Here are the factorizations:1. x3 + 343 = (x + 17)(x2 - 17x + 343)2. x3 + 216 = (x + 15)(x2 - 15x + 216) 3. x3 - 64 = (x - 8)(x2 + 8x + 64)4. x3 - 512 = (x - 22)(x2 + 22x + 512

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Sum and Difference of Cubes. The sum or difference of two cubes can be factored into a product of a binomial times a trinomial. That is, x3 + y3 = (x + y)(x2 − xy + y2) and x3 − y3 = (x − y)(x2 + xy + y2) .

Education

PERFECT CUBES.
= 1 × 1 × 1 = 1
= 2 × 2 × 2 = 8
= 3 × 3 × 3 = 27
= 4 × 4 × 4 = 64
= 5 × 5 × 5 = 125
= 6 × 6 × 6 = 216
= 7 × 7 × 7 = 343
= 8 × 8 × 8 = 512
= 9 × 9 × 9 = 729
= 10 × 10 × 10 = 1000
13
23
33
43
53
63
73
83
93
103

SUM OF TWO CUBES
𝒂𝟑
+ 𝒃𝟑
= (𝒂 + 𝒃) (𝒂𝟐
−𝒂𝒃 + 𝒃𝟐
)
DIFFERENCE OF TWO CUBES
𝒂𝟑
− 𝒃𝟑
= (𝒂 − 𝒃) (𝒂𝟐
+𝒂𝒃 + 𝒃𝟐
)

SUM OF TWO CUBES
𝒂𝟑 + 𝒃𝟑= (𝒂 + 𝒃) (𝒂𝟐−𝒂𝒃 + 𝒃𝟐)
FACTORS OF SUM OR DIFFERENCE OF TWO CUBES
opposite in sign
Same positive
Always positive

FACTORS OF SUM OR DIFFERENCE OF TWO CUBES
S -SAME
O -OPPOSITE
A -ALWAYS
P -POSITIVE
SQUARE
MULTIPLY
SQUARE
X
Factor 𝒙𝟑
+125
( )𝟑
( )𝟑
𝒙 𝟓
𝒂 𝒃
+
𝒂𝟑
+ 𝒃𝟑
= (𝒂 + 𝒃) (𝒂𝟐
−𝒂𝒃 + 𝒃𝟐
)
𝒙𝟑
+ 𝟏𝟐𝟓 = ( )( )
+ − +
𝒙 + 𝟓 𝒙𝟐 − (𝒙)(𝟓) + 𝟓𝟐
𝒙𝟑
+ 125= (𝒙 + 𝟓) (𝒙𝟐
−𝟓𝒙 + 𝟐𝟓)✓
SUM OF TWO CUBES
Example 1:

FACTORS OF SUM OR DIFFERENCE OF TWO CUBES
S -SAME
O -OPPOSITE
A -ALWAYS
P -POSITIVE
SQUARE
MULTIPLY
SQUARE
X
Factor 𝒙𝟑
+8
( )𝟑
( )𝟑
𝒙 𝟐
𝒂 𝒃
+
𝒂𝟑
+ 𝒃𝟑
= (𝒂 + 𝒃) (𝒂𝟐
−𝒂𝒃 + 𝒃𝟐
)
𝒙𝟑
+ 8= ( )( )
+ − +
𝒙 + 𝟐 𝒙𝟐 − (𝒙)(𝟐) + 𝟐𝟐
𝒙𝟑
+ 8= (𝒙 + 𝟐) (𝒙𝟐
−𝟐𝒙 + 𝟒) ✓
SUM OF TWO CUBES
Example 2 :

FACTORS OF SUM OR DIFFERENCE OF TWO CUBES
S -SAME
O -OPPOSITE
A -ALWAYS
P -POSITIVE
SQUARE
MULTIPLY
SQUARE
X
Factor 𝒙𝟑
+64
( )𝟑
( )𝟑
𝒙 𝟒
𝒂 𝒃
+
𝒂𝟑
+ 𝒃𝟑
= (𝒂 + 𝒃) (𝒂𝟐
−𝒂𝒃 + 𝒃𝟐
)
𝒙𝟑
+ 64= ( )( )
+ − +
𝒙 + 𝟒 𝒙𝟐 − (𝒙)(𝟒) + 𝟒𝟐
𝒙𝟑
+ 64= (𝒙 + 𝟒) (𝒙𝟐
−𝟒𝒙 + 𝟏𝟔) ✓
SUM OF TWO CUBES
Example 3 :

FACTORS OF SUM OR DIFFERENCE OF TWO CUBES
S -SAME
O -OPPOSITE
A -ALWAYS
P -POSITIVE
SQUARE
MULTIPLY
SQUARE
X
Factor 𝒙𝟑
+27
( )𝟑
( )𝟑
𝒙 𝟑
𝒂 𝒃
+
𝒂𝟑
+ 𝒃𝟑
= (𝒂 + 𝒃) (𝒂𝟐
−𝒂𝒃 + 𝒃𝟐
)
𝒙𝟑
+ 𝟐𝟕 = ( )( )
+ − +
𝒙 + 𝟑 𝒙𝟐 − (𝒙)(𝟑) + 𝟑𝟐
𝒙𝟑
+ 3= (𝒙 + 𝟑) (𝒙𝟐
−𝟑𝒙 + 𝟗) ✓
SUM OF TWO CUBES
Example 4:

