10
Mathematics
Quarter 1 - Module 8
FACTORING POLYNOMIALS AND
ILLUSTRATING POLYNOMIAL EQUATIONS
How I am
going to
factor
polynomials?
What is a
polynomial
equation?
(x +4)(x–2)
Department of Education ● Republic of the Philippines
Mathematics- Grade 10
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Quarter 1 - Module 8: FACTORING POLYNOMIALS and
ILLUSTRATING POLYNOMIAL EQUATIONS
Fifth Edition, 2021
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10
Mathematics
Quarter 1 - Module 8
FACTORING POLYNOMIALS and
ILLUSTRATING POLYNOMIAL
EQUATIONS
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Table of Contents
Page
COVER PAGE
COPYRIGHT PAGE
TITLE PAGE
TABLE OF CONTENTS
OVERVIEW
PRE- ASSESSMENT
Lesson 1. Factoring Polynomials (Day 1- 4)
Pre- Assessment
Prior Knowledge
Activity 1. Find My Products.
Presentation
Describing Polynomials
Factoring Special Products
Factoring Polynomials by Grouping
Concept Development
Factoring Polynomials with the use of GCF/ monomial factor
Factoring Polynomials by using Quadratic Formula
Factoring Polynomials by Inspection
Factoring Polynomials by AC Method
Factoring Special Products
Factoring Perfect Square Trinomial
Factoring by Grouping
Factoring by using Synthetic Division
Activities
Guided/Controlled Practice: Activity 1. What’s The Other Factor?
Independent Practice: Activity 2. Find My Factors.
Guided/Controlled Practice: Activity 3. Identify Me.
Independent Practice: Activity 4. Find My Special Factors.
Activity 5. Fill Me In!
Assessment
Activity 1. What’s My Pattern?
Application
Activity 1. Let’s Apply!
Activity 2. Complete Me!
6
6
7
10
11
12
13
15
18
19
19
19
19
20
20
20
20
20
20
21
Lesson 2. Illustrating Polynomial Equations (Day 1-2)
Pre - Assessment
Prior Knowledge
Activity 1. Play and Learn!
Identifying Several Types of Polynomial
Activity 2. What Makes Me True?
Presentation
Activity 1. Match Me With My Equations.
22
22
23
23
23
24
24
25
Concept Development
1
2
3
3
4
4
4
4
26
Describing Polynomials
Writing the Polynomial in Standard Form
Activities
Guided/Controlled Practice: Activity 1. Math Henyo
(Textify Me)
Independent Practice: Activity 2. Combine My Parts.
Assessment
Activity 1. Lead Me to the Formula.
Activity 2. Follow My Destiny!
Application
Activity 1. Make it Real!
Activity 2. Construct Me.
Independent Assessment: Activity 2. Relate then Connect!
Activity 4. Which is Which?
POST - ASSESSMENT
GENERALIZATION/SYNTHESIS
ANSWER KEY
REFERENCES
26
27
27
27
28
28
28
29
29
30
30
30
30
32
Overview
What comes to your mind when you hear the word(s) polynomials or polynomial
equations? Have you ever wondered if you actually use it in a real- life situations?
Every activity in this module is uniquely designed to help you show where and when
polynomials and polynomial equation are used in the real world.
The scope of this module permits it to be used in the new learning situations.
The language used recognizes the diverse vocabulary level of students. The lessons
and activities are arranged to cater your needs following the standard sequence of the
course.
The module is divided into two lessons namely:

Lesson 1 – Factoring Polynomials

Lesson 2 – Illustrating Polynomial Equations
After going through this module, you are expected to:
Learning Competency 1: factors polynomials (M10AL-Ih-1) (4 days)
Sub-tasks: 1. Find the factors of polynomials by using different strategies (by
inspection, AC method, Quadratic Formula, Synthetic Division, and
by applying the Factor Theorem.
2. Find the factors of the following special products;
a. Perfect Square Trinomial
b. Difference Between Two Squares
c. Sum and Difference of Two Cube
Learning Competency 2: illustrates polynomial equations (M10AL-Ii-1) ( 2 days )
Sub-tasks: 1. illustrate, identify and define polynomial equations.
2. solve problems in real-life situation.
MATERIALS NEEDED:
Algebra Tiles
Calculator
Video Links
References
1
Pre-Assessment
Directions: Find out how much you already know about polynomials and
polynomial equations. Choose the letter that you think best answers the
question. Please answer all items. Take note of the items that you missed to
answer correctly and find the right answer as you go through this module.
1. What is the greatest common factor of 20xy2, 16x2y3, and 36x3y4?
A. 4xy
B. 4xy2
2
3
C. 4x y
D. 4x2y2
2. Which of the following is the factored form of x3 + 3x2 – 10x - 24?
A. (x + 4) (x -3) (x + 2)
B. (x - 4) (x -3) (x + 2)
C. (x - 4) (x -3) (x - 2)
D. (x + 4) (x + 3) (x - 2)
3. Factor P(x) = x4 + x3 +x2 +x.
A. x (x +1)(x2 +1)
C. x (1)(x2 +1)
B. x (x - 1)(x2 +1)
D. x (-1)(x2 +1)
4. Find a cubic polynomial equation with roots -2,2, and 4.
A. x3 + 4x2 – 4x + 16 =0
B. 10x3 - x2 – x + 16 =0
C. x3 - 4x2 – x + 16 =0
D. x3 -4x2 – 4x + 16 =0
5. Find the factors of 3x4 + x3 -52x2 -124x -80.
A. (x – 2) (x +2) (3x + 4) (x-5)
B. (x + 2) (x +2) (3x + 4) (x-5)
C. (x – 2) (x -2) (3x + 4) (x-5)
D. (x – 2) (x +2) (3x - 4) (x-5)
6. Which of the following represents a polynomial equation?
A. x2 – 4 = 0
B. √x +5 = 0
C. y - 2 - 4 = 0
D. x >2
7. Which of the following statements is NOT true about a polynomial?
A. It contains a plus or minus sign between each term.
B. It contains a constant term or it can be a constant term.
C. It contains a variable in the denominator
D. It has an exponent or degree.
8. What do you call this equation, x2 + 5x + 3 = 0?
A. linear
B. monomial
C. trinomial
D. quadratic
9. What theorem states that any polynomial of degree n has n roots?
A. Factor Theorem
B. Fundamental Theorem of Algebra
C. Rational Root Theorem
D. Zero Product Property
10. Which of the following polynomials has a multiplicity of 3 roots?
A. (X +2)3 (X-1)2
B. (X +2)2 (X-1)
C. (X +2) (X-1)
D. X (X +3)
If you are done, try to check your pre-assessment. Answer key is provided on
page 36 of this module. If you get a perfect score or 8 out of 10 without any help from
others, you may skip this module lesson. But, if you missed few items or more than
three items you may continue doing all the activities at your own desire. You may
submit your rating sheet to your teacher for this pre-assessment.
2
LESSON
1
FACTORING POLYNOMIALS
After working on this lesson, you should be able to:
1. find the factors of polynomials by using different strategies such as:
a. by Inspection, AC method, Quadratic Formula, Synthetic Division, and
Factor Theorem
2. find the factors of the following special products;
a. Perfect Square Trinomial
b. Difference Between Two Squares
c. Sum and Difference of Two Cube
Materials needed:
Paper, pencil, bond papers/mathematics activity notebook
Pre – Assessment
Let us find out first what you already know about factoring polynomials.
Try to answer all items. Take note of the terms/equations that you were not able
to answer correctly and revisit them as you go through this section for selfcorrection.
1. What is the GCF of 5k3p – 3kp2 + k3p5?
A. k
B. kp
C. 3kp
D. 5kp
2. Find the factor of 15x2- 5x using the GCF.
A. 3x (5x2 – 1)
B. 5x (3x2 – 1)
C. 3x (5x – 1)
D. 5x (3x – 1)
3. Calculate the product of 2x ( b + 5) ( b – 5 ).
A. 2bx – 50
B. 2bx – 50x
C. 2b2x - 50
D. 2b2x – 50x
4. Which of the following polynomial is NOT factorable?
A. x2- 5x + 4
B. x2- 4x + 4
C. x2 + 4x + 4
D. x2 + 2x + 1
5. Decide which of the following is the complete factorization of 12x2 – 8x – 15?
A.(4x + 3)(3x – 5) B. (6x - 5)(2x + 3) C. .(6x + 5)(2x – 3) D. cannot be factored
Please check if your answers are correct through the Answer Key on page 36.
3
Prior knowledge
Factoring is a useful skill in real life. Common applications include:
dividing something into equal pieces, exchanging money, comparing prices,
understanding time and making calculations during travel.
Welcome to factoring polynomials! Before you start learning how to factor
polynomials, let’s see what you’ve learned so far when you were in grade 7 to grade
9. This activity will help you understand the lesson. Match the following factors to their
product.
Activity 1. Find My Product
A (Factors)
B (Product)
2
a. m -4mn -21n2
1. 2(x-14)
2. 2x (3x2 – 2x + 1)
b.x2 -9x +18
3. (x + 4) (x + 3)
c. 6x3 - 4x2 + 2x
4. (x - 3) (x – 6)
d. x2 + 7x+12
5. (m – 7n) (m + 3n)
e. 2x - 28
Well, how was it? Do you think you fared well? Please check your answers if they
are correct using the answer key on page __. Did you get a good score? If all your
answers are correct, very good! You may still study the module to review what you
already know. Who knows, you might learn more new things as well.
