1.
Mathematics
Quarter 2 - Module 1
Illustrating Polynomial Functions
Department of Education ● Republic of the Philippines
10

2.
Mathematics- Grade 10
Alternative Delivery Mode
Quarter 2 - Module 1: Illustrating Polynomial Functions
First Edition, 2020
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Published by the Department of Education
Secretary: Leonor Magtolis Briones
Development Team of the Module
Author/s: Cristine Mae C. Baguio
Reviewers: Rhodel A. Lamban, PhD
Elbert R. Francisco, PhD
Alicia P. Micayabas, PhD
Illustrator and Layout Artist: Cristine Mae C. Baguio
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3.
Mathematics
Quarter 2 - Module 1
ILLUSTRATING POLYNOMIAL FUNCTIONS
This instructional material was collaboratively developed and reviewed
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Department of Education at bukidnon@deped.gov.ph.
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Department of Education-Division of Bukidnon ● Republic of the Philippines
10
i

4.
Table of Contents
PAGE
COVER PAGE
COPYRIGHT PAGE
TITLE PAGE i
TABLE OF CONTENTS ii
WHAT THIS MODULE IS ABOUT iv
Note to the Teacher/Facilitator
Note to the Parents/Guardian
Note to the Learner
Module Icons
WHAT I NEED TO KNOW 1
WHAT I KNOW (Pretest) 2
LESSON 1: Definition of Polynomial Function
What I Need to Know 4
What I Know 5
What’s In 7
What’s New 7
What is it 8
What’s More 9
Guided/Controlled Practice
Independent Practice
What I Have Learned 10
What I Can Do 11
Assessment 12
Guided Assessment
ii

5.
Independent Assessment
Additional Activities 14
LESSON 2: Writing Polynomial Functions in Standard Form
What I Need to Know 15
What I Know 16
What’s In 18
What’s New 21
What is it 22
What’s More 26
Guided/Controlled Practice
Independent Practice
What I Have Learned 28
What I Can Do 28
Assessment 29
Guided Assessment
Independent Assessment
Additional Activities 31
SUMMARY 32
ASSESSMENT (Post-Test) 33
KEY TO ANSWERS 35
REFERENCES 39
iii

6.
What This Module is About
For the Facilitator:
Welcome to the Mathematics Grade 10 Alternative Delivery Mode Module
entitled “Illustrating Polynomial Functions”.
This module was collaboratively designed, developed and reviewed by
educators both from public and private institutions to assist you, the teacher or
facilitator in helping the learners meet the standards set by the K to 12 Curriculum
while overcoming their personal, social, and economic constraints in schooling.
This learning resource hopes to engage the learners into guided and
independent learning activities at their own pace and time. Furthermore, this also aims
to help learners acquire the needed 21st century skills while taking into consideration
their needs and circumstances.
As a facilitator you are expected to orient the learners on how to use this
module. You also need to keep track of the learners' progress while allowing them to
manage their own learning. Furthermore, you are expected to encourage and assist
the learners as they do the tasks included in the module.
You may prepare your own related activities if you feel that the activities
suggested here are not appropriate to the level and contexts of students (examples,
slow/fast learners, and localized situations/examples).
Notes to the Parents/Guardians:
Welcome to the Mathematics Grade 10 Alternative Delivery Mode Module
entitled “Illustrating Polynomial Functions”.
This Module was designed and developed to cater the academic needs of the
learners in this trying time. Teaching and learning process do not only happen inside
the four corners of the classroom but also in your respective homes. We hope that
you will cooperate, provide encouragement and show full support to your children in
answering all the activities found in this module.
iv

7.
Notes to the Learners:
Welcome to the Mathematics Grade 10 Alternative Delivery Mode Module
entitled “Illustrating Polynomial Functions”.
This module was intended to provide you with fun and meaningful opportunities
for guided and independent learning at your own pace and time.
This module was designed and written with you in mind. The scope of this
module permits it to be used in many different learning situations. The language used
recognizes the diverse vocabulary level of students. The lessons are arranged to
follow the standard sequence of the course. But the order in which you read them can
be changed to correspond with the textbook you are now using.
This module has the following parts and corresponding icons:
What I Need to Know This will give you an idea of the skills or
competencies you are expected to learn in the
module.
What I Know This part includes an activity that aims to check
what you already know about the lesson to take.
If you get all the answers correct (100%), you may
decide to skip this module.
What’s In This is a brief drill or review to help you link the
current lesson with the previous one.
What’s New In this portion, the new lesson will be introduced to
you in various ways such as a story, a song, a
poem, a problem opener, an activity or a situation.
What is It This section provides a brief discussion of the
lesson. This aims to help you discover and
understand new concepts and skills.
What’s More This comprises activities for independent practice
to solidify your understanding and skills of the
topic. You 156
What I Have Learned This includes questions or blank
sentence/paragraph to be filled into process what
you learned from the lesson.
v

8.
What I Can Do This section provides an activity which will help
you transfer your new knowledge or skill into real
life situations or concerns.
Assessment This is a task which aims to evaluate your level of
mastery in achieving the learning competency.
Additional Activities In this portion, another activity will be given to you
to enrich your knowledge or skill of the lesson
learned. This also tends retention of learned
concepts.
Answer Key This contains answers to all activities in the
module.
At the end of this module you will also find:
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any part of the
module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer What I Know before moving on to the other activities
included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not
hesitate to consult your teacher or facilitator. Always bear in mind that you are not
alone.
We hope that through this material, you will experience meaningful learning and
gain deep understanding of the relevant competencies. Remember, your academic
success lies in your own hands! You can do it!
References This is a list of all sources used in developing this
module.
vi

9.
What I Need to Know
In this module, you need to recall what you have learned about
polynomials like the degree, coefficients, constant terms, factoring, etc.
The module is divided into two lessons, namely:
• Lesson 1: Definition of Polynomial Function
• Lesson 2: Writing Polynomial Functions in Standard Form and in Factored
Form
After you go through this module, you are expected to:
1. illustrates polynomial functions (M10AL-IIa-1);
2. write polynomial function in standard form and in factored form.
1

10.
What I Know
Directions: Choose the letter that best answers each question.
1. Which of the following is the value of 𝑛 in 𝑓(𝑥) = 𝑥𝑛
if f is a polynomial
function?
A. √2 B. 2 C. −2 D.
1
2
2. Which of the following is NOT a polynomial function?
A. 𝑓(𝑥) = 0
B. 𝑓(𝑥) = 1
C. 𝑓(𝑥) = 𝑥2
+ 𝑥 + 1
D. 𝑓(𝑥) = −
1
2𝑥
3. Which of the following is a polynomial function?
i. 𝑓(𝑥) = 𝑥−3
+ 2𝑥 + 1 ii. 𝑓(𝑥) = 𝑥2
+ 𝑥 + 1 iii. 𝑓(𝑥) = √2 𝑥2 + √𝑥
A. i only B. ii only C. i and ii D. i and iii
4. What is the leading coefficient of 𝑓(𝑥) = 𝑥2
+ 4𝑥3
+ 1?
A. 1 B. 2 C. 3 D. 4
5. What is the constant term of the polynomial function in number 4?
A. 1 B. 2 C. 3 D. 4
6. What is the standard form of 𝑓(𝑥) = (5𝑥 − 3)(25𝑥2
+ 15𝑥 + 9)?
A. −125𝑥3
− 27
B. 125𝑥3
− 27
C. −125𝑥3
+ 27
D. 125𝑥3
+ 27
7. What is the leading term of number 6?
A. −27 B. 27 C. 125𝑥3
D. −125𝑥3
2

11.
8. What is the constant term of the polynomial in number 6?
A. −3 B. −9 C. 27 D. −27
9. Given that 𝑓(𝑥) = 2𝑥−2𝑛
+ 8𝑥2
, what value should be assigned to 𝑛 to make 𝑓
a function of degree 3?
A. −
2
3
B. −
3
2
C.
2
3
D.
3
2
10.How should the polynomial function 𝑓(𝑥) = 𝑥4
− 8𝑥2
+
𝑥
2
+ 4𝑥3
+
1
2
be written
in standard form?
A. 𝑓(𝑥) = −8𝑥2
+ 4𝑥3
+
1
2
+ 𝑥4
+
𝑥
2
B. 𝑓(𝑥) =
𝑥
2
+
1
2
− 8𝑥2
+ 4𝑥3
+ 𝑥4
C. 𝑓(𝑥) = 𝑥4
+ 4𝑥3
− 8𝑥2
+
𝑥
2
+
1
2
D. 𝑓(𝑥) =
1
2
+ 4𝑥3 − 8𝑥2 +
𝑥
2
+ 𝑥4
11. What is the leading coefficient of number 10?
A. −8 B. 1 C.
1
2
D. −4
12.What is the constant term of the polynomial in number 10?
A. −8 B. 1 C.
1
2
D. -4
13. How should 𝑓(𝑥) = 𝑥4
+ 𝑥3
+ 𝑥2
+ 𝑥 be written in factored form?
A. 𝑓(𝑥) = 𝑥(𝑥 + 1)(𝑥2
+ 1)
B. 𝑓(𝑥) = 𝑥(1)(𝑥2
+ 1)
C. 𝑓(𝑥) = 𝑥(𝑥 − 1)(𝑥2
+ 1)
D. 𝑓(𝑥) = 𝑥(−1)(𝑥2
+ 1)
14. What is the factored form of 𝑓(𝑥) = 𝑥3
+ 3𝑥2
− 4𝑥 − 12?
A. 𝑓(𝑥) = (𝑥 + 2)(𝑥 − 2)(𝑥 + 3)
B. 𝑓(𝑥) = (𝑥 + 2)(𝑥 + 2)(𝑥 + 3)
C. 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 2)(𝑥 + 3)
D. 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 2)(𝑥 − 3)
15.What is the factored form of 𝑦 = 9𝑥3
− 3𝑥2
+ 81𝑥 − 27?
A. 𝑦 = −3(𝑥2
+ 9)(3𝑥 − 1)
B. 𝑦 = 3(𝑥2
+ 9)(3𝑥 − 1)
C. 𝑦 = 3(𝑥2
− 9)(3𝑥 − 1)
D. 𝑦 = 3(𝑥2
+ 9)(3𝑥 + 1)
3

