2. Drill: Square the following number
1. 42
2. 62
3. 102
4. 92
5. 12
6. 32
7. 82
8. 22
9. 122
10. 72
3. So the answer in the drill
16, 100, 36, 81, 1, 9, 64, 4, 144 and 49 are all
examples of square number or PERFECT
SQUARE Number
A square number or perfect square is an
integer that is the square of an integer; in other
words, it is the product of some integer with
itself.
since it can be written as the product of same number.
4. Example 1:
in Perfect square number 16, the root of 16 is 4 since
16 can be express or written as 42 or 4● 4 .
The number that is being square is the root of
the perfect square number.
Example 2:
The root of 81 is 9, since 81 can be express or
written as 92 or 9● 9 .
5. Drill: Tell if the number is a perfect square number? If it is
give the square root of the number.
1. 4
2. 6
3. 10
4. 9
5. 100
6. 36
7. 81
8. 16
9. 64
10. 144
6. Factoring Difference of Two Squares is the product of
sum and difference of two binomials.
Lesson Proper:
(x+1) is the sum of binomial
(x – 1) is the difference of binomial
(x+1)multiply by (x -1) the product is (x2 -1)
Note: In sum and difference two binomial, they have the same
first and
last terms.
7. Factoring Difference of Two Squares is the product of
sum and difference of two binomials.
Lesson Proper:
So (x+1) is the sum of binomial and
(x – 1) is the difference of binomial are the
factors of the product (x2 -1) since
(x+1)(x-1) = x2 - 1
Note: In sum and difference two binomial, they have the same
first and
last terms.
8. Factoring Difference of Two Squares is finding the sum and
difference of two binomial whose product is the difference of
the squares of the binomial.
Lesson Proper:
In x2 - 1 is the difference of two squares because x2 – 12
finding the factors of x2 – 1 that is
(x+1) is the sum of binomial and
(x – 1) is the difference of binomial
So
x2 – 1= (x+1)(x-1)
9. In Factoring Difference of Two Squares remember the following
characteristic to factor using difference of two squares.
Lesson Proper:
2. The first term is a perfect square or can be written in
squares.
3. They have subtraction (minus) in the middle
sign.
4. The last term is also a perfect square or can
be written in squares.
1. It has two terms.
10. Example1: x2 – 4 can be factor using factoring difference of two
squares?
Lesson Proper:
2. The first term is a perfect square or can be written in
squares
3. They have subtraction (minus) in the middle
sign.
4. The last term is also a perfect square or can
be written in squares.
1. It has two terms.
. x2
22
11. Example 2: m4 + 16 can be factor using factoring difference of
two squares?
Lesson Proper:
2. The first term is a perfect square or can be written in
squares
3. They have subtraction (minus) in the middle
sign.
4. The last term is also a perfect square or can
be written in squares.
1. It has two terms.
. (m2)2
42
12. Example 3: a6 - 8 can be factor using Factoring Difference of
two squares?
Lesson Proper:
2. The first term is a perfect square or can be written in
squares
3. They have subtraction (minus) in the middle
sign.
4. The last term is also a perfect square or can
be written in squares.
1. It has two terms.
. (2m3)2
NO, 8 is not a perfect square
13. Example 4: 10n2 - 25 can be factor using Factoring Difference of
two squares?
Lesson Proper:
2. The first term is a perfect square or can be written in
squares
3. They have subtraction (minus) in the middle
sign.
4. The last term is also a perfect square or can
be written in squares.
1. It has two terms.
. No since 10 is not a perfect square but (n)2
52
14. Steps Using Factoring Difference of Two Squares
Lesson Proper: Method 1
2. Find the square root of the second term
3. Write the roots as the sum and difference of
two binomial (a+b)(a-b)
1. Find the square root of the first term.
15. Example1: Find the factors of x2 - 4
Lesson Proper: Method 1
2. Find the square root of the second term
3. Write the roots as the sum and difference of
two binomial (a+b)(a-b)
1. Find the square root of the first term.
The square root of x2 is x
The square root of 4 is 2
(x + 2)(x – 2)
So the factor of x2 –
4 is (x+2)(x – 2)
x2 – 4 = (x+2)(x – 2)
Note: square root of the variable copy the variable and dive the exponent by 2
16. Example 2: Find the factors of 16 m4 - 36
Lesson Proper: Method 1
2. Find the square root of the second term
3. Write the roots as the sum and difference of
two binomial (a+b)(a-b)
1. Find the square root of the first term.
The square root of 16m4 is 4m2
The square root of 36 is 6
(4m2 + 6)(4m2 – 6)
So the factor of 16x4 –
36 is (4m2 +6)(4m2 – 6)
16x4 – 36 = (4m2 +6)(4m2 – 6)
17. Find the factors of the following:
Try This: Method 1
1. a2 - 100
2. 9y4 – 49 z2
18. Steps Using Factoring Difference of Two Squares
Lesson Proper: Method 2
2. Write the factors as (a + b) and (a – b)
1. Write the difference of two square using the formula
a2 – b2 = (a+b)(a – b) meaning express the first and last
term in square form.
19. Find the factors of x2 - 4
Example: Method 2
2. Write the factors as (a + b) and (a – b)
1. Write the difference of two square using the formula
a2 – b2 = (a+b)(a – b) meaning express the first and last
term in square form. x2 – 22 since 4 is 22
In x2 – 22 , x2 = a2 and 22 = b2 therefore a = x and b = 2
So (a+b)(a-b) = ( x + 2)(x – 2)
Factors of x2 – 4 is ( x + 2)(x – 2)
20. Find the factors of 16m4 - 25
Example: Method 2
1. 16m4 – 25 = (4m2)2 - 52
2. a2 = (4m2)2 and b2 = 52 ,
Factors of 16m4 – 25 is
(4m2 + 5)(4m2 – 5)
a = 4m2 and b = 5
So (a+b)(a-b) = (4m2 + 5)(4m2 – 5)
16m4 – 25 = (4m2 + 5)(4m2 – 5)
21. Find the factors of the following:
Try This: Method 2
1. b2 - 81
2. 36p6 – 64q2
22. Find the factors of x2 - 25
Practice :
a. ( x + 5 ) ( x - 5 )
b. ( x - 5 ) ( x - 5 )
c. ( x + 5 ) ( x + 5 )
d. Prime
23. What are the factors of x2 - 16?
Practice :
a. Prime
b. ( x - 4) ( x + 4)
c. ( x - 8) ( x + 8)
d. ( x - 4) ( x - 4)
24. What are the factors of 4x2 - 64y2?
Practice :
a. Prime
b. ( 2x - 32y) (2 x + 32y)
c. ( 2x - 8y) (2 x - 8y)
d. ( 2x - 8y) (2 x + 8y)
25. What are the factors of 25x2y2 - 100?
Practice :
a. ( 5xy - 10) (5xy + 10)
b. ( 5xy + 10) (5xy + 10)
c. Prime
d. ( 5xy - 50) (5xy + 50)
26. What are the factors of x2 + 81?
Practice :
a. ( x - 9 ) ( x + 9 )
b. ( x + 9 ) ( x + 9 )
c. ( x - 9 ) ( x - 9 )
d. Prime