If you got a low score, don’t feel bad. This only means that this module is right for
you. It will help you understand some important concepts that you can apply in your
daily life. If you study this module carefully, you will learn the answers to all the items
in the test and a lot more! Are you ready?
You may now go to the next page to begin Lesson 1.
Presentation
Polynomials
Polynomial expression P(x) is an expression of the form P(x) = 𝒂𝒏 𝒙𝒏 +
𝒂𝒏−𝟏 𝒙𝒏−𝟏 + 𝒂𝒏−𝟐 𝒙𝒏−𝟐 + … + 𝒂𝟎 , where the nonnegative integer n is called degree of
polynomial 𝒂𝒏 is the leading coefficient, 𝒂𝟎 is the constant term and coefficients
𝒂𝟎 , 𝒂𝟏,… 𝒂𝒏 are real numbers.
We can rewrite polynomial expressions in a simpler form through its factors
through factorization or factoring. A factor of polynomial P(x) is any polynomial which
divides evenly into P(x). For example, x-5 is a factor of the polynomial x2-25. And the
factors of x2-25 are (x-5) (x +5).
4
You have multiple factoring options to choose from when finding the factors of
polynomials. The greatest common factor, factoring by inspection, AC method, and
the quadratic formula are some of the techniques in finding factors of a polynomial. If
the polynomial is a trinomial, you can use the FOIL method for multiplying binomials
backward by checking your answer.
Factoring Special Products
There are some special products that can be used in getting the factors of
polynomials by identifying them. Special products are called "special" because they
do not need long solutions in finding the factors. These special products can be
recognized through their patterns. The different types of special products are perfect
square trinomial, difference of two squares, and sum and difference of two cubes.
Different Patterns for Special products:
1) Perfect Square Trinomial
A2 + 2AB + B2 = (A + B )2 Sum of Squares
A2 – 2AB + B2 = (A - B )2 Difference of Squares
2. Difference of Two Squares
A2 – B2 = (A + B) (A – B)
3. Sum and Difference of Two Cubes
A3 + B3 = (A + B) (A2 - AB+ B2)
Sum of Two Cubes
A3 - B3 = (A - B) (A2 + AB+ B2)
Difference of Two Cubes
Factoring Polynomials by Grouping
Factoring by grouping is a method of factoring that works on four-term
polynomials that have a specific pattern to them. It requires the original polynomial to
have a specific pattern that not all four term polynomials will have. If you do the
factorization in step three and the two groups don't have a common factor, then you
need to go back to square one and try a different approach.
Another technique in finding the factors of the polynomials specially if the
polynomial is quadratic and the factors are not obvious or factorable, then you can use
quadratic formula.
Recall in your lesson 1 that the quadratic formula.
𝑥=
−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎
Factoring by grouping
There’s a factor for every root, and vice versa. The factor theorem states that
(x−r) is a factor if and only if r is a root. It implies also that you can go back and forth
between the roots of a polynomial and the factors of a polynomial. In other words, if
you know one, you know the other. At times, your teacher or your textbook may ask
you to factor a polynomial with a degree higher than two. If you can find its roots, you
can find its factors.
5
In symbols, the factor theorem states that if x – c is a factor of the
polynomial f(x), then f(c) = 0. The variable c is a zero, a root or a solution, whatever
you want to call it (the terms all mean the same thing)
Concept Development
Let’s explore!
In this lesson you will use the Greatest Common Factor (GCF), factoring by
Inspection, AC method, and quadratic formula in finding the factors of polynomials.
GCF is the largest polynomial that will divide evenly into that polynomial.
Inspection factoring is just taking the factors of the last term and finding a set of
factors that will create a sum of the middle term. And Quadratic Formula is a
formula that gives the solutions of the quadratic equation ax2 + bx + c = 0 and that is
usually written in the form of
𝑥=
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
.
To further understand how to factor polynomials, eight sample problems are
provided below with their step-by-step solutions. Study each problem carefully.
Let’s Solve the following examples;
Example 1: Factor 4x4 - 8x3
Steps
Step 1. Arrange the
polynomial in descending
order.
Figure/Expression
4x4
-
Arrange the polynomial
according to their
degree or exponents.
8x3
Step 2. Look for the
greatest factor common to
every term
4𝑥 4 − 8𝑥 3
Step 3. Divide the
polynomial by the factor
found in step 1. The
quotient is the other factor
4𝑥 4 − 8𝑥 3
Step 4. Express the
polynomial as the product
of the two factors.
2𝑥 3
2𝑥 3
Discussion
= 2𝑥 − 4
Hence the factors of
4x4 - 3x3 = 2𝑥 3 (2𝑥 − 4)
6
Since 2x3 is common
to all terms, divide the
polynomial by 2x3.
2𝑥 3 (2𝑥 − 4) are
factors of
4x4 - 8x3.
Check mentally by
multiplying the factors
using distributive
property.
Example 2: Factor 6x3 - 4x2+ 2x.
Steps
Figure/Expression
Step 1. Arrange the
polynomial in descending
order.
6x3
Step 2. Look for the
greatest factor common to
every term
Step 3. Divide the
polynomial by the factor
found in step 1. The
quotient is the other factor
Step 4. Express the
polynomial as the product
of the two factors.
-
4x2+
2x
6𝑥 3 −4𝑥 2 + 2𝑥
2𝑥
6𝑥 3 −4𝑥 2 + 2𝑥
2𝑥
= 3𝑥2 −
2𝑥 + 1
Hence the factors of
6x3 - 4x2+ 2x
= 𝟐𝒙 (𝟑𝒙𝟐 − 𝟐𝒙 + 𝟏)
Example 3. Factor 3𝑥 2 − 2𝑥 + 1 using quadratic formula.
Steps
Figure/Expression
Step 1. Arrange
the polynomial
in descending
order.
Step 2. Look
for the greatest
factor common
to every term.
Step 3. Use
quadratic
formula in
finding the
factors of
polynomials.
3𝑥 2 − 2𝑥 + 1
Discussion
Write the polynomial in
standard form
Since 2x is common to
all terms, divide the
polynomial by 2x.
2𝑥 (3𝑥 2 − 2𝑥 + 1) are
factors of
6x3 - 4x2+ 2x
Check mentally by
multiplying the factors
using distributive
property.
Discussion
The polynomial is in
descending order
There is no GCF. Then
transform the
3𝑥 − 2𝑥 + 1
polynomial expression
3𝑥 2 − 2𝑥 + 1 = 0
to polynomial equation.
Transform the
3𝑥 2 − 2𝑥 + 1 = 0
expression to
From the equation, a =3, b = -2,
polynomial equation.
c = 1.
Substitute in the quadratic formula Then use the quadratic
formula since the
−𝑏±√𝑏2 −4𝑎𝑐
expression is
𝑥=
2𝑎
quadratic. Find the
−(−2)±√(−2)2 −4(3)(−1)
roots of quadratic
𝑥=
2(3)
expression. X1= 1 and
2 ±√16
x2 = -1/3.
2
𝑥=
𝑥=
6
2 ±√16
6
=2±
4
6
Therefore;
2+4 6
x1 =
= =1 ;
6
6
2 − 4 −2 −1
x2 =
=
=
6
6
3
7
Step 4.
Transform the
roots to its
factors.
X1= 1
1( x= 1)
x=1
x-1 = 1-1
x-1 = 0
x2 = -1/3
3( x = -1/3)
3x= -1
3x + 1= -1 + 1
3x + 1 = 0
Get the two roots. If one of
the roots has a
denominator greater than
1, multiply the denominator
by it equation to have a
factors that is not in fraction
form. Then apply addition
property of equality.
Check by multiplying the
factors using FOIL Method.
Step 5. Express
Hence the factors of
the polynomial
3𝑥 2 − 2𝑥 + 1.= (x -1) (3x + 1 )
as the product of
the two factors.
Note: You can still factor 𝟑𝒙𝟐 − 𝟐𝒙 + 𝟏 by using quadratic formula.
2
−𝑏±√𝑏 −4𝑎𝑐
2𝑎
Use quadratic formula 𝑥 =
in finding the factors of ax2 + bx +c when
the factors are not obvious and if it is not factorable.
Example 4. Factor 2𝑥 2 − 11𝑥 − 21 using quadratic formula.
Steps
Figure/Expression
Step 1. Arrange
the polynomial in
descending
order.
Step 2. Look for
the greatest
factor common
to every term.
Step 3. Use
quadratic
formula in
finding the
factors of
polynomials.
2𝑥 2 − 11𝑥 − 21
Discussion
The polynomial is in
descending order
2𝑥 2 − 11𝑥 − 21
2𝑥 2 − 11𝑥 − 21 = 0
There is no GCF. Then
transform the polynomial
expression to polynomial
equation.
2
Transform the expression
2𝑥 − 11𝑥 − 21 = 0
to polynomial equation.
From the equation, a =2, b = -11
Then use the quadratic
c = -21.
Substitute in the quadratic formula formula since the
expression is quadratic.
−𝑏 ± √𝑏 2 − 4𝑎𝑐
Find the roots of
𝑥=
quadratic expression. X1=
2𝑎
7and
−(−11) ± √(−11)2 − 4(2)(−21)x2 = -3/2.
