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Department of Education
Factors completely different types of
polynomials
Module 1
MA. PAZ B. SOLAR
WRITER
DOROTHY A. MENDOZA, PhD EDITOR
EMELITA D. BAUTISTA EdD VALIDATOR
MATHEMATICS 8
Schools Division Office – Muntinlupa City Student Center for Life Skills Bldg., Centennial Ave., Brgy. Tunasan, Muntinlupa City
(02) 8805-9935 / (02) 8805-9940
MATHEMATICS First Quarter – Module 1:
Factors completely different types of
polynomials
8
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Quarter 1, Module 1:
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Printed in the Philippines by ________________________
Department of Education – Bureau of Learning Resources (DepEd-BLR)
Office Address: _____________________________________________
_____________________________________________
Telefax: _____________________________________________
E-mail Address: _____________________________________________
Development Team of the Module
Authors: MA. PAZ B. SOLAR
Editors/Reviewers: DOROTHY A. MENDOZA, PhD
Validator: EMELITA D. BAUTISTA, EdD
Illustrator:
Layout Artist:
Management Team:
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JHS
This instructional material was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at [email protected].
We value your feedback and recommendations.
1
Introductory Message
For the facilitator:
The COVID-19 crisis presented us a gift. We have the gift of knowing what has
worked in the past, and the capacity to prepare for the future. In this light, this module
is intended to ensure learning continuity under the new ‘normal’ in education amidst
the COVID-19 pandemic. It is designed to provide a general background in factoring
completely different types of polynomials ( polynomials with common monomial factor,
difference of two squares , sum and difference of two cubes, perfect square
trinomials, and general trinomials )
As the learning facilitator, you are requested to orient your learners on the proper
use of this module and assist para-teachers such as parents, elder siblings, and other
significant adults to understand their vital role in optimizing this learning material
towards developing mutual responsibility for children’s success under the distant
learning set up.
Finally, please do not forget to remind the learners to use separate sheets or
notebook in answering the pre-test, self-check exercises, and post-test.
For the learner:
In the beginning of each lesson, you will review related concepts. Then, you will
discover what the learning episode is about. After the presentation of concepts, you will
do self-check exercises that will lead you to an application task. Finally, you will be
guided in managing your takeaways.
2
Mathematics has always been defined by its disciplines - by its areas of focus,
study, training, specialties, and subject matters. From this perspective, you will be
guided using the D-I-S-C-I-P-L-I-N-E Model (Dedicate, Inform, Seek, Create, Impart,
Participate, Link, Interpret, Nurture, and Elaborate) in this journey. The following are
the standard symbols (icons) used to represent some parts of the module:
DEDICATE (What I Need to Know) This part contains the learning objectives covered by the material. It also introduces the topic/content of the module briefly.
INFORM (What I Know)
This is the pre-assessment. It is given to check what you know about the lesson you are about to take.
SEEK (What’s In) This part connects the current lesson with the previous lesson by allowing you to go over concepts that you have learned previously.
CREATE (What’s New) It is in this part that the new lesson is introduced through a story, a poem, song, situation, or an activity.
IMPART (What is It)
This part provides a brief discussion of the lesson. It will help you to
a better understanding about the concept.
PARTICIPATE (What’s More)
In this part, you will be asked to do enrichment activities that are
designed to reinforce or refine your understanding.
LINK (What Have I Learned)
This part offers a question, fill in the blank sentence/paragraph to
enable you to process what you have learned from the lesson.
3
INTERPRET (What I Can Do) This part presents an activity that will allow you to transfer the skills/knowledge you gained or learned into real-life concerns/situations.
NURTURE (Assessment)
This evaluates your level of mastery in achieving the learning objectives, validates the concepts and provides more opportunities to
deepen the learning.
ELABORATE (Additional Activities)
This part provides an activity in any form that can increase the
strength of your responses and encourages repetitions of
actions/learning.
Since the module highlight a specific lesson, it will also help you appreciate the
use of what you learn in daily life and expand your understanding of the community
you live in and beyond.
