Mathematics 10 Quarter 2 Module 1: Graphs of Polynomial Functions

Mathematics Quarter 2 – Module 1: Graphs of Polynomial Functions

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CO_Q2_Mathematics 10_ Module 1

Mathematics – Grade 10 Alternative Delivery Mode Quarter 2 – Module 1: Graphs of Polynomial Functions First Edition, 2020

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Writer’s Name: Grezel B. Limbog Co-Writer’s Name: Vic Jomar M. Laderas Reviewer’s Name: Bryan A. Hidalgo

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Mathematics

Quarter 2 – Module 1: Graphs of Polynomial Functions

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

What I Need to Know

This module was designed and written with you in mind. It is here to help you graph polynomial functions. 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 and answer this module is dependent on your ability.

After going through this module, you are expected to:

a. describe and interpret the graphs of polynomial functions; and b. graph the polynomial functions

What I Know

Choose the letter of the correct answer and write it on a separate sheet of paper.

1. Which of the following is a polynomial function?
   A. $y = x^{-3} + 2x^2 + x – 1$
   C. $f(x) = \sqrt{2x^3} + 5$
   B. $y = \frac{3}{x} – 2x^3 + 5x – 4$
   D. $f(x) = x^{\frac{2}{3}} + 1$

   
   

2. Find the zeros of $P(x) = (x + 4)^3(x – 5)^6$.
   A. ${-4 \text{ multiplicity } 3, 5 \text{ multiplicity } 6}$
   B. ${4 \text{ multiplicity } 6, -5 \text{ multiplicity } 3}$
   C. ${-4 \text{ multiplicity } 5, -5 \text{ multiplicity } 5}$
   D. ${4 \text{ multiplicity } 3, 5 \text{ multiplicity } 6}$

   
   

3. What is the factored form of $f(x) = x^3 – 2x^2 – 3x$?
   A. $f(x) = (x – 3)(x + 1)$
   C. $f(x) = x^2(x + 3)(x – 1)$
   B. $f(x) = x(x – 3)(x + 1)$
   D. $f(x) = (x + 3)(x – 1)$

   
   

4. Determine the degree of $f(x) = (x + 3)(x – 4)^2(x + 4)^3(x + 1)^5$.
   A. 9 B. 10 C. 11 D. 12

   
   

5. Given that $P(x) = 2x^3 + 3x^2 – 4x – 24$, what is the value of $P(3)$?
   A. 42 B. 43 C. 44 D. 45

   
   

6. Which of the following is one of the factors of $P(x) = x^3 + x^2 – 6x – 6$?
   A. $x + 1$ B. $x – 1$ C. $x + 5$ D. $x – 5$

   
   

7. What is the leading coefficient of $P(x) = (2x + 2)(5x – 2)(x + 3)$?

   
   

A. 10 B. −1 C. −2 D. −21

8. On the Cartesian plane, where is the turning point of the graph of $f(x) = x^4 + 1$ located?
   A. above the x-axis
   B. on the x-axis
   C. below the x-axis
   D. on the y-axis

   
   

9. What determines the end behavior of the polynomial function?
   A. vertical line test
   B. horizontal line test
   C. leading coefficients test
   D. multiplicity test

   
   

10. How do you describe the behavior of the graph if the degree is even and the leading coefficient is negative?
    A. The graph rises to the left and falls to the right.
    B. The graph falls to the left and rises to the right.
    C. The graph falls on both sides.
    D. The graph rises on both sides.

    
    

11. Describe the behavior of the graph of the function $P(x) = (x + 1)(x – 1)(x – 3)$.
    A. The graph rises on the left and falls on the right.
    B. The graph falls on the left and rises on the right.
    C. The graph falls on both sides.
    D. The graph rises on both sides.

