K to 10 NTOT.pptx

Regional Training Of Trainers on K to 10
Mathematics 8
May 15 – May 26, 2018
Cagayan De Oro City
VANISSA L. BOLINGOT
Malibud National High School
Gingoog City Division
Reference: Mark Vidallo – Makati Science High School

Put the following 5 animals in order
of your preference:
Cow, Tiger, Sheep, Horse, Pig
QUESTION 1

Write one word that describes each
one of the following:
Dog, Rat, Coffee, Sea
QUESTION 2

Think of someone, who also knows you
and is important to you, which you can
relate them to the following colors.
Write just one name for each color. Do
not repeat the names.
Yellow, Orange, Red, White, Green
QUESTION 3

This will define your priorities in life.
Cow Signifies CARREER
Tiger Signifies PRIDE
Sheep Signifies LOVE
Horse Signifies FAMILY
Pig Signifies MONEY
ANSWER:1

Your description of dog implies your own
personality
Your description of cat implies the personality of
your partner
Your description of rat implies the personality of
your enemies
Your description of coffee is how you interpret
sex
Your description of the sea implies your own life.
ANSWER:2

Yellow: Someone you will never forget
Orange: Someone you consider your true friend
Red: Someone that you really love
White: Your twin soul
Green: Someone that you will always remember for the rest of
your life.
ANSWER:3

L eader
P resenter
M aterials manager
O verseer
N ote taker

DEPARTMENT OF EDUCATION
BUREAU OF CURRICULUM DEVELOPMENT

Factoring Techniques
“Non-routine Problems”
Operations on Rational Expression
& Complex Fraction
“Partial Fractions”
Linear Equation
Key Contents

FACTORING TECHNIQUES
36𝑐𝑦5 − 56𝑐2𝑦3𝑧
A. Common Monomial Factor:
36𝑐𝑦5 − 56𝑐2𝑦3𝑧
36𝑐𝑦5
− 56𝑐2
𝑦3
𝑧
4𝑐𝑦3
(9𝑦2
− 14𝑐𝑧)
4𝑐𝑦3(9𝑦2 − 14𝑐𝑧)

B. Difference of Two Squares:
𝑥2 − 𝑦2 = (𝑥 + 𝑦)(𝑥 − 𝑦)
(2𝑥 − 3)2
− 𝑦 + 2 2
= [ 2𝑥 − 3 + 𝑦 + 2 ][ 2𝑥 − 3 − 𝑦 + 2 ]
(2𝑥 − 3)2
− 𝑦 + 2 2
= [ 2𝑥 − 3 + 𝑦 + 2 ][ 2𝑥 − 3 − 𝑦 + 2 ]
(2𝑥 − 3)2
− 𝑦 + 2 2
= [ 2𝑥 − 3 + 𝑦 + 2 ][ 2𝑥 − 3 − 𝑦 + 2 ]
(2𝑥 − 3)2
− 𝑦 + 2 2
= [ 2𝑥 − 3 + 𝑦 + 2 ][ 2𝑥 − 3 − 𝑦 + 2 ]
(2𝑥 + 𝑦 − 1)(2𝑥 − 𝑦 − 5)

C. Perfect Square Trinomial:
𝑥2
± 2𝑥𝑦 + 𝑦2
= (𝑥 ± 𝑦)2
𝑥4
− 14𝑥2
𝑦 + 49𝑦2
= (𝑥2
− 7𝑦)2
𝑥4
− 14𝑥2
𝑦 + 49𝑦2
= (𝑥2
− 7𝑦)2
𝑥4
− 14𝑥2
𝑦 + 49𝑦2
= (𝑥2
− 7𝑦)2
𝑥4
− 14𝑥2
𝑦 + 49𝑦2
= (𝑥2
− 7𝑦)2

C. Simple Trinomial:
𝑥2 + 2𝑥 − 35 = (𝑥 + 7)(𝑥 − 5)
𝑥2
+ 2𝑥 − 35 = (𝑥 + 7)(𝑥 − 5)
𝑥2
+ 2𝑥 − 35 = (𝑥 + 7)(𝑥 − 5)
𝑥2 + 2𝑥 − 35 = (𝑥 + 7)(𝑥 − 5)

D. General Trinomial by Grouping:
2𝑥2 + 7𝑥 + 5 = 2𝑥2 + 5𝑥 + 2𝑥 + 5
2𝑥2 + 7𝑥 + 5 = 2𝑥2 + 5𝑥 + 2𝑥 + 5
2𝑥2
+ 7𝑥 + 5 = 2𝑥2
+ 5𝑥 + 2𝑥 + 5
2𝑥2 + 7𝑥 + 5 = 𝑥(2𝑥 + 5) + 1(2𝑥 + 5)
2𝑥2
+ 7𝑥 + 5 = (2((2𝑥 + 5)(𝑥 + 1)