12.
Lesson
1
DEFINITION OF
POLYNOMIAL FUNCTIONS
What I Need to Know
This lesson is good for one (1) day. You may skip this if you can get a perfect
score in What I Know.
At the end of the lesson, you should be able to:
1. illustrates polynomial function;
2. identify polynomial function; and
3. determine the degree, the leading term and coefficient and the constant term.
4

13.
What I Know
Directions: Choose the letter that best answers each question.
1. Which of the following is a monomial or a sum of monomials?
A. constant term
B. degree
C. leading term
D. polynomial
2. What function is 𝑦 = 𝑥3
+ 2𝑥 + 1?
A. Linear Function
B. Polynomial Function
C. Quadratic Function
D. Rational Function
3. What is the value of 𝑛 in 𝑓(𝑥) = 𝑥𝑛
if f is a polynomial function?
A. √3 B. 3 C. −3 D.
1
3
4. Which of the following is NOT a polynomial function?
A. 𝑃(𝑥) = 𝑎𝑥 + 𝑏
B. 𝑃(𝑥) =
𝑝(𝑥)
𝑞(𝑥)
C. 𝑃(𝑥) = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐
D. 𝑃(𝑥) = 𝑎𝑥4
+ 𝑏𝑥3
+ 𝑐𝑥4
+ 𝑑𝑥 + 𝑒
For numbers 5-8, use the given function 𝑓(𝑥) = 3𝑥3
+ 6𝑥2
+
𝑥
2
+ 2𝑥4
and choose
your answers below:
A. 0 B. 2 C. 4 D. 6
5. Which of the choices is the leading coefficient of the function?
6. What is the constant term of the function?
7. What is the degree of the function?
8. Which is not a coefficient of the function?
5

14.
9. What type of polynomial function is 𝑓(𝑥) = 3𝑥3
+ 6𝑥2
+
𝑥
2
+ 2𝑥4
?
A. Cubic Polynomial Function
B. Quadratic Polynomial Function
C. Quartic Polynomial Function
D. Zero Polynomial Function
10. What type of polynomial function is 𝑃(𝑥) = (𝑥 + 2)(𝑥 − 2)?
A. Cubic Polynomial Function
B. Quadratic Polynomial Function
C. Quartic Polynomial Function
D. Zero Polynomial Function
For numbers 11-14, use the polynomial function in number 10.
11.What is the leading term of the function?
A. 𝑥2 B. 2𝑥2
C. 3𝑥2
D. 4𝑥2
12.What is the constant term of the function?
A. – 4 B. – 2 C. 0 D. 2
13.What is the degree of the function?
A. 0 B. 1 C. 2 D. 3
14.Which is the leading coefficient of the function?
A. – 4 B. – 2 C. 0 D. 1
15.Given that 𝑓(𝑥) = 𝑥−3𝑛
+ 2𝑥2
, what value should be assigned to 𝑛 to make 𝑓 a
polynomial function of degree 4?
A. −
4
3
B. −
3
4
C.
2
3
D.
3
2
6

15.
What’s In
You have learned in the last module that to solve problems involving
polynomials, you must follow steps to have an easy solution.
Start this module by recalling your knowledge on the concept of polynomial
expressions.
• The word polynomial is derived from Greek words “poly” which means many and
“nominal” which means terms, so polynomial means many terms.
• Polynomials are composed of constants (numbers), variables (letters) and
exponents such as 2 in x2
. The combination of numbers, variables and
exponents is called terms.
• Example:
2𝑥3
+ 𝑥2
+ 1 There are three (3) terms in this expression: 2𝑥3
, 𝑥2
& 1, where 1
is the constant, x is the variable and 3 and 2 are the exponents.
This knowledge will help you understand the formal definition of polynomial
function.
What’s New
Let’s explore!
Directions: Complete the table below. State your reason if it is not a polynomial.
Expression Polynomial or Not Reason/s
1. 10𝑥
2. 𝑥3
− 2√5𝑥 + 𝑥
3. −2020𝑥
4. 𝑥
2
3 + 3𝑥 + 1
5.
1
𝑥2 +
2
𝑥3 +
3
𝑥4
6. 𝜋
7. 3𝑥√2
+ √3𝑥2
8. 𝑥3
+ 2𝑥 + 1
9. −2𝑥−3
+ 𝑥3
10.1 − 4𝑥2
Did you complete the table correctly? Do you remember when an expression
is a polynomial? A polynomial is an expression of one or more algebraic terms each
of which consists of a constant multiplied by one or more variable raised to a
nonnegative integral power.
7

16.
What Is It
A polynomial function is a function of the form
𝑃(𝑥) = 𝑎𝑛𝑥𝑛
+ 𝑎𝑛−1𝑥𝑛−1
+ 𝑎𝑛−2𝑥𝑛−2
+ ⋯ + +𝑎1𝑥 + 𝑎0, 𝑎𝑛 ≠ 0,
where 𝑛 is a nonnegative integer , 𝑎0, 𝑎1, … , 𝑎𝑛 are real numbers called coefficients
(numbers that appear in each term) , 𝑎𝑛𝑥𝑛
is the leading term, 𝑎𝑛 is the leading
coefficient, and 𝑎0 is the constant term (number without a variable). The highest
power of the variable of 𝑃(𝑥) is known as its degree.
There are various types of polynomial functions based on the degree of the
polynomial. The most common types are:
• Zero Polynomial Function (degree 0): 𝑃(𝑥) = 𝑎𝑥0
= 𝑎
• Linear Polynomial Function (degree 1): 𝑃(𝑥) = 𝑎𝑥1
+ 𝑏 = 𝑎𝑥 + 𝑏
• Quadratic Polynomial Function (degree 2): 𝑃(𝑥) = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐
• Cubic Polynomial Function (degree 3): 𝑃(𝑥) = 𝑎𝑥3
+ 𝑏𝑥2
+ 𝑐𝑥 + 𝑑
• Quartic Polynomial Function (degree 4): 𝑃(𝑥) = 𝑎𝑥4
+ 𝑏𝑥3
+ 𝑐𝑥2
+ 𝑑𝑥 + 𝑒
where 𝑎, 𝑏, 𝑐, 𝑑 and 𝑒 are constants.
Other than 𝑃(𝑥) , a polynomial function can be written in different ways,
like the following:
𝑓(𝑥) = 𝑎𝑛𝑥𝑛
+ 𝑎𝑛−1𝑥𝑛−1
+ 𝑎𝑛−2𝑥𝑛−2
+ ⋯ + 𝑎1𝑥 + 𝑎0,
𝑦 = 𝑎𝑛𝑥𝑛
+ 𝑎𝑛−1𝑥𝑛−1
+ 𝑎𝑛−2𝑥𝑛−2
+ ⋯ + 𝑎1𝑥 + 𝑎0,
Example:
Degree of
the
Polynomial
Type of
Function
Leading
Term
Leading
Coefficient
Constant
Term
1. 𝑦 = 8𝑥4
− 4𝑥3
+
2𝑥 + 22
4 Quartic 8𝑥4
8 22
2. 𝑦 = 3𝑥2
+ 6𝑥3
+ 2𝑥 3 Cubic 6𝑥3
6 0
8

17.
What’s More
Let’s do this…
A. Directions: Complete the table below. If the given is a polynomial function, give
the degree, leading coefficient and its constant term. If it is not, then just give the
reason.
Polynomial
Function
or Not
Reason Degree
Leading
Coefficient
Constant
Term
1. 𝑓(𝑥) = 0
2. 𝑓(𝑥) = 𝑥2
− √2𝑥 + 𝑥
3. 𝑓(𝑥) = −𝑥
4. 𝑓(𝑥) = 𝑥
3
4 + 2𝑥 + 2
5. 𝑓(𝑥) =
3
√3𝑥
6. 𝑦 = √5𝑥
7. 𝑦 = 3𝑥 + 𝑥2
8. 𝑦 = −𝑥−1
9. 𝑦 = 1 + 2𝑥 + 𝑥3
10.𝑦 = 1 − 4𝑥2
11.𝑃(𝑥) = 2020
12.𝑃(𝑥) = −√𝑥 + 𝑥
13.𝑃(𝑥) =
3𝑥
√4
14.𝑃(𝑥) = 𝑥 + 2
15.𝑃(𝑥) =
3
𝑥−1
9

