation ( )5

2

11

2= +g x x , where g(x) is the distance the paintball has traveled. The

ball is released at the same time the paintball is launched.

1. Graph each equation on the same coordinate plane.

2. How many points of intersection exist?

3. How long will it take the paintball to cross the path the ball traveled?

Lesson 2A.4.1: Solving Systems of Equations Graphically

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Use the graph to estimate the solution(s), if any, to the system of equations.

–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

–80

–70

–60

–50

–40

–30

–20

–10

10

20

30

40

50

60

f(x)

g(x)

1. Determine the number of times the graphs of f(x) and g(x) intersect.

2. Estimate the points of intersection.

continued

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Example 2

Use a graph to estimate the real solution(s), if any, to the system of equations ( ) 4 1

( ) 13

= +

= +

f x x

g x x.

Verify that any identified coordinate pairs are solutions.

Example 3

Create a table to approximate the real solution(s), if any, to the system ( ) 4 1

( ) 3 14 3

= + +

= + +

f x x

g x x x.

Example 4

The following table shows values for two functions, f(x) and g(x). Based on the table, what conclusion can you draw about the solutions to the system of equations f(x) and g(x)?

x –4 –3 –2 –1 0 1 2 3 4f(x) –10 –7 –4 –1 2 5 8 11 14g(x) 55 25 11 7 7 5 –5 –29 –73

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Problem-Based Task 2A.4.1: Advertise Here! An advertising agency paid for the right to post signs along a footbridge at a new racetrack. The footbridge passes over the new racetrack’s signature S-curve, providing multiple placements for advertising directly above the track. The signs need to be posted in the proper positions so that fans watching the race on TV will be able to read them as the cars race through the S-curve and under the bridge.

The location of the S-curve in the racetrack can be described by the equation 1

303 2703 2( )( )= − −f x x x x . The location of the footbridge can be described by the linear equation

g(x) = 2 – 3x. What are the coordinates of the proper positions of the advertised signs?

What are the coordinates of

the proper positions of the advertised signs?

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For problems 1–3, use the graphs to estimate the solution(s), if any, to the system of equations. Approximate solutions to the nearest tenth, if necessary.

1.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

–27

–24

–21

–18

–15

–12

–9

–6

–3

3

6

9

12

15

18

21

24

27f(x)

g(x)

2.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

–14

–12

–10

–8

–6

–4

–2

2

4

6

8

10

12

14

16

18f(x)

g(x)

Practice 2A.4.1: Solving Systems of Equations Graphically

continued

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3.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10–2

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

f(x)

g(x)

For problems 4–7, graph each system of equations. Use the graph to estimate the real solution(s), if any exist. Round solutions to the nearest tenth, if necessary.

4. ( ) 2 2 10 6

( ) 3 9

3 2= − − −=− +

f x x x x

g x x

5. ( ) 5 5 4

( ) 4 6 64 3 2

= − −

=− + − − +

f x x

g x x x x x

6. ( ) 2

( ) 3

( 2 )

3 2

=

= + +

f x

g x x x x

x

7. ( ) 7

( ) 2 2 4 53 2

=− +

= − + +

f x x

g x x x x

continued

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For problems 8–10, each table shows values for two functions in a system of equations. Based on the table, what conclusions can you draw about the number of solutions and the x-values of any solutions?

8. x –3 –2 –1 0 1 2 3 4 5f(x) –120 –60 –24 –6 0 0 0 6 24g(x) –240 –140 –72 –30 –8 0 0 –2 0

9. x –3 –2 –1 0 1 2 3 4 5f(x) –40 –8 2 2 4 20 62 142 272g(x) 5 4 3 2 2 3 4 5 6

10. x –4 –3 –2 –1 0 1 2 3 4f(x) 10.003 10.015 10.062 10.25 11 14 26 74 266g(x) 731 179 11 11 35 11 –61 –109 11

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UNIT 2A • POLYNOMIAL RELATIONSHIPSLesson 5: Geometric Series



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Warm-Up 2A.5.1Alicia’s little sister found a tennis ball at the playground. Her sister then climbed to the top of the jungle gym and dropped the ball from a height of 10 feet. With each new bounce, the ball reached only 40% of the height of the previous bounce.

