Republic of the Philippines
Department of Education Regional Office IX, Zamboanga Peninsula
Mathematics Quarter 2 - Module 1: Polynomial Functions
Zest for Progress
Zeal of Partnership
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Name of Learner: ___________________________
Grade & Section: ___________________________
Name of School: ___________________________
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What I Need to Know
In this module you will learn the concepts of polynomial functions. Polynomial functions are mathematical models used to represent more
complicated situations in physics, economics, meteorology, ecology, biology, and others. They are formed using real numbers just like the linear and
quadratic functions. In fact, linear and quadratic functions are special types
of polynomial functions.
After going through this module, you are expected to:
1. define and illustrate polynomial functions; and
M10AL – IIa – 1 2. describe and interpret the graphs of polynomial
functions.
Lesson 1 Polynomial Functions
Let us start this module by recalling your knowledge on the concept of
polynomial expressions. This knowledge will help you understand the formal
definition of a polynomial function. Also, you need to revisit the lessons and
your knowledge on factoring polynomials and solving polynomial equations.
Your knowledge of these topics will help you sketch the graph of polynomial
functions manually. You may also use graphing utilities/tools in order to have
a clearer view and a more convenient way of describing the features of the
graph and focus on polynomial functions of degree 3 and higher.
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What’s In
A. Which is Which?
Determine whether each of the following is a polynomial expression
or not. Write P if it is a polynomial and NP if it is not. Write your answer
on a separate sheet of paper.
1. √𝑥 + 4
2. 12𝑥𝑦𝑧2
3. 𝑥2
3 + 2𝑥1
4 + 2
4. 1
2𝑥3+
2
3𝑥−
3
𝑥2
5. 5. 𝑥3 + 5𝑥 − 1
6. 6. 5𝑥3 − 4𝑥√2 + 𝑥
7. 7. 4𝑑−2 − 3
Questions:
1. Did you answer each item correctly?
2. When can an expression be considered polynomial?
What’s New
Consider the following mathematical statements:
1. y = 5x3 - x + 2
2. P(x) = x2 + 2x -1
Question:
1. How do these mathematical statements differ from polynomial
expressions?
𝑓(𝑥) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + 𝑎𝑛−2𝑥𝑛−2 + ⋯ + 𝑎1𝑥 + 𝑎0
𝑦 = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + 𝑎𝑛−2𝑥𝑛−2 + ⋯ + 𝑎1𝑥 + 𝑎0
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What is it
The mathematical statements in the previous activity are examples of
polynomial functions.
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, 𝑎𝑛𝑥𝑛 is the leading term, 𝑎𝑛 is the leading coefficient, and
𝑎0 is the constant term.
The terms of a polynomial may be written in any order. However, if they
are written in decreasing powers of x, we say the polynomial function is in
standard form.
Other than P(x), a polynomial function may also be denoted by f(x).
sometimes, a polynomial function is represented by a set P of ordered pairs
(x, y). Thus, a polynomial function can be written in different ways, like the
following.
Polynomials may also be written in factored form and as a product of
irreducible factors, that is, a factor that can no longer be factored using
coefficients that are real numbers. Here are some examples.
a. 𝑦 = 𝑥4 + 2𝑥3 − 𝑥2 + 14𝑥 − 56 in factored form 𝑦 = (𝑥2 + 7)(𝑥 − 2)(𝑥 + 4) b. 𝑦 = 𝑥4 − 2𝑥3 − 13𝑥2 − 10𝑥 in factored form 𝑦 = 𝑥(𝑥 − 5)(𝑥 + 2)(𝑥 + 1) c. 𝑓(𝑥) = 6𝑥3 + 45𝑥2 + 66𝑥 − 45 in factored form 𝑦 = 3(2𝑥 − 1)(𝑥 + 3)(𝑥 + 5)
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Activity1:
A. Fix and Move Them, then Fill Me Up
Consider the given polynomial functions then fill in the table below.