Please handle this module with utmost care and use separate sheets or notebook
in answering the activities.
Happy learning!
This module was designed and written with you in mind. It is here to help you
(what the students need to learn). 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.
After going through this module, you are expected to:
1. Factors completely different types of polynomials.
a. Polynomials with common monomial factor.
b. Difference of two squares.
c. Sum and difference of two cubes.
d. Perfect square trinomials
e. and general trinomials.
4
Directions: Choose the letter of the best answer. Use a separate sheet of paper for
your answers.
1. What is the Greatest Common Factor (GCF) of 12 and 20?
a. 5 b. 4 c. 3 d. 2
2. It is the process of finding the factors of an expression.
a. Monomial Factor b. GCF c. Factoring d. Prime
Number
3. What is the GCF of 8x2 + 16x?
a. 8x b. 4x c. 8x2 d. 4x2
4. It is a number greater than 1 which has only two positive factors: 1 and itself.
a. Factor b. GCF c. Composite Number d. Prime Number
5. Find the GCF of 12x5y4 – 16x3y4 + 28x6.
a. 4x3 b. 4x4 c. 6x3 d. 4x5
6. An algebraic expression that represents a sum of one or more terms containing
whole number exponents on the variables are called ___________.
a. Monomial b. Binomial c. Trinomial d. Polynomial
7. What do you call a polynomial with one term?
a. Monomial b. Binomial c. Trinomial d. Polynomial
8. A polynomial with three terms is called __________?
a. Monomial b. Binomial c. Trinomial d. Polynomial
9. What do you call a polynomial with two terms?
a. Monomial b. Binomial c. Trinomial d. Polynomial
10. Which of the following terms is an example of monomial?
a. 7x2y3 b. x2 – 4 c. 6x3 – 7x + 5 d. x2 + 8y2 – x + y
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Lesson
1 (Q1W1 TOPIC)
What is Factoring?
The process of finding the
factors of an expression is called
factoring, expression which are
divisors of a certain number,
larger a number goes the
tendency of increasing factors are always positive, factoring is also
the reverse process of multiplication. A prime number is a number
greater than 1 which has only two positive factors: 1 and itself for
example, 1, 3, 7, 11, 199 and the list goes on.
6
Factoring Binomials
Factor the following Binomials:
1) 2ab + 4c 2) t3 – ts
3) 3n – 6m 4) 5y2 + y
5) qr2 – r3 6) 6t + 12uv
7) 3p2 – 15q 8) 3a – a4
9) 16uv + 8uw 10) 9m3 + 18n+
7
Factoring Polynomials.
Each quadratic equation is of the form x2 + bx + c, where the coefficient of the
squared term is 1. If the coefficient or the constant is not a perfect square, the
trinomial cannot be factored into a square of a binomial. It may, however, be possible
to factor it into the product of two different binomials.
The FOIL method is useful when factoring polynomials such as
x2 + 3x + 2.
F O I L
a. (x + 1) (x + 2) = x2 + 2x + x + 2
= x2 + 3x + 2
b. Another way to factor x2 + 3x + 2, think of FOIL in reverse order.
1. The first term, x2, is the result of x times x. Thus, the first term of each
binomial is x.
(x + ___) (x + ___)
2. The coefficients of the middle term and the last term of the trinomial are two
numbers whose product is 2 and whose sum is 3, respectively. Those numbers are 1
and 2. Thus, the factors are (x + 1) (x + 2).
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Factoring Cubes:
Let x and y be real numbers, variables, or algebraic expressions.
Factoring Sum of Two Cubes x3 + y3 = (x + y) (x2 – xy + y2)
Factoring Difference of Two Cubes x3 – y3 = (x – y) (x2 + xy + y2)
If we think of the sum of two cubes as the cube of a First quantity plus the cube of a
Last quantity, we have the formula
F3 + L3 = ( F + L )( F2 – FL + L2 )
That is, to factor the cube of a First quantity plus the cube of a Last quantity, we
multiply the First plus the Last by:
- the First squared
- minus the First times the Last
- plus, the Last squared
Note: Expressions like a3b6 is also a perfect cube because it can be written as
(ab2 )3
The cubes of the numbers from 1 to 10 are:
1, 8, 27, 64, 125, 216, 343, 512, 729 and 1 000.