18.
B. Directions: Identify whether the following is a polynomial function or not. If the
given is a polynomial function, give the degree of polynomial, the type of
polynomial function, the leading term and its constant term.
1. 𝑦 = 𝑥
2. 𝑦 = 3𝑥 + 4𝑥2
3. 𝑦 = −𝑥−10
4. 𝑦 = 12 + 6𝑥 + 𝑥2
5. 𝑦 = 10 − 5𝑥2
6. 𝑃(𝑥) =
1
2
7. 𝑃(𝑥) = −√𝑥 + 3𝑥2
8. 𝑃(𝑥) =
1
2
𝑥2
− 3
What I Have Learned
A. Directions: Fill in the blank with the choices provided in the box.
A __________(1)__________ is a function which involves only
________(2)____________ integer powers or only positive integer exponents. The
_________(3)_______ of any polynomial is the highest power present in it. In the
____(4)_____ polynomial function 𝑦 = 4 + 2𝑥 + 𝑥3
, __(5)_____ is the leading term, 4
is the ___(6)_____, 1 is the ___(7)______, and ___(8)____ is the degree.
polynomial function cubic nonnegative
constant term leading coefficient degree
3 1 𝑥3
9. 𝑃(𝑥) = 𝑥5
− 𝑥4
− 𝑥 + 2
10.𝑃(𝑥) =
1
4𝑥
+ 3
11.𝑓(𝑥) =
1
2
√𝑥
12.𝑓(𝑥) =
5
8
𝑥
13.𝑓(𝑥) =
2𝑥+2
3
14.𝑓(𝑥) = 𝑥2
15.𝑓(𝑥) =
𝑥−3
𝑥+2
10

19.
B. Directions: Complete the table below. If the given is a polynomial function, give
the degree, leading coefficient and its constant term. If it is not, then just give
the reason.
Polynomial
Function
or Not
Reason Degree Leading
Term,
Coefficient
Constant
Term
9. 𝑦 = 20
10.𝑦 = √𝑥 + 18
11.𝑓(𝑥) = −1991𝑥
12.𝑓(𝑥) = 𝑥
1
2 + 𝑥 − 1
13.𝑓(𝑥) =
5
√5𝑥
14.𝑦 = √4𝑥
15.𝑦 = 20 − 𝑥 + 𝑥2
What I Can Do
Directions: Give five polynomial functions of different degree of polynomial.
Identify the degree of polynomial, the type of polynomial, the leading coefficient and
its constant term.
Polynomial Functions
Degree of
Polynomial
Type of
Polynomial
Leading
Coefficient
Constant
Term
1.
2.
3.
4.
5.
11

20.
Assessment
Directions: Choose the letter that best answers each question.
1. Which of the following is the term with number without variable?
A. constant term
B. degree
C. leading term
D. polynomial
2. What function is 𝑦 = 𝑥4
+ 1?
A. Linear Function
B. Quadratic Function
C. Quartic Function
D. Rational Function
3. What is the value of 𝑛 in 𝑓(𝑥) = 𝑥𝑛
if f is a polynomial function?
A. √3 B. 3 C. −3 D.
1
3
4. Which of the following is NOT a polynomial function?
A. 𝑃(𝑥) = 𝑎𝑥 + 𝑏
B. 𝑃(𝑥) =
𝑝(𝑥)
𝑞(𝑥)
C. 𝑃(𝑥) = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐
D. 𝑃(𝑥) = 𝑎𝑥4
+ 𝑏𝑥3
+ 𝑐𝑥4
+ 𝑑𝑥 + 𝑒
5. Given that 𝑓(𝑥) = 𝑥−3𝑛
+ 2𝑥2
, what value should be assigned to 𝑛 to make 𝑓 a
polynomial function of degree 4?
A. −
4
3
B. −
3
4
C.
4
3
D.
3
4
12

21.
For numbers 6-9, use the given function 𝑓(𝑥) = 5𝑥3 + 𝑥2 + 3𝑥 + 15 and choose your
answers below:
A. 1 B. 3 C. 5 D. 15
6. Which of the choices is the leading coefficient of the function?
7. What is the constant term of the function?
8. What is the degree of the function?
9. Which is not a coefficient of the function?
10.What type of polynomial function is 𝑓(𝑥) = 5𝑥3
+ 𝑥2
+ 15 ?
A. Cubic Polynomial Function
B. Quadratic Polynomial Function
C. Quartic Polynomial Function
D. Zero Polynomial Function
11.What type of polynomial function is 𝑓(𝑥) = (𝑥 + 2)(2𝑥 − 8)?
A. Cubic Polynomial Function
B. Quadratic Polynomial Function
C. Quartic Polynomial Function
D. Zero Polynomial Function
For numbers 12-15, use the polynomial function in number 11.
12.What is the leading term of the function?
A. 𝑥2 B. 2𝑥2
C. 4𝑥2
D. 8𝑥2
13.What is the constant term of the function?
A. – 24 B. – 16 C. – 8 D. – 4
14.What is the degree of the function?
A. 0 B. 1 C. 2 D. 3
15.Which is the leading coefficient of the function?
A. 2 B. 4 C. 6 D. 8
13

22.
Additional Activity
Directions: Give two examples for each type of polynomials. Identify the degree
of polynomial, the leading term and the constant term.
14

23.
Lesson Writing Polynomial Functions
2
In Standard Form and in
Factored Form
What I Need to Know
This lesson is good for one (1) day. You may skip this if you can get a perfect
score in What I Know.
At the end of the lesson, you should be able to:
1. write polynomial functions in standard form; and
2. write polynomial functions in factored form
15

24.
What I Know
Directions: Choose the letter that best answers each question.
1. What is the product of (𝑥 + 2)(𝑥 + 5)?
A. 𝑥2
+ 3𝑥 + 10
B. 𝑥2
− 3𝑥 + 10
C. 𝑥2
+ 7𝑥 + 10
D. 𝑥2
+ 3𝑥 − 3
2. What is the product of (𝑥 + 2)(𝑥2
− 2𝑥 + 4)?
A. 𝑥3
− 8 B. 𝑥3
+ 8 C. 𝑥3
− 4 D. 𝑥3
+ 4
3. What term has the highest exponent in 𝑓(𝑥) = −2𝑥4
+ 𝑥6
+ 3𝑥 + 1?
A. −2𝑥4
B. 𝑥6
C. 3𝑥 D. 1
4. What is the constant term in number 3?
A. −2𝑥4
B. 𝑥6
C. 3𝑥 D. 1
5. What is the standard form of the polynomial function in number 3?
A. 𝑓(𝑥) = 𝑥6
− 2𝑥4
+ 3𝑥 + 1
B. 𝑓(𝑥) = 1 + 𝑥6
− 2𝑥4
+ +3𝑥
C. 𝑓(𝑥) = 𝑥6
− 2𝑥4
+ 1 + 3𝑥
D. 𝑓(𝑥) = −2𝑥4
+ 3𝑥 + 𝑥6
+ 1
6. What should be the order of terms of the polynomial function in standard form?
A. constant term, term with highest exponent, term/s with lower exponent
B. constant term, term/s with lower exponent, term with highest exponent
C. term with highest exponent, constant term, term/s with lower exponent
D. term with highest exponent, term/s with lower exponent, constant term
7. What is the standard form of 𝑦 = 8𝑥2
+ 4𝑥 + 3𝑥6
+ 3?
A. 𝑦 = 3𝑥6
+ 3 + 8𝑥2
+ 4𝑥
B. 𝑦 = 3𝑥6
+ 8𝑥2
+ 3 + 4𝑥
C. 𝑦 = 3𝑥6
+ 8𝑥2
+ 4𝑥 + 3
D. 𝑦 = 3𝑥6
+ 3 + 4𝑥 + 8𝑥2
16

25.
8. What is the standard form of 𝑦 = 20𝑥 + 14𝑥2
+ 2𝑥3
?
A. 𝑦 = 2𝑥3
+ 20𝑥 + 14𝑥2
B. 𝑦 = 14𝑥2
+ 20𝑥 + 2𝑥3
C. 𝑦 = 2𝑥3
+ 14𝑥2
+ 20𝑥
D. 𝑦 = 14𝑥2
+ 2𝑥3
+ 20𝑥
9. What is the factored form of the polynomial function in number 8?
A. 𝑦 = 2𝑥(𝑥 + 5)(𝑥 − 2)
B. 𝑦 = 2𝑥(𝑥 + 5)(𝑥 + 2)
C. 𝑦 = 5𝑥(𝑥 + 2)(𝑥 + 2)
D. 𝑦 = 2𝑥(𝑥 + 5)(𝑥 + 5)
10.How should the polynomial function 𝑓(𝑥) = 𝑥3
+ 8 be written in factored form?
A. 𝑓(𝑥) = (𝑥 + 2)(𝑥2 + 2𝑥 + 4)
B. 𝑓(𝑥) = (𝑥 − 2)(𝑥2 + 2𝑥 + 4)
C. 𝑓(𝑥) = (𝑥 + 2)(𝑥2 − 2𝑥 + 4)
D. 𝑓(𝑥) = (𝑥 − 2)(𝑥2 − 2𝑥 + 4)
11. How should 𝑓(𝑥) = 𝑥3
− 64 be written in factored form?
A. 𝑓(𝑥) = (𝑥 − 4)(𝑥2
+ 4𝑥 + 16)
B. 𝑓(𝑥) = (𝑥 + 4)(𝑥2
+ 4𝑥 + 16)
C. 𝑓(𝑥) = (𝑥 − 4)(𝑥2
− 4𝑥 + 16)
D. 𝑓(𝑥) = (𝑥 + 4)(𝑥2
− 4𝑥 + 16)
12.What is the factored form of 𝑦 = 1 − 4𝑥2
?
A. 𝑦 = (1 + 2𝑥)(1 + 2𝑥)
B. 𝑦 = (1 − 2𝑥)(1 + 2𝑥)
C. 𝑦 = (2𝑥 + 1)(1 + 2𝑥)
D. 𝑦 = (2𝑥 + 1)(2𝑥 − 1)
13.How should 𝑦 = −10 + 3𝑥 + 𝑥2
be written in standard form?
A. 𝑦 = 𝑥2
+ 3𝑥 − 10
B. 𝑦 = 𝑥2
−10 + 3𝑥
C. 𝑦 = −10 + 3𝑥 + 𝑥2
D. 𝑦 = 3𝑥 − 10 + 𝑥2
17