Polynomial Function
Polynomial
Function in Standard Form
Degree Leading
Coefficient Constant
Term
1. 𝑓(𝑥) = 2 − 11𝑥 + 2𝑥2
2. 𝑓(𝑥) =2𝑥3
3+
5
3+ 15𝑥
3. 𝑦 = 𝑥(𝑥2 − 5)
4. 𝑦 = −𝑥(𝑥 + 3)(𝑥 − 3)
5. 𝑦 = (𝑥 + 4)(𝑥 +1)(𝑥 − 1)2
B. Do you miss me? Here I Am Again
Factor each polynomial completely using any method.
1. (𝑥 − 1)(𝑥2 − 5𝑥 + 6)
2. (𝑥2 + 𝑥 − 6)(𝑥2 − 6𝑥 + 9)
3. (2𝑥2 − 5𝑥 + 3)(𝑥 − 3)
4. 𝑥3 + 3𝑥2 − 4𝑥 − 12
5. 2𝑥4 + 7𝑥3 − 4𝑥2 − 27𝑥 − 18
Questions:
1. Did you get the answers correctly?
2. What method(s) did you use?
The preceding task is very important for you since it has something to
do with the x – intercepts of a graph. These are the x – values when y = 0, thus, the point(s) where the graph intersects the x – axis can be determined.
To recall the relationship between factors and x – intercepts, consider
this example:
Find the intercepts of 𝑦 = 𝑥3 − 4𝑥2 + 𝑥 + 6.
Solution:
To find the x – intercept/s, set y = 0. Use the factored form. That is,
𝑦 = 𝑥3 − 4𝑥2 + 𝑥 + 6 𝑦 = (𝑥 + 1)(𝑥 − 2)(𝑥 − 3) Factor completely
0 = (𝑥 + 1)(𝑥 − 2)(𝑥 − 3) Equate y = 0
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𝑥 + 1 = 0 𝑜𝑟 𝑥 − 2 = 0 𝑜𝑟 𝑥 − 3 = 0 Equate each factor to 0
𝑥 = −1 𝑥 = 2 𝑥 = 3 to determine x
The x – intercepts are −1, 2, 𝑎𝑛𝑑 3. This means the graph will pass
through (−1,0), (2,0), 𝑎𝑛𝑑 (3,0).
Finding the y – intercept is more straight forward. Simply set x = 0 in
the given polynomial. That is,
𝑦 = 𝑥3 − 4𝑥2 + 𝑥 + 6
𝑦 = 03 − 4(0)2 + 0 + 6
𝑦 = 0
The y – intercept is 6. This means the graph will also pass through (0,6).
Activity 3: What is the destiny of my behavior?
Given the polynomial function 𝑦 = (𝑥 + 4)(𝑥 + 2)(𝑥 − 1)(𝑥 − 3), complete the
table below. Answer the questions that follow.
Value
of x
Value
of y
Relation of y value to 0:
𝑦 > 0, 𝑦 = 0, 𝑜𝑟𝑦 < 0? Location of the point (x,y):
above the x – axis, on the x – axis, or below the x – axis
-5 144 𝑦 > 0 above the x – axis
-4
-3
-2 0 𝑦 = 0 on the x – axis
0
1
2
3
4
Questions:
1. At what point(s) does the graph pass through the x – axis?
2. If 𝑥 < −4, what can you say about the graph?
3. If −4 < 𝑥 < −2, what can you say about the graph?
4. If −2 < 𝑥 < 1, what can you say about the graph?
5. If 1 < 𝑥 < 3, what can you say about the graph?
6. If 𝑥 > 3, what can you say about the graph?
This table may be transformed into a simpler one that will instantly
help you in locating the curve. We call this the table of signs.
The roots of the polynomial function 𝑦 = (𝑥 + 4)(𝑥 + 2)(𝑥 − 1)(𝑥 −
3) are 𝑥 = −4, −2, 1, 𝑎𝑛𝑑 3. These are the only values of x where the graph
will cross the x – axis. These roots partition the number lines into interval.
Test values are then chosen from within each interval.