1. What was the height of the ball after the first bounce?

2. What was the height of the ball after the second bounce?

3. What was the height of the ball after the third bounce?

4. What was the height of the ball after the fourth bounce?

5. Graph the number of bounces versus the height of each bounce in feet.

6. Does the graph show a pattern? Explain.

Lesson 2A.5.1: Geometric Sequences

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Find the missing terms in the sequence that follows.

{1.2, 3.6, 10.8, 32.4, a5, a

6, a

7, …}

1. Determine whether there exists in the sequence a common ratio, r.

2. Use the common ratio and the fourth term to find the next term in the sequence.

3. Use the common ratio and the fifth term to find the next term in the sequence.

4. Use the common ratio and the sixth term to find the next term in the sequence.

5. Summarize your results.

continued

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Example 2

Find the specified term, an, given the first term and the common ratio.

a1 = 128,

1

2r = , n = 8

Example 3

Write the recursive formula for the geometric sequence that follows, then determine the missing terms in the sequence.

{0.32, 0.48, 0.72, 1.08, a5, a

6, a

7, …}

Example 4

Given the geometric sequence that follows, write the explicit formula and find the ninth term.

{32,768, 8192, 2048, 512, …, a9, …}

Example 5

You’re planning to experiment on a certain bacteria strain. It takes 1 hour for a cell of this strain to double. If you begin the experiment with 64 cells, how many cells will the experiment have exactly 1 day later?

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Problem-Based Task 2A.5.1: New Restaurant BuzzA new restaurant has just opened in Kylie’s town. Wanting to support her community, Kylie went to the restaurant for dinner. Her experience was great—not only did she have a short wait for her food, but when Kylie complimented the manager on the excellent meal and service, the manager gave her a coupon for a discount on her next meal. Kylie was so pleased, she promised she would tell her friends to try the new restaurant.

Since restaurant owners often rely on word of mouth to promote their restaurant, happy customers can be an important factor in the success of a business. Assume that Kylie will tell 4 friends about her great experience within 1 week, and that each of those friends will tell 4 other people within the next week, continuing on with that pattern of 4 new people being told about the experience each week. Use this pattern to explain why restaurant owners might rely on the compliments of customers to help promote business. Use mathematics to justify your answer and write a formula to predict the number of people who will be told about Kylie’s positive experience in week n.

Use mathematics to justify your

answer and write a formula to predict

the number of people who will be told about Kylie’s positive experience

in week n.

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Practice 2A.5.1: Geometric SequencesFind the missing terms in each sequence and state the common ratio, r.

1.

3

64,

3

16,

3

4,3,

2. {96, 48, 24, 12, …}

Find the specified term, an, using the given information.

3. a1 = 26,244,

1

3r = , n = 11

4. 1

321a = , r = 2, n = 10

5. 9

82a = , 2

3r = , n = 7

Write the explicit formula for the given sequence and find the specified term.

6. {8, 20, 50, 125, …, a8, …}

7.

96

27,

24

9, 2,

3

2, , ,6a

continued

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For problems 8–10, read each scenario and use the geometric sequence it describes to answer the questions.

8. With each successive bounce, new tennis balls bounce to about 53% of the previous bounce height when dropped on a hard surface. If Byron drops a tennis ball from a height of 10 feet, how high will the ball go on the fifth bounce? What is the explicit formula that shows how high the tennis ball will be on the nth bounce?

9. Dagmar is cooling a pot of boiling water in the freezer as part of a science experiment. Water boils at 212ºF. There’s a thermometer in the pot, so Dagmar can see that the temperature decreases by a different number of degrees every minute. After 1 minute, the temperature is 190.8ºF; after 2 minutes it is 171.72ºF; and after 3 minutes it is 154.548ºF. If this pattern continues, what would be the temperature of the water after 10 minutes? What is the explicit formula to determine the temperature after n minutes?

