1. Factoring the Sum and Difference of Cubes

2. Compare/contrast sum & difference of cubes with difference of squares.
Essential Question
Compare/contrast sum & difference of cubes with difference of squares.

3. Sum of Two Cubes 𝑎 3 + 𝑏 3 =(𝑎+𝑏)( 𝑎 2 −𝑎𝑏+ 𝑏 2 ) Sum (addition) sign

4. Difference of Two Cubes
𝑎 3 − 𝑏 3 =(𝑎−𝑏)( 𝑎 2 +𝑎𝑏+ 𝑏 2 )
Difference (subtraction) sign
Two Cubes

5. Use this pattern: 𝑎 3 + 𝑏 3 = (𝑎+𝑏) ( 𝑎 2 ) − 𝑎𝑏 + 𝑏 2
( 𝑎 ) − 𝑎𝑏
+ 𝑏 2
Write a new binomial without the exponents.
Use the new binomial to create the trinomial.
1. Square the first and last terms of the binomial to create the first and last terms of the trinomial.
2. Multiply the terms of the binomial to create the middle term of the trinomial.
3. Sign of the 2nd term is opposite of the binomial.

6. Example 1 =𝑥 3 + 2 3 =(𝑥+2)( ) 𝑎 3 + 𝑏 3 =(𝑎+𝑏)( 𝑎 2 −𝑎𝑏+ 𝑏 2 )
Factor: 𝑥 3 +8
Example 1
Write as a sum of
2 cubes: =𝑥
Write the binomial
without the cubes:
=(𝑥+2)( )

7. =(𝑥+2)( ) =(𝑥+2)( 𝑥 2 +4) =(𝑥+2)( 𝑥 2 2𝑥+4) =(𝑥+2)( 𝑥 2 −2𝑥+4)
𝑎 3 + 𝑏 3 =(𝑎+𝑏)( 𝑎 2 −𝑎𝑏+ 𝑏 2 )
=(𝑥+2)( )
Square the first and last terms:
=(𝑥+2)( 𝑥 )
Multiply the terms in the binomial:
=(𝑥+2)( 𝑥 𝑥+4)
=(𝑥+2)( 𝑥 2 −2𝑥+4)
Opposite signs:

8. Example 2 =(2𝑥) 3 − 3 3 =(2𝑥−3)( ) 𝑎 3 − 𝑏 3 =(𝑎−𝑏)( 𝑎 2 +𝑎𝑏+ 𝑏 2 )
Factor: 𝑥 3 −27
Example 2
Write as the difference of
2 cubes:
=(2𝑥) 3 − 3 3
Write the binomial
without the cubes:
=(2𝑥−3)( )

9. =(2𝑥−3)( ) =(2𝑥−3)( 4𝑥 2 +9) =(2𝑥−3)(4 𝑥 2 6𝑥+9) =(2𝑥−3)(4 𝑥 2 +6𝑥+9)
𝑎 3 + 𝑏 3 =(𝑎+𝑏)( 𝑎 2 −𝑎𝑏+ 𝑏 2 )
=(2𝑥−3)( )
The sign of the last term in the trinomial is always positive!
Square the first and last terms:
=(2𝑥−3)( 4𝑥 )
Multiply the terms in the binomial:
=(2𝑥−3)(4 𝑥 𝑥+9)
=(2𝑥−3)(4 𝑥 2 +6𝑥+9)
Opposite signs:

10. Hint
Don’t try to factor the trinomial after factoring the sum or difference of two cubes.
= 2𝑥−3 4 𝑥 2 +6𝑥+9
If the greatest common factor has already been taken out, the resulting trinomial cannot be factored using integers.

11. Example 3 =( 𝑥 2 ) 3 + (5𝑦) 3 =( 𝑥 2 +5𝑦)( )
𝑎 3 + 𝑏 3 =(𝑎+𝑏)( 𝑎 2 −𝑎𝑏+ 𝑏 2 )
Factor: 𝑥 𝑦 3
Example 3
Write as a sum of
2 cubes:
=( 𝑥 2 ) 3 + (5𝑦) 3
Write the binomial
without the cubes:
=( 𝑥 2 +5𝑦)( )

12. 𝑎 3 + 𝑏 3 =(𝑎+𝑏)( 𝑎 2 −𝑎𝑏+ 𝑏 2 )
=( 𝑥 2 +5𝑦)( )
Square the first and last terms:
=( 𝑥 2 +5𝑦)( 𝑥 𝑦 2 )
Multiply the terms in the binomial:
=( 𝑥 2 +5𝑦)( 𝑥 𝑥 2 𝑦+25 𝑦 2 )
Opposite signs:
=( 𝑥 2 +5𝑦)( 𝑥 4 −5 𝑥 2 𝑦+25 𝑦 2 )

13. Example 4 =(1) 3 − 6 𝑥 3 𝑦 3 =(1−6 𝑥 3 𝑦)( )
𝑎 3 − 𝑏 3 =(𝑎−𝑏)( 𝑎 2 +𝑎𝑏+ 𝑏 2 )
Factor: −216 𝑥 9 𝑦 3
Example 4
Write as the difference of
2 cubes:
=(1) 3 − 6 𝑥 3 𝑦 3
Write the binomial
without the cubes:
=(1−6 𝑥 3 𝑦)( )

14. 𝑎 3 + 𝑏 3 =(𝑎+𝑏)( 𝑎 2 −𝑎𝑏+ 𝑏 2 )
=(1−6 𝑥 3 𝑦)( )
Square the first and last terms:
=(1−6 𝑥 3 𝑦)( 𝑥 6 𝑦 2 )
Multiply the terms in the binomial:
=(1−6 𝑥 3 𝑦)(1 6 𝑥 3 𝑦+36 𝑥 6 𝑦 2 )
Opposite signs:
=(1−6 𝑥 3 𝑦)(1+6 𝑥 3 𝑦+36 𝑥 6 𝑦 2 )