The table of signs and the rough sketch of the graph of this function
can now be constructed, as shown below.
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The Table of Signs
Intervals
𝑥 < −4 −4 < 𝑥 < −2 −2 < 𝑥 < 1 1 < 𝑥 < 3 𝑥 > 3
Test value -5 -3 0 2 4
(𝑥 + 4) − + + + +
(𝑥 + 2) − − + + +
(𝑥 − 1) − − − + +
(𝑥 − 3) − − − − +
𝑦 = (𝑥 + 4)(𝑥 + 2) (𝑥 − 1)(𝑥 − 3)
+ − + − +
Position of the curve
relative to the x - axis above below above below above
The Graph of 𝒚 = (𝒙 + 𝟒)(𝒙 + 𝟐)(𝒙 − 𝟏)(𝒙 − 𝟑).
We can now use the information from the table of signs to construct a
possible graph of the function. At this level, though, we cannot determine the
turning points of the graph, we can only be certain that the graph is correct
with respect to intervals where the graph is above, below, or on the x-axis.
The arrow heads at both ends of the graph signify that the graph
indefinitely goes upward.
In this activity, you learned how to sketch the graph of
polynomial functions using the intercepts, some points, and the
position of the curves determined from the table of signs.
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What’s More
Activity 4: Follow my Path
Shown below are polynomial functions and their corresponding graphs. Study each figure and answer the questions that follow.
Case 1
The graph on the right is defined by
𝑦 = 2𝑥3 − 7𝑥2 − 7𝑥 + 12
or in factored form
𝑦 = (2𝑥 + 3)(𝑥 − 1)(𝑥 − 4)
Questions:
a. Is the leading coefficient a positive
or a negative number?
b. Is the polynomial of even degree or
odd degree?
c. Observe the end behaviors of the
graph on both sides. Is it rising or
falling to the left or to the right?
Case 2
The graph on the right is defined by
𝑦 = −𝑥5 + 3𝑥4 + 𝑥3 − 7𝑥2 + 4
or in factored form
𝑦 = −(𝑥 + 1)2(𝑥 − 1)(𝑥 − 2)2
Questions:
a. Is the leading coefficient a positive
or a negative number?
b. Is the polynomial of even degree or
odd degree?
c. Observe the end behaviors of the
graph on both sides. Is it rising or
falling to the left or to the right?
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Case 3
The graph on the right is defined by
𝑦 = 𝑥4 − 7𝑥2 + 6𝑥 or in factored form
𝑦 = 𝑥(𝑥 + 3)(𝑥 − 1)(𝑥 − 2)
Questions:
a. Is the leading coefficient a positive
or a negative number?
b. Is the polynomial of even degree or
odd degree?
c. Observe the end behaviors of the
graph on both sides. Is it rising or
falling to the left or to the right?
Case 4
The graph on the right is defined by
𝑦 = −𝑥4 + 2𝑥3 + 13𝑥2 − 14𝑥 − 24
or in factored form
𝑦 = −(𝑥 + 3)(𝑥 + 1)(𝑥 − 2)(𝑥 − 4)
Questions:
a. Is the leading coefficient a positive
or a negative number?
b. Is the polynomial of even degree or
odd degree?
c. Observe the end behaviors of the
graph on both sides. Is it rising or
falling to the left or to the right?
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What I Have Learned
Now, complete this table. In the last column, draw a possible graph for
the function, showing how he function behaves. (You do not need to place your
graph on the xy – plane). The first one is done for you.
Summarize your findings from the four cases above. What do you observe if:
1. the degree of the polynomial is odd and the leading coefficient is
positive? 2. the degree of the polynomial is odd and the leading coefficient is
negative?
3. the degree of the polynomial is even and the leading coefficient is positive?
4. the degree of the polynomial is even and the leading coefficient is negative?
You have now illustrated The Leading Coefficient Test. You should
have realized that this test can help you determine the end behaviors of the
graph of a polynomial function as x increases or decreases without bound.
