Republic of the Philippines

Department of Education Regional Office IX, Zamboanga Peninsula

Mathematics Quarter 2 - Module 2:

Problems Involving Polynomial Functions

Zest for Progress Zeal of Partnership

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Name of Learner: ___________________________

Grade & Section: ___________________________

Name of School: ___________________________

Math Module – Grade 10

Alternative Delivery Mode

Quarter 2 – Module 2: Problems Involving Polynomial Functions

First Edition, 2020

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Development Team of the Module Management Team: SDS: Ma. Liza R. Tabilon, Ed.D, CESO V ASDS: Dr. Ma. Judelyn J. Ramos ASDS: Dr. Armando P. Gumapon ASDS: Dr. Judith V. Romaguera CID CHIEF: Lilia E. Abello, Ed.D EPS-LRMS: Evelyn C. Labad EPS-MATH: Ismael K. Yusoph

PSDS: MA. THERESA M. IMPERIAL

mailto:zn.division@deped.gov.ph

What I Need to Know In this module you will apply the concepts of polynomial functions to real – life situations. You learned already that polynomial functions are

mathematical models used to represent more complicated situations in

physics, economics, meteorology, ecology, biology, and others. You are now in

your last year in junior high school. In this level and in the higher levels of

your education, you might ask the question: What are math problems and

solutions for? In what other fields are the mathematical concepts like

functions used? How are these concepts applied?

The ultimate goal of this module is for you to answer these questions:

How are polynomial functions related to other fields of study? How are these

used in solving real-life problems and in decision making?

As you go through this module, you are expected to solve real-life

problems (like area and volume, deforestation, revenue-advertising expense

situations, etc.) that can be modelled with polynomial functions.

Lesson 1

Problems Involving Polynomial Functions

Let us start this module by recalling some concepts of polynomial

functions you’ve previously learned. Your knowledge of these concepts will

help you apply your skills solving problems that can be modelled through

polynomial functions.

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What’s In

For each given polynomial function, describe or determine the

following, then sketch the graph. You may use graphing utilities to help

you sketch the graph.

a. leading term

b. leading coefficient c. degree of polynomial function

d. end behaviors e. x – intercepts

points on the x - axis

f. sketch

1. 𝒇(𝒙) = 𝒙𝟒 − 𝟓𝒙𝟐 + 𝟒

What’s New

A. Investigate deeper and decide wisely:

Apply the concepts of polynomial functions to answer the questions in

each problem. Use a calculator when needed.

1. Look at the pictures below. What do these tell us? Filipinos need to take the problem of deforestation seriously.

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The table below shows the forest cover of the Philippines in relation

to

its total land area of approximately 30 million hectares.

Year 1990 1920 1960 1970 1987 1998

Forest Cover (%)

70 60 40 34 23.7 22.2

Source: Environmental Science for Social Change, Decline of the Philippine

forest.

A cubic polynomial that best models the data is given by

𝒚 =𝟐𝟔𝒙𝟑 − 𝟑𝟓𝟎𝟎𝒙𝟐 − 𝟑𝟗𝟏𝟑𝟎𝟎𝒙 + 𝟔𝟗𝟕𝟏𝟕𝟎𝟎𝟎

𝟏𝟎𝟎𝟎𝟎𝟎𝟎; 𝟎 ≤ 𝒙 ≤ 𝟗𝟖

where y is the percent forest cover x years from 1900.

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Questions/Tasks:

a. Using the graph, what is the approximate forest cover during

the year 1940?

b. Compare the forest cover in 1987 (as given in the table) to the forest

cover given by the polynomial function. Why are these values

not exactly the same?

c. Do you think you can use the polynomial to predict the forest cover in

the year 2100? Why or why not?

Were you surprised that polynomial functions have real and

practical uses? What do you need to solve these kinds of problems?.

What Is It

The goal of this section is to check if you can apply polynomial functions

to real-life problems and produce a concrete object that satisfies the

conditions given in the problem.

Activity: Make Me Useful by Producing Something

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Read the problem carefully and answer the questions that follow.

You are designing candle-making

kits. Each kit contains 25 cubic

inches of candle wax and a mold for

making a pyramid-shaped candle

with a square base. You want the

height of the candle to be 2 inches

less than the edge of the base.

Questions/Tasks:

1. What should the dimensions of your candle mold be? Show a mathematical

procedure in determining the dimensions.

2. Use a sheet of cardboard as sample material in preparing a candle mold

with such dimensions. The bottom of the mold should be closed. The height

of one face of the pyramid should be indicated.