    
    

12. What are the x-intercepts of the graph of $y = (x – 3)(x + 1)(x – 1)$?
    A. 3, 1, 1 B. −3, 1, 1 C. 3, 1, −1 D. −3, −1, −1

    
    

13. Determine the number of turning/s of the graph of $P(x) = (x – 3)(x + 1)(x – 1)$?
    A. 4 B. 3 C. 2 D. 1

    
    

14. Which of the following best describes the graph of the polynomial function $y = x^3 – 4x^2 + 3x – 12$?
    A. Graph falls to the left and rises to the right.
    B. Graph rises to the left and falls to the right.
    C. Graph rises on both sides.
    D. Graph falls on both sides.

    
    

15. Which polynomial function in factored form is represented by the given graph?

    
    

    [chart: line graph of a cubic polynomial function on a Cartesian plane; no number or title printed. Values are not labeled on the chart.]

    
    

    A. $P(x) = (x – 3)(x – 2)(x – 1)$
    B. $P(x) = (x + 3)(x + 2)(x + 1)$
    C. $P(x) = (x – 3)(x + 2)(x – 1)$
    D. $P(x) = (x + 3)(x – 2)(x + 1)$

    
    

[transcriber’s note: the axis labels on the chart are too small to read reliably, so no values are transcribed for it.]

Lesson 1: Graphs of a Polynomial Function

In the previous module, you learned about determining polynomial functions. Refresh your mind by answering the activity below.

What’s In

Direction: Classify each function as polynomial or not by completing the table below. If the function is a polynomial, identify the degree and the leading coefficient. Write your answer on a separate sheet of paper.

Function	Function	Function	Function
$P(x) = x + x^2 + x^3$	$P(x) = x^{\frac{1}{2}} + 2$
$P(x) = x^4 – \frac{1}{8}x$	$P(x) = 4x – 2 + 2x – 3$
$P(x) = \frac{3}{x}$	$P(x) = 12$
$f(x) = 9\sqrt{x} + 2x$	$P(x) = 5x + 1$
$f(x) = x^{\frac{4}{2}}$	$P(x) = x^3 + x^4 + x^6$
$f(x) = 5 – \frac{4}{x^2}$	$P(x) = 5 – x^{11}$

[transcriber’s note: the table shown above is the printed function grid; its header row was not printed. The answer table printed on the page has the headers below and four empty rows.]

Questions:

Polynomial Function	Degree	Leading Coefficient	Non-Polynomial Function

1. Describe a polynomial function.
2. What are the characteristics of non-polynomial functions?

What’s New

After recalling polynomial functions, you will now see polynomial functions and its corresponding graphs. Study the graph and the polynomial function of each figure and complete the table in the next page.

[chart: graph of a polynomial function on a Cartesian plane; no number or title printed. Values are not labeled on the chart.]

$P(x) = (x + 3)(x + 2)(x + 1)$

[chart: graph of a polynomial function on a Cartesian plane; no number or title printed. Values are not labeled on the chart.]

$P(x) = (x + 2)(x + 1)(x – 1)(x – 2)$

[chart: graph of a polynomial function on a Cartesian plane; no number or title printed. Values are not labeled on the chart.]

$P(x) = -(x + 1)^2(x – 1)(x – 2)^2$

[chart: graph of a polynomial function on a Cartesian plane; no number or title printed. Values are not labeled on the chart.]

$P(x) = -(x + 3)(x + 1)(x – 2)(x – 4)$

Did you correctly identify the leading coefficient, degree and the behavior of the graph? If yes, then congratulations! You should have recognized that this can help you determine the behavior of the graph of a polynomial function as x increases or decreases without bound.

Polynomial Function	Leading Coefficient (a_n > 0 or a_n < 0)	Degree (Even or Odd)	End-Behavior of the Graph: Left Tail (rises or Falls)	End-Behavior of the Graph: Right Tail (rises or falls)
1. P(x) = (x + 3)(x + 2)(x + 1)
2. P(x) = (x + 2)(x + 1)(x − 1)(x − 2)
3. P(x) = −(x + 1)² (x − 1) (x − 2)²
4. P(x) = −(x + 3)(x + 1)(x − 2)(x − 4)

What is It

From the activity in the What’s new, your answers should be:

