1. 5.5: Factoring the Sum and Difference of Two Cubes

2. Objectives Factor the sum of two cubes.
Factor the difference of two cubes.
Factor a polynomial involving the sum or difference of two cubes.

3. Factoring the sum and difference of two cubes
Recall: x2 – y2 = (x + y)(x – y)
In this section, we will look at
x3 + y3
x3 – y3

4. Factor the sum and difference of two cubes
We note:
(x + y)(x2 – xy + y2) = x3 – x2y + xy2 + x2y– xy2 + y3
= x3 + y3
(x - y)(x2 + xy + y2) = x3 + x2y + xy2 - x2y -xy2 -y3
= x3 - y3
So, we can conclude:
x3 + y3 = (x + y)(x2 – xy + y2)
x3 - y3 = (x - y)(x2 + xy + y2)

5. Example Useful to know cubes of 1, 2, 3, 4, etc. Factor: x3 + 8
Solution: x3 + 8 = x = (x + 2)(x2 – x  ) = (x + 2)(x2 – 2x + 4) n 1 2 3 4 5 6 7 8 9 10 n3 27 64
125
216
343
512
729
1000

6. Example
Factor: a3 – 64b3
Solution: a3 – 64b3 = a3 – (4b) = (a – 4b)(a2 + a(4b) + (4b)2)
= (a – 4b)(a2 + 4ab + 16b2)

7. Example
Factor: –2t t 2
Solution: –2t t 2 = –2t 2(t 3 – 64)
= –2t 2(t 3 – 43)
= –2t 2(t – 4)(t 2 + 4t + 16)

8. Your Turn Factor: –2x 5 + 54x 2 (Hint: First, factor the GCF)
Solution: –2x5 – 54x 2 = –2x2(x )
= –2x 2(x )
= –2x 2(x + 3)(x 2 – (3)x + 32) = –2x 2(x + 3)(x 2 – 3x + 9)