26.
14.How should 𝑦 = −10 + 3𝑥 + 𝑥2
be written in factored form?
A. 𝑦 = (𝑥 + 5)(𝑥 + 2)
B. 𝑦 = (𝑥 + 5)(𝑥 − 2)
C. 𝑦 = (𝑥 − 5)(𝑥 + 2)
D. 𝑦 = (𝑥 − 5)(𝑥 − 2)
15.What is the standard form of 𝑦 = (3𝑥 + 1)(2𝑥 − 7)?
A. 𝑦 = 6𝑥2
+ 19𝑥 − 7
B. 𝑦 = 6𝑥2
− 19𝑥 − 7
C. 𝑦 = 6𝑥2
− 23𝑥 − 7
D. 𝑦 = 6𝑥2
+ 19𝑥 − 7
18

27.
What’s In
A polynomial function is a function of the form
𝑃(𝑥) = 𝑎𝑛𝑥𝑛
+ 𝑎𝑛−1𝑥𝑛−1
+ 𝑎𝑛−2𝑥𝑛−2
+ ⋯ + +𝑎1𝑥 + 𝑎0, 𝑎𝑛 ≠ 0.
The terms of a polynomial may be written in any order. However, if they are
written in decreasing powers of x, then the polynomial function is in standard form.
Before you proceed, try to recall the following.
Types of Special Products
1. Square of Binomial
This special product results into Perfect Square Trinomial (PST).
(𝑎 + 𝑏)2
= 𝑎2
+ 2𝑎𝑏 + 𝑏2
(𝑎 − 𝑏)2
= 𝑎2
− 2𝑎𝑏 + 𝑏2
Example: (2𝑥 − 3)2
= 4𝑥2
− 12𝑥 + 9
2. Product of Sum and Difference of Two Terms
This results to Difference of Two Squares.
(𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2
− 𝑏2
Example: (𝑥 + 2)(𝑥 − 2) = 𝑥2
− 4
3. Square of Trinomial
This would result to six (6) terms.
(𝑎 + 𝑏 + 𝑐)2
= 𝑎2
+ 𝑏2
+ 𝑐2
+ 2𝑎𝑏 + 2𝑎𝑐 + 2𝑏𝑐
Example: (2𝑥 + 3𝑦 + 4𝑧)2
= 4𝑥2
+ 9𝑦2
+ 16𝑧2
+ 12𝑎𝑏 + 16𝑎𝑐 + 24𝑏𝑐
4. Product of Binomials
The result is a General Trinomial. F.O.I.L (First, Outer, Inner, Last) method is
usually used.
(𝑎 + 𝑏)(𝑐 + 𝑑) = 𝑎𝑐 + (𝑏𝑐 + 𝑎𝑑) + 𝑏𝑑
Example: (𝑥 + 2)(𝑥 + 3) = 𝑥2
+ (2𝑥 + 3𝑥) + 6
= 𝑥2
+ 5𝑥 + 6
19

28.
5. Product of Binomial and Trinomial
The result is a Sum or Difference of Two Cubes.
(𝑎 + 𝑏)(𝑎2
− 𝑎𝑏 + 𝑏2
) = 𝑎3
+ 𝑏3 (𝑎 − 𝑏)(𝑎2
+ 𝑎𝑏 + 𝑏2) = 𝑎3
− 𝑏3
Example: (𝑥 + 2)(𝑥2
− 2𝑥 + 4) = 𝑥3
+ 8
Methods of Factoring
Method When is it Possible Example
1. Factoring out
the Greatest
Common
Factor
(GCF)
If each term in the
polynomial has a
common factor.
2𝑥2
+ 8𝑥
The common factor of both terms is
2x.
2𝑥2
+ 8𝑥 = 𝟐𝒙(𝒙 + 𝟒)
2. The Sum-
Product
Pattern (A-C
Method)
If the polynomial is of the
form 𝑥2
+ 𝑏𝑥 + 𝑐 and
there are factors of 𝑐
that if added will get 𝑏.
𝑥2
+ 5𝑥 + 6
The factors of 6 that if added will get 5
are 2 and 3.
𝑥2
+ 5𝑥 + 6 = (𝒙 + 𝟐)(𝒙 + 𝟑)
3. Grouping
Method
If the polynomial is of the
form 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 and
there are factors of 𝑎𝑐
that if added will get 𝑏.
Steps:
• Split up middle
term.
• Group the terms.
• Factor out GCFs
of each group.
• Factor out the
common
binomial.
2𝑥2
+ 9𝑥 − 5
The factors of 𝑎𝑐 = (2)(−5) = −10
that if added will get 9 are 10 and −1.
• Split up middle term
2𝑥2
+ 9𝑥 − 5 = 2𝑥2
+ 10𝑥 − 1𝑥 − 5
• Group the terms (make sure to
group the terms with common
factors)
= (2𝑥2
− 1𝑥) + (10𝑥 − 5)
• Factor out GCFs of each group
= 𝑥(2𝑥 − 1) + 5(2𝑥 − 1)
• Factor out the common binomial
= (𝟐𝒙 − 𝟏)(𝒙 + 𝟓)
20

29.
4. Perfect
Square
Trinomials
If the first and last terms
are perfect squares and
the middle term is twice
the product of their roots.
4𝑥2
+ 12𝑥 + 9
The first and last terms are perfect
squares: √4𝑥2 = 2𝑥 √9 = 3
The middle term is twice the product
of their roots: 2(2𝑥)(3) = 12𝑥
4𝑥2
+ 12𝑥 + 9 = (𝟐𝒙 − 𝟑)𝟐
5. Difference of
Squares
If the expression
represents a difference
of two squares
𝑥2
− 4
Square roots of the terms:
√𝑥2 = 𝑥 √4 = 2
𝑥2
− 4 = (𝒙 + 𝟐)(𝒙 − 𝟐)
What’s New
Directions: Complete the table below.
Polynomial Function
Term with
highest
exponent
Term/s with lower
exponents in
descending order
Constant
term
1. 𝑦 = −4𝑥2
+ 𝑥4
− 45
2. 𝑦 = 6𝑥2
+ 4𝑥 + 3𝑥3
3. 𝑦 = 5𝑥4
− 5 − 2𝑥 + 𝑥3
4. 𝑦 = 9𝑥2
− 11𝑥4
+ 2
5. 𝑦 = −8𝑥2
+ 2𝑥3
+ 6𝑥
21

30.
What Is It
Writing Polynomial Function in Standard Form
When giving a final answer, you must write the polynomial function in standard
form. Standard form means that you write the terms by decreasing exponents.
Here’s what to do:
1. Write the term with the highest exponent first.
2. Write the terms with lower exponents in descending order.
3. Remember that a variable with no exponent has an understood exponent of 1.
4. A constant term always comes last.
Examples: Write the following polynomial functions in standard form.
1. 𝑦 = 1 + 2𝑥 + 𝑥5
− 4𝑥3
+ 2𝑥4
+ 5𝑥2
Term with
highest
exponent
Term/s with lower
exponents in
descending order
Constant
term
Standard form
𝑥5 2𝑥4
,
−4𝑥3
,
5𝑥2
,
2𝑥
1 𝒚 = 𝒙𝟓
+ 𝟐𝒙𝟒
− 𝟒𝒙𝟑
+ 𝟓𝒙𝟐
+ 𝟐𝒙 + 𝟏
2. 𝑓(𝑥) = 5𝑥 + 9𝑥2
− 3𝑥8
Often, the polynomial function does not contain all of the exponents. You still
follow the same procedure listing the highest exponent first (8) then the next (2)
and finally the term with just a variable (understood exponent of 1).
Term with
highest
exponent
Term/s with lower
exponents in
descending order
Constant
term
Standard form
−3𝑥8
9𝑥2
,
5𝑥
0 𝒇(𝒙) = −𝟑𝒙𝟖
+ 𝟗𝒙𝟐
+ 𝟓𝒙
22