E. Sum and Difference of Cubes:
𝑥3
± 𝑦3
= (𝑥 ± 𝑦)(𝑥2
∓ 𝑥𝑦 + 𝑦2
)
125𝑥3
− 64𝑦3
= (𝑥 ± 𝑦)(𝑥2
∓ 𝑥𝑦 + 𝑦2
)
125𝑥3
− 64𝑦3
= (5𝑥 − 4𝑦)(𝑥2
∓ 𝑥𝑦 + 𝑦2
)
125𝑥3
− 64𝑦3
= (5𝑥 − 4𝑦)(𝑥2
∓ 𝑥𝑦 + 𝑦2
)
125𝑥3
− 64𝑦3
= (5𝑥 − 4𝑦)(25𝑥2
+ 20𝑥𝑦 + 16𝑦2
)
125𝑥3
− 64𝑦3
= (5𝑥 − 4𝑦)(25𝑥2
+ 20𝑥𝑦 + 16𝑦2
)
125𝑥3
− 64𝑦3
= (5𝑥 − 4𝑦)(25𝑥2
+ 20𝑥𝑦 + 16𝑦2
)

F. Grouping:
𝑥𝑦 − 𝑦 + 𝑥 − 1
𝑥𝑦 − 𝑦 + 𝑥 − 1
𝑦(𝑥 − 1) + (𝑥 − 1)
(𝑥 − 1) + 1(𝑥 − 1)
(𝑥 − 1)(𝑦 + 1)

G. Factoring 𝒙𝒏 ± 𝒚𝒏:
If n is an even number, then 𝒙𝒏 − 𝒚𝒏 can be considered as
the difference of two squares.
𝑥8
− 𝑦8
= 𝑥4 2
− 𝑦4 2
𝑥8
− 𝑦8
= 𝑥4 2
− 𝑦4 2
= (𝑥4
−𝑦4
)(𝑥4
+ 𝑦4
)
= (𝑥2
−𝑦2
)(𝑥2
+ 𝑦2
)(𝑥4
+ 𝑦4
)
= (𝑥 − 𝑦)(𝑥 + 𝑦)(𝑥2
+ 𝑦2
)(𝑥4
+ 𝑦4
)

H. Factoring 𝒙𝒏 ± 𝒚𝒏:
If n is a multiple of 3, then 𝒙𝒏 ± 𝒚𝒏 can be considered as the
sum and difference of two cubes.
𝑥6
+ 𝑦6
= 𝑥2 3
+ 𝑦2 3
= (𝑥2
+ 𝑦2
)(𝑥4
− 𝑥2
𝑦2
+𝑦4
)

I. Factoring 𝒙𝒏 ± 𝒚𝒏:
If n is odd and not a multiple of 3, then
𝒙𝒏
+ 𝒚𝒏
= (𝒙 + 𝒚)(𝒙𝒏−𝟏
− 𝒙𝒏−𝟐
𝒚 + 𝒙𝒏−𝟑
𝒚𝟐
− ⋯ + 𝒚𝒏−𝟏
)
𝒙𝒏 − 𝒚𝒏 = (𝒙 − 𝒚)(𝒙𝒏−𝟏 + 𝒙𝒏−𝟐𝒚 + 𝒙𝒏−𝟑𝒚𝟐 + ⋯ + 𝒚𝒏−𝟏)

Factor 𝑥5
− 32𝑦5
𝑥5
− 32𝑦5
= 𝑥 5
− (2𝑦)5
= (𝑥 − 2𝑦)(𝑥4
+ 𝑥3
(2𝑦) + 𝑥2
2𝑦 2
+ 𝑥(2𝑦)3
+ (2𝑦)4
)
= (𝑥 − 2𝑦)(𝑥4
+ 2𝑥3
𝑦 + 4𝑥2
𝑦2
+ 8𝑥𝑦3
+ 16𝑦4
)

Activity 1

3𝑥2
+ 3
3𝑥
= −
11𝑥
3𝑥
NON-ROUTINE FACTORING TECHNIQUES
1. If 3𝑥2
+ 11𝑥 + 3 = 0, what is 𝑥 +
1
𝑥
?
Solution:
3𝑥2
+ 3 = −11𝑥
𝑥 +
1
𝑥
= −
11
3

2. Simplify:
2017 2020 4039 + 2024
20182

2017 2020 4039 + 2024
20182
Solution:
Let 𝑥 = 2018
𝑥 − 1 𝑥 + 2 2𝑥 + 3 + (𝑥 + 6)
𝑥2
𝑥2
(2𝑥 + 5)
𝑥2
𝑥2
(2𝑥 + 5)
𝑥2
𝑥 − 1 𝑥 + 2 2𝑥 + 3 + (𝑥 + 6)
𝑥2
𝑥 − 1 𝑥 + 2 2𝑥 + 3 + (𝑥 + 6)
𝑥2
𝑥 − 1 𝑥 + 2 2𝑥 + 3 + (𝑥 + 6)
𝑥2

