1. Objectives Factor the sum and difference of two cubes.
Factor by grouping.

2. Example 1: Factoring by Grouping
Factor: x3 – x2 – 25x + 25.
(x3 – x2) + (–25x + 25)
Group terms.
Factor common monomials from each group.
x2(x – 1) – 25(x – 1)
Factor out the common binomial (x – 1).
(x – 1)(x2 – 25)
Factor the difference of squares.
(x – 1)(x – 5)(x + 5)

3. Check It Out! Example 2
Factor: x3 – 2x2 – 9x + 18.
(x3 – 2x2) + (–9x + 18)
Group terms.
Factor common monomials from each group.
x2(x – 2) – 9(x – 2)
Factor out the common binomial (x – 2).
(x – 2)(x2 – 9)
Factor the difference of squares.
(x – 2)(x – 3)(x + 3)

4. Check It Out! Example 3
Factor: 2x3 + x2 + 8x + 4.
(2x3 + x2) + (8x + 4)
Group terms.
Factor common monomials from each group.
x2(2x + 1) + 4(2x + 1)
Factor out the common binomial (2x + 1).
(2x + 1)(x2 + 4)
(2x + 1)(x2 + 4)

5. Just as there is a special rule for factoring the difference of two squares, there are special rules for factoring the sum or difference of two cubes.

6. Example 4: Factoring the Sum or Difference of Two Cubes
Factor the expression.
4x x
4x(x3 + 27)
Factor out the GCF, 4x.
4x(x3 + 33)
Rewrite as the sum of cubes.
Use the rule a3 + b3 = (a + b)  (a2 – ab + b2).
4x(x + 3)(x2 – x  )
4x(x + 3)(x2 – 3x + 9)

7. Example 5: Factoring the Sum or Difference of Two Cubes
Factor the expression.
125d3 – 8
Rewrite as the difference of cubes.
(5d)3 – 23
(5d – 2)[(5d)2 + 5d  ]
Use the rule a3 – b3 = (a – b)  (a2 + ab + b2).
(5d – 2)(25d2 + 10d + 4)

8. Check It Out! Example 6
Factor the expression.
8 + z6
Rewrite as the difference of cubes.
(2)3 + (z2)3
(2 + z2)[(2)2 – 2  z + (z2)2]
Use the rule a3 + b3 = (a + b)  (a2 – ab + b2).
(2 + z2)(4 – 2z + z4)

9. Check It Out! Example 7
Factor the expression.
2x5 – 16x2
2x2(x3 – 8)
Factor out the GCF, 2x2.
Rewrite as the difference of cubes.
2x2(x3 – 23)
Use the rule a3 – b3 = (a – b)  (a2 + ab + b2).
2x2(x – 2)(x2 + x  )
2x2(x – 2)(x2 + 2x + 4)

10. Example 4: Geometry Application
The volume of a plastic storage box is modeled by the function V(x) = x3 + 6x2 + 3x – 10. Identify the values of x for which V(x) = 0, then use the graph to factor V(x).
V(x) has three real zeros at x = –5, x = –2, and x = 1. If the model is accurate, the box will have no volume if x = –5, x = –2, or x = 1.

11. Example 4 Continued
One corresponding factor is (x – 1). 1
–10
Use synthetic division to factor the polynomial. 1 7 10 1 7 10
V(x)= (x – 1)(x2 + 7x + 10)
Write V(x) as a product.
V(x)= (x – 1)(x + 2)(x + 5)
Factor the quadratic.

12. P(1) ≠ 0, so x – 1 is not a factor of P(x).
Lesson Quiz
Is the binomial a factor of the polynomial?
1. x – 1; P(x) = 3x2 – 2x + 5
P(1) ≠ 0, so x – 1 is not a factor of P(x).
2. x + 2; P(x) = x3 + 2x2 – x – 2
P(2) = 0, so x + 2 is a factor of P(x).
Factor:
3. x3 + 3x2 – 9x – 27
(x + 3)(x + 3)(x – 3)
4. x3 + 3x2 – 28x – 60
(x + 6)(x – 5)(x + 2)
5. 64p3 – 8q3
8(2p – q)(4p2 + 2pq + q2)