2(2)
𝑥=
𝑥=
11 ±√289
2(2)
11 ±17
4
= 11 ±
17
4
11 + 17 28
x1 =
=
=7 ;
4
4
11 − 17 −6 −3
x2 =
=
=
4
4
2
8
Step 4.
Transform the
roots to its
factors.
X1= 7
1( x= 7)
x=7
x-7 = 7-7
x-7 = 0
x2 = -3/2
2( x = -3/2)
2x= -3
2x + 3= -3 + 3
2x + 3 = 0
Step 5. Express Hence the factors of
the polynomial 2𝑥 2 − 11𝑥 − 21= (x -7) (2x + 3
as the product of )
the two factors.
Get the two roots. If one of
the roots has a denominator
greater than 1, multiply the
denominator by it equation
to have a factors that is not
in fraction form. Then apply
addition property of equality.
Check by multiplying the
factors using FOIL Method.
In continuation of example 2, the factors of 6x3 - 4x2+ 2x are 2x (x -1) (3x + 1).
The next technique in finding the factors is factoring through inspection.
Sydney made up this riddle for Vince “I’m thinking of two numbers”. The product
of the numbers is -24, and the sum of the numbers is -2. Vince tried a few combinations
in his head before coming up with the answer: “The numbers must be -6 and +4.” Is
any other combination possible?
The situation above is an example of finding the factors by inspection. To fully
understand let us study example five and six.
Example 5. Factor x2 + 3x – 10 using inspection.
(x + a)(x + b) = x2 + bx + ax + ab
3x
-10
Figure/Expression
Steps
Step 1. Arrange
the polynomial
in descending
order.
Step 2. Look for
the greatest
factor common
to every term
Step 3. Place
the factors for
the first term of
the trinomial in
the front of each
set of
parentheses.
x2
The polynomial is in descending
order.
+ 3x – 10
(
(x +
)
Discussion
(
)
) (x -
)
There is no GCF. Therefore, the
factoring process is begun by
opening two sets of parentheses.
The factors of the first term x2 is
(x) (x). Since the last term is
negative, it indicates that they
don’t have the same sign.
9
Step 4. Find the
factors of the
third term.
Step 5. Express
the polynomial
as the product
of the two
factors.
Find the product of
Factors of -10 = (5) (-2) -10. (5) (-2) =10. Since the
(x + 5) (x - 2 )
middle term is positive 3x, which
2
(x + 5) (x - 2 )= x – 2x + means that the highest factor
5x -10
must be of positive sign and the
other factor is negative. Use
3x FOIL Method to get the sum of
the outer and inner term, -2x
and 5x, must be 3x.
Hence the factors of
Check by multiplying the factors
x2 + 3x – 10 = (x + 5) (x - using FOIL Method.
2 )
Example 6. Factor x2 - 7x – 18 using inspection.
(x + a)(x + b) = x2 + bx + ax + ab
-7x
-18
Figure/Expression
Steps
Step 1. Arrange
the polynomial in
descending order.
Step 2. Look for
the greatest factor
common to every
term
Step 3. Place the
factors for the first
term of the
trinomial in the
front of each set of
parentheses.
Step 4. Find the
factors of the third
term.
The polynomial is in
descending order
x2 - 7x – 18
(
(x +
)
(
)
) (x -
)
There is no GCF.
Therefore, the factoring
process is begun by
opening two sets of
parentheses.
The factors of the first term
x2 is (x) (x). Since the last
term is negative, it
indicates that they don’t
have the same sign.
Factors of -18 = (-9) (2)
(x - 9) (x + 2 )
(x - 9) (x + 2 ) = x2 + 2x – 9x -18
-7x
Step 5. Express
the polynomial as
the product of the
two factors.
Discussion
Hence the factors of
x2 - 7x – 18 = (x - 9) (x + 2 )
Find the product of
-18. (-9) (2) = -18. Since
the middle term is positive 7x, which means that the
highest factor must be of
negative sign and the other
factor is positive. Use FOIL
Method to get the sum of
the outer and inner term, 2x
and -9x, must be
-7x.
Check by multiplying the
factors using FOIL Method.
10
Factoring through inspection is tedious and a long process for some polynomial’s
expressions. Thus, alternative way of technique would be very beneficial. Another way
of factoring is through grouping or AC method.
Closely look at the given steps and compare them with trial and error of factoring
by inspection.
Example 7. Factor a2 – 7ab + 12b2 using AC method.
Steps
Figure/Expression
Step 1. Arrange the
polynomial in
descending order.
Step 2. Look for the
greatest factor
common to every term
a2
– 7ab +
12b2.
a2 – 7ab + 12b2.
Step 3. Find the
product of the leading
term and the last term.
a2 – 7ab + 12b2.
(a2)(12b2) = 12 a2b2
Step 4. Find the
factors of 12 a2b2
whose sum is -7ab
(-4ab) + (– 3ab) = -7ab
The polynomial is in
descending order.
There is no GCF.
Therefore, the factoring
process is begun by
opening two sets of
parentheses.
a2 – 7ab + 12b2.
(a2)(12b2) = 12a2b2
Step 5. Rewrite the
a2 -4ab – 3ab + 12b2
trinomial as a four-term
expression by
replacing the middle
term with the sum of
the factors.
Step6. Group terms
a2 -4ab – 3ab + 12b2
with common factors
a (a -4b) – 3b(a - 4b)
and get the GCF of the
first and second group.
Step7. Express the
polynomial as the
product.
Discussion
(a -4b) (a -3b)
Hence; The factors of
a2 – 7ab + 12b2
= (a -4b) (a -3b)
11
(-4a)(-3b) = 12 a2b2
Add the two product the
answer is the middle term
Replace – 7ab by
-4ab – 3ab.
a (a -4b) =a2 -4ab
-3b(a - 4b) = - 3ab + 12b2
Check mentally by
multiplying the factors
using distributive
property.
Factor out the common
binomial factor and write
the remaining factor as a
sum or difference of the
common monomial
factors.
Example 8. Factor 6b2 – 5b - 6 using AC method.
Steps
Figure/Expression
Step 1. Arrange the
polynomial in
descending order.
Step 2. Look for the
greatest factor
common to every term
6b2
The polynomial is in
descending order
– 5b - 6
6b2 – 5b - 6
Step 3. Find the
product of the leading
term and the last term.
6b2 – 5b - 6
(6b2)(-6) = -36 b2
Step 4. Find the
factors of -36 b2 whose
sum is -5b
(-9b) + (4b) = -5b
Step 5. Rewrite the
6b2 -9b + 4b -6
trinomial as a four-term
expression by
replacing the middle
term with the sum of
the factors.
Step6. Group terms
6b2 -9b + 4b -6
with common factors
3b (2b -3) + 2 (2b -3)
and get the GCF of the
first and second group.
Step7. Express the
polynomial as the
product.
Discussion
(2b - 3) (3b + 2)
Hence; The factors of
6b2 – 5b - 6 = (2b - 3) (3b +
2)
There is no GCF.
Therefore, the factoring
process is begun by
opening two sets of
parentheses.
6b2 – 5b – 6
.
(6b2)(-6) = -36 b2
(-9b) (4b) = -36 b2
Add the two product the
answer is the middle term
Replace – 5b by
-9b + 4b.
3b (2b -3) = 6b2 -9b
2 (2b -3) = 4b -6
Check mentally by
multiplying the factors
using distributive
property.
Factor out the common
binomial factor and write
the remaining factor as a
sum or difference of the
common monomial
factors.
Perfect Square Trinomial
When you square a binomial, the product is called a perfect square trinomial.
(x + 4) 2 = (x + 4) (x + 4) = x2 + 8x + 16
square of the 1st term
twice the product
square of the last term
of the two terms
In general, polynomials that are perfect square trinomials factor as follows:
(x + y)2 = x2+ 2xy + y2
(x– y)2 = x2 – 2xy + y2
Sum of Squares
Difference of Squares
12
As we can see the square of the binomial is always a trinomial. When a
trinomial is the square of a binomial, we call it a perfect square trinomial. A perfect
square trinomial can always be identified because two of its terms are squares, and
the remaining term is twice the product of their square roots.
The square terms of a perfect square trinomial are always positive. The
remaining term may be positive or negative, depending on whether the binomial is the
sum or the difference of two numbers.
Example 1: Factor x2 - 8x + 16.
Steps
Step 1. Arrange the
polynomial in
descending order.
Step 2. Look for the
greatest factor
common to every
term
Figure/Expression
x2
x2
Discussion
Write the polynomial in
standard form
- 8x + 16
- 8x + 16
Step 3. Examine the
trinomial x2 – 8x +
16.
Get the square root
of the first and last
term. Then multiply
the square root of
the first and last
term by 2.
x2 – 8x + 16
√𝑥 2 = x; √16 = 4, and 2(x)(4)
=8x
Step 4. Identify the
special product
Difference of Squares
Hence the factors of
Factor x2 - 8x + 16 = (x -4 )2 or
(x-4) (x-4)
Difference of Squares
There is no GCF.
Therefore the factoring
process is begun by
opening two sets of
parentheses.
The first term is a
perfect square, the last
term is also a perfect
square, the middle
terms are twice the
product of the square
root of the first term
and the second term
and the sign is
negative.
Check mentally by
using the patterns or
FOIL method.