DIFFERENCE OF TWO CUBES
𝒂𝟑 − 𝒃𝟑= (𝒂 − 𝒃) (𝒂𝟐+𝒂𝒃 + 𝒃𝟐)
FACTORS OF SUM OR DIFFERENCE OF TWO CUBES
opposite in sign
Same negative
Always positive

DIFFERENCE OF TWO CUBES
FACTORS OF SUM OR DIFFERENCE OF TWO CUBES
Example 3 :

TRY TO ANSWER.
Direction: Factor the following:
1. 𝒙𝟑+343
2. 𝒙𝟑
+216
3. 𝒙𝟑 − 64
4. 𝒙𝟑
− 512

Here are the factorizations:1. x3 + 343 = (x + 17)(x2 - 17x + 343)2. x3 + 216 = (x + 15)(x2 - 15x + 216) 3. x3 - 64 = (x - 8)(x2 + 8x + 64)4. x3 - 512 = (x - 22)(x2 + 22x + 512

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Here are the factorizations:1. x3 + 343 = (x + 17)(x2 - 17x + 343)2. x3 + 216 = (x + 15)(x2 - 15x + 216) 3. x3 - 64 = (x - 8)(x2 + 8x + 64)4. x3 - 512 = (x - 22)(x2 + 22x + 512

2. PERFECT CUBES. = 1 × 1 × 1 = 1 = 2 × 2 × 2 = 8 = 3 × 3 × 3 = 27 = 4 × 4 × 4 = 64 = 5 × 5 × 5 = 125 = 6 × 6 × 6 = 216 = 7 × 7 × 7 = 343 = 8 × 8 × 8 = 512 = 9 × 9 × 9 = 729 = 10 × 10 × 10 = 1000 13 23 33 43 53 63 73 83 93 103

3. SUM OF TWO CUBES 𝒂𝟑 + 𝒃𝟑 = (𝒂 + 𝒃) (𝒂𝟐 −𝒂𝒃 + 𝒃𝟐 ) DIFFERENCE OF TWO CUBES 𝒂𝟑 − 𝒃𝟑 = (𝒂 − 𝒃) (𝒂𝟐 +𝒂𝒃 + 𝒃𝟐 )

4. SUM OF TWO CUBES 𝒂𝟑 + 𝒃𝟑= (𝒂 + 𝒃) (𝒂𝟐−𝒂𝒃 + 𝒃𝟐) FACTORS OF SUM OR DIFFERENCE OF TWO CUBES opposite in sign Same positive Always positive

5. FACTORS OF SUM OR DIFFERENCE OF TWO CUBES S -SAME O -OPPOSITE A -ALWAYS P -POSITIVE SQUARE MULTIPLY SQUARE X Factor 𝒙𝟑 +125 ( )𝟑 ( )𝟑 𝒙 𝟓 𝒂 𝒃 + 𝒂𝟑 + 𝒃𝟑 = (𝒂 + 𝒃) (𝒂𝟐 −𝒂𝒃 + 𝒃𝟐 ) 𝒙𝟑 + 𝟏𝟐𝟓 = ( )( ) + − + 𝒙 + 𝟓 𝒙𝟐 − (𝒙)(𝟓) + 𝟓𝟐 𝒙𝟑 + 125= (𝒙 + 𝟓) (𝒙𝟐 −𝟓𝒙 + 𝟐𝟓)✓ SUM OF TWO CUBES Example 1:

6. FACTORS OF SUM OR DIFFERENCE OF TWO CUBES S -SAME O -OPPOSITE A -ALWAYS P -POSITIVE SQUARE MULTIPLY SQUARE X Factor 𝒙𝟑 +8 ( )𝟑 ( )𝟑 𝒙 𝟐 𝒂 𝒃 + 𝒂𝟑 + 𝒃𝟑 = (𝒂 + 𝒃) (𝒂𝟐 −𝒂𝒃 + 𝒃𝟐 ) 𝒙𝟑 + 8= ( )( ) + − + 𝒙 + 𝟐 𝒙𝟐 − (𝒙)(𝟐) + 𝟐𝟐 𝒙𝟑 + 8= (𝒙 + 𝟐) (𝒙𝟐 −𝟐𝒙 + 𝟒) ✓ SUM OF TWO CUBES Example 2 :