1. For the polynomial function P(x) = (x + 3)(x + 2)(x + 1), the degree is 3 (odd), the leading coefficient is 1(positive) and the graph falls to the left and rises to the right.
2. For the polynomial function P(x) = (x + 2)(x + 1)(x − 1)(x − 2), the degree is 4 (even), the leading coefficient is 1(positive) and the graph rises both to the left and right.
3. For the polynomial function P(x) = −(x + 1)² (x − 1) (x − 2)², the degree is 5 (odd), the leading coefficient is −1(negative) and the graph rises to the left and falls to the right.
4. For the polynomial functionP(x) = −(x + 3)(x + 1)(x − 2)(x − 4), the degree is 4 (even), the leading coefficient is −1(negative) and the graph falls both to the left and right.

Noticed that the degree and the leading coefficient of the polynomial functions determine the end-behavior of the graph.

There are four cases of the Leading Coefficient Test:

Given a polynomial function in standard form,

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + aₙ₋₂xⁿ⁻² + … + a₁x + a₀

The table below shows the behavior of the graph of polynomial function in standard form.

	Leading Coefficient	Degree	End-Behavior of the Graph
Case 1	Positive	Odd	[illustration: cubic graph with positive leading coefficient] Falls left rises right
Case 2	Negative	Odd	Rises left [illustration: cubic graph with negative leading coefficient] Falls right
Case 3	Positive	Even	Rises left [illustration: quartic graph with positive leading coefficient] Rises right
Case 4	Negative	Even	[illustration: quartic graph with negative leading coefficient] Falls left Falls right

Summary of the Leading Coefficient Test

1. Case 1 showed that 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. Case 2 showed that 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. Case 3 showed that 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. Case 4 showed that 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.

Here are the steps involved in graphing polynomial functions:

1. Write the function in factored form.
2. Determine the end-behavior of the graph of a given polynomial function using the Leading Coefficient Test.
3. Find the zeros of the polynomial function and their multiplicity.
   • If (x − a)ᵏ is a factor of the polynomial function, then a is a zero of multiplicity k. The multiplicity refers to the exponent of the factors of the polynomial.
   • If the multiplicity of the zeros is even, then the graph touches the x-axis or tangent to the x-axis.
   • If the multiplicity of the zeros is odd, then the graph crosses the x-axis.
4. Construct a table of values for x and P(x).
5. Plot the points and draw a smooth continuous curve to connect the points.
6. Make sure that the graph follows the end behavior as found in the above step.

Example 1

Describe the behavior of the graph by completing the table.

Function	Factored Form	Degree	Leading Coefficient	End-behavior of the graph: Left Tail	End-behavior of the graph: Right Tail
1. P(x) = x³ − 2x² − 3x
2. P(x) = −x³ + x² − 12x

Solution:

1. Factor the given function using the factoring techniques you have learned:

   • x³ − 2x² − 3x → x(x² − 2x − 3) → x(x + 1)(x − 3)
   • −x³ + x² − 12x → −x(x² − x + 12) → −x(x − 4)(x + 3)

2. The degree of the polynomial function in one variable can be easily identified when it is written in standard form. The degree is the highest exponent of the function. In this case, the degree of P(x) = x³ − 2x² − 3x is 3 and the degree of P(x) = −x³ + x² − 12x is 3.

   
   

3. The leading coefficient is the numerical coefficient of the term with the highest degree. The leading coefficient of P(x) = x³ − 2x² − 3x is 1 and for P(x) = −x³ + x² − 12x is −1.

   
   

4. Using the Leading Coefficient Test, the graph of P(x) = x³ − 2x² − 3x is falling at the left and rising at the right while the graph of P(x) = −x³ + x² − 12x is rising at the left and falling at the right.

   
   

Example 2

Given the function in factored form, P(x) = (x + 2)²(x + 1)³(x − 1)⁴(x − 2). Complete the table below by finding the zeros, multiplicity of the zeros, characteristic of its multiplicity and behavior of the graph.