Example 2: Factor 8x3 + 24x2 + 18x
Steps
Figure/Expression
Step 1. Arrange the
polynomial in
descending order.
8x3 + 24x2 + 18x
Step 2. Look for the
greatest factor
common to every term
2x (4x2 +12x + 9)
13
Discussion
Write the polynomial in
standard form
Since 2x is common
to all terms of
8x3 + 24x2 + 18
Step 3. Examine the
trinomial 4x2+12x + 9.
Get the square root of
the first and last term.
Then multiply the
square root of the first
and last term by 2.
Step 4. Identify the
special product
4x2+12x + 9
√4𝑥 2 = 2x , √9 = 3, and
2(2x)(3) = 12x and the sign
is positive (+).
Sum of Squares
4x2 +12x + 9 = (2x + 3) (2x
+3) or
(2x + 3)2
Hence the factors of
8x3 + 24x2 + 18x =
2x (2x + 3)2 or
2x (2x + 3) (2x +3)
The first term is a perfect
square, the last term is
also a perfect square,
the middle terms are
twice the product of the
square root of the first
term and the second
term and the sign is
positive.
Check by using the
patterns or FOIL
method, the apply the
distributive property.
Difference of Two Squares
The product of the sum and the difference of the same two terms is the
difference of two squares.
( x + 9)(x – 9) = (x) 2 – (9) 2 = x2 – 81
square of the 1st term
square of the last term
x2 – 81 is a difference of two squares.
In general, (A- B) (A + B) = A2 – B2, the factors of A2 – B2 are (A- B) (A + B).
Let us examine 9y2 - 4. Now √9𝑦 2 = 3y, and √4 = 2. Hence 9y2 - 4= (3y + 2) (3y –2).
Example 3: Factor 12h2 -75.
Steps
Step 1. Arrange the
polynomial in
descending order.
Figure/Expression
12h2 -75
Write the polynomial
in standard form.
Step 2. Look for the
3 (4h2 – 25)
greatest factor
common to every term.
Step 3. Examine the
trinomial 4h2 – 25.
Get the square root of
the first and last term.
Then multiply the
square root of the first
and last term by 2.
Step 4. Identify the
special product.
Discussion
Since 3 is common
to all terms of
12h2 -75.
4h2 – 25
√4ℎ2 = 2h √25 = 5,
Sum and Difference
4h2 – 25 = (2h -5) (2h +
The first term is a
perfect square; the
last term is also a
perfect square.
5)
Sum and Difference
Hence the factors of
12h2 -75= 3 (2h -5) (2h + 5)
or 3 (2h -5)2
14
Check by using the
patterns or FOIL
method, the apply the
distributive property.
Sum and Difference of Two Cubes
The sum of two cubes has two factors, one binomial and one trinomial. The
binomial factor is the sum of the cube roots of the given terms while the trinomial
factor is made up the sum of the square of the first term of the binomial factor, the
negative of the product of the first and the second terms of the binomial factor, and
the square of the second term of the binomial factor.
A3 + B3 = (A + B)(A2 – AB + B2)
The difference of two cubes has two factors, one binomial and one trinomial.
The binomial factor is the difference of the cube roots of the given terms while the
trinomial factor is made up the sum of the square of the first term of the binomial
factor, the positive of the product of the first and the second terms of the binomial
factor, and the square of the second term of the binomial factor.
A3 - B3 = (A - B)(A2 + AB + B2)
EXAMPLES 4 . Factor a3 + 8.
Solution: The given expression can be written as (a)3 + (2)3,
by factoring using the pattern
A3 + B3 = (A + B) (A2 – AB + B2), we have
= (a + 2) (a2 – (a)(2) +22)
= (a + 2) (a2 -2a + 4), simplifying further
Therefore, the factors of a3 + 8 are (a + 2)(a2 -2a + 4)
EXAMPLES 5. Factor 2x4 – 16x
Solution: First, look for the greatest factor common to every term.
Since 2x is common to all terms, divide the polynomial by 2x.
2x4 – 16x = 2x (x3 - 8).Then factor the difference of two cubes.
A3 - B3 = (A - B)(A2 + AB + B2), we have
(x3 – 8) = (x - 2) (x2 + (x)(2) +22)
= (x - 2) (x2 +2x + 4), simplifying further
Therefore, the factors of 2x4 – 16x are 2x (x- 2) (x2 +2x + 4).
Example 1: Factor x3 + x2 – x – 1
Steps
Figure/Expression
Step 1. Arrange the
x3 + x2 – x -1
polynomial in
descending order.
Step 2. Group the first (x3 + x2) + (–x – 1)
two terms together and
then the last two
terms.
15
Discussion
Write the polynomial in
standard form.
Be careful with the negative
sign in the second group.
(x3 + x2) + (–x – 1)
Since the operation now is
addition.
Step 3. Find the GCF
and factor it out.
x2(x + 1) – 1(x + 1)
Step 4. Factor it out
and find the factors of
(x2 -1).
Step 4. Express the
polynomial as the
product.
(x + 1) (x2 -1)
(x + 1)(x + 1)(x – 1),
x2(x + 1) = (x3 + x2)
and
-1 (x +1) = – x -1
x2 – 1 is a difference of
squares and factor again.
Hence, the factors of
x3 + x2 – x – 1 = (x + 1)
(x + 1)(x – 1),
Check mentally by
multiplying the factors
Example 2: Factor x3 + x2 - 3x - 3
Steps
Figure/Expression
Step 1. Arrange the
x3 + x2 - 3x - 3
polynomial in
descending order.
Step 2. Group the first (x3 + x2) + (-3x - 3)
two terms together and
then the last two
terms.
Step 3. Find the GCF
and factor it out.
x2(x + 1) - 3(x + 1)
Step 4. Factor it out
and find the factors of
(x2 - 3).
(x + 1) (x2 - 3)
From the equation x2 – 3 = 0 , a = 1,
Discussion
Write the polynomial in
standard form.
Be careful with the
negative sign in the
second group.
(x3 + x2) + (-3x-3). Since
the operation now is
addition.
x2(x + 1) = (x3 + x2)
and
- 3(x + 1) = - 3x - 3
Since (x2 - 3) is not
factorable. You can
use quadratic formula
in finding the factors of
a quadratic polynomial.
b=0
c=3
Substitute in the quadratic formula.
𝑥=
−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎
𝑥=
𝑥=
𝑥=
−(0)±√(−0)2 −4(1)(−3)
2(1)
0±√12
2
0±2√3
2
=±
2√3
2
𝑥 = ±√3
Therefore, the factors x3 + x2 - 3x - 3 of are (x+1), (x- √𝟑, and (x+ √𝟑)
16
Example 3: Factor 3x3 + 15x2 +2x + 10
Steps
Figure/Expression
Step 1. Arrange the
polynomial in
descending order.
Step 2. Group the first
two terms together and
then the last two terms
Step 3. Find the GCF
and factor it out.
3x3
+
15x2
+2x + 10
Discussion
The polynomial is
already in order
(3x3 + 15x2) + (2x + 10) No need to change the
sign since the second
group are all positive
3x2 (x + 5) + 2(x + 5)
3x2 (x + 5)= (3x3+15x2)
and
2(x + 5) = (2x + 10)
2
Step 4. Factor it out
(x + 5) (3x +2)
3x2 +2 is not factorable.
and find the factors of
You can use quadratic
2
3x +2.
formula in finding the
factors of a quadratic
polynomial.
From the equation 3x2 +2 = 0 , a = 3,
b=0
c=2
Substitute in the quadratic formula.
𝑥=
−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎
𝑥=
𝑥=
𝑥=
𝑥=
−(0)±√(−0)2 −4(3)(2)
2(3)
0±√−24
6
0±2√−6
6
√−6
± 3
=±
2√−6
6
Note that √−1 = 𝑖, an imaginary number, not real
Therefore, the factors x3 + x2 - 3x - 3 of are (x+5), (x-
√𝟔
𝒊),
𝟑
and (x+
√𝟔
𝒊)
𝟑
Illustrative Examples:
Example 1. Find the factors of P(x) where the zeros are 3, -2, -1, and 1.
Solution: Applying the zero product property.
x= 3 ;
x = -2 ;
x = -1 ;
x =1
(x – 3) =0
(x + 2) = 0
(x – 1)=0
(x + 1)=0
Thus, the factors of P(x) = (x – 3) (x + 2) (x – 1) (x + 1)
Example 2. Solve for the other factors of 𝑃(𝑥) = 𝑥 3 − 2𝑥 2 − 3𝑥 + 10, given that
(x + 2) is one of its factors.
Solution:
By Factor Theorem, 𝑥 + 2 is a factor of 𝑥 3 − 2𝑥 2 − 3𝑥 + 10. Then,
𝑥 3 − 2𝑥 2 − 3𝑥 + 10 = (𝑥 + 2) • 𝑄(𝑥)
Determine 𝑄(𝑥) using synthetic division.
-2
1 -2 -3 10
-2
8 -10
1 -4
5
0
2
𝑄(𝑥) = 𝑥 − 4𝑥 + 5
17
The equation 𝑥 2 − 4𝑥 + 5 = 0 is a depressed equation of 𝑃(𝑥).
A depressed equation of P is an equation which has a degree less than that of P.
The depressed equation is a quadratic trinomial. Use this to find the other zeros. Since
𝑥 2 − 4𝑥 + 5 is not factorable, use the quadratic formula in finding the values of x.