31.
3. 𝑦 = 𝑥( 𝑥2
− 5)
With a factored form of a polynomial function, you must find the product first.
In finding the product of a monomial and a binomial, recall the Distributive
Property.
Multiply the monomial to the first
term of the binomial
𝑥( 𝑥2) = 𝑥1+2
= 𝑥3
Multiply the monomial to the
second term of the binomial 𝑥( −5) = −5𝑥
Arrange the exponents in
descending order.
Therefore, the standard form is 𝒚 = 𝒙𝟑
− 𝟓𝒙
4. 𝑓(𝑥) = −𝑥( 𝑥 − 4)( 𝑥 + 4)
Use the special product, Sum and Difference of two terms, in answering this
function.
Get the product of the sum and
difference of two terms.
( 𝑥 − 4)( 𝑥 + 4) = 𝑥2
− 16
Multiply -x to the product. −𝑥(𝑥2
− 16) = 𝑥3
+ 16𝑥
Thus, the polynomial function in
standard form becomes
𝒇(𝒙) = 𝒙𝟑
+ 𝟏𝟔𝒙.
23

32.
Writing Polynomial Function in Factored Form
We will focus on polynomial functions of degree 3 and higher, since linear and
quadratic functions were already taught in previous grade levels. The polynomial
function must be completely factored.
Examples: Write the following polynomial functions in factored form.
1. 𝑦 = 64𝑥3
+ 125
This is of the form 𝑎3
+ 𝑏3
which is called the sum of cubes. The factored
form of 𝑎3
+ 𝑏3
is (𝑎 + 𝑏)(𝑎2
− 𝑎𝑏 + 𝑏2
). To factor the polynomial function follow
the steps below:
Find 𝑎 and 𝑏
(𝑎 is the cube root of the first term)
(𝑏 is the cube root of the second term)
𝑎 = 4𝑥
𝑏 = 5
Substitute the values of 𝑎 and 𝑏 in
(𝑎 + 𝑏)(𝑎2
− 𝑎𝑏 + 𝑏2
)
𝑦 = (4𝑥 + 5)[(4𝑥)2
− (4𝑥)(5) + (5)2
]
So the factored form is 𝒚 = (𝟒𝒙 + 𝟓)(𝟏𝟔𝒙𝟐
− 𝟐𝟎𝒙 + 𝟐𝟓)
2. 𝑦 = 3𝑥3
+ 6𝑥2
+ 4𝑥 + 8
This is of the form 𝑎𝑥3
+ 𝑏𝑥2
+ 𝑐𝑥 + 𝑑. This can be easily factored if
𝑎
𝑏
=
𝑐
𝑑
.
To factor the polynomial function, follow the steps below:
Group the terms (𝑎𝑥3
+ 𝑏𝑥2
) + (𝑐𝑥 + 𝑑) 𝑦 = (3𝑥3
+ 6𝑥2
) + (4𝑥 + 8)
Factor 𝑥2
out of the first group of terms.
Factor the constants out of both groups.
𝑦 = 𝑥2
(3𝑥 + 6) + (4𝑥 + 8)
𝑦 = 3𝑥2
(𝑥 + 2) + 4(𝑥 + 2)
Add the two terms by adding the coefficients 𝑦 = 3𝑥2
(𝑥 + 2) + 4(𝑥 + 2)
𝑦 = (3𝑥2
+ 4)(𝑥 + 2)
So, the factored form is 𝒚 = (𝟑𝒙𝟐
+ 𝟒)(𝒙 + 𝟐)
24

33.
3. 𝑦 = 45𝑥3
+ 18𝑥2
− 5𝑥 − 2
This is of the form 𝑎𝑥3
+ 𝑏𝑥2
+ 𝑐𝑥 + 𝑑. Follow the steps below:
Group the terms (𝑎𝑥3
+ 𝑏𝑥2
) + (𝑐𝑥 + 𝑑) 𝑦 = (45𝑥3
+ 18𝑥2
) + (−5𝑥 − 2)
Factor 𝑥2
out of the first group of terms.
Factor the constants out of both groups.
𝑦 = 𝑥2(45𝑥 + 18) + (−5𝑥 − 2)
𝑦 = 9𝑥2(5𝑥 + 2) − (5𝑥 + 2)
Add the two terms by adding the
coefficients
𝑦 = 9𝑥2(5𝑥 + 2) − 1(5𝑥 + 2)
𝑦 = (9𝑥2
− 1)(5𝑥 + 2)
This can be further factored as a difference
of two squares 𝑦 = (3𝑥 + 1)(3𝑥 − 1)(5𝑥 + 2)
So, the factored form is 𝒚 = (𝟑𝒙 + 𝟏)(𝟑𝒙 − 𝟏)(𝟓𝒙 + 𝟐)
4. 𝑦 = 81𝑥4
− 16
This is of the form 𝑎4
− 𝑏4
. We can factor a difference of fourth powers
(and higher powers) by treating each term as the square of another base, using the
power to a power rule. Follow the steps below:
Treat 𝑎4
as (𝑎2
)2
and 𝑏4
as (𝑏2
)2
(𝑎2
)2
− (𝑏2
)2 𝑦 = (9𝑥2
)2
− (4)2
It shows difference of two squares, factor it.
(𝑎2
)2
− (𝑏2)2
= (𝑎2
+ 𝑏2
)(𝑎2
− 𝑏2
) 𝑦 = (9𝑥2
+ 4)(9𝑥2
− 4)
This can be further factored as a difference
of squares 𝑦 = (9𝑥2
+ 4)(3𝑥 + 2)(3𝑥 − 2)
So, the factored form is 𝒚 = (𝟗𝒙𝟐
+ 𝟒)(𝟑𝒙 + 𝟐)(𝟑𝒙 − 𝟐)
5. 𝑦 = 𝑥4
− 4𝑥2
− 45
This is of the form 𝑎𝑥4
+ 𝑏𝑥2
+ 𝑐. In similar manner, we can factor some
trinomials of degree four by treating 𝑥4
as (𝑎2
)2
. Follow the steps below:
Treat 𝑎4
as (𝑎2
)2
(𝑥2
)2
− 𝑏(𝑥2
) − 𝑐 𝑦 = (𝑥2
)2
− 4(𝑥2
) − 45
Let 𝑥2
= 𝑥, thus, it shows a quadratic
trinomial: 𝑎𝑥2
+ 𝑏𝑥 + 𝑐
Factor it:
𝑦 = 𝑥2
− 4𝑥 − 45
𝑦 = (𝑥 − 9)(𝑥 + 5)
Put it back. (Substitute 𝑥 = 𝑥2
) 𝑦 = (𝑥2
− 9)(𝑥2
+ 5)
This can be further factored as a difference
of squares 𝑦 = (𝑥 + 3)(𝑥 − 3)(𝑥2
+ 5)
So, the factored form is 𝒚 = (𝒙 + 𝟑)(𝒙 − 𝟑)(𝒙𝟐
+ 𝟓)
25

34.
What’s More
A. Directions: Complete the table below.
Polynomial Function
Term with
highest
exponent
Term/s with lower
exponents in
descending order
Constant
term
Standard
form
1. 𝑓(𝑥) = 4 + 4𝑥4
+ 8𝑥
2. 𝑓(𝑥) = (𝑥 + 2)(𝑥 − 2)
3. 𝑦 = 1 + 2𝑥 + 𝑥3
4. 𝑦 = −5 + 5𝑥10
+ 5𝑥5
5. 𝑓(𝑥) = 𝑥2
− 9𝑥5
+ 6
B. Directions: Write the factored form of the following polynomial functions by
completing the table:
1. 𝑦 = 343𝑥3
+ 27
Find 𝑎 and 𝑏
(𝑎 is the cube root of the first term)
(𝑏 is the cube root of the second term)
𝑎 = _____
𝑏 = _____
Substitute the values of 𝑎 and 𝑏 in
(𝑎 + 𝑏)(𝑎2
− 𝑎𝑏 + 𝑏2
) 𝑦 = (__ + __)[(__)2
− 2(__)(__) + (__)2
]
So, the factored form is 𝒚 = (__ + __)(__𝟐
− __ + __)
2. 𝑦 = 27𝑥3
− 8
Find 𝑎 and 𝑏
(𝑎 is the cube root of the first term)
(𝑏 is the cube root of the second term)
𝑎 = _____
𝑏 = _____
Substitute the values of 𝑎 and 𝑏 in
(𝑎 − 𝑏)(𝑎2
+ 𝑎𝑏 + 𝑏2
) 𝑦 = (__ − __)[(__)2
+ 2(__)(__) + 1(__)2
]
So, the factored form is 𝒚 = (__ − __)(__𝟐
+ __ + __)
26

