Q1 week 1 (common monomial,sum & diffrence of two cubes,difference of two squares)

ROSE MARIEL F. MAITEM
Mathematics Teacher, THE COLLEGE OF MAASIN
Week 1:
Mathematics 8
Quarter One: Patterns and Algebra
Topic: Factor of Polynomials
Let us recall the distributive property which state that if a, b, and c are real
numbers, then
𝑎𝑏 + 𝑎𝑐 = 𝑎(𝑏 + 𝑐)
If we keep the distributive property in mind, it will not be difficult to factor a
polynomial having a common monomial factor like 2𝑥3
+ 𝑥2
− 7𝑥.
We can easily observe that the three terms of 2𝑥3
+ 𝑥2
− 7𝑥 have the common factor
𝑥. That is,
2𝑥3
+ 𝑥2
− 7𝑥 = (2𝑥2)(𝑥) + (𝑥)(𝑥) − 7(𝑥)
Thus, according to the distributive property, we may write this as
2𝑥3
+ 𝑥2
− 7𝑥 = (2𝑥2)(𝑥) + (𝑥)(𝑥) − 7(𝑥)
2𝑥3
+ 𝑥2
− 7𝑥 = 𝑥(2𝑥2
+ 𝑥 − 7)
Let’s Try!
a. 10𝑥3
+ 9𝑥2
+ 4𝑥
b. 3𝑥6
+ 9𝑥4
+ 12𝑥2
A polynomial whose terms have a common monomial factor
may be factored by identifying this common factor and
applying the distributive property of multiplication over addition.
Factors 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).
MELCs & Codes:
M8AL-Ia-b-1
Common
Monomial Factor
Objective:
Factor completely polynomials with
common monomial factors
Introduction

Week 1:
Mathematics 8
Otherwise, Greatest Common Factor (GCF) may apply to ensure that the
polynomial factor is irreducible or a prime polynomial.
Greatest Common Factor (GCF)
The greatest common factor is the largest integer, monomial, or
multinomial that a set of numbers or polynomial have in
common.
For instance, Find the GCF of 12𝑥3
𝑦2
, 8𝑥𝑦2
, 𝑎𝑛𝑑 4𝑥2
𝑦2
.
Solution:
Express each as a product of prime factors.
12𝑥3
𝑦2
= 2 • 2 • 3 • x • x • x • y • y
8𝑥𝑦2
= 2 • 2 • 3 • x • y • y
4𝑥2
𝑦2
= 2 • 2 • x • x • y • y
2 2 x y y
The GCF of these three monomials is 2 • 2 • x • y • y = 4𝑥𝑦2
.
Notice that the degree of the GCF is equal to or less than the degree of the
expression with the lowest degree.
Let’s Try!
a. Find the GCF of 24𝑎2
𝑏3
𝑐3
, 30𝑎3
𝑏𝑐4
, 𝑎𝑛𝑑 48𝑎𝑏2
𝑐2
.
Factoring is the reverse process of multiplication. When a number or a polynomial
is factored, it is written as a product of two or more factors. A polynomial is said to be
factored into prime factors if it expresses as the product of two or more irreducible or
prime polynomials of the same type.
A polynomial is factored completely if each of its factors can no longer be
expressed as a product of two other polynomials of lower degree and that the
coefficients have no common factors without introducing a fraction, 1 or -1. If each
term of a polynomials is divisible by the same monomial, this monomial is referred to as
a Common Monomial Factor.
Common Monomial Factoring
1. Find the greatest common factors (GCF) of the terms in the
polynomials. This is the first factor.
2. Divide each term by the GCF to get the other factor.

Week 1:
Mathematics 8
v
Example 1. Factors each expression
a. 7𝑥2
− 7𝑦
b. 8𝑥3
− 16𝑥4
+ 48𝑥7
c. 2(𝑎 + 𝑏) − 𝑥(𝑎 + 𝑏)
Solution:
a. The GCF of 7𝑥2
and −7𝑦is 7.
7𝑥2
− 7𝑦 = 7 (
7𝑥2
7
+
−7𝑦
7
)
= 7(𝑥2
− 𝑦)
b. The GCF of 8𝑥3
, −16𝑥4
and 48𝑥7
is 8𝑥3
.
8𝑥3
= 2 • 2 • 2 • x • x • x
16𝑥4
= 2 • 2 • 2 • x • x • x • x • 2
48𝑥7 = 2 • 2 • 2 • x • x • x • x • 2 • 3 • x • x • x
2 • 2 • 2 • x • x • x
GCF is 2 • 2 • 2 • x • x • x = 8𝑥3
Factor by Common Monomial:
8𝑥3
− 16𝑥4
+ 48𝑥7
= 8𝑥3
(
8𝑥3
8𝑥3 −
16𝑥4
8𝑥3 +
48𝑥7
8𝑥3 )
= 8𝑥3(1 − 2𝑥 + 6𝑥4)
c.    baxba 2   xba  2
Give the GCF of the given monomials.
1. 85,70,45
2. 6635
, yxyx
3. 524435
26,110,98 yxyxyx
4. yzxyxyz 55,55,15 2