\
Note: A quadratic formula is helpful in finding the roots of ax2 + bx +c if the
factors are not obvious or if it is not factorable.
From the equation, a = 1,
b = -4,
Substitute in the quadratic formula.
𝑥=
c = 5.
−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎
𝑥=
𝑥=
𝑥=
−(−4)±√(−4)2 −4(1)(5)
2(1)
4±√16−20
2
4±√−4
2
𝑥 = 4±
x1 =
2𝑖
2
= 4±
√−4
2
Note that √−1 = 𝑖, an imaginary number, not real
4 + 2𝑖
4 − 2𝑖
= 2 + 𝑖 ; x2 =
= 2−𝑖
2
4
The zeros of 𝑃(𝑥) are -2, 2 + 𝑖, and 2 – 𝑖
Thus, 𝑥 3 − 2𝑥 2 − 3𝑥 + 10 = (x + 2) (x – (2 + 𝑖) ) (x + (2 – 𝑖) ) or
(x + 2) (x – 2 - 𝑖) (x + 2 – 𝑖)
Example 3. Solve for the other zeros of P(x) = 𝑥 4 − 𝑥 3 − 11𝑥 2 + 9𝑥 + 18, given that
one zero is -3.
Solution:
By Factor Theorem, (x + 3) is a factor of 𝑥 4 − 𝑥 3 − 11𝑥 2 + 9𝑥 + 18. Then,
𝑥 4 − 𝑥 3 − 11𝑥 2 + 9𝑥 + 18 = (x + 3) • Q(x).
Determine Q(x) using synthetic division,
-3
1 -1 -11 9 18
-3 12 -3 -18
1 -4
1
6
0
3
2
Q(x) = 𝑥 − 4𝑥 + 𝑥 + 6.
The equation 𝑥 3 − 4𝑥 2 + 𝑥 + 6 = 0 is the first depressed equation of P(x).
To find the other zeros, try factors of c in the first depressed equation using
synthetic division. For this example, c = 6, the possible factors are {±1, ±2,
±3, ±6}. Try -1,
-1
1 -4 1 6
-1 5 -6
1 -5 6 0
So, x + 1 is a factor of 𝑥 3 − 4𝑥 2 + 𝑥 + 6. This implies that
𝑥 3 − 4𝑥 2 + 𝑥 + 6 = (x + 1)(𝑥 2 − 5𝑥 + 6)
18
The equation 𝑥 2 − 5𝑥 + 6 = 0 is the second depressed equation of P(x).
There are now two zeros known, -3 and -1. Find the other zeros by factoring
the second depressed equation which is a quadratic trinomial.
𝑥 2 − 5𝑥 + 6 = 0
(𝑥 − 2)(𝑥 − 3) = 0
Therefore the zeros of factors of P(x) = 𝑥 4 − 𝑥 3 − 11𝑥 2 + 9𝑥 + 18
(x +1) (x + 3) (x – 2) (x - 3) .
Example 4. Solve the other factors of P(x) = 𝑥 3 − 6𝑥 2 − 𝑥 + 30, given that (x + 2) is
a factor of the polynomial.
Solution:
By Factor Theorem, (x - 2) is a factor of 𝑥 3 − 6𝑥 2 − 𝑥 + 30
Then,𝑥 3 − 6𝑥 2 − 𝑥 + 30 = (x + 2) • Q(x).
Determine Q(x) using synthetic division,
-2
1 -6 -1 30
-2 16 -30
1 -8 15 0
Q(x) = 𝑥 2 − 8𝑥 + 15.
The equation 𝑥 2 − 8𝑥 + 15. = 0 is the first depressed equation of P(x).
Since 𝑥 2 − 8𝑥 + 15 is a quadratic trinomial.
𝑥 2 − 8𝑥 + 15 = 0
(𝑥 − 3)(𝑥 − 5) = 0
Therefore the factors of P(x) = 𝑥 3 − 6𝑥 2 − 𝑥 + 30
(x +2) (x -3 ) (x – 5) .
ACTIVITIES
Let’s go beyond
Guided/Controlled Practice: ACTIVITY 1. WHAT’S THE OTHER FACTOR?
Find the factors of the following polynomial expressions:
1. 2m3y – 12m2y4 = 2m2y (_______)
2. 33w3y2 + 11w2y2 = _____(3w + 1)
3. 9cd4 - 6c2d2 – 3c3d = 3cd (___________)
Independent Practice: ACTIVITY 2. FIND MY FACTORS.
Find the factors of the following polynomial expressions:
4. x2 + 7x -18
5. 8b2 + 26b + 15
Guided/Controlled Practice: ACTIVITY 1. IDENTIFY ME?
Direction: Identify the special products involve.
1. 4z2 – 4z + 1
2. x2 + 12x + 36
3. 16c2 – 64
4. y2 - 81
5. a3 + 125
19
Independent Practice: ACTIVITY 4. FIND MY SPECIAL FACTORS.
Direction: Factor each of the following special products.
1. a3 – 125 = (a-5) (_________)
2. 122 - 52 = (12 – 5) (_____)
3. 2x4 -16x = 2x(____) (____)
Guided/Controlled Practice: ACTIVITY 5. FILL ME IN!
Direction: Find the factors of the following polynomials.
A. Find the following factors of P(x) given the following zeros. ( 2points)
1. -2, -3, and 1 = ( x+2) (x +3) (___)
2. -3, -5, and -1 = ( x + 3) ( x + 5) (___)
Score
11-15
6-10
1-5
Meaning
You have gained complete knowledge on the
given exercises.
You have gained adequate learning on the given
exercises.
You can rework on the given exercises.
ASSESSMENT
Activity 1 What’s My Pattern?
A Identify the pattern/s of special products involve:
1) (x – 6)2 = x2 – 12x + 36____________________________________________
2) (5x + 3y)2 = 25x2 + 30xy + 9y2_______________________________________
3) (x + 5)(x – 5) = x2 – 25 ____________________________________________
4) (5m -3) (5m + 3) = 25m2 -9 __________________________________________
5) (a - 4)(a2 + 4a + 16) = a3 – 64 _______________________________________
It is said that if a polynomial P(x) of degree n, with real coefficients has at most n real
roots. Consider the following polynomial equations. At most how many roots does
each have?
6. 6x4 - 3x3- 24x2 + 12x = 0_________________________.
7.𝑥 4 − 2𝑥 3 − 7𝑥 2 + 8𝑥 + 12 = 0_____________________
APPLICATION (Performance Tasks)
ACTIVITY 1. LET’S APPLY! Answer each of the following completely.
1. If 9x2 + 30x + 25 represent the area of a square, find the binomial that represents
the length of a side of the square.
2.One of the roots of the polynomial equation is x4 +4x3 -16x-16 = 0 is -2. Find the
other factors
20
Lesson
2
Illustrating Polynomial Equations
After going through this module, you should be able to demonstrate
understanding of key concepts of polynomial equations, formulate real-life problems
involving these concepts and solve those using variety of strategies. Furthermore, you
should be able to investigate mathematical relationships in various situations involving
polynomial equations. This lesson is good for 2 days.
LEARNING OUTCOMES:
At the end of the 2 days session, this module will help you:
1. illustrate, identify and define polynomial equations.
2. solve problems in real-life situation.
MATERIALS NEEDED:
Algebra Tiles, Calculator, Video Links, and References
Pre – Assessment
Direction: Let us determine how much you already know about polynomial
equations. Take this test. Read and understand the questions below. Select the
best answer to each item then write your choice on your answer sheet. Do not write
anything on this Module.
1. The following represents polynomial equation EXCEPT…
A. x 2 – 4x = 0
B. 4x +5 = 0
C. y - 4 = 0
D.√x -½ >2
2. Which of the following equations has three terms?
A. x +2
B. (x +1) (x +2) C. x(x2 +2)
D. x2(x2 +2)
3. Which of the following does NOT belong to the group?
A cubic equation
B. linear equation
C. quadratic equation
D. quadratic inequality
4. . What do you call this equation x2 + 5x + 3 = 0 ?
A. binomial
B. monomial
C. trinomial
D. quartic
5 . What theorem states that any polynomial of degree n has n roots?
A. Factor Theorem
B. Fundamental Theorem of Algebra
C. Rational Root Theorem
D. Zero Product Property
If you are done, try to check your pre-assessment. Answer key is provided
on page ___ of this module. If you get a perfect score or 5 out of 5 in the preassessment without any help from others, you may skip this module lesson. But, if
you missed few items or more than three items you may continue doing all the
activities at your own desire. You may submit your rating sheet to your teacher for
this pre-test.
21
Prior knowledge
Let us start this lesson by recalling a linear and quadratic equation. The
knowledge and skills in doing this activity will help you a lot in understanding the
polynomial equation. In going over this lesson, you will be able to identify a
polynomial equation and represent it in different ways.
ACTIVITY 1. PLAY AND LEARN!
STEPS:
1. Choose any number.
2. Multiply your number by a constant 3.
3. Add the sum of your number and 8 to the number you got when you
multiplied.
4. Divide by the sum of your number and 2.
5. The answer is always 4.
Let’s Try This!
1.
2.
3.
4.