35.
3. 𝑦 = 𝑥3
+ 3𝑥2
− 4𝑥 − 12
Group the terms (𝑎𝑥3
+ 𝑏𝑥2
) + (𝑐𝑥 + 𝑑) 𝑦 = (__3
+ __2
) + (__ − __)
Factor 𝑥2
out of the first group of terms.
Factor the constants out of both groups. 𝑦 = 𝑥2(__ + __) − __(__ + __)
Add the two terms by adding the coefficients 𝑦 = (__2
− __)(__ + __)
This can be further factored as a difference of
squares
𝑦 = (__ + __)(__ − __)(__ + __)
So, the factored form is 𝑦 = (__ + __)(__ − __)(__ + __)
4. 𝑦 = 𝑥4
− 5𝑥2
+ 4
Treat 𝑎4
as (𝑎2
)2
(𝑥2
)2
− 𝑏(𝑥2
) + 𝑐 𝑦 = (__)2
− __(__2
) + __
Let 𝑥2
= 𝑥 , thus, it shows a quadratic
trinomial: 𝑎𝑥2
+ 𝑏𝑥 + 𝑐
Factor it:
𝑦 = 𝑥2
− __ + __
𝑦 = (𝑥 − __)(𝑥 − __)
Put it back. (Substitute 𝑥 = 𝑥2
) 𝑦 = (𝑥2
− __)(𝑥2
− __)
This can be further factored as a difference
of Two squares 𝑦 = (𝑥 + __)(𝑥 − __)(𝑥 + __)(𝑥 − __)
So, the factored form is 𝒚 = (𝒙 + __)(𝒙 − __)(𝒙 + __)(𝒙 − __)
C. Directions: Match the following polynomial functions into its standard/factored
forms. Numbers 6-10 have two answers which is it’s standard and factored form.
Column A
____1. 𝑓(𝑥) = 2 − 𝑥4
+ 8𝑥
____2. 𝑓(𝑥) = (𝑥 + 5)(𝑥 + 1)
____3. 𝑓(𝑥) = 6 − 2𝑥
____4. 𝑓(𝑥) = −16 + 5𝑥8
− 5𝑥3
____5. 𝑓(𝑥) = 𝑥2
− 9𝑥5
+ 6
____6. 𝑦 = 𝑥 − 2𝑥2
+ 𝑥3
____7. 𝑦 = −100 + 𝑥2
____8. 𝑓(𝑥) = 4 + 5𝑥 + 𝑥2
____9. 𝑦 = 16 + 𝑥2
+ 8𝑥
____10. 𝑦 = 1 − 4𝑥2
Column B
A. 𝑓(𝑥) = −9𝑥5
+ 𝑥2
+ 6
B. 𝑓(𝑥) = −𝑥4
+ 8𝑥 + 2
C. 𝑓(𝑥) = 5𝑥8
− 5𝑥3
− 16
D. 𝑓(𝑥) = −2𝑥 + 6
E. 𝑓(𝑥) = 𝑥2
+ 6𝑥 + 5
F. 𝑦 = 𝑥3
− 2𝑥2
+ 𝑥
G. 𝑦 = −4𝑥2
+ 1
H. 𝑦 = 𝑥(1 − 𝑥)(1 − 𝑥)
I. 𝑦 = 𝑥2
+ 5𝑥 + 4
J. 𝑦 = (𝑥 − 10)(𝑥 + 10)
K. 𝑦 = 𝑥2
+ 8𝑥 + 16
L. 𝑦 = (1 − 2𝑥)(1 + 2𝑥)
M. 𝑦 = 𝑥2
− 100
N. 𝑦 = (𝑥 + 4)2
O. 𝑓(𝑥) = (𝑥 + 4)(𝑥 + 1)
27

36.
What I Have Learned
A. Directions: Fill in the blanks with the correct word/s to complete each
statement.
_______(1)________ means that you write the terms by decreasing exponents.
Steps in writing this form:
1. Write the term with the ____(2)_________ first.
2. Write the terms with lower exponents in ____(3)_________ order.
3. Remember that a variable with no exponent has an understood exponent of (4).
4. A ______(5)_________ always comes last.
B. Direction: Factor the following:
1. 𝑦 = 𝑥4
− 512𝑥
2. 𝑦 = 9𝑥3
− 36𝑥2
+ 4𝑥 − 16
What I Can Do
Directions: Write the standard form of the polynomial functions that is found in nature.
1. The intensity of light emitted by a firefly can be determined by
𝐿(𝑡) = 10 + 0.3𝑡 + 0.4𝑡2
− 0.01𝑡3
.
2. The total number of hexagons in a honeycomb can be modeled by the
function 𝑓(𝑟) = 1 + 3𝑟2
− 3𝑟.
28

37.
Assessment
Directions: Choose the letter that best answers each question.
1. What is the product of (𝑥 + 3)(𝑥 + 3)?
A. 𝑥2
+ 3𝑥 + 9
B. 𝑥2
− 3𝑥 + 9
C. 𝑥2
+ 6𝑥 + 9
D. 𝑥2
− 6𝑥 + 9
2. What is the product of (𝑥 − 2)(𝑥2
+ 2𝑥 + 4)?
A. 𝑥3
+ 8 B. 𝑥3
− 8 C. 𝑥3
− 4 D. 𝑥3
+ 4
3. What term has the highest exponent in 𝑓(𝑥) = 𝑥4
+ 5𝑥7
+ 3𝑥?
A. 𝑥4
B. 5𝑥7
C. 3𝑥 D. 0
4. What is the constant term in number 3?
A. 𝑥4
B. 5𝑥7
C. 3𝑥 D. 0
5. What is the standard form of the polynomial function in number 3?
A. 𝑓(𝑥) = 5𝑥7
+ 𝑥4
+ 3𝑥
B. 𝑓(𝑥) = 5𝑥7
+ 3𝑥 + 𝑥4
C. 𝑓(𝑥) = 𝑥4
+ 5𝑥7
+ 3𝑥
D. 𝑓(𝑥) = 3𝑥 + 5𝑥7
+ 𝑥4
6. What should be the order of terms of the polynomial function in standard form?
A. term with highest exponent, term/s with lower exponent, constant term
B. term with highest exponent, constant term, term/s with lower exponent
C. constant term, term with highest exponent, term/s with lower exponent
D. constant term, term/s with lower exponent, term with highest exponent
7. What is the standard form of 𝑦 = 8𝑥2
+ 4𝑥 + 3𝑥6
+ 3?
A. 𝑦 = 3𝑥6
+ 3 + 8𝑥2
+ 4𝑥
B. 𝑦 = 3𝑥6
+ 8𝑥2
+ 4𝑥 + 3
C. 𝑦 = 3𝑥6
+ 8𝑥2
+ 3 + 4𝑥
D. 𝑦 = 3𝑥6
+ 3 + 4𝑥 + 8𝑥2
29

38.
8. What is the standard form of 𝑦 = 6𝑥 + 12𝑥2
+ 2𝑥3
?
A. 𝑦 = 2𝑥3
+ 6𝑥 + 12𝑥2
B. 𝑦 = 12𝑥2
+ 6𝑥 + 2𝑥3
C. 𝑦 = 2𝑥3
+ 12𝑥2
+ 6𝑥
D. 𝑦 = 12𝑥2
+ 2𝑥3
+ 6𝑥
9. What is the factored form of the polynomial function 𝑦 = 2𝑥3
+ 14𝑥2
+ 20𝑥
A. 𝑦 = 2𝑥(𝑥 + 5)(𝑥 − 2)
B. 𝑦 = 2𝑥(𝑥 + 5)(𝑥 + 2)
C. 𝑦 = 5𝑥(𝑥 + 2)(𝑥 + 2)
D. 𝑦 = 2𝑥(𝑥 + 5)(𝑥 + 5)
10.How should 𝑓(𝑥) = 𝑥3
− 64 be written in factored form?
A. 𝑓(𝑥) = (𝑥 − 4)(𝑥2
+ 4𝑥 + 16)
B. 𝑓(𝑥) = (𝑥 + 4)(𝑥2
+ 4𝑥 + 16)
C. 𝑓(𝑥) = (𝑥 − 4)(𝑥2
− 4𝑥 + 16)
D. 𝑓(𝑥) = (𝑥 + 4)(𝑥2
− 4𝑥 + 16)
11.How should the polynomial function 𝑓(𝑥) = 𝑥3
+ 8 be written in factored form?
A. 𝑓(𝑥) = (𝑥 + 2)(𝑥2 + 2𝑥 + 4)
B. 𝑓(𝑥) = (𝑥 − 2)(𝑥2 + 2𝑥 + 4)
C. 𝑓(𝑥) = (𝑥 + 2)(𝑥2 − 2𝑥 + 4)
D. 𝑓(𝑥) = (𝑥 − 2)(𝑥2
− 2𝑥 + 4
12.What is the factored form of 𝑦 = −4𝑥2
+ 1?
A. 𝑦 = (1 + 2𝑥)(1 + 2𝑥)
B. 𝑦 = (1 − 2𝑥)(1 + 2𝑥)
C. 𝑦 = (2𝑥 + 1)(1 + 2𝑥)
D. 𝑦 = (2𝑥 + 1)(2𝑥 − 1)
30

39.
13.How should 𝑦 = −10 + 3𝑥 + 𝑥2
be written in standard form?
A. 𝑦 = 𝑥2
+ 3𝑥 − 10
B. 𝑦 = 𝑥2
−10 + 3𝑥
C. 𝑦 = −10 + 3𝑥 + 𝑥2
D. 𝑦 = 3𝑥 − 10 + 𝑥2
14.How should 𝑦 = 𝑥2
−10 + 3𝑥 be written in factored form?
A. 𝑦 = (𝑥 + 5)(𝑥 + 2)
B. 𝑦 = (𝑥 + 5)(𝑥 − 2)
C. 𝑦 = (𝑥 − 5)(𝑥 + 2)
D. 𝑦 = (𝑥 − 5)(𝑥 − 2)
15.What is the standard form of 𝑦 = (3𝑥 + 1)(2𝑥 − 7)?
A. 𝑦 = 6𝑥2
− 19𝑥 − 7
B. 𝑦 = 6𝑥2
+ 19𝑥 − 7
C. 𝑦 = 6𝑥2
− 23𝑥 − 7
D. 𝑦 = 6𝑥2
+ 19𝑥 − 7
Additional Activity
Directions: Give 3 situations where polynomial function is found and write their
standard form.
31