If we think of the difference of two cubes as the cube of a
First quantity minus the cube of the Last quantity, we have the formula:
F3 – L3 = ( F – L )(F2 + FL + L2 )
This indicates that to factor the cube of a First quantity minus the cube of the Last
quantity, we multiply the First minus the Last by:
- the First squared
- plus, the First times the Last
- plus, the Last squared
9
Illustrative Examples:
A. Factoring the Sum of two Cubes
Factor each completely:
a. a3 + 64
Solution:
a3 + 4
Using the formula:
F3 + L3 = ( F + L )( F2 – FL + L2 )
a3 + 43 = ( a + 4 )( a2 – (a)(4) + 42 )
= ( a + 4 )( a2 – 4a + 16)
Let’s check :
( a + 4 )(a2 – 4a + 16) = ( a + 4 )a2 – ( a + 4)4a + (a + 4) 16
= a3 + 4a2 – 4a2 – 16a + 16a + 64
= a3 + 0 + 0 + 64
= a3 + 64
8b3 + 27c3
Cube root of 8b3 = 2b (2b X 2b X 2b) = 8b3
Cube root of 27c3 = 3c (3c X 3c X 3c) = 27c3
Solution:
8b3 + 27c3
(2b)3 + (3c)3
Cube root of 64 = 43
4 x 4 = 16
10
= F3 + L3 = ( F + L ) ( F2 – FL + L2 )
= (2b)3 + (3c)3 = ( 2b + 3c) [ (2b)2 – (2b)(3c) + (3c)2 ]
= (2b + 3c) [ (2b)(2b) – 6bc + (3c)(3c) ]
= ( 2b + 3c) ( 4b2 – 6bc + 9c2 )
= (2b + 3c)(4b2) – (2b + 3c)(6bc) + (2b + 3c)(9c2)
= 8b3 + (12b2c – 12b2c) + (-18bc2 + 18bc2) + 27c3
= 8b3 + 0 + 0 + 27c3
= 8b3 + 27c3
B. Factoring the Difference of Two Cubes
Factor of each completely:
1. 27c3 – d3
Cube root of 27c3 = 3c (3c X 3c X 3c) = 27c3
Cube root of d3 = d (d X d X d) = d3
Solution:
27c3 – d3
( 3c )3 – ( d )3
= F3 – L3 = ( F – L ) ( F2 + F L + L2 )
(3c)3 – (d)3 = ( 3c – d ) [ (3c)2 + (3c)(d) + ( d )2 ]
= ( 3c – d ) [ ( 3c)(3c) + 3cd + (d)(d) ]
= ( 3c – d ) (9c2 + 3cd + d2)
= (3c – d )(9c2) + (3c – d )(3cd) + (3c – d )(d2)
= 27c3 – 9c2d + 9c2d – 3cd2 + 3cd2 – d3
= 27c3 + (-9c2d + 9c2d) + (-3cd2 + 3cd2 ) – d3
= 27c3 + 0 + 0 - d3
= 27c3 – d3
11
2. 64 – p6
Cube root of 64 = 4 (4 X 4 X 4) = 64
Cube root of p6 = p2 (p2 X p2 X p2) = p6
Solution:
64 – p6
(4 )3 – (p2 )3
= F3 – L3 = ( F – L ) ( F2 + F L + L2 )
( 4 )3 – (p2)3 = ( 4 – p2 ) [ (4)2 + ( 4)(p2) + (p2)2 ]
= (4 – p2) [ (4 X 4) + 4p2 + p4]
= (4 – p2) (16 + 4p2 + p4)
= (4 – p2) (16) + (4 – p2) (4p2) + (4 – p2) (p4)
= 64 – 16p2 + 16p2 – 4p4 + 4p4 – p6
= 64 – ( -16p2 + 16p2) + (-4p4 + 4p4) – p6
= 64 – 0 + 0 - p6
= 64 – p6
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Directions: Factor the Difference of Two Perfect Squares:
1. 44a2 – 169b2 6. – 64a2 + (9/25) b2
2. 1 – 0.09a2 7. x4 – 256
3. 16x2 – 121 6. (x + y)4 – z4
Directions: Factor the following by taking the difference of squares:
1. x2 – 9 4. 4x2 – 25
2. a2 – 1 5. a2b2 – 16
3. 49 – x2 6. a4 – b4
Directions: Factor the Following completely.