Choose 5
Multiply 5 by 3 = 5x3=15
Add (5+8) + 15 =28
Divide 28 by (5+2) =7
Let x=5
x(3) =3x
(X+8)+3x = 4x+8
4x+8 the value of
x=4
X+2
Did you enjoy the activity? Try another one if you have time, you just
follow the steps above. The expression (4x + 8) and (x + 2) at the right of the
table describes a polynomial.
From the previous modules, you have learned how to derive a
polynomial equation by finding the product of two terms or just one term.
Fundamental Theorem of Algebra states that any polynomial of degree n has
n roots. A polynomial has several types such as:
1. Monomial Equations or Linear Equations
An equation which has only one variable term. It can be expressed
as ax +b = 0, where a and b are real numbers.
Example: 2x + 1= 0
2. Binomial Equations or Quadratic Equations
An equation which has only two variable terms. It can be expressed
in the algebraic form as ax2 + bx + c = 0.
Example: 5x2 + 2x + 1 = 0
3. Trinomial or Cubic Equations
An equation which has only three variable terms.
Example: x3 + 2x2 + x +4 = 0
4. Quartic Polynomials
A polynomial of degree 4. It has also 4 real roots.
Example: x4 + x3 + 2x2 + x +4
5. Other polynomials with larger degrees.
Example: x20 – 1= 0
22
Observe that the different types presented is in the form of
a0 xn + a1 xn-1 +……..+ an-2 x2 + an-1 x + an.
Therefore, a polynomial must NOT have the following:
a. Negative exponent
b. Variable in the denominator
c. Fractional exponent
Some examples are shown below.
Examples
1. x+2 = 0
5
2. x2 + = 0
Polynomial /NOT
Polynomial
Polynomial
NOT Polynomial
Reasons
Linear Equation
It has a variable in the
𝑥
denominator
3. y -3 + y + 2 = 0
NOT Polynomial
It has a negative exponent.
4. √x – 5 = 0
NOT Polynomial
The exponent of x is ½
2
5. ½ x - 1 = 0
Polynomial
Quadratic/ Binomial
Equation
You can watch also this video link: www.eHow Education.com// How to
describe a polynomial equation for more information.
ACTIVITY 2. WHAT MAKES ME TRUE?
Directions: Determine whether each expression is a polynomial or NOT.
1. x2 + 2x + 1 = 0
6. ½ x2 - 1 = 0
5
2. 5x3 + 𝑥 = 0
7. ¾ x4 y3 -21x = 0
3. 5x -2 + 5 = 0
8. 9 + √x – 3 = 0
x
4. 10 + 2x + 1 = 0
9. √y + 2 = 0
5. -x2 + 10 = 0
10.¼ x3 + 2x = 0
How did you find the activity? Were you able to describe and illustrate
polynomials?
If you get a perfect score or 8 out of 10 in this activity without any help
from others, you may skip this module lesson. But, if you missed few items
or more than three items you may continue doing all the activities at your own
desire or try another activity in the next section.
Presentation
Let’s explore!
Based on Mendel’s experiment in genetic breeding, certain traits result from
the pairing of two genes, one from the male parent and one from the female. We
can use expressions to summarize the possible outcomes of an experiment. For
example, suppose a white gumamela flower has a genotype WW, a red gumamela
flower has a genotype RR, and a pink gumamela flower has a genotype WR Each
letter represent one of two genes that make up the characteristics.
The two bred flowers offspring can be expressed using an algebra model
called a Punnet square.
23
R
PARENT # 1
RW
RR
WW
WR
W
W
R
PARENT # 2
The sum of the possible results for four offspring can be written as WW +
WR + WR + RR; that is, one white, two pink and one red-gumamela flower.
Suppose we substitute x for W and y for R. The result would be a sum of four
monomials, xx + xy + xy + yy, or x2 +2xy + y2. The two monomials xy and xy can
be combined because they are like terms. Like terms are two monomials that are
the same, or differ only by their numerical coefficients.
The expression x2 + 2xy + y2 is called a polynomial. A polynomial is a
monomial or a sum of monomials. A polynomial must be in the form of a0 xn + a1
xn-1 +……..+ an-2 x2 + an-1 x + an.
For more details, refer to this video link: https://MathHelp.com./Solving
Polynomial Equations
ACTIVITY 1. MATCH ME WITH MY EQUATIONS?
Directions. Refer to the previous presentation on types of Polynomial equations in
order to answer the following activity. Put the letter of the given equation
in the diagram below where you think it belongs.
a. X5 -X4 –X + 2 = 0
b. 5X3 + 3X2 –X + 1 = 0
c. ¾ X + 5 = 0
d. 7X3 + 4X -12 = 0
e.-2X3 + 7X – 2 = 0
f. ½ X2 –X + 2 = 0
g. 5X2 –2X + 3 = 0
h. 2X4 - 1 = 0
i. 5X – 3 = 0
j. X20 – 1 = 0
Monomial equation
(ax + c = 0)
Binomial/quadratic equation
(ax2 +bx +c = 0)
__________
_________
________
_______
_______
other polynomials with
higher degree
( an-1xn-1+ an-2xn-2 +……)
________
________
Trinomial/cubic
polynomial equation
(ax3 + ax2 + ax + c = 0)
___________
__________
_______
24
Quartic polynomial
Equation
(ax3+ ax3 + ax2 + ax + c = 0)
________
________
In the activity you have just done, were you able to identify different types
of polynomial equations? Were you able to describe each type? These
equations have common characteristics and you will learn more of these in the
succeeding activities in the next section.
Concept Development
Before doing the tasks ahead, read and understand first some important
notes on polynomial equations and the examples presented.
Enumerate
types of
polynomial
Step 2
Show the
formula
STEPS
Name
based on
no. of
terms
Step 1
DISCUSSION
A polynomial is one term or
the sum or difference of two or
more terms.
3x +4 1 linear
2
binomial
From the examples, a
polynomial can be name by its
2x2+2 2 quadratic 3
Trinomial
x+1
/cubic
degree and terms.
4x3
3 cubic
1
monomial
Recommended link
5
0 constant 1
monomial
www.study.com//Forming
Polynomial equation with roots
Based on the formula a
n
n-1
2
a0 x + a1 x +……..+ an-2 x + an-1 x polynomial has the following
+ an.
properties:
a. NO negative exponent
b. NO variable in the
denominator
c. NO fractional exponent
No. of
terms
DESCRIBING
POLYNOMIALS
FIGURE/ EXPRESSION/
EQUATION
Polynomi
al
degree
Name
using
degree
STEPS
FIGURE/ EXPRESSION/ EQUATION
Show the
TERMS
Example1.
x3 - 4x+2x2 +7
x3
3 degree
1 degree
- 4x
2x2 2 degree
7
0 degree
Step 3
Polynomial
Standard form
Describe
examples
Example 1. 2x-5
Example 2.
3x4 – 4 + 2x2
Example 3.
- 2x +5 - 4x2 + x3
Example 4.
2x + x8 - 1
2x-5
3x4 + 2x2 – 4
x3 - 4x2 - 2x +5
x8 + 2x -1
25
DISCUSSION
For a term that has only one
variable the. degree of term is
the exponent of the variable. The
degree of the constant is 0.
The degree of the
polynomial is the same as the
degree of the term with the
highest degree.
The example in the chart is
NOT in standard form.
The first example has two
terms. which is already in
standard form.
In the 2nd example, the
highest degree is 4. Write it as
the first term, then you must
interchange the 2nd and 3rd term
to have a standard form. Thus,
the degree is arranged from
highest to lowest. Same process
for the examples 3 and 4
STEPS
WRITING
POLYNOMIAL IN
STANDARD
FORM
Step 1
Identify the
number of
terms and the
highest
degree.
Step 2
Arrange the
terms with
degrees from
highest to
lowest
FIGURE/ EXPRESSION/
EQUATION
Polynomial
Standard form
Example 1. 2x-5
2x-5
Example 2.
3x4 – 4 + 2x2
3x4 + 2x2 – 4
Example 3.
- 2x +5 - 4x2 + x3
x3 - 4x2 - 2x +5
Example 4.
2x + x8 - 1
x8 + 2x -1
Example 2. 3x4 – 4 + 2x2
2nd term
3rd term
interchange
Standard form 3x4 + 2x2 – 4
DISCUSSION
To transform polynomial
into a standard form,
identify the number of terms
and arrange the degrees in
decreasing order. In the first
example it has two terms.
which is already in standard
form.
In the 2nd example, the
highest degree is 4. Write it
as the first term, then you
must interchange the 2nd
and 3rd term to have a
standard form. Thus, the
degree is arranged from
highest to lowest. Same
process for the examples 3
and 4. For more details
watch a video link through
www.
eHowEducation.com//
How to describe a
polynomial equation.
Now that you have a deeper understanding of the topic, you are ready to
do the tasks in the next part.
Activities
Your goal in this section is to take a closer look at some aspects of this topic. You
are going to think deeper and test further your understanding of solving problems in
polynomials that involve a real life situation. After doing the following activities, you
should be able to answer this question: “How do polynomial equations facilitate in
solving real-life problems and in making decisions?”
You will be given an activity that test if you have understood the previous lesson
by performing the tasks leading to the formula of polynomial equation.
Guided/Controlled Practice: ACTIVITY 1. Math Henyo (TEXTTIFY ME) 5 points.
Direction: CREATE the polynomial illustrated by the statements below.
1.
2.
3.
4.
5.