40.
Summary
A polynomial function is a function of the form
𝑃(𝑥) = 𝑎𝑛𝑥𝑛
+ 𝑎𝑛−1𝑥𝑛−1
+ 𝑎𝑛−2𝑥𝑛−2
+ ⋯ + +𝑎1𝑥 + 𝑎0, 𝑎𝑛 ≠ 0,
where 𝑛 is a nonnegative integer , 𝑎0, 𝑎1, … , 𝑎𝑛 are real numbers called coefficients
(numbers that appear in each term) , 𝑎𝑛𝑥𝑛
is the leading term (has the highest
degree), 𝑎𝑛 is the leading coefficient, and 𝑎0 is the constant term (number without
a variable). The highest power of the variable of 𝑃(𝑥) is known as its degree.
When giving a final answer, you must write the polynomial function in standard
form. Standard form means that you write the terms by decreasing exponents.
Here’s what to do:
1. Write the term with the highest exponent first.
2. Write the terms with lower exponents in descending order.
3. Remember that a variable with no exponent has an understood exponent of 1.
4. A constant term always comes last.
In writing polynomial function in Factored Form, make sure that it is factored
completely. The following questions might help you to factor the polynomial functions
completely.
1. Is there a common factor?
2. Is there a difference of squares?
3. Is there a perfect square trinomial?
4. Is there an expression of the form 𝑥2
+ 𝑏𝑥 + 𝑐?
5. Are there factors of 𝑎𝑐 that add up to 𝑏?
32

41.
Assessment: (Post-Test)
Directions: Choose the letter that best answers each question.
1. Which of the following is the value of 𝑛 in 𝑓(𝑥) = 𝑥𝑛
if 𝑓 is a polynomial function?
A. √2 B. −2 C. 2 D.
1
2
2. Which of the following is NOT a polynomial function?
A. 𝑓(𝑥) = 2021
B. 𝑓(𝑥) = 19
C. 𝑓(𝑥) = 𝑥2
− 𝑥
D. 𝑓(𝑥) = √3 𝑥2
3. Which of the following is a polynomial function?
i. 𝑓(𝑥) = 𝑥3
+ 2𝑥 + 1 ii. 𝑓(𝑥) = 𝑥2
+ 𝑥 + 1 iii. 𝑓(𝑥) = √2 𝑥2 + √𝑥
A. i only B. ii only C. i and ii D. i and iii
4. What is the leading term of 𝑓(𝑥) = 𝑥2
+ 4𝑥3
+ 1?
A. x B. 2 C. 3 D. 4𝑥3
5. What is the constant term of the polynomial function in number 4?
A. 1 B. 2 C. 3 D. 4
6. What is the standard form of 𝑓(𝑥) = (5𝑥 + 3)(25𝑥2
− 15𝑥 + 9)?
A. −125𝑥3
− 27
B. 125𝑥3
− 27
C. −125𝑥3
+ 27
D. 125𝑥3
+ 27
7. What is the leading term of number 6?
A. 27 B. −27 C. 125𝑥3
D. −125𝑥3
8. What is the constant term of the polynomial in number 6?
A. 27 B. −27 C. 125𝑥3
D. −125𝑥3
33

42.
9. Given that 𝑓(𝑥) = 2𝑥−2𝑛
+ 8𝑥2
, what value should be assigned to 𝑛 to make 𝑓 a
function of degree 5?
A. −
2
5
B. −
5
2
C.
2
5
D.
5
2
10. How should the polynomial function 𝑓(𝑥) = 𝑥4
− 8𝑥2
+
𝑥
2
+
1
2
+ 4𝑥3
be written in
standard form?
A. 𝑓(𝑥) = −8𝑥2
+
1
2
+ 4𝑥3
+ 𝑥4
+
𝑥
2
B. 𝑓(𝑥) =
𝑥
2
− 8𝑥2
+
1
2
+ 4𝑥3
+ 𝑥4
C. 𝑓(𝑥) = 𝑥4
+ 4𝑥3
− 8𝑥2
+
𝑥
2
+
1
2
D. 𝑓(𝑥) = 4𝑥3
+
1
2
− 8𝑥2
+
𝑥
2
+ 𝑥4
11. What is the leading coefficient of number 10?
A. −8 B. 1 C.
1
2
D. −4
12. What is the constant term of the polynomial in number 10?
A. −8 B. 1 C.
1
2
D. −4
13. What is the factored form of 𝑦 = 9𝑥3
− 3𝑥2
+ 81𝑥 − 27?
A. 𝑦 = −3(𝑥2
+ 9)(3𝑥 − 1)
B. 𝑦 = 3(𝑥2
+ 9)(3𝑥 − 1)
C. 𝑦 = 3(𝑥2
− 9)(3𝑥 − 1)
D. 𝑦 = 3(𝑥2
+ 9)(3𝑥 + 1)
14. How should 𝑓(𝑥) = 𝑥4
+ 𝑥3
+ 𝑥2
+ 𝑥 be written in factored form?
A. 𝑓(𝑥) = 𝑥(𝑥 + 1)(𝑥2
+ 1)
B. 𝑓(𝑥) = 𝑥(1)(𝑥2
+ 1)
C. 𝑓(𝑥) = 𝑥(𝑥 − 1)(𝑥2
+ 1)
D. 𝑓(𝑥) = 𝑥(−1)(𝑥2
+ 1)
15. What is the factored form of 𝑓(𝑥) = 𝑥3
+ 3𝑥2
− 4𝑥 − 12?
a. 𝑓(𝑥) = (𝑥 + 2)(𝑥 − 2)(𝑥 + 3)
b. 𝑓(𝑥) = (𝑥 + 2)(𝑥 + 2)(𝑥 + 3)
c. 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 2)(𝑥 + 3)
d. 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 2)(𝑥 − 3)
34

43.
Answer Key
What
I
Know
(Pre-test)
on
page
2
1.
B
6.
B
11.
B
2.
D
7.
C
12.
C
3.
B
8.
D
13.
A
4.
D
9.
B
14.
A
5.
A
10.
C
15.
B
What’s
New
on
page
7
Expression
Polynomial
or
Not
Reason/s
1.
Polynomial
2.
Not
The
variable
of
one
term
is
inside
the
radical
sign.
3.
Polynomial
4.
Not
The
exponent
of
the
variable
is
not
a
whole
number.
5.
Not
The
variables
appear
in
the
denominator.
6.
Polynomial
7.
Not
The
exponent
of
the
variable
is
not
a
whole
number.
8.
Polynomial
9.
Not
It
has
a
negative
exponent.
10.
Polynomial
What’s
more
(A)
on
page
9
Polynomial
Function
or
Not
Reason
Degree
Leading
Coefficient
Consta
nt
Term
1.
Polynomial
0
None
0
2.
Not
The
variable
of
one
term
is
inside
the
radical
sign.
3.
Polynomial
1
-1
0
4.
Not
The
exponent
of
the
variable
is
not
a
whole
number.
5.
Not
The
variable
appears
in
the
denominator.
6.
Polynomial
1
√5
0
7.
Polynomial
2
1
0
8.
Not
It
has
a
negative
exponent.
9.
Polynomial
3
1
1
10.
Polynomial
2
-4
1
11.
Polynomial
0
None
2020
12.
Not
The
variable
of
one
term
is
inside
the
radical
sign.
13.
Polynomial
1
3
√4
0
14.
Polynomial
1
1
2
15.
Polynomial
1
3
0
Lesson
1:
What
I
Know
on
page
5
1.
D
6.
A
11.
A
2.
C
7.
C
12.
A
3.
B
8.
C
13.
C
4.
B
9.
C
14.
D
5.
B
10.
B
15.
A
35

44.
What’s
More
(B)
on
page
10
Polynomial
Function
or
Not
Degree
of
Polynomial
Type
of
Polynomial
Leading
Term
Constant
Term
1.
Polynomial
1
Linear
x
0
2.
Polynomial
2
Quadratic
4x
2
0
3.
Not
4.
Polynomial
2
Quadratic
x
2
12
5.
Polynomial
2
Quadratic
-5x
2
10
6.
Polynomial
0
Zero
None
1
2
7.
Not
8.
Polynomial
2
Quadratic
1
2
x
2
−3
9.
Polynomial
5
Quintic
𝑥
5
2
10.
Not
11.
Not
12.
Polynomial
1
Linear
5
8
x
0
13.
Polynomial
1
Linear
2𝑥
3
2
3
14.
Polynomial
2
Quadratic
𝑥
2
0
15.
Not
What
I
Have
Learned
on
page
10
1.
polynomial
function
2.
nonnegative
3.
degree
4.
cubic
5.
x
3
6.
constant
term
7.
leading
coefficient
8.
3
Polynomial
Function
or
Not
Reason
Degree
Leading
Term,
Coefficient
Constant
Term
9.
Polynomial
0
None
20
10.
Not
The
variable
of
one
term
is
inside
the
radical
sign.
11.
Polynomial
1
−1991𝑥,
−1991
None
12.
Not
The
exponent
of
the
variable
is
not
a
whole
number.
13.
Not
The
variables
appear
in
the
denominator.
14.
Polynomial
1
√4𝑥
or
2𝑥,
√4
or
2
None
15.
Polynomial
2
𝑥
2
,
1
20
What
I
Can
Do
on
page
11
Answers
may
vary.
Additional
Activity
on
page
14
Answers
may
vary.
Assessment
on
page
12
1.
A
6.
C
11.
B
2.
C
7.
D
12.
B
3.
B
8.
B
13.
B
4.
B
9.
D
14.
C
5.
A
10.
A
15.
A
36