𝑥 − 1 𝑥 + 2 2𝑥 + 3 + (𝑥 + 6)
𝑥2
= 2𝑥 + 5
2(2018) + 5
2(2020 − 2) + 5
2 2020 − 2(2) + 5
4041

3. If 𝑟 + 𝑠 − 𝑎 − 𝑏 = 2 and 𝑟𝑠 + 𝑎 + 𝑏 + 2 = 0 , find the
value of (𝑟 + 1)(𝑠 + 1).

𝑟 + 𝑠 − 𝑎 − 𝑏 = 2
Solution:
v 𝑟 + 𝑠 = 𝑎 + 𝑏 + 2
𝑟𝑠 + 𝑎 + 𝑏 + 2 = 0 𝑟𝑠 + (𝑟 + 𝑠) = 0
Upon expanding the unknown,
𝑟 + 1 𝑠 + 1 = 𝑟𝑠 + 𝑟 + 𝑠 + 1 = 1
𝑟𝑠 + 𝑎 + 𝑏 + 2 = 0

4. Given: 𝑥 =
1
2
7 + 5 and 𝑦 =
1
2
7 − 5 , find
the numerical value 𝑥2
+ 𝑦2
.

Solution:
𝑥2
+ 𝑦2
= (𝑥 + 𝑦)2
− 2𝑥𝑦
The expanded form of the unknown is
(𝑥 + 𝑦)2
=
1
2
7 + 5 +
1
2
7 − 5
2
= ( 7)2
= 7
(𝑥 + 𝑦)2
=
1
2
7 + 5 +
1
2
7 − 5
2
= ( 7)2
= 7
(𝑥 + 𝑦)2
=
1
2
7 + 5 +
1
2
7 − 5
2
= ( 7)2
= 7
(𝑥 + 𝑦)2
=
1
2
7 + 5 +
1
2
7 − 5
2
= ( 7)2
= 7
−2𝑥𝑦 = −2
7
2
+
5
2
7
2
−
5
2
= −2
7
4
−
5
4
= −1
−2𝑥𝑦 = −2
7
2
+
5
2
7
2
−
5
2
= −2
7
4
−
5
4
= −1

𝑥2
+ 𝑦2
= 6

5. Factor completely:
243𝑥10
+ 32𝑦5

8𝑝2
+ 2𝑚𝑝 − 2𝑛𝑝 − 3𝑚2
+ 6𝑚𝑛 − 3𝑛2

7. Solve for x in the equation
(𝑥 + 1) + 2 𝑥 + 1 + 3 𝑥 + 1 + ⋯ + 10 𝑥 + 1 = 110

8. Find 𝑥2
+ 𝑦2
if 𝑥𝑦 + 𝑥 + 𝑦 = 90 and 𝑥2
𝑦 + 𝑥𝑦2
= 2025.

9. If a, b and c are positive numbers such that
𝑎 + 𝑏 + 𝑐 = 10 and
1
𝑎+𝑏
+
1
𝑏+𝑐
+
1
𝑐+𝑎
=
4
5
, what
is the value of
𝑐
𝑎+𝑏
+
𝑎
𝑏+𝑐
+
𝑏
𝑐+𝑎
?

Given:
𝑎 + 𝑏 + 𝑐 = 10 and
1
𝑎 + 𝑏
+
1
𝑏 + 𝑐
+
1
𝑐 + 𝑎
=
4
5
𝑐
𝑎 + 𝑏
+
𝑎
𝑏 + 𝑐
+
𝑏
𝑐 + 𝑎
Find:
Observe that
𝑎 = 10 − 𝑏 + 𝑐 , 𝑏 = 10 − 𝑎 + 𝑐 and 𝑐 = 10 − 𝑎 + 𝑏

The expression
𝑐
𝑎+𝑏
+
𝑎
𝑏+𝑐
+
𝑏
𝑐+𝑎
can be written as
10 − (𝑎 + 𝑏)
𝑎 + 𝑏
+
10 − (𝑏 + 𝑐)
𝑏 + 𝑐
+
10 − (𝑐 + 𝑎)
𝑐 + 𝑎
10
𝑎 + 𝑏
+
10
𝑏 + 𝑐
+
10
𝑐 + 𝑎
− 3
10
𝑎 + 𝑏
−
(𝑎 + 𝑏)
𝑎 + 𝑏
+
10
𝑏 + 𝑐
−
(𝑏 + 𝑐)
𝑏 + 𝑐
+
10
𝑐 + 𝑎
−
(𝑐 + 𝑎)
𝑐 + 𝑎
10
𝑎 + 𝑏
−
(𝑎 + 𝑏)
𝑎 + 𝑏
+
10
𝑏 + 𝑐
−
(𝑏 + 𝑐)
𝑏 + 𝑐
+
10
𝑐 + 𝑎
−
(𝑐 + 𝑎)
𝑐 + 𝑎