Zeros	Multiplicity of Zero	Characteristic of the Multiplicity	Behavior of the Graph

Solution:

• To find the zeros of P(x) set P(x) = 0 and solve for the values of x. That is,
  (x + 2)² = 0, (x + 1)³ = 0, (x − 1)⁴ = 0, (x − 2) = 0

  
  

  The zeros are the x – intercepts of the polynomial function. It is where the graph crosses or touches the x – axis.

  
  

• The exponent of each factor is the multiplicity of the zero. Hence, the zeros −2 is of multiplicity 2, −1 is of multiplicity 3, 1 is of multiplicity 4, and 2 is of multiplicity 1.

  
  

• To determine the characteristic of the multiplicity, just indicate whether the exponent of the factor is even or odd.

  
  

• If the multiplicity of the zero is even, it means that the graph will just touch the x – axis at the zero while if odd it will cross the x – axis at the zero.

  
  

Zeros	Multiplicity of Zero	Characteristic of the Multiplicity	Behavior of the Graph
−2	2	Even	touches the x-axis at −2
−1	3	Odd	crosses the x-axis at −1
1	4	Even	touches the x-axis at 1
2	1	odd	crosses the x-axis at 2

Example 3

Complete the table below.

Polynomial Function	Sketch	Degree	Number of Turning Points
1. P(x) = x⁴ − 2x² − 15	[illustration: graph of a quartic function with two valleys and a peak between them]
2. P(x) = x⁵ + x³ − 2x + 1	[illustration: graph of a quintic function with a peak and a valley]

Solution:

Polynomial Function	Sketch	Degree	Number of Turning Points
1. P(x) = x⁴ − 2x² − 15	[illustration: graph of a quartic function with two valleys and a peak between them]	4	3
2. P(x) = x⁵ + x³ − 2x + 1	[illustration: graph of a quintic function with a peak and a valley]	5	2

Take note: Quartic functions like P(x) = x⁴ − 2x² − 15 have odd number of turning points while quintic functions like P(x) = x⁵ + x³ − 2x + 1 has even number of turning points. The number of turning points is at most (n − 1), where n is the degree of the polynomial function.

Example 4

Make a table of values for x and P(x) of the polynomial function:

P(x) = (x − 1)³(x + 2)²

Solution: The zeros of the given polynomial are 1 and −2. These zeros divide the x – axis in to three intervals: (∞, −2), (−2, 1) and (1, ∞). Hence, make a table of values by choosing an arbitrary x – value from each interval so that you can see the behavior of the graph from each interval. Include the zeros of the polynomial function in your table of values.

x	−3	−2	−1	0	1	2
P(x)

Substitute the selected values of x to the given polynomial function. Then simplify.

• If x = −3: P(-3) = (-3 – 1)³ ( -3 + 2)²
  P(-3) = (-4)³ (-1)² = (-64) (1) = -64
• If x = −2: P(-2) = (-2 – 1)³ ( -2 + 2)²
  P(-2) = (-3)³ ( 0 )² = 0
• If x = −1: P(-1) = (-1 – 1)³ ( -1 + 2)²
  P(-1) = (-2)³ ( 1 )² = -8
• If x = 0: P(0) = (0 – 1)³ ( 0 + 2)²
  P(0) = (-1)³ ( 2 )² = – 4
• If x = 1: P(1) = (1 – 1)³ ( 1 + 2)²
  P(1) = (0)³ ( 3 )² = 0
• If x = 2: P(2) = (2 – 1)³ ( 2 + 2)²
  P(2) = (1)³ ( 4 )² = 16

x	−3	−2	−1	0	1	2
P(x)	−64	0	−8	−4	0	16

Example 5

Sketch the graph of the polynomial function P(x) = x(x − 3)(x + 1).

Solution: Let us use the steps in graphing the polynomial function

• Step 1. Factor the polynomial function. The given function is already in factored form.
• Step 2. The x – intercepts are -1, 0 and 3. Plot these values to create intervals.
• Step 3. The degree is 3 which is odd and the leading coefficient is positive 1. Thus, the graph falls on the left and rises on the right.
• Step 4. Fill in the table of values for P(x) by using x – values in each Interval

x	−2	−1	0	1	2	3
P(x)	−10	0	0	−4	−6	0

• Step 5. Plot all the points and connect the points with a smooth curve.
• Step 6. The number of turning points is 2.