10. Sadhana is analyzing the salary she’s been offered for a new job. Sadhana’s prospective employer is offering her $30,000 the first year, with a guaranteed minimum raise of 2% each year. What will be Sadhana’s minimum salary after 4 years, 5 years, and 7 years? What is the explicit formula to show her minimum salary in n years?

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Warm-Up 2A.5.2Farid is a runner who recently had knee surgery. His doctor has recommended that he slowly build up to his goal of running at least 5 miles per day. The doctor told Farid to start by running a half mile each day, then increase his distance by 20% each week.

1. What is the explicit formula for Farid’s running program?

2. How many miles per day will Farid be running by the end of week 5?

3. How many weeks will it take Farid to get back to his goal of running at least 5 miles each day?

Lesson 2A.5.2: Sum of a Finite Geometric Series

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Expand the series shown in the given summation notation.

41

21

6 1

k

k

∑

=

−

1. Determine the number of terms in the expanded series.

2. Substitute values for k.

3. Simplify and apply the exponents.

4. Simplify the terms.

continued

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Example 2

Sum the series by expansion.

2(3) 1

1

5k

k∑ −

=

Example 3

Derive the formula for the sum of a finite geometric series, which is given by 1

11 1

1

S ara r

rnk

n

k

n

∑ ( )= =

−−

−

=

.

Example 4

Sum the given series using the sum formula for a finite geometric series.

3(2)1

61

k

k∑=

−

Example 5

Seline is applying for a $10,000 college loan to be repaid over 10 years. The annual percentage rate

(APR) on the loan is 3%. However, the interest on the loan is compounded monthly. The amount Seline

will owe can be found using the formula for the sum of an amortized loan, 1

11

1

P Aik

n k

∑= +

=

−

, where:

P = loan amount (principal)

A = monthly payment amount

i = monthly interest rate of the loan

n = number of monthly payments

Use the formula to determine Seline’s monthly payment amount, A.

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Problem-Based Task 2A.5.2: Which Refinancing Option Is Best?Ten years ago, your aunt bought a house when the annual percentage rate (APR) was 6%. She took out a 30-year fixed-rate mortgage for $250,000. Now, interest rates have gone down to 4.4% for a 30-year fixed-rate mortgage and 3.5% for a 15-year fixed-rate mortgage. Your aunt wants to refinance her mortgage so that she pays a lower interest rate on the rest of what she owes. She estimates that she owes about $210,000 on her original mortgage. For a refinanced mortgage, she can choose from the 15-year loan at 3.5% interest, or the 30-year loan at 4.4% interest. Which option would you recommend to your aunt? Justify your recommendation using mathematics.

Recall that a mortgage is an amortized loan, in which monthly payments are split between

principal and interest. The formula for the sum of an amortized loan is given by 1

11

1

P Aik

n k

∑= +

=

−

.

Which option would you

recommend to your aunt?

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Expand the series.

1. 241

4

1

1

5 k

k∑

−

=

Evaluate the series using expansion.

2. 2 5 1

1

4k

k∑ ( ) −

=

Evaluate the series using the sum formula.

3. 121

3

1

1

8 k

k∑

−

=

Find the first term of the series given the sum, the common ratio, and the number of terms.

4. Sn = 20,000, r = 0.996, n = 48

Read each scenario and use the given information to answer the questions.

5. You found out that one of your favorite TV shows has been canceled. That day, you post the news to 5 friends on your social media page. The next day, those 5 friends each post to 5 of their friends’ pages. How many people will have shared the news after 6 days if this pattern continues?

6. Santo catches a contagious cough at the beginning of the school day. Within 1 hour, he has infected 4 other people in the school. If this pattern continues, how many people will be infected by the end of the school day, which is 5 hours after Santo arrived?