45.
Lesson
2
What
I
Know
on
page
16
1.
C
6.
D
11.
A
2.
B
7.
C
12.
B
3.
B
8.
C
13.
A
4.
D
9.
B
14.
B
5.
A
10.
C
15.
B
What’s
New
on
page
21
Polynomial
Function
Term
with
highest
exponent
Term/s
with
lower
exponents
in
descending
order
Constant
term
1.
𝑥
4
−4𝑥
2
−45
2.
3𝑥
3
6𝑥
2
;
4𝑥
0
3.
5𝑥
4
𝑥
3
;
−2𝑥
−5
4.
−11𝑥
4
9𝑥
2
2
5.
2𝑥
3
−8𝑥
2
;
6𝑥
0
What’s
more
(A)
on
page
26
Polynomial
Function
Term
with
highest
exponent
Term/s
with
lower
exponents
in
descending
order
Constant
term
Standard
form
1.
4𝑥
4
8𝑥
4
𝑓
(
𝑥
)
=
4𝑥
4
+
8𝑥
+
4
2.
𝑥
2
0
−4
𝑓
(
𝑥
)
=
𝑥
2
−
4
3.
𝑥
3
2𝑥
1
𝑦
=
1
+
2𝑥
+
𝑥
3
4.
5𝑥
10
5𝑥
5
−5
𝑦
=
5𝑥
10
+
5𝑥
5
−
5
5.
−9𝑥
5
𝑥
2
6
𝑓
(
𝑥
)
=
𝑥
2
−
9𝑥
5
+
6
What’s
more
(B)
on
page
26
1.
𝑦
=
343𝑥
3
+
27
Find
𝑎
and
𝑏
(𝑎
is
the
cube
root
of
the
first
term)
(𝑏
is
the
cube
root
of
the
second
term)
𝑎
=
7𝑥
𝑏
=
3
Substitute
the
values
of
𝑎
and
𝑏
in
(𝑎
+
𝑏)(𝑎
2
−
𝑎𝑏
+
𝑏
2
)
𝑦
=
(
7𝑥
+
3
)
[(7𝑥)
2
−
2(7𝑥)(3)
+
(
3
)
2
]
So
the
factored
form
is
𝒚
=
(
𝟕𝒙
+
𝟑
)
(𝟒𝟗𝒙
𝟐
−
𝟐𝟏𝒙
+
𝟗)
2.
𝑦
=
27𝑥
3
−
8
Find
𝑎
and
𝑏
(𝑎
is
the
cube
root
of
the
first
term)
(𝑏
is
the
cube
root
of
the
second
term)
𝑎
=
3𝑥
𝑏
=
2
Substitute
the
values
of
𝑎
and
𝑏
in
(𝑎
−
𝑏)(𝑎
2
+
𝑎𝑏
+
𝑏
2
)
𝑦
=
(
3𝑥
−
2
)
[(3𝑥)
2
+
(3𝑥)(2)
+
(
2
)
2
]
So
the
factored
form
is
𝒚
=
(
𝟑𝒙
−
𝟐
)
(𝟗𝒙
𝟐
+
𝟔𝒙
+
𝟒)
37

46.
38
3.
𝑦
=
𝑥
3
+
3𝑥
2
−
4𝑥
−
12
Group
the
terms
(𝑎𝑥
3
+
𝑏𝑥
2
)
+
(𝑐𝑥
+
𝑑)
𝑦
=
(𝑥
3
+
3𝑥
2
)
+
(−4𝑥
−
12)
Factor
𝑥
2
out
of
the
first
group
of
terms.
Factor
the
constants
out
of
both
groups.
𝑦
=
𝑥
2
(
𝑥
+
3
)
+
(−4𝑥
−
12)
𝑦
=
𝑥
2
(
𝑥
+
3
)
−
4(𝑥
+
3)
Add
the
two
terms
by
adding
the
coefficients
𝑦
=
(𝑥
2
−
4)(𝑥
+
3)
This
can
be
further
factored
as
a
difference
of
squares
𝑦
=
(𝑥
+
2)(𝑥
−
2)(𝑥
+
3)
So
the
factored
form
is
𝒚
=
(𝒙
+
𝟐)(𝒙
−
𝟐)(𝒙
+
𝟑)
4.
𝑦
=
𝑥
4
−
5𝑥
2
+
4
Treat
𝑎
4
as
(𝑎
2
)
2
(𝑥
2
)
2
−
𝑏(𝑥
2
)
+
𝑐
𝑦
=
(𝑥
2
)
2
−
5(𝑥
2
)
+
4
Let
𝑥
2
=
𝑥,
thus,
it
shows
a
quadratic
trinomial:
𝑎𝑥
2
+
𝑏𝑥
+
𝑐
Factor
it:
𝑦
=
𝑥
2
−
5𝑥
+
4
𝑦
=
(𝑥
−
4)(𝑥
−
1)
Put
it
back.
(Substitute
𝑥
=
𝑥
2
)
𝑦
=
(𝑥
2
−
4)(𝑥
2
−
1)
This
can
be
further
factored
as
a
difference
of
Two
squares
𝑦
=
(𝑥
+
2)(𝑥
−
2)(𝑥
+
1)(𝑥
−
1)
So
the
factored
form
is
𝒚
=
(𝒙
+
𝟐)(𝒙
−
𝟐)(𝒙
+
𝟏)(𝒙
−
𝟏)
What’s
more
(C)
on
page
27
What
I
Have
Learned
on
page
28
1.
B
6.
F,
H
2.
E
7.
M,
J
3.
D
8.
I,
O
4.
C
9.
K,
N
5.
A
10.
G,
L
What
I
Can
Do
on
page28
1.
𝐿
(
𝑡
)
=
−0.01𝑡
3
+
0.4𝑡
2
+
0.3𝑡
+
10
2.
𝑓
(
𝑟
)
=
3𝑟
2
−
3𝑟
+
1
Assessment
on
page
29
1.
C
6.
A
11.
C
2.
B
7.
B
12.
B
3.
B
8.
C
13.
A
4.
D
9.
B
14.
B
5.
A
10.
A
15.
A
Additional
Activity
on
page
31
Answers
may
vary.
1.
𝑦
=
𝑥
4
−
512𝑥
=
𝑥(𝑥
3
−
512)
=
𝑥(𝑥
−
8)(𝑥
2
+
8𝑥
+
64)
2.
𝑦
=
9𝑥
3
−
36𝑥
2
+
4𝑥
−
16
=
(9𝑥
3
−
36𝑥
2
)
+
(4𝑥
−
16)
=
9𝑥
2
(
𝑥
−
4
)
+
4
(
𝑥
−
4
)
=
(9𝑥
2
+
4)
(
𝑥
−
4
)
Assessment
(Post-Test)
on
page
33
1.
C
6.
D
11.
B
2.
D
7.
D
12.
C
3.
C
8.
A
13.
B
4.
D
9.
B
14.
A
5.
A
10.
C
15.
A

47.
References
• Admin, Unknown. “Polynomial Functions- Definition, Formula, Types
and Graph With Examples.” BYJUS. BYJU'S, January 7, 2020.
https://byjus.com/maths/polynomial-functions/.
• Admin, Unknown. “Polynomial Functions- Definition, Formula, Types
and Graph With Examples.” BYJUS. BYJU'S, January 7, 2020.
https://byjus.com/maths/polynomial-functions/.
• Gloag, Andrew, Melissa Kramer, and Anne Gloag. “Polynomials in
Standard Form.” CK. CK-12 Foundation, November 20, 2019.
https://www.ck12.org/c/algebra/polynomials-in-standard-
form/lesson/Polynomials-in-Standard-Form-BSC-ALG/.
• “Polynomial.” Merriam-Webster. Merriam-Webster. Accessed June 23,
2020. https://www.merriam-webster.com/dictionary/polynomial.
• “Writing Polynomials in Standard Form.” Math. Accessed June 23,
2020.
https://www.softschools.com/math/algebra/topics/writing_polynomials_i
n_standard_form/.
• SparkNotes. SparkNotes. Accessed June 23, 2020.
https://www.sparknotes.com/math/algebra2/factoring/section2/.
• SparkNotes. SparkNotes. Accessed June 23, 2020.
https://www.sparknotes.com/math/algebra2/factoring/section3/.
39

48.
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Development Center (LRMDC)
DepEd Division of Bukidnon
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Telefax: ((08822)855-0048
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