I am a polynomial with three terms written in standard form.
The coefficients of my variable x is 1.
My constant term is -2.
I have two real roots (positive and negative)
Create Me!
Answer:
______________________________________________
26
Independent Practice: ACTIVITY 2. COMBINE MY PARTS. (5 points each)
Direction: Write a mathematical equation of each from the model below.
Given
x2
-x2
x
-x
1
-1
1._____________________ 2._____________________ 3. _____________________________
Now that you know the important ideas about the topic, lets go deeper
by moving to the next section.
Assessment
Your goal in this section is to apply your learning to real-life situations.
You will be given practical task which will demonstrate your understanding of
illustrating polynomial equations.
ACTIVITY 1. LEAD ME TO THE FORMULA!
Directions: Use the situation in the box to answer the questions that follow.
Mr. Opalla, one of the farmers in town would like to enclose his rectangular garden
whose length is 4 more than twice its width. The area of the garden is 240m2.
length (2x + 4)
width ( x)
A=240m2.
1. How would you represent the length of the garden?
A. x
B. 2x
C. 2x-4
D. 2x+4
2. What mathematical sentence would represent the width of the garden?
A. x
B. 2x
C. 2x-4
D. 2x+4
3. What equation will you use in finding the dimensions of the garden?
A. x + (x+4)=240 B. x - (x+4)=240 C. x (x+4)=240
D. x (x+4)=240
4. Which of the following strategies is appropriate in finding the dimensions of
the garden?
A. Factor Theorem
B. Rational Root Theorem
C. Fundamental Theorem of Algebra D. Zero Product Property
5. How would you describe the equations formulated from the situation above?
A. Linear equation
B. Polynomial equation
C. Linear inequality
D. all of the above
27
In the activity you have just done, were you able to identify and describe
polynomial equation and its parts? Was there any point in your life that you realized
that you actually use polynomial equations in solving real-life problems and in making
right decisions? Before moving to the next section, let us review first the previous
lessons. Examples of how to do it is found in the section of Concept Development.
ACTIVITY 2. FOLLOW MY DESTINY!
Direction: Write the polynomial equation in standard form.
POLYNOMIAL
STANDARD FORM
1. ½ X2 –X + 2 = 0
1.
2. 3X2 + 5X3 + 3X4–X + 1 = 0
2.
3. 2 X2 + X4 + 4X+1= 0
3.
4. – X2 + 5X -10 = 0
4.
5. 6X-2x2 + 3x4 + 2 = 0
5.
ACTIVITY 3. DESCRIBE ME IN MANY WAYS!
Direction: Supply the missing word to make the statement true.
1. A _______________ is a monomial or a sum of monomials.
2. The monomials that make up the polynomials are called the _________.
3. The _______________is the sum of the exponents of its leading/first term
variables.
4. The degree of the polynomial is the same with the _______________
according to Karl Friedrich Gauss (1777-1895).
5. An equation which has only one variable term is _____________.
APPLICATION (Performance Tasks)
Have you asked yourself on how an architect was able to divide the space
of a building in order to place all amenities the owner’s want? How a carpenter
uses minimal materials to fence the rectangular garden? Or how students in
Grade 10 identify polynomials by applying the different strategies and
theorems?
You have learned how to illustrate, describe and define polynomial
equation. Your knowledge and skills will be of great help to deepen your
understanding for further application of the concepts
28
ACTIVITY 1. MAKE IT REAL!
Direction: Refer to the figure below and answer the questions that follow.
Questions:
1
1. Suppose the area of square 1 is 4y2 square units
and the perimeter of square 2 is 4y, what is the
is the area of square 3? __________________
2
2. If the area of square 3 is 9x2 what is the dimension
of square 2? ___________________________
How do you solve a problem? Do you first decide what the problem really
is, and take a series of steps to improve the situations? In Algebra, that exactly
happens. To solve problems, many people including pharmacists, marine
biologists and money managers, all use simple equations and step-by-step
methods. You can use polynomial equations and formulas to model a variety of
real world problem.
ACTIVITY 2. CONSTRUCT ME.
Direction: Suppose you were trying to model the product (x + 2) (x + 3).
a. Draw a rectangle to represent each type of monomial in the
product (5 points).
b. Write an example of a polynomial equation / product (5 points).
Independent Assessment: ACTIVITY 3 RELATE THEN CONNECT.
Directions: Determine whether or Not each expression is a polynomial. Then state
the degree. (5 points).
Polynomial Expression
1. √𝑥 + 3 = 0
2. X4 -3x3 = 0
Polynomial or NOT
Degree
Every day you are required to make decisions. Some may be simple as
what you want to eat for breakfast or what you should wear to school. Have you
ever realized that those quantities can be mathematically represented to come
up with practical decisions?
ACTIVITY 4. WHICH IS WHICH?
Direction: Use mathematical sentences below to answer the questions that follow.
1.
2.
3.
4.
5.
2x2 + 2x + 1 = 0
5
5x4 + 𝑥 = 0
8x -3 + 5 = 0
√y + 2 = 0
–x6 + 7 = 0
6. ½ x3 - 1 = 0
7. X3 + √x – 8 = 0
8. ¼ x -3 = 0
1
9. 𝑥 - 2 = 0
10.X20 – 4 = 0
29
Questions:
1. Which of the given mathematical sentence are polynomial? (3points).
____________________________________________________________
2. How do you describe polynomial equation? (3points).
____________________________________________________________
3. Which of the given are NOT polynomial? Why? (3points).
____________________________________________________________
4. How would you describe those mathematical sentences which are not
polynomial equations? (3points).
____________________________________________________________
5. How are those mathematical sentences which are not polynomial equations
different from those equations which are polynomial?
(3points).
Congratulations! You have finished the activities in lessons 1 and 2. You
are a great Learner!
Best of luck for the next lesson!
Generalization/ Synthesis
This lesson was about Factoring Polynomials and Illustrating
Polynomial Equations. In Lesson 1, you were able to find the factors of
polynomials by using different strategies such as by Inspection, AC method,
Quadratic Formula, Synthetic Division, and Factor Theorem. In lesson 2, you
were also able to illustrates, identifies, and describes polynomial equations;
More importantly, you were given a chance to formulate and solve real-life
problems, and demonstrate your understanding of the lesson by doing
some practical tasks.
You have learned the following:
Steps in factoring polynomials:
1.Arrange the polynomial to descending order.
2.Look for greatest common factor.
3.Look for the number of terms in remaining polynomial. If it is four terms use try
to use factoring by grouping, if three terms factor into product of two binomials,
and if it has two terms use the difference of squares or sum and difference of
squares. You can use quadratic formula in finding the factors of quadratic
polynomials if the factors are not obvious and not factorable.
TYPES OF POLYNOMIAL EQUATION
1. Monomial Equations or Linear Equations
An equation which has only one variable term. It can be expressed as
ax +b = 0, where a and b are real numbers. Example: 2x + 1= 0
2. Binomial Equations or Quadratic Equations
An equation which has only two variable terms. It can be expressed in
the algebraic form as ax2 + bx + c = 0. Example: 5x2 + 2x + 1 = 0
30
3. Trinomial or Cubic Equations
An equation which has only three variable terms.
Example: x3 + 2x2 + x +4 = 0
4. Quartic Polynomials
A polynomial of degree 4. It has also 4 real roots.
Example: x3 + 2x2 + x +4
5. Other polynomials with or more than 4 degree
Example: x20 – 1= 0
LIST OF THEOREMS IN THIS MODULE
1. Fundamental Theorem of Algebra
Any polynomial of degree n has n roots. The degree of a polynomial with one
variable is the largest exponent of that variable. A polynomial of degree 3 will have 3
roots (places where the polynomial is equal to zero. A polynomial of degree 4 will have
4 roots and soon.
2. Zero Product Property
States that if ab = 0, then either a = 0 or b = 0 (or both). A Product of factors is zero if
and only if one or more of the factors is zero.
3. Factor Theorem
States that a polynomial f(x) has a factor (x-r) if and only if f(r) = 0. Then r is the root.
Finding the root is the same as finding the factor of a polynomials It helps us analyze
polynomial equations. It tells us how the zeros/roots of a polynomial are related to the
factors. It is also helpful in verifying the factors of a certain polynomial
expressions/equations.
GLOSSARY OF TERMS
Degree of a Polynomial- the highest degree of a term in a polynomial
Factor Theorem - the polynomial P(x) has x-r as a factor if and only if P(r) =0
Special Products - products that are "special" because they do not need long
solutions in finding the factors. It can easily be recognized because of their patterns.
Synthetic Division – a short method in dividing polynomial expressions using only
the coefficient of the terms.
monomial – An expression that is a number, a variable, or the product of a number
and one or more variables. Examples: 5c, -a, 17x3, ½x4y3z2
like terms – Are two monomials that are the same or differ only by their numerical
coefficients. Examples: xy and xy, x3 and 6x3
equation - A sentence in mathematics that contain an equal sign.
Examples: x = 0, 2X-5= 0, 12x3 + 5x2 ˗ 2 = 0
polynomials – A monomial or the sum of monomials.
Examples: 12x +4, 3x3 + 2x2 ˗ 2
degree of a polynomial– An exponent in the leading term
polynomial equation – An equation of one or more than one term
theorem – A statement that needs to be proven.