Practice 2A.5.2: Sum of a Finite Geometric Series

continued

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7. Nhu is a biologist who is conducting a study about a species of butterfly called the Common Buckeye. She estimates that in the spring, the number of butterflies living in her study area will increase rapidly. For every butterfly in the area, 2 new butterflies hatch each week. If approximately 50 butterflies were counted during the first week of the season, how many butterflies will there be by the twelfth week?

8. Lydia is a childbirth assistant. The first year of her career, she attended the births of 2 babies. Every year after that, the number of births Lydia attended has been double that of the previous year. After 6 years of working as a childbirth assistant, how many births has she attended in total?

9. Your sister just graduated from college and wants to buy a new car. She found a car for $14,000. The annual percentage rate is 4%, with the interest compounded monthly. What will be her monthly car payment if she takes out a 3-year loan for that amount?

10. TaNisha is purchasing a new home. She wants to borrow $150,000 for 15 years at an annual percentage rate of 4.5%, with the interest compounded monthly. How much will she pay each month for this mortgage?

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Warm-Up 2A.5.3Mario is a land surveyor and stakes out property markers using a rubber mallet and wooden stakes. In one spot, Mario was able to pound the stake 3 inches into the ground on the first swing of his mallet. However, each of his next 5 swings only drove the stake 20% deeper into the ground.

1. Write an explicit formula of the geometric sequence for the distance Mario drove the stake into the ground on each swing.

2. Write the related series for the total distance Mario drove the stake into the ground.

3. Evaluate the series you wrote for problem 2.

4. Mario needs to drive the stake 4 inches into the ground. Do you think he’ll be able to accomplish that if he keeps hitting the stake? Explain your reasoning.

Lesson 2A.5.3: Sum of an Infinite Geometric Series

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Derive the formula for the sum of infinite series, 1

1Sa

rn = −, if 1r < . Base the derivation on the sum

formula for a finite series.

1. Write the formula for the sum of a finite series.

2. Distribute the first term in the numerator.

3. Rewrite the fraction as the difference of two fractions.

4. Substitute an infinity symbol for the exponent, n.

5. Analyze the effect of raising a value of 1r < to an infinite power.

6. Simplify the formula using the process of raising r, where 1r < , to a power of ∞ .

continued

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Example 2

Determine the sum, if a sum exists, of the following geometric series.

41

21

1

k

k

∑

=

∞ −

Example 3

Determine the sum, if a sum exists, of the following geometric series. Justify the result.

63

21

1

k

k

∑

=

∞ −

Example 4

Use the formula for an infinite geometric series to rewrite 0.6 as a fraction.

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Problem-Based Task 2A.5.3: Will It Overflow or Not?Luca bought a water dispenser that was advertised as holding 5 gallons, but he’s not sure if that’s true. Luca experiments with the container’s capacity by first pouring 200 fluid ounces into it. Then he pours 100 ounces more, 50 ounces more, and so on. At what point, if any, will the container overflow? Recall that 1 gal = 128 fl oz. Explain your reasoning.

At what point, if any, will the

container overflow?

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continued

Determine the sum of each series, if a sum exists. If no sum exists, write “No sum exists.”

1. 3

4

1

21

1

k

k

∑

=

∞ −

2. 5

6

3

51

1

k

k

∑

=

∞ −

3.

1

10

1

9

10

81+ + +

4. 0.37 + 0.0037 + 0.000037 + …

Use the formula for an infinite geometric series to rewrite each repeating decimal as a fraction.

5. 0.7

6. 0.345

Use the following information to solve problems 7 and 8.

An experiment to determine the rate of decay for a certain radioactive isotope starts out with 1,000 mg of the isotope. After 1 hour, only 995 mg remain.

7. Write an infinite geometric series to model the rate of decay of the isotope over time.

8. Evaluate the series from problem 7.

Practice 2A.5.3: Sum of an Infinite Geometric Series

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Use the information that follows to solve problems 9 and 10.