3. Write your solution in one of the faces of your output (mold).

Rubric for Mathematical Solution

Score Descriptor

4

The problem is correctly modeled with a polynomial

function, appropriate mathematical concepts are used in

the solution, and the correct final answer is obtained.

3

The problem is correctly modeled with a polynomial

function, appropriate mathematical concepts are

partially used in the solution, and the correct final

answer is obtained.

2

The problem is not properly modeled with a polynomial

function, other alternative mathematical concepts are

used in the solution, and the correct final answer is

obtained.

1

The problem is not properly modeled with a polynomial

function, a solution is presented but the final answer is

incorrect.

Criteria for Rating the Output:

The mold has the needed dimensions and parts.

The mold is properly labeled with the required length of parts.

The mold is durable.

The mold is neat and presentable.

Point/s to be given:

4 points if all items in the criteria are evident

3 points if any three of the items are evident

2 points if any two of the items are evident

1 point if any of the items is evident

What’s More 1. The members of a group of packaging designers of a gift shop are looking

for a precise procedure to make an open rectangular box with a volume of 560 cubic inches from a 24-inch by 18-inch rectangular piece of material.

The main problem is how to identify the side of identical squares to be cut from the four corners of the rectangular sheet so that such box can be

made.

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Question/Task:

Suppose you are chosen as the leader and you are tasked to lead in

solving the problem. What will you do to meet the specifications needed for the box? Show a mathematical solution.

Were you surprised that polynomial functions have real and

practical uses? What do you need to solve these kinds of problems? Enjoy learning as you proceed to the next section.

What I Have Learned Read and analyze the situation below. Then, answer the questions or perform

the required task.

An open box with dimensions 2 inches by 3 inches by 4 inches needs to be

increased in size to hold five times as much material as the current box.

(Assume each dimension is increased by the same amount.)

Task:

a. Write a function that represents the volume V of the new box.

b. Find the dimensions of the new box.

c. Using hard paperboard, make the two boxes - one with the original dimensions and another with the new dimensions.

d. On one face of the bigger box, write your mathematical solution in getting the new dimensions.

Additional guidelines:

1. The boxes should look presentable and are durable enough to hold any dry material such as sand, rice grains, etc.

2. Consider the rubric below.

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Rubric for Rating the Output:

Score Descriptor

3

Polynomial function is correctly presented as model,

appropriate mathematical concepts are used in the solution,

and the correct final answer is obtained.

2

Polynomial function is correctly presented as model,

appropriate mathematical concepts are partially used in the

solution, and the correct final answer is obtained.

1

Polynomial function is not correctly presented as model, other

alternative mathematical concepts are used in the solution,

and the final answer is incorrect.

Criteria for Rating the Output (Box):

Each box has the needed dimensions.

The boxes are durable and presentable.

Point/s to be Given: 3 points if the boxes have met the two criteria

2 points if the boxes have met only one criterion 1 point if the boxes have not met any of the criteria

What I Can Do

Read and analyze the situation below. Then, answer the question and

perform the tasks that follow.

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Karl Benedic, the president of Mathematics Club, proposed a project-

to put up a rectangular Math Garden whose lot perimeter is 36 meters. He

was soliciting suggestions from the members for feasible dimensions of the

lot.

Suppose you are a member of the club, what will you suggest to Karl

Benedic if you want a maximum lot area? You must convince him through a

mathematical solution.

Consider the following guidelines:

1. Make an illustration of the lot with the needed labels.

2. Solve the problem. Hint: Consider the formulas P = 2l + 2w for perimeter

and A = lw for the area of the rectangle. Use the formula for P and the

given information in the problem to express A in terms of either l or w.

3. Make a second illustration that satisfies the findings in the solution made

in number 2.

4. Submit your solution on a sheet of paper with recommendations.

Rubric for Rating the Output

Score Descriptors

4

The problem is correctly modeled with a quadratic

function, appropriate mathematical concepts are fully

used in the solution, and the correct final answer is

obtained.

3

The problem is correctly modeled with a quadratic

function, appropriate mathematical concepts are partially

used in the solution, and the correct final answer is

obtained.

2

The problem is not properly modeled with a quadratic

function, other alternative mathematical concepts are used

in the solution, and the correct final answer is obtained.

1

The problem is not totally modeled with a quadratic

function, a solution is presented but has incorrect final

answer.

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Assessment Directions: Answer the following problems. Show your solutions on a separate

sheet of paper.

1. Consider this Revenue-Advertising Expense situation.

A drugstore that sells a certain brand of vitamin capsule estimates that

the profit P (in pesos) is given by 𝑃 = −50𝑥3 + 2400𝑥2 − 2000, 0 ≤ 𝑥 ≤ 32

where x is the amount spent on advertising (in thousands of pesos). An

advertising agency provides four (4) different advertising packages with

costs listed below. Which of these packages will yield the highest revenue

for the company?