With the technology readily available at your fingertips, you can use downloadable apps in graphing like Desmos and Geogebra. These will help you in graphing polynomial functions easily and efficiently.

[chart: graph of a cubic polynomial function curve on a coordinate plane; labeled points: (−1,0), (0,0), (3,0), (1,−4), (2,−6). Values are not labeled on the chart beyond these points.]

What’s More

To enhance more of your skills in graphing polynomial function, perform the next activities.

Assessment 1

Direction: Complete the table below by identifying the factored form, leading coefficient, degree and describe the end-behavior of the graph.

Function	Factored Form	Sign of the Leading Coefficient	Degree	End behavior of the graph: Left	End behavior of the graph: Right
P(x) = (x − 1)(x² − 5x + 6)
P(x) = (2x² − 5x + 3)(x − 3)
P(x) = x³ − 6x² + 5x + 12
P(x) = 2x⁴ − 3 − 12x² + 7x + 6

Assessment 2

A. Given the function P(x) = x(x − 2)(x + 1)(x − 3)² complete the table below.

Zeros	Multiplicity of Zero	Characteristic of the Multiplicity	Behavior of the Graph

B. Given the function P(x) = x(x − 2)(x + 1)(x − 3)² compute for the value of P(x) that corresponds to each value of x.

X	−2	−1	0	1	2	3
P(x)

Assessment 3

Indicate the degree and determine the number of turning points.

Polynomial Function	Degree	Number of Turning Points
1. P(x) = (x − 2)(x + 1)
2. P(x) = x(x + 1)³
3. P(x) = x⁵ − 6x⁴ − 4x³
4. P(x) = x(x − 2)(x + 1)(x + 3)
5. P(x) = (x + 2)(x − 1)(x − 3)²

Assessment 4

Directions: Sketch the graph of the following polynomial functions. Follow the steps that were discussed from pages 8 and 11. Use a graphing paper for your graphs.

a) P(x) = x(x − 1)²(x + 2)³(x + 3) b) P(x) = 2x³ + 5x² + x − 2

What I Have Learned

Complete each sentence by filling in the blanks. Here are the steps in sketching the graph of a polynomial function.

1. Write the polynomial function in ____________ form.
2. Describe the end-behavior of the graph of the given polynomial function using the _______________________.
3. Find the _________ of the polynomial function and their multiplicity.
4. Construct a _________________ for x and 𝑃(𝑥).
5. Plot the points and draw a __________ ________________ curve to connect the points.

What I Can Do

Solve.

A box with no lid was created from piece of cardboard 25 cm long and 15 cm wide. Equal squares are cut from each corner of the cardboard and the sides are folder up.

a) Write a polynomial function (in standard form) to represent the volume of the box. b) Graph the polynomial function.

Assessment

Choose the letter of the correct answer and write it on a separate sheet of paper.

1. What are the end-behaviors of the graph 𝑃(𝑥) = −5𝑥 + 2𝑥³ + 3𝑥⁵ – 7?
   a. rises to the left and falls to the right
   b. falls to the left and rises to the right
   c. rises to both directions
   d. falls to both directions

   
   

2. If you will draw the graph of 𝑦 = 𝑥(𝑥 + 2)², how will you sketch it with respect to the x-axis?
   a. Sketch is crossing both (-2, 0) and (0, 0).
   b. Sketch is crossing (-2, 0) and is tangent at (0, 0).
   c. Sketch is tangent at (-2, 0) and crossing (0, 0).
   d. Sketch is tangent at both (-2, 0) and (0, 0).