31
Post Assessment
Directions. Let us determine how much you alreay know about factoring
polynomials and polynomial equations. Take this test. Read and understand the
questions below. Select the best answer to each item then write your choice on your
answer sheet. Do not write anything in this Module.
1. Which of the following is the sum and difference of two squares?
A. y2 + 24y+ 144
B. 27 - r3
C. b2 + 10b + 24
D. 16c2 - 9
2
2. What are the factors of 3x -x -10?
A. (3x + 5) (x - 2)
B. (3x + 5) (x + 2)
C. (x + 2) (3x-5)
D. (3x - 5) (x - 2)
3. Which of the following is TRUE?
A. x4 – 4x2y2 + 2y4 = (x + 2y2)2
B. 36x3 + 20xy + 49y2 = (6x + 7)2
C. 100x2 -50xy + 25y2 = (10x -5y)2
D. (x+3) (x + 3) = x2-9
4. What is the greatest common factor of 14x3y, 7x2y, and -7xy?
A. 7xy
B. -7xy
C. 7x2y
D. -7x2y
5. Which of the following is the factored form of x3 + 3x2 – 4x - 12?
A. (x + 3) (x -2) (x + 2)
B. (x + 3) (x + 2) (x + 2)
C. (x - 4) (x -3) (x - 2)
D. x - 4) (x -3) (x - 2)
6. Which of the following is NOT an example of a polynomial?
A. x2 – 4 = 0
B. √x +5 = 0
-2
C. y - 4 = 0
D. x
7. From the equation x3 - 4 x2 + 5x + 3 = 0, what is the highest degree of a
polynomial?
A. 0
B. 1
C. 2
D. 3
8. Rewrite (x+1) (x+1) in standard form?
A. 5(X +2) = 0
B. X(Y-6) = 0
C. (X-3) (X+2) =0 D. X2 + 2X + 1=
0
Given
x2
X
1
9. What polynomial is shown by this set?
A. 3X2 + 3X + 1= 0
B. 3X2 + 2X + 1= 0
C. 2X2 + 3X + 2=0
D. 2X2 + 2X + 2= 0
10. Which of the following mathematician discovered the relationship between
the number of roots which are the same with the number of the degree of
a polynomial equation?
A. Blaise Pascal
B. James Gregory
C. Karl Friedrich Gauss
D. Rene Descartes
If you are done, try to check your posttest. Answer key is provided on
page of this module. If you get a perfect score or 8 out of 10 in the post test
without any help from others, you are now ready to study the next module. But if
you missed few items or more than 5 items you may review all the activities at
your own desire. You may submit your rating sheet to your teacher for this post
assessment
32
Answer Key
PRE- ASSESSMENT (PAGE 2)
1.
2.
3.
4.
5.
B
D
D
D
C
LESSON 1
Pre – Assessment page 3 Prior Knowledge page 4
1. B
2. D
3. D
4. A
5. C
6. A
10.
7. CB
8. D
9. B
10. B
ACTIVITIES page 19
Activity 1
What’s the
other factor?
Activity 2 page 19
Find My Factors
.
1. 2m2y(m-y3)
2. 11w2y2(3w+1)
3. 3cd(3d3-2cd-1)
1. e
2. c
3. d
4. b
5. a
Activity 3 page 19
Identify Me.
Activity 4 page 20
Find My Special Factors
1. Difference of Two Squares
2. Sum of Two Squares
3. Sum and Difference
4.Sum and Difference
5.Sum of Two Cubes
4.(x+9) (x-2)
5 (4b +3) (2b+5)
1.(a-5)(a2 +5a+25)
2.(12-5)(12+5)
3.2x(x-2)(x2+2x+24)
Activity 5A page 20
Fill Me In!
1.(x +2)(x+3)(x-1)
2.(x+3)(x+5)(x+1)
APPLICATION page 20
Activity 1.Let’s Apply!
Complete Me
Activity 1B
What’s My Pattern?
1.(3x+5)2
6.4
7.4
2. (x+2)(x+2)
(x+2) (x-2)
X2 + X - 2
1. X2 + 5X - 6 = 0
2. –X2 – 5X + 6 = 0
3. -4X2 + 6X + 3 = 0
Activity 2 page 29
ASSESSMENT page 28
Activity 1.Lead Me to
Activity 2 page 29
he Formula
Follow My Destiny
Activity 3 page 29
Describe Me in Many Ways
1. Polynomial
2. Terms
3. Degree of the
polynomials
4. Number of Roots
5. Binomial Equation
1. D
2. C
3. D
4. A
5. B
6. Polynomial
7. Polynomial
8. NOT
9 NOT
10 Polynomial
ACTIVITIES page 27
Activity 2.Combine My Parts
Monomial Binomial Trinomial
c, i
f, g
b, d , e
1. Polynomial
2. NOT
3. NOT
4. NOT
5. Polynomial
ACTIVITIES page 27
Activity 1.Math Henyo
Presentation page 25
Match Me With My Equations
quartic
LESSON 2
Pre – Assessment page 22 Prior Knowledge page 24
Which is Which?
other polynomial
with higher degrees
a, j
1. Difference of Squares
2. Sum of
Squares
Complete
Me
3. Sum and Difference
4.Sum and Difference
5.Difference of Two Cubes
h
ASSESSMENT page 20
Activity 1A
What’s My Pattern?
1. 1/2 X2 - X + 2 = 0
2. 3X4 + 5X3 + 3X2 – X + 1 = 0
3. X4 + 2X2 + 4X + 1 = 0
4. –X2 + 5X – 10 = 0
5. 3X4 – 2X2 + 6X + 2 = 0
1. D
2. A
3. D
4. A
5. B
33
Answer key
Application page 30
Activity 3. Relate Then Connect
1.
Not
½
Polynomial
Application page 30
Activity 2. Construct Me.
2.
Application page 30
Activity 1. Make it Real!
4
a. 9X2
b. 1
the exponent of X is
b. X2 + 5X + 6 = 0
Application page 30-31
Activity 4 Which is Which? Page 30
X+2
a.
X+3
1.D
2.A
3.C
4.A
5.A
1. 1,4,5,6,7,10,11 &15
2. A polynomial is a monomial or a sum of monomials.
3. 2,8,9,12,13 &14
4. Not a polynomial because it has:
1 a variable in the denominator,
2 a fractional exponent,
3 negative exponent
5. Not polynomial equations, cannot be solved by using
different theorems mentioned in the module.
POST- ASSESSMENT (PAGE 32
6. B
7. D
8. D
9. C
10.A
References
Deped Instructional Materials That Can be Used As Additional
Resources for the Lesson Equations.
EASE Modules Year II Modules 1, 2 and 3
BEAM Mathematics 8 Module 4 pp. 1-55
Learners Module for Mathematics 9 pp. 1-113
Learners Module for Mathematics 10 pp. 83-87
References and Website Links Used In This Module
References:
Department of Education- Instructional Materials Council Secretariat
(DepEd-IMCS).2015. Grade 9 Mathematics Learning Module. 2nd ed. Philippines. Rex
Book Store INC.
Department of Education- Instructional Materials Council Secretariat (DepEdIMCS).2015. Grade 10 Mathematics Learning Module. 2nd ed. Philippines. Rex
Book Store INC.
Special Products and Factor. Retrieved May 18, 2020.
White Crane Education. Retrieved May 25, 2020.
Algebra Lab. Retrieved May 20, 2020.
34
Allan E Bellman, et Al. Algebra 2 Prentice Hall Mathematics, Pearson Prentice
Hall, New Jersey USA, 2004
Robbie Bonneville, Cindy J. Boyd, Eva Gates, Beatrice Moore-Harris, and
Melissa McClure, Algebra 2 (Integration, Applications and Connections).
McGraw Hill, Companies Inc., New York, New York, 1998.
Website Links for References and for Learning Activities
BrownMath.com. Algebra Polynomial solving. (2002). Retrieved May 23, 2020 from
http://www.BrownMath.com/Algebra/Polynomial Solving by Stan Brown
Paul’s Online Notes.com. Algebra Polynomial Functions, Roots of Polynomial.
(2003). https://www.Paul’s Online Notes.com/Algebra/Polynomial
Functions/Zeroes/Roots of Polynomial by Paul Dawkins
Varsity Tutors.com. Zero Product Property. (2007).
https://www.VarsityTutors.com/The Zero Product Property
MathIsfun.com. Fundamental Theorem of Algebra. (2017).
https://www.MathIsFun.com/Fundamental theorem of Algebra
Tutorial.math.lamar.edu.com Zeroes of Polynomials.
https://tutorial.math.lamar.edu/classes.com/Alg/zeroesOfPolynomials.aspx
Website Links for Videos
MathHelp, “Solving Polynomial Equations”October 23, 2007, video, 2:50,
https://MathHelp.com./Solving Polynomial Equations, Retrieved May 22, 2020
Brian McLogan, “Finding all the roots of a polynomial:How to find all the roots of a
polynomial by factoring December 11, 2015, video, 3:21
http://www.freemathvideos.com/ How to find all the roots of a polynomial by
factoring. Retrieved May 22, 2020
For inquiries and feedback, please write or call:
Department of Education –Learning Resources Management and
Development Center(LRMDC)
DepEd Division of Bukidnon
Sumpong, Malaybalay City, Bukidnon
Telefax:
((08822)855-0048
E-mail Address:
bukidnon@deped.gov.ph
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