A. Package A: Php 8,000.00

B. Package B: Php 16,000.00

C. Package C: Php 32,000.00

D. Package D: Php 48,000.00

2. A car manufacturer determines that its profit, P, in thousands of pesos,

can be modeled by the function 𝑃(𝑥) = 0.00124𝑥4 + 𝑥 − 3, where x

represents the number of cars sold. What is the profit at 𝑥 = 150

A. Php 75.28 C. Php 3,000,000.00

B. Php 632,959.50 D. Php 10,125,297.00

3. A demographer predicts that the population, P, of a town t years from now

can be modeled by the function P(t) = 6t4 – 5t3 + 200t + 12 000. What will

the population of the town be two (2) years from now? A. 12 456 C. 1 245 600

B. 124 560 D. 12 456 000

4. A gymnast dismounts the uneven parallel bars. Her height, h, depends

on the time, t, that she is in the air as follows:

h = -16t2 + 8t + 8

a) How long will it take the gymnast to reach the ground?

b) When will the gymnast be 8 feet above the ground?

5. The dimensions of a pool: height = x – 2 meters, length = 2x + 5 meters,

and width = 2x – 1 meters. If the volume of the pool is 182 cubic meters,

what is the value of x?

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References

1. Mathematics Learner’s Module

2. Teachers Guide in Mathematics

3. Jose-Dilao, S., Orines, F. B., & Bernabe, J. G. (2003). Advanced

Algebra, Trigonometry and Statistics. Quezon City, Philippines: JTW Corporation

4. Marasigan, J. A., Coronel, A. C., & Coronel, I. C. (2004). Advanced

Algebra with Trigonometry and Statistics. Makati City, Philippines: The Bookmark, Inc.

https://www.onlinemathlearning.com/polynomial-equation-

word-problem.html

https://www.youtube.com/watch?v=QYD388w9BlI

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https://www.onlinemathlearning.com/polynomial-equation-word-problem.htmlhttps://www.onlinemathlearning.com/polynomial-equation-word-problem.htmlhttps://www.youtube.com/watch?v=QYD388w9BlI

What’s In:

1.

a. 𝑥4 b. 1

c. degree 4

d. rising to the left and right

e. −2, 2, −1, 𝑎𝑛𝑑 1 (−2, 0), (2, 0), (−1, 0), 𝑎𝑛𝑑 (1, 0)

2.

a. 𝑥3 b. 1 c. degree 3

d. falling to the left, rising to the right

e. 3, 2, 1 (2, 0), (1, 0), 𝑎𝑛𝑑 (3, 0)

3.

a. −2𝑥4 b. −2 c. 𝑑𝑒𝑔𝑟𝑒𝑒 4 d. falling to the left, falling to the right

e. 4, 2, 1, 𝑎𝑛𝑑 −1

2

(4, 0), (2, 0), (1, 0), 𝑎𝑛𝑑 (−1

2 0)

Answer Key

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What’s New

Activity: Investigate Deeper and Decide Wisely

a. 50% b. The value given by the table is 23.7%. The polynomial gives a

value of 26.3%. The given polynomial is the cubic polynomial

that best fits the data. We expect it to give a good

approximation of the forest cover but it may not necessarily

produce the exact values. c. The domain of the function is [0,98]. Since year 2100

corresponds to x = 200, we cannot use the function to predict

forest cover during this year. Moreover, if x = 200, the

polynomial predicts a forest cover of 59.46%. This is very

unrealistic unless major actions are done to reverse the trend.

You can find other data that can be modelled by a polynomial.

Use the regression tool in MS Excel or GeoGebra to determine

the best fit polynomial for the data.

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What Is It

Activity: Make Me Useful by Producing Something

Answers to the Questions

Solution:

Let x be the side of the square base of the pyramid. So,

Area of the base (B): 𝐵 = 𝑥2 Height of the pyramid (h): ℎ = 𝑥 − 2

Working Equation: 𝑉 =1

3𝐵ℎ

𝑉(𝑥) =1

3𝑥2(𝑥 − 2)

25 =1

3𝑥2(𝑥 − 2)

75 = 𝑥3 − 2𝑥2 → 𝑥3 − 2𝑥2 − 75 = 0 (𝑥 − 5)(𝑥2 + 3𝑥 + 15)

The only real solution to the equation is 5. So, the side of the square

base is 5 inches long and the height of the pyramid is 3 inches.

What’s More

s