1. Lesson 8-8 Warm-Up

2. “Factoring by Grouping” (8-8)
How can you