5. 453435
48,48,32 yxyxyx 
Activities

Week 1:
Mathematics 8
Learning Insights
Factor the following polynomials.
1. 𝑎2
𝑏𝑐 + 𝑎𝑏2
𝑐 + 𝑎𝑏𝑐2
2. 4𝑚2
𝑛2
− 4𝑚𝑛3
3. 25𝑎 + 25𝑏
4. 3𝑥2
+ 9𝑥𝑦
5. 2𝑥2
𝑦 + 12𝑥𝑦
A. Reflect on your participation in doing all the activities in this
lesson and complete the following statements:
• I learned that I...
• I was surprised that I...
• I noticed that I...
• I discovered that I...
• I was pleased that I...
Exercise
JOURNAL WRITING:
“Common Monomial Factor”
Description: This journal will enable you to reflect about the
topic and activities you underwent.
Instruction: Reflect on the activities you have done in this
lesson by completing the following statements. Write your
answers on the space provided for.
References:
Orines, Fernando B. Mathematics 8. Next Century second Edition
Bureau of Secondary Education. Distance Learning Module Mathematics 2
Escaner, Jose Maria L. IV PhD. et.al (2013) K to 12 spiral math 8

Week 1:
Mathematics 8
Learning Insights
Name: ____________________ Grade & Section: ______________
Instruction: Reflect on your participation in doing all the
activities in this lesson and complete the following
statements. Write your answers on the space provided for.
 I learned that I...
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 I was surprised that I...
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 I noticed that I...
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 I discovered that I…
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_________________________________________________________________
_________________________________________________________________
_________________________________________________________________
 I was pleased that I...
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“JOURNAL WRITING”
Common Monomial Factor

Week 1:
Mathematics 8
Let’s recall the special pattern 𝑎2
− 𝑏2
, which is the result when the sum of two
terms is multiplied by the difference of the same two terms. In other words, when the
two binomials have the form (𝑎 + 𝑏) and (𝑎 − 𝑏), you can easily get the product as
(𝑎2
− 𝑏2) which is the difference of 2 perfect squares.
For example, (𝑥 + 5)(𝑥 − 5) = 𝑥2
− 25. Therefore, whenever you encounter a
binomial that has the form 𝑎2
− 𝑏2
, you can do the reverse process where in the given
terms are both perfect squares.
Say,
𝑥2
− 25 = (𝑥)2
− (5)2
= (𝑥 + 5)(𝑥 − 5)
Factoring the Difference of two squares is a special type of factoring, a problem that
is often used in mathematics
trinomials).
MELCs & Codes:
M8AL-Ia-b-1
Factors of Difference
of Two Squares
Objective:
difference of two squares.
Introduction
Factors of Difference of Two Squares
1. Get the principal square root of each of the two squares.
2. Using these square roots, form two factors: a sum and a
difference.

Week 1:
Mathematics 8
This pattern can be generalized as follows:
A binomial that is the difference between two squares, 𝑎2
− 𝑏2
,
for any real numbers, a and b, can be factored as the product
of the sum (𝑎 + 𝑏) and the difference (𝑎 − 𝑏) of the terms that
are being squared:
𝑎2
− 𝑏2
= (𝑎 + 𝑏) (𝑎 − 𝑏)
Examples 1. Factor the following polynomials
a. 𝑥2
− 36
b. 4𝑎2
− 9𝑏2
Solution:
a. 𝑥2
− 36
𝑥2 − 36 = (𝑥)2 − (6)2 , therefore, a=x and b=6
𝑎2
− 𝑏2
= (𝑎 + 𝑏) (𝑎 − 𝑏) use difference of two squares pattern
𝑥2
− 62
= (𝑥 + 6) (𝑥 − 6), by substitution
𝑥2
− 36 = (𝑥 + 6) (𝑥 − 6)
b. 4𝑎2
− 9𝑏2
4𝑎2
− 9𝑏2
= (2𝑎)2
− (3𝑏)2
, therefore, a=2a and b=3b
𝑎2
− 𝑏2
= (𝑎 + 𝑏) (𝑎 − 𝑏) use difference of two squares pattern
(2𝑎)2
− (3𝑏)2
= (2𝑎 + 3𝑏) (2𝑎 − 3𝑏), by substitution
4𝑎2
− 9𝑏2
= (2𝑎 + 3𝑏) (2𝑎 − 3𝑏)
Give the square root of each.
1. 644
c
2. 8116 2
b
3. 462
10036 mkj 
4.
24
4 ed
Activities

