1. 6.5 Factoring Cubic Polynomials
2/13/2013

2. Perfect Cubes
1000 = 103
729 = 93
512 = 83
343 = 73
216 = 63
125 = 53
64 = 43
27 = 33
8 = 23
1 = 13

3. The sum of two cubes:
The difference of two cubes:

4. Example 1 a. Factor . x 3 + 64 b. Factor . 8p 3 – q 3 SOLUTION x 3 +
Factor the Sum or Difference of Two Cubes a.
Factor
x 3 + 64 b.
Factor
8p 3 –
q 3
SOLUTION
Write as sum of two
cubes.
x 3 + 64 = 43 a. ( ) 4 x +
x 2 4x – 42 =
Use special product
pattern. ( ) 4 x +
x 2 4x – 16 =
Simplify.

5. [ ] Example 1 b. 8p 3 – q 3 ( ) 2p 3 = = – ( ) q 2p + q2 2pq 2p 2 = –
Factor the Sum or Difference of Two Cubes b.
8p 3 –
q 3 ( )
2p 3 =
Write as difference of two cubes. = – ( ) q 2p + q2
2pq [ ]
2p 2
Use special product pattern. = – ( ) q 2p + q2
2pq
4p2
Simplify. 5

6. Checkpoint
Factor the polynomial.
ANSWER ( ) 1 x +
x 2 – 1.
x 3 + 1 2.
125x 3 + 8 ( ) 2 5x +
25x 2
10x – 4 3.
x 3
216 – ( ) 6 x +
x 2 6x 36 –

7. Finding Greatest Common Factor (GCF)
Find the GCF of the terms in the polynomial:
𝑥 3 +𝑥 2
3 𝑥 3 −18
2 𝑥 3 +8 𝑥 2 −12𝑥
𝑥 2 3 2x

8. Example 2 a. Factor x 3 5x 2 6x. + – b. Factor 16x 4 2x. – SOLUTION
Factor Polynomials
a. Factor
x 3
5x 2
6x. + –
b. Factor
16x 4
2x. –
SOLUTION
Factor common
monomial.
x 3
5x 2 6x + – = ( ) x
x 2 5x 6 a.
Factor trinomial. = x ( ) 3 – 2
Factor common
monomial. = ( ) 2x
8x 3 1 – b.
16x 4
Use special product
pattern. = ( ) 2x 1 –
4x 2 + 8

9. Checkpoint Factor the polynomial. ANSWER x ( ) 1 – 3 + 4. x 3 + 2x 2 –
Factor Polynomials
Factor the polynomial.
ANSWER x ( ) 1 – 3 + 4.
x 3 +
2x 2 – 3x 5.
2x 3
10x 2 8x – + 2x ( ) 4 x – 1 6.
3x 4
24x + 3x ( ) 2 x +
x 2 2x 4 – 7.
54x 4
16x – 2x ( ) 2 3x
9x 2 6x 4 + –

10. Homework:
WS 6.5 #1-18 skip #8

11. Example 4 Factor the polynomial. a. x 2 ( x – 1 ) – 9 ( x – 1 ) b. x 3
Factor by Grouping
Factor the polynomial. a.
x 2 ( x – 1 ) – 9 ( x – 1 ) b.
x 3 –
2x 2 –
16x + 32
SOLUTION
Use
distributive
property. a.
x 2 ( ) 1 x – 9 =
Difference
of two
squares = ( ) 3 x – + 1 11

12. Example 4 b. = ( ) x 3 – 32 2x 2 16x + = ) x 2 – ( 2 + 16 x = ) – 16 (
Factor by Grouping b. = ( )
x 3 – 32
2x 2
16x +
Group
terms.
Factor each group. = )
x 2 – ( 2 + 16 x
Use distributive property. = ) – 16 ( 2 x
x 2
Difference
of two
squares = ( ) 4 x – + 2 12

13. Factor the polynomial by grouping. ANSWERS
Checkpoint
Factor by Grouping
Factor the polynomial by grouping.
ANSWERS ( ) 2 x – + 6 8.
x 2 ( ) 6 x + 4 – 9.
x 3
4x 2
25x –
100 + ( ) 5 x – + 4
10.
x 3
3x 2 4x 12 + ( ) 3 x + 4
x 2

14. Example 5 Solve 2x 3 14x 2 – = 24x. SOLUTION 2x 3 14x 2 – = 24x + ( )
Solve a Cubic Equation by Factoring
Solve
2x 3
14x 2 – =
24x.
SOLUTION
2x 3
14x 2 – =
24x +
Rewrite in standard form. ( )
x 2 7x – = 12 +
Factor common monomial. 2x ( ) 4 x – 3 2x =
Factor trinomial.
Use zero product property. 4 x – 3 2x = or x = 0, 4, 3
Solve for x. 14

15. Example 6 Solve x 3 6x 2 – = 2x. + 12 SOLUTION x 3 6x 2 – = + 12 2x (
Solve a Cubic Equation by Factoring
Solve
x 3
6x 2 – =
2x. + 12
SOLUTION
Rewrite in standard form.
x 3
6x 2 – = + 12 2x ( )
x 3
6x 2 – = 12 +
Group terms. 2x ( ) 6 x –
x 2 =
Factor each group. + 2 ( ) 6 x – =
Use distributive property. 2
x 2 6 x – =
Use zero product property. 2
x 2 or
x2 = 2
x = 6 15

16. Solve the equation by factoring. ANSWER
Checkpoint
Solve a Cubic Equation by Factoring
Solve the equation by factoring.
ANSWER
4, 0, 1 –
13.
x 3 +
3x 2 = 4x
14.
3x 3
30x – = 9x
13, 0 + –
15.
x 3
2x 2 3x + – 6 = 2, – 3 +
16.
x 3
7x 2 5x = – 35
5, 7 + –