Fractals are very detailed geometric figures made up of repeated, self-similar patterns. Each small section (or stage) of the figure is the same shape as the entire figure. For example, the following illustration shows stages 0 through 4 of the Koch snowflake. The original stage, stage 0, is constructed of an equilateral triangle. Each segment of the triangle is divided into thirds and new equilateral triangles are constructed to create stage 1. This is repeated over and over again. For problems 9–10, assume that the length of each segment of the equilateral triangle in stage 0 is 3 cm long. (Drawings are not to scale.)

Stage 0 Stage 1 Stage 2 Stage 3

9. Write a geometric series for the area of the stages of the Koch snowflake. (Hint: Use the area

formula for an equilateral triangle: 3

4

2

As

= .)

10. Evaluate the series from problem 9.

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UNIT 2A • POLYNOMIAL RELATIONSHIPSStation Activities Set 1: Polynomial Functions

CCSS IP Math III Teacher ResourceU2A-318

© Walch Education

Name: Date:

Station 1Work with your group to answer the questions about each polynomial function.

1. f x x x x x( ) = + − + +3 2 6 75 3 2

a. How many roots does this function have?

b. Is 3 a zero of this function?

c. What are all possible rational zeros?

d. What is f(–1)?

2. f x x x( ) = + +3 2 14


b. What are all possible rational zeros?

c. Is 13

a zero of this function?

d. Is −13

a zero of this function?

e. Is –1 a zero of this function?

3. g x x x x( ) = − − + +2 3 5 103 2



c. Is 1 a zero of this function?

d. What is g(5)?

4. g x x x x( ) = + + −6 5 44 2 1



c. What is g(–1)?

d. What is g(1)?

149


CCSS IP Math III Teacher Resource© Walch EducationU2A-319

Name: Date:

Station 2Work with your group to answer the questions about each polynomial function. Use a calculator to compute values.

1. g x x x x( ) = − + − +2 2 7 103 2

g( )2 =

2. f x x x x x( ) = − + − −4 6 3 47 5 4 3

f (4) =

3. f x x x x x x x x( ) = − − + − + + −2 7 2 58 7 6 5 4 3

f ( )2 =

4. g x x x( ) = − +4 5 2

g( )2 =

5. f x x x x x x( ) = − + − − +3 2 2 4 56 5 4 3 2

f ( )5 =

6. g x x x x x( ) = − + − −18 10 2 6 35 3 2

g( )5 =

7. h x x x x x( ) = − + − −4 3 2 104 3 2

h( )2 =

150



© Walch Education

Name: Date:

Station 3Work with your group to answer the questions about each polynomial function. Show all your work.

1. g x x x( ) = − +4 2 47

a. How many roots does the function have?

b. What is g (1)?

c. What is g (–1)?

2. f x x x x x x( ) = + − − − +2 5 3 16 5 3 2

a. How many roots does the function have?

b. What is f (6)?

c. What is f (1)?

Factor the following polynomials. Use any method you like.

3. x x x x4 3 24 53 60 108 0− − + + =

4. x x x x x5 4 3 22 15 20 44 48 0− − + + − =

5. If x +( )2 is a factor of 3 10 14 20 16 04 3 2x x x x− − + + =, what are the other factors?

6. If 3 5 is a root of 4 7 182 315 90 04 3 2x x x x− − + + =, what are the other roots?

7. If ( )1 2− i is a root of x x x3 25 11 15 0− + =– , what are the other roots?

8. If ( )2 2+ i is a root of 2 11 26 16 16 04 3 2x x x x− + =– – , what are the other roots?

151



Name: Date:

Station 4Work with your group to answer the questions about each polynomial function. Then use a graphing calculator to find the graph of the function. Sketch the graphs.

1. f(x) = x x x3 23 4 12 0+ − − =

a. Factor the polynomial to find the zeros of the function.

b.