   
   

3. Determine the number of turning points of 𝑃(𝑥) = (2 − 𝑥)(𝑥 + 2)(𝑥 + 4)?
   a. 0
   b. 1
   c. 2
   d. 3

   
   

4. If 𝑥 − 2 is a factor of 𝑉(𝑥) = 𝑥³ − 𝑥² − 4𝑥 + 4, what is the other factor?
   a. x + 1
   b. x + 2
   c. x + 3
   d. x + 4

   
   

5. Find the zeros of 𝑃(𝑥) = (𝑥 − 1)²(𝑥 + 2)³.
   a. {1 multiplicity 2, −2 multiplicity 3}
   b. {−1 multiplicity 2, 2 multiplicity 3}
   c. {2 multiplicity 1, 3 multiplicity 2}
   d. {1 multiplicity 3, −2 multiplicity 2}

   
   

6. Which of the following is one of the factors of 𝑃(𝑥) = 𝑥³ + 2𝑥² − 5𝑥 − 6?
   a. 𝑥 + 1
   b. 𝑥 − 1
   c. 𝑥 + 5
   d. 𝑥 – 5

   
   

7. What are the zeros of the polynomial function 𝑃(𝑥) = (𝑥 + 5)(𝑥 − 3)
   a. {−5, −3}
   b. {5, −3}
   c. {−5, 3}
   d. {5, 3}

   
   

8. Given 𝑃(𝑥) = 3𝑥³ + 3𝑥² − 5𝑥 − 24. What is the value of 𝑃(3)?
   a. 65
   b. 69
   c. 67
   d. 70

   
   

9. How do you describe the end behavior of the graph if the degree is odd and the leading coefficient is negative?
   a. The graph falls to the left and rises to the right.
   b. The graph rises to the left and falls to the right.
   c. The graph falls on both sides.
   d. The graph rises on both sides.

   
   

10. Which of the following could be the graph of the polynomial function, 𝑦 = 𝑥³ − 4𝑥² + 3𝑥 − 12?
    a. The graph falls to the right and rises to the left.
    b. The graph falls to the left and rises to the right.
    c. The graph falls on both sides.
    d. The graph rises on both sides.

    
    

11. Which polynomial function in factored form is represented by the given graph?

    
    

[chart: graph of a polynomial function; item 11 graph] transcriber’s note: the axis tick labels on this graph are too small to read, so no values are transcribed.

a. 𝑃(𝑥) = (𝑥 − 3)(𝑥 − 2)(𝑥 − 1)
b. 𝑃(𝑥) = (𝑥 + 3)(𝑥 + 2)(𝑥 + 1)
c. 𝑃(𝑥) = (𝑥 − 3)(𝑥 + 2)(𝑥 − 1)
d. 𝑃(𝑥) = (𝑥 + 3)(𝑥 − 2)(𝑥 + 1)

12. The graph of a polynomial function rises to the left and falls to the right when its degree is _____________ and its leading coefficient is __________?
    a. even, positive
    b. even, negative
    c. odd, positive
    d. odd, negative

    
    

13. What are the zeros of 𝑃(𝑥) = (𝑥 – 3)(𝑥 + 1) (𝑥 − 1)?
    a. 3, 1, 1
    b. -3, 1, -1
    c. 3, -1, 1
    d. 2, 1, -1

    
    

14. What kind of test uses the leading term of the polynomial function to determine the right-hand and left-hand behaviors of the graph?
    a. Number Line Test
    b. Constant Term Test
    c. Leading Coefficient Test
    d. Multiplicity

    
    

15. What is the factored form of 𝑓(𝑥) = 𝑥³ + 𝑥² − 30𝑥?
    a. 𝑓(𝑥) = 𝑥(𝑥 − 6)(𝑥 + 5)
    b. 𝑓(𝑥) = 𝑥(𝑥 + 6)(𝑥 − 5)
    c. 𝑓(𝑥) = 𝑥²(𝑥 − 6)(𝑥 + 5)
    d. 𝑓(𝑥) = 𝑥²(𝑥 + 6)(𝑥 − 5)

    
    

Additional Activity

Complete the table and sketch the graph. Use a graphing paper.