Week 1:
Mathematics 8
Learning Insights
Factor the following polynomials
1. 9𝑎2
− 1
2. 81 − 625𝑧6
3. 121𝑔2
− 169ℎ4
4. ( 𝑥2
− 1)2
− 𝑥2
Exercise
JOURNAL WRITING:
“Difference of Two Squares”
Description: This journal will enable you to reflect about the topic
and activities you underwent.
References:

Week 1:
Mathematics 8
Learning Insights
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Difference of Two Squares

Week 1:
Mathematics 8
Sum or Difference of two cubes
  2233
yxyxyxyx 
  2233
yxyxyxyx 
Observe how the factors of 33
yx  are obtained by introducing arbitrary terms
without affecting the given expression and by using grouping techniques.
3333
0 yxyx  Renaming 0 as sum of yx2
and yx2

3223
yyxyxx 
   3223
yyxyxx  Grouping the 1st two terms and the last two terms
   222
yxyyxx  Bringing out the common monomial factors
in each group
    yxyxyyxx  2
Factoring the difference of two squares
    yxyxyx  2
Factoring out  ,yx  a common binomial factor
  22
yxyxyx 
Follow the same process for 33
yx  to obtain
  2233
yxyxyxyx 
trinomials).
MELCs & Codes:
M8AL-Ia-b-1
Sum and Difference
of Two Cubes
Objective:
sum & difference of two cubes.
Introduction

Week 1:
Mathematics 8
Steps in factoring the sum or difference of two cubes
1. Get the cube root of each cubed terms
2. Taking the operation between the cubes, use the cube roots in
step 1 to obtain a binomial factors.
3. Form the trinomial factor as follow:
a. Square the first cube root.
b. Multiply the two cubes roots. The sign of the product is opposite
the sign between the cubes.
c. Square the second cube root.
Examples 1. 83
x
Solution:
Rename each terms as a sum of two cubes. Then, apply the formula for sum
of cubes.
   333
28  xx
        22
222  xxx
  422 2
 xxx
Example 2. 612
64yx 
Solution:
Note that the binomial is a difference of squares.
   3234612
464 yxyx 
        22242424
444 yyxxyx 
  424824
1644 yyxxyx 
   424822
16422 yyxxyxyx 
Factors the following expression.
1.
33
ax  3.
33
125 vu  5. 126
jh 
2. 127 3
z 4. 333
rqp 
Activities

Week 1:
Mathematics 8
Learning Insights
Factor the following expressions.
1. 6
8
27
y 3.
3
125216 d 5.
63
125.0008.0 dc 
2.
366
8cba  4.
36
648 ts 
Exercise
JOURNAL WRITING:
“Sum or Difference of Two Cubes”
Description: This journal will enable you to reflect about the topic
and activities you underwent.
References:

Week 1:
Mathematics 8
Learning Insights
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Sum or Difference of Two Cubes

Week 1:
Mathematics 8
A. Give the GCF of the given monomials.
1. 77,49,35
2. 24927
, zyxzyx
3. 244337
72,42,18 yzxzyxzyx
4. 3273
36,18,9 yxyyx 
5. 453434
168,84,24 qxyxyx
B. Factor each expression.
1. 621459 x
2.
432332
544236 zyzyzy 
3.
2275
4088104 vuuvvu 
4. mkhhkgjhg 3723232

5.
344645
710535 bababa 
“ASSESSMENT”
Common Monomial Factors

Week 1:
Mathematics 8
Factor the following polynomials
1.
44
7527 wv 
2.
82
49.0 gf
3. 23
3625 hjh 
4. 1
25
2

a
5. 296,1256 2
y
6.
45
483 tut 
7.
32
5424 mmk 
8. 1282 16
x
9. 1002
n
m
10. 50200
9850 yx 
“ASSESSMENT”
Difference of Two Squares

Week 1:
Mathematics 8
A. Factor the following expressions.
1.
3
64 w
2. 16
t
3. 83
x
4.
6
125 a
5. 216963
fed
6. 1216 3
e
7.
33
64
1
8 gf 
8.
123
8 xw 
9. 2764 9
z
10. 36
001.0 fe 
“ASSESSMENT”
Sum or Difference of Two Cubes

Q1 week 1 (common monomial,sum & diffrence of two cubes,difference of two squares)

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