Sample polynomial
Function
Leading Coefficient:
𝑛 > 0 𝑜𝑟 𝑛 <0
Degree: Even
or Odd
Behavior of the Graph:
Rising or Falling
Possible
Sketch
Left-
hand
Right-
hand
1. 𝑦 = 2𝑥3 − 7𝑥2 − 7𝑥 + 12 𝑛 > 0 odd falling rising
2. 𝑦 = −𝑥5 + 3𝑥4 + 𝑥3 −7𝑥2 + 4
3. 𝑦 = 𝑥4 − 7𝑥2 + 6𝑥
4. 𝑦 = −𝑥4 + 2𝑥3 + 13𝑥2 −14𝑥 − 24
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What I Can Do Activity 5: It’s Your Turn, Show Me
For each of the following function below, give
a. the x – intercepts
b. the intervals obtained when the x – intercepts are used
to partition the number line
c. the table of signs
d. a sketch of the graph
1. 𝑦 = (2𝑥 + 3)(𝑥 − 1)(𝑥 − 4)
2. 𝑦 = 𝑥4 − 26𝑥2 + 25
Questions:
a. What happens to the graph as x decreases without bound?
b. For which interval(s) is the graph (i) above and (ii) below the x-axis?
c. What happens to the graph as x increases without bound?
d. What is the leading term of the polynomial function?
e. What are the leading coefficient and the degree of the function?
Assessment Multiple Choice:
Directions: Read the questions carefully. Choose the letter of the correct
answer.
1. Which of the following could be the value of n in the equation
𝒇(𝒙) = 𝒙𝒏 𝒊𝒇 𝒇 𝒊𝒔 𝒂 𝒑𝒐𝒍𝒚𝒏𝒐𝒎𝒊𝒂𝒍 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏?
A. −3 𝐵. 0 𝐶. 1
2 𝐷. √5
2. Which of the following is NOT a polynomial function?
A. 𝒇(𝒙) = ∏ C. 𝒇(𝒙) = −𝒙 + √𝟓𝒙𝟑
B. 𝒇(𝒙) = −𝟐
𝟑𝒙𝟑 + 𝟏 D. 𝒇(𝒙) = 𝒙
𝟏
𝟓 − 𝟐𝒙𝟐
3. In the polynomial function 𝑓(𝑥) = 𝟔𝒙𝟑 + 𝟕𝒙𝟒 + 𝟏𝟎 + 𝟒𝒙𝟐 − 𝟑𝒙, what is the
leading coefficient?
A. 7 B. 6 C. 4 D. -3
4. The graph of 𝑝(𝑥) = (𝑥 + 1)(2𝑥 − 3)(𝑥 + 2) crosses the x – axis __________. A. once B. twice C. thrice D. four time
5. What is the degree of the polynomial function defined by 𝑓(𝑥) = 3𝑥𝑛−2 +2𝑥3𝑛−5 − 4𝑥2𝑛+1 𝑖𝑓 𝑛 = 3? A. 10 B. 7 C. 3 D. 1
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6. How should the polynomial function 𝒇(𝒙) =𝟏
𝟐𝒙 − 𝒙𝟐 + 𝟏𝟏𝒙𝟒 + 𝟐𝒙𝟑 be written
in standard form?