10
𝑎 + 𝑏
+
10
𝑏 + 𝑐
+
10
𝑐 + 𝑎
− 3
10
1
𝑎 + 𝑏
+
1
𝑏 + 𝑐
+
1
𝑐 + 𝑎
− 3
10
4
5
− 3 = 5

10. If a, b and c be real numbers with
𝑎𝑏
𝑎+𝑏
=
1
3
,
𝑏𝑐
𝑏+𝑐
=
1
4
𝑐𝑎
𝑐+𝑎
=
1
5
, then what is the value of ,
𝑎𝑏𝑐
𝑎𝑏+𝑏𝑐+𝑐𝑎
?

𝑥6
+ 𝑥2
− 2

Activity 2
Non-routine Problems
Group Work

Activity 3
Factoring By Grouping
Group Work

RATIONAL EXPRESSION
Definition
If 𝑃 𝑥 and 𝑄 𝑥 are polynomials and 𝑄 𝑥 ≠ 0,
then
𝑃(𝑥)
𝑄(𝑥)
is a rational expression in x where 𝑃 𝑥 and
𝑄 𝑥 are the numerator and denominator, respectively
of the expression.

Activity 4
Simplifying Rational Expression
Moving Test

MULTIPLICATION & DIVISION OF RATIONAL EXPRESSION
1. Perform the indicated operations and simplify:
𝑥2
− 𝑦2
2𝑥2 + 𝑥𝑦 − 3𝑦2
∙
6𝑥2
+ 13𝑥𝑦 + 6𝑦2
𝑥 + 𝑦

𝑎3
− 𝑏3
2𝑎2 + 4𝑎𝑏 + 2𝑏2
÷
𝑎3
+ 𝑎2
𝑏 + 𝑎𝑏2
𝑎2 − 𝑏2
÷
3(2𝑎2
− 3𝑎𝑏 + 𝑏2
)
6𝑎 + 6𝑏

𝑎5
− 1
𝑎 + 1
∙
𝑎2
+ 1
𝑎4 − 1
÷
𝑎4
+ 𝑎3
+ 𝑎2
+ 𝑎 + 1
𝑎4 − 𝑎3 + 𝑎 − 1

ADDITION & SUBTRACTION OF RATIONAL EXPRESSION
2𝑥
𝑥2 − 5𝑥 − 6
+
1
𝑥2 − 6𝑥
+
6𝑥 + 4
𝑥3 − 5𝑥2 − 6𝑥

1
3𝑎 + 4
−
𝑎 − 7
3𝑎2 + 13𝑎 + 12
−
4
𝑎2 + 4𝑎 + 3

6. If
𝑎
𝑥+1
+
𝑏
(𝑥+1)2 +
𝑐
(1+𝑥)3 =
𝑥2+3𝑥+3
(𝑥+1)3 , determine the
value of a, b and c.

COMPLEX FRACTION
7. The fraction
36
79
=
1
𝑎+
1
𝑏+
1
𝑐
where a, b and c are
natural numbers. Find the numerical value of
a + b + c .

COMPLEX FRACTION
8. Simplify
1+
1
1+
1
𝑥
1+
1
𝑥+1

COMPLEX FRACTION
1 −
1
𝑥 − 1
+
𝑥
𝑥 + 1
1
𝑥 + 1
+
𝑥
1 − 𝑥

Activity 5
Operations on Rational Expression and
Complex Expression
Group Work

Activity 6
Operations on Rational Expression and
Complex Expression
Quiz Bee (Knock-Out Format)

MISCELLANEOUS PROBLEMS ON LINEAR EQUATION
1. What is the equation of the line through
(– 2, 6) with x-intercept thrice the y-
intercept.

2. Give the slope-intercept form of the line
whose y-intercept is twice the x-intercept
and is passing through (2, – 3).

3. Find the equation of the line intersecting
the line 2𝑦 − 5𝑥 = 11 at its y-intercept
such that these two lines are
perpendicular.

4. A right triangle has its right angle at (– 4, 1)
and the equation of one of its legs is
2𝑥 − 3𝑦 + 11 = 0. Find the equation of the
other leg.

5. Find the values of k and m so that the system of
linear equation
3𝑥 − 𝑘𝑦 = −5
7𝑦 − 4𝑥 = 𝑚
has (– 2, 1) as the only solution.

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