1. x3 – 125
2. d3 + 27
3. 216g3 + t3
4. 8p3 + 64
5. u3 – 64
6. 64c3 – d3
7. y3 – 125
8. 8b3 + 512
9. a3 – 125
10. b3 + 27
11. 216h3 + j3
12. 8q3 + 64
13. s3 – 64
14. x² + 22x + 121
15. x² + 10x + 25
1
Directions: Factor the following Completely
1) 9x2 − 1 2) 4n2 − 49
3) 36k2 − 1 4) p2 − 36
5) 2x2 − 18 6) 196n2 − 144
7) 180m2 − 5 8) 294r2 − 150
9) 150k2 − 216 10) 20a2 − 45
11) 3n2 − 75 12) 24x3 − 54x
13) a2 − 25b2 14) 4x2 + 49y2
15) 25x2 + 16y2 16) 6a2 + 96b2
17) x2 − 9y2 18) 49x2 − 25y2
19) 9x2 − 16y2 20) 54v2 − 6u2
21) 36a4 − 25b4 22) 2x4r − 72y4r
23) 125m4 − 20n4 24) 216x4ay − 6y5a
25) 4x4 − 144y4 26) 4x4m − 36y4m
27) 7x4 − 28y4 28) 7x4 − 343y4
29) 16m6 − n6 30) 64x6 − y6
14
Directions: Choose the letter of the best answer.