A. 𝒇(𝒙) = 𝟏𝟏𝒙𝟒 + 𝟐𝒙𝟑 +𝟏
𝟐𝒙 − 𝒙𝟐 C. 𝒇(𝒙) = 𝟏𝟏𝒙𝟒 + 𝟐𝒙𝟑 − 𝒙𝟐 +
𝟏
𝟐𝒙
B. 𝒇(𝒙) = −𝒙𝟐 +𝟏
𝟐𝒙 + 𝟐𝒙𝟑 + 𝟏𝟏𝒙𝟒 D. 𝒇(𝒙) =
𝟏
𝟐𝒙 − 𝒙𝟐 + 𝟐𝒙𝟑 + 𝟏𝟏𝒙𝟒
7. Which polynomial function in factored form represents the given graph?
A. 𝒚 = (𝟐𝒙 + 𝟑)(𝒙 − 𝟏)𝟐
B. 𝒚 = −(𝟐𝒙 + 𝟑)(𝒙 − 𝟏)𝟐
C. 𝒚 = (𝟐𝒙 + 𝟑)𝟐(𝒙 − 𝟏)
D. 𝒚 = −(𝟐𝒙 + 𝟑)𝟐(𝒙 − 𝟏)
8. Which of the following should be the graph of 𝒚 = 𝒙𝟒 − 𝟓𝒙𝟐 + 𝟒?
A. B. C. D.
9. Given that 𝒇(𝒙) = 𝟕𝒙−𝟑𝒏 + 𝒙𝟐 , what value should be assigned to n to
make f a function of degree 7?
A. −𝟕
𝟑 B. −
𝟑
𝟕 C.
𝟑
𝟕 D.
𝟕
𝟑
10. Your friend Myrna asks your help in drawing a rough sketch of the graph
of 𝒚 = −(𝒙𝟐 + 𝟏)(𝟐𝒙𝟒 − 𝟑) by means of the Leading Coefficient Test. How will
you explain the behavior of the graph?
A. The graph is falling to the left and rising to the right.
B. The graph is rising to both left and right.
C. The graph is rising to the left and falling to the right.
D. The graph is falling to both left and right.
Development Team of the Module
Writer: Arlene T. Ordeniza
Editors: Elma Marie L. Reso-or, Rizaldo Calunsag
Reviewer: Ismael K. Yusoph, Ellen A. Olario, Armil O. Turtor
Management Team:
Ma. Liza R. Tabilon
Ma. Judelyn J. Ramos
Armando P. Gumapon
Judith Romaguera
Lilia E. Abello
Evelyn C. Labad
Ma. Theresa M. Imperial
Nilda Y. Galaura
References
1. Mathematics Learner’s Module 10
2. Teachers Guide in Mathematics 10
3. Oriones, F. et. al., Advanced Algebra for Fourth Year High
School. Quezon City, Philippines: Phoenix Publishing House,
Inc.
4. Litong, D., Fourth Year Contemporary Math. Makati City,
Philippines: Salesiana Publishers, Inc.
Website Link
1. https://www.bing.com/videos/search?q=Graphing+Polynomi
al+Functions
2. https://www.youtube.com/watch?v=miTyZ...
3. https://www.youtube.com/watch?v=6aKes_9Ktjl
What’s In:
A.Which is Which?
1.Not polynomial because the variable of
one term is inside the radical sign
2.Polynomial
3.Not polynomial because the exponents of
the variables are not whole numbers
4.Not polynomial because the variables are
in the denominator
5.Polynomial
6.Polynomial
7.Not polynomial because the exponent of
the variable is negative
Activity 2
B. Do you miss me? Here I Am
Again
A.(𝑥−1)(𝑥−3)(𝑥−2)
B.(𝑥+3)(𝑥−2)(𝑥−3)(𝑥−3)
C.(2𝑥−3)(𝑥−1)(𝑥−3)
D.(𝑥+2)(𝑥−2)(𝑥+3)
E.(2𝑥+3)(𝑥+1)(𝑥−2)(𝑥+3)
Activity 1
Fix and Move Them, then Fill Me Up
What Is It Activity 3: What is the destiny of my behavior
Assessment
1.B 2.D 3.A 4.C 5.B 6.C 7.B 8.A 9.A 10.D
Answer Key
What’s More:
Activity 4: Follow My Path
What I Have Learned
Summary Table
Synthesis: (The Leading Coefficient Test)
1.If the degree of the polynomial is odd and the leading coefficient is positive, then the graph falls to the left and rises to the right.
2.If the degree of the polynomial is odd and the leading coefficient is negative, then the graph rises to
the left and falls to the right.
3.If the degree of the polynomial is even and the leading coefficient is positive, then the graph rises to
the right and also rises to the left.
4.If the degree of the polynomial is even and the leading coefficient is negative, then the graph falls to the left and also falls to the right.