–20 –18 –16 –14 –12 –10 –8 –6 –4 –2 2 4 6 8 10 12 14 16 18 20

201816141210

86420

–2–4–6–8

–10–12–14–16–18–20

y

x

2. f(x) = 3 5 49 11 30 04 3 2x x x x+ − + + =

a. Factor the polynomial to find the zeros of the function.

b.

–20 –18 –16 –14 –12 –10 –8 –6 –4 –2 2 4 6 8 10 12 14 16 18 20

100

0

–100

–200

–300

–400

y

x

continued

152



© Walch Education

Name: Date:

3. f(x) = x3 – 4x2 + x – 4

a. Factor the polynomial to find the zeros of the function given that one of the zeros is –i.

b.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

109876543210

–1–2–3–4–5–6–7–8–9

–10–11–12–13–14–15–16–17–18–19–20

y

x

4. f(x) = x x x x6 5 4 25 3 3 0− + − + =

a. State all possible rational roots and find all rational roots.

b.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

y

x1 2 3 4 5 6 7 8 9 10

-800

-700

-600

-500

-400

-300

-200

-100

continued

153



Name: Date:

5. f(x) = 10 19 10 20 05 3 2x x x x+ − + − =

a. State all possible rational roots and find all rational roots.

b.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

3028262422201816141210

86420

–2–4–6–8

–10

y

x

6. f(x) = 9 9 37 37 4 4 05 4 3 2x x x x x+ + + + + =

a. State all possible rational roots and find ALL roots. (Hint: Use the quadratic formula after finding a rational root.)

b.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

y

x

154

UNIT 2A • POLYNOMIAL RELATIONSHIPSStation Activities Set 2: Sequences and Series


© Walch Education

Name: Date:

Station 1Working in pairs, practice reading summation notation and writing out the terms of a series. List the terms of each series.

1. 31

kk

n

=∑

2. 41

k

k

n

=∑

3. a dkk

n

11

+=∑

4. Ar k

k=

∞

∑1

5. 6 13

8

( )kk

−=∑

6. k k

k

2

2

5

2+=∑

155



Name: Date:

Station 2

With a partner, discuss and develop a formula to calculate the value of r k

k m

n−

=∑ 1 easily when 1 < m < n.

1. What similar formulas do you already know?

2. Write out the terms of the series given above and of similar ones that you know.

3. Compare the terms in the two sequences.

4. Based on the formula you already know, find a new one to calculate r k

k m

n−

=∑ 1 .

156



© Walch Education

Name: Date:

Station 3At this station, you will find a set of cards labeled with odd numbers from 1 to 19. Pull the notecards as directed below to answer the questions.

1. First, pull out the card that has the number 1 on it. If this is the only term in a series, what is the total sum?

2. Pull out the card that has the number 3 on it and place it next to the “1” card. What is the total sum?



5. Repeat the process until you have added all the cards. Write the new total sum each time that you add a new card.

6. What do you notice about the numbers in each sum?

7. Write out the series in summation notation.

8. Write a formula to find the value of the series of the first n odd numbers.

157



Name: Date:

Station 4At this station, you will find two sets of cards. On one set of cards, write the numbers 1 through 5. On the second set of cards, write the cube of each number from the first set of cards. Follow the instructions to form two rows of cards.

1. Place the card with the number 1 on the table. Below it, place its cube. If those are the only terms of two different series, what is each total sum? (Determine the sum for each series separately.)

2. Place the card with the number 2 on the table next to the “1” card. Below it, place its cube. What is the total sum of each series?

3. Repeat the process with the number 3 and its cube. What is the total sum of each series?



6. Compare each sum of original numbers with the corresponding sum of cubed numbers. What do you notice?

7. Write out the series of cubed numbers in summation notation.

8. Write a formula to find the value of the series of the first n cubed numbers.

158

student workbook with scaffolded practice unit 2a · 2017. 8. 9. · program workbook pages...

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