1. A polynomial with two terms is called _______
A. Monomial B. Binomial C. Trinomial D.
Polynomial
2. Which of the following is an example of a binomial?
A. a2 + 4 B. a3 + 2a2 – 2 C. xy D. y2 – 3y + 9
3. What is the factor of m2 – 4 ?
A. ( m + 2 )( m + 2) B.( m – 2)(m – 2) C. (m + 2)(m – 2) D.( 2 + m)( 2 – m)
4. The factor of 9z2 – 100 is _____
A. ( 3 + 10)(3+ 10) C. ( 3z + 10)(3z + 10)
B. ( 3 – 10)( 3 – 10) D. ( 3z– 10) (3z + 10)
5. Which of the following is NOT an example of binomial?
A. p2 + 6p – 4 B. b2 – 25 C. -16y2 + 25 D. 4k2
– 121
6. What is the factors of -49h2 + 1?
A. (7h + 1)(7h + 1) C. (-7h – 1)(7h – 1)
B. (7m – 1)(7m – 1) D. (7m – 1)(7m + 1)
7. What is the factor of 9d2 – 49 ?
A. (3d + 7) ( 3d + 7) C. ( 3d – 7)( 3d – 7)
B. ( 3d + 7) (3d – 7) D. (-3d + 7)( 3d – 7)
8. The factor of 1 – 81x2 is what?
A. ( 1 – 9x )( 1 + 9x) C. ( -9x – 1)( 9x – 1 )
B. ( 1 + 9x) ( 1 + 9) D. ( 9x + 1)( 9x + 1 )
9. The factor of 9r2 – 25s4 is what ?
A. ( 3r + 5s)( 3r + 5s) C. ( 3r + 5s2 ) ( 3r – 5s2 )
B. ( 3r + 5us)( 3r – 5s ) D. ( 3r – 5s2 ) ( 3r – 5s2 )
10. What is the factor of 81 – 4t6u4
A. ( 9 – 2tu )( 9 – 2tu) C. ( 9 + 2t3u2 )( 9 + 2t3u2)
B. ( 9 + 2tu)( 9 + 2tu) D. ( 9 + 2t3u2 )( 9 – 2t3u2)
15
What I Know
1.B
2.C
3.A
4.D
5.A
6.D
7.A
8.C
9.B
10.A
Assessment
1.B
2.A
3.C
4.D
5.A
6.C
7.B
8.A
9.C
10.D
What’s New
1.2(ab + 2c)
2.t(t2 – s)
3.3(n – 2m)
4.y(5y + 1)
5.r2(q – r)
6.6(t + 2uv)
7.3(p2 – 5q)
8.a(3 – a3)
9.8u(2v + w)
10.9(m3 + 2n3)
What I can do.
1) (3x + 1)(3x − 1) 2) (2n + 7)(2n − 7)
3) (6k + 1)(6k − 1) 4) (p + 6)(p − 6)
5) 2(x + 3)(x − 3) 6) 4(7n + 6)(7n − 6)
7) 5(6m + 1)(6m − 1) 8) 6(7r + 5)(7r − 5)
9) 6(5k + 6)(5k − 6) 10) 5(2a + 3)(2a − 3)
11) 3(n + 5)(n − 5) 12) 6x(2x + 3)(2x − 3)
13) (a + 5b)(a − 5b) 14) Not factorable
15) Not factorable 16) 6(a2 + 16b2)
17) (x + 3y)(x − 3y) 18) (7x + 5y)(7x − 5y)
19) (3x + 4y)(3x − 4y) 20) 6(3v + u)(3v − u)
21) (6a2 + 5b2)(6a2 − 5b2) 22) 2r(x2 + 6y2)(x2 − 6y2)
23) 5(5m2 + 2n2)(5m2 − 2n2) 24) 6ay(6x2 + y2)(6x2 − y2)
25) 4(x2 + 6y2)(x2 − 6y2) 26) 4m(x2 + 3y2)(x2 − 3y2)
27) 7(x2 + 2y2)(x2 − 2y2) 28) 7(x2 + 7y2)(x2 − 7y2)
29) (4m3 + n3)(4m3 − n3) 30) (2x + y)(2x − y)(4x2 − 2xy + y2)(4x2 + 2xy +
y2)
16
What’s More:
1.(12a + 13b) (12a - 13b)
2.(1 + 0.3a) (1 - 0.3a)
3.(4x + 11) (4x - 11)
4.[(3/5)b + 8a] [(3/5)b - 8a]
5.(x2 + 16) (x + 4) (x – 4)
6.[(x + y)2 + z2] (x + y + z) (x + y - z)
7.(x + 3) (x - 3)
8.(b) (a + 1) (a - 1)
9.(c) (7 + x) (7 - x)
10.(d) (2x + 5) (2x - 5)
11.(e) (ab + 4) (ab - 4)
12.(f) (a2 + b2) (a + b) (a - b)
What have I learned?
1.( x – 5 ) (x2 + 5x + 25)
2.( d + 3) (d2 – 3d + 9)
3.( 6g + t) ( 36g2 – 6gt + t2)
4.( 2p + 4) (4p2 – 8p + 16)
5.( u – 4 ) (u2 + 4u + 16)
6.( 4c – d ) (16c2 + 4cd + d2)
7.( y – 5 ) ( y2 + 5y + 25 )
8.( 2b + 8 ) ( 4b2 – 16b + 64)
9.1. ( a – 5 a2 + 5a + 25)
10.( b + 3) (b2 – 3b + 9 )
11.(6h + j) ( 36h2 – 6hj + j2 )
12.(2q + 4) ( 4q2 – 8q + 16)
13. ( s – 4 ) ( s2 + 4s + 16)
14.(x + 11) ²
15.(x + 5) ²