Polynomial Functions	Factored Form	Degree	Sign of the Leading Coefficient	End-Behavior of the Graph	Number of Turning Points
1. 𝑃(𝑥) = (𝑥 − 2)(𝑥 + 1)(𝑥 + 3)
2. 𝑃(𝑥) = −𝑥³ − 4𝑥² + 7𝑥 + 10

Answer Key

transcriber’s note: this page is printed upside down. The answer key is transcribed in reading order; cells read from the rotated image are marked [?] where the reading is uncertain.

What I Know

1. C
2. A
3. B
4. C
5. D
6. A
7. A
8. B
9. C
10. C
11. B
12. C
13. C
14. A
15. B

What’s In

Polynomial Function	Degree	Leading Coefficient	Non-Polynomial Function
P(x) = x³ + x² + x	3	1	x/3 [?]
P(x) = x³/3 [?]	3	1/3 [?]
P(x) = x⁴ − x/8 [?]	4	1 [?]
f(x) = x²/4 [?]
f(x) = 5 − x/4 [?]
P(x) = 12 [?]	0	12
P(x) = x⁵ + 1 [?]	5	1
P(x) = x⁶ + x⁴ + 3x [?]	6	1
P(x) = 5 − x¹¹ [?]	11	−1

transcriber’s note: several rows of this table were read from the rotated image, and the rows are not in a confirmed order. Verify against the printed page.

What’s More

Assessment 1

Factored Form	Sign of the Leading Coefficient	Degree	End-Behavior of the graph
P(x) = (1 − x)(2 − x)(3 − x) [?]	Positive	3	Falls left and Rise right
P(x) = (2 − x)(3 − x)(3 − x)… [?]	Positive	3	Falls left and Rise right
P(x) = (x + 1)(x − 3)(x − 4) [?]	Positive	3	Falls left and Rise right
P(x) = (2x − 1)(x − 2)(x + 1)(x − 1)… [?]	Positive	4	Rise left and Rise right

Assessment 2

Zeros	Multiplicity (c) of Zero	Characteristic of the Multiplicity	Behavior of the graph
0	1	Odd	Crosses the x-axis
2	1	Odd	Crosses the x-axis
−1	1	Odd	Crosses the x-axis
3	2	Even	Touches the x-axis

x	−2	−1	0	1	2
P(x)	−200	0	0	−8	0

transcriber’s note: this unlabeled table of values is printed below Assessment 2 on the page; its header row is the x values, and its header and cell alignment were read from the rotated image.

What’s More

Assessment 3

Degree	Number of Turning Points
4	3
4	3
4	3
5	4
4	3
2	1

transcriber’s note: the rows of this table were read from the rotated image; verify against the printed page.

Assessment 4

1. 𝑃(𝑥) = 𝑥(𝑥 − 1)²(𝑥 + 2)³

[chart: graph of a polynomial function; Assessment 4, item 1]

2. 𝑃(𝑥) = 2𝑥³ + 5𝑥² − 𝑥 − 2 [?]

[chart: graph of a polynomial function; Assessment 4, item 2]

What I Have Learned

1. factored
2. Leading Coefficient Test
3. zeros
4. table of values
5. smooth continuous

What I Can Do

a) 𝑃(𝑥) = 4𝑥³ − 80𝑥² + 375𝑥

b) Graph

[chart: graph of P(x) = 4x³ − 80x² + 375x; What I Can Do part b]

Assessment

1. B
2. C
3. C
4. B
5. A
6. A
7. C
8. B
9. B
10. B
11. D
12. D
13. C
14. C
15. B

Additional Activity

Polynomial Functions	Degree	Sign of the Leading Coefficient	Behavior of the Graph	Number of Turning Points
1.	3	positive	falls left, rises right	2
2.	3	negative	Rises left, falls right	2

[chart: graph of a polynomial function; Additional Activity, item 1]

[chart: graph of a polynomial function; Additional Activity, item 2]

References

Callanta, Melvin M. et.al, Mathematics – Grade 10 Learner’s Module. Pasig City, REX Bookstore, Inc. 2015.

Capul, Erist A. et.al, Next Generation Math. Makati City, Diwa Learning Systems, Inc. 2015.

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