7. FACTORS OF SUM OR DIFFERENCE OF TWO CUBES S -SAME O -OPPOSITE A -ALWAYS P -POSITIVE SQUARE MULTIPLY SQUARE X Factor 𝒙𝟑 +64 ( )𝟑 ( )𝟑 𝒙 𝟒 𝒂 𝒃 + 𝒂𝟑 + 𝒃𝟑 = (𝒂 + 𝒃) (𝒂𝟐 −𝒂𝒃 + 𝒃𝟐 ) 𝒙𝟑 + 64= ( )( ) + − + 𝒙 + 𝟒 𝒙𝟐 − (𝒙)(𝟒) + 𝟒𝟐 𝒙𝟑 + 64= (𝒙 + 𝟒) (𝒙𝟐 −𝟒𝒙 + 𝟏𝟔) ✓ SUM OF TWO CUBES Example 3 :

8. FACTORS OF SUM OR DIFFERENCE OF TWO CUBES S -SAME O -OPPOSITE A -ALWAYS P -POSITIVE SQUARE MULTIPLY SQUARE X Factor 𝒙𝟑 +27 ( )𝟑 ( )𝟑 𝒙 𝟑 𝒂 𝒃 + 𝒂𝟑 + 𝒃𝟑 = (𝒂 + 𝒃) (𝒂𝟐 −𝒂𝒃 + 𝒃𝟐 ) 𝒙𝟑 + 𝟐𝟕 = ( )( ) + − + 𝒙 + 𝟑 𝒙𝟐 − (𝒙)(𝟑) + 𝟑𝟐 𝒙𝟑 + 3= (𝒙 + 𝟑) (𝒙𝟐 −𝟑𝒙 + 𝟗) ✓ SUM OF TWO CUBES Example 4:

9. DIFFERENCE OF TWO CUBES 𝒂𝟑 − 𝒃𝟑= (𝒂 − 𝒃) (𝒂𝟐+𝒂𝒃 + 𝒃𝟐) FACTORS OF SUM OR DIFFERENCE OF TWO CUBES opposite in sign Same negative Always positive

10. FACTORS OF SUM OR DIFFERENCE OF TWO CUBES S -SAME O -OPPOSITE A -ALWAYS P -POSITIVE SQUARE MULTIPLY SQUARE X Factor 𝒙𝟑 −8 ( )𝟑 ( )𝟑 𝒙 𝟐 𝒂 𝒃 − 𝒂𝟑 − 𝒃𝟑 = (𝒂 − 𝒃) (𝒂𝟐 +𝒂𝒃 + 𝒃𝟐 ) 𝒙𝟑 −8 = ( )( ) − + + 𝒙 − 𝟐 𝒙𝟐 + (𝒙)(𝟐) + 𝟐𝟐 𝒙𝟑 − 8= (𝒙 − 𝟐) (𝒙𝟐 +𝟐𝒙 + 𝟒) ✓ DIFFERENCE OF TWO CUBES Example 1 :

11. FACTORS OF SUM OR DIFFERENCE OF TWO CUBES S -SAME O -OPPOSITE A -ALWAYS P -POSITIVE SQUARE MULTIPLY SQUARE X Factor 𝒙𝟑 − 27 ( )𝟑 ( )𝟑 𝒙 𝟑 𝒂 𝒃 − 𝒂𝟑 − 𝒃𝟑 = (𝒂 − 𝒃) (𝒂𝟐 +𝒂𝒃 + 𝒃𝟐 ) 𝒙𝟑 −𝟐𝟕 =( )( ) − + + 𝒙 − 𝟑 𝒙𝟐 + (𝒙)(𝟑) + 𝟑𝟐 𝒙𝟑 − 27= (𝒙 − 𝟑) (𝒙𝟐 +𝟑𝒙 + 𝟗) ✓ DIFFERENCE OF TWO CUBES Example 2:

12. DIFFERENCE OF TWO CUBES FACTORS OF SUM OR DIFFERENCE OF TWO CUBES Example 3 :

13. TRY TO ANSWER. Direction: Factor the following: 1. 𝒙𝟑+343 2. 𝒙𝟑 +216 3. 𝒙𝟑 − 64 4. 𝒙𝟑 − 512

Here are the factorizations:1. x3 + 343 = (x + 17)(x2 - 17x + 343)2. x3 + 216 = (x + 15)(x2 - 15x + 216) 3. x3 - 64 = (x - 8)(x2 + 8x + 64)4. x3 - 512 = (x - 22)(x2 + 22x + 512

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Similar to Here are the factorizations:1. x3 + 343 = (x + 17)(x2 - 17x + 343)2. x3 + 216 = (x + 15)(x2 - 15x + 216) 3. x3 - 64 = (x - 8)(x2 + 8x + 64)4. x3 - 512 = (x - 22)(x2 + 22x + 512

Similar to Here are the factorizations:1. x3 + 343 = (x + 17)(x2 - 17x + 343)2. x3 + 216 = (x + 15)(x2 - 15x + 216) 3. x3 - 64 = (x - 8)(x2 + 8x + 64)4. x3 - 512 = (x - 22)(x2 + 22x + 512 (20)

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Here are the factorizations:1. x3 + 343 = (x + 17)(x2 - 17x + 343)2. x3 + 216 = (x + 15)(x2 - 15x + 216) 3. x3 - 64 = (x - 8)(x2 + 8x + 64)4. x3 - 512 = (x - 22)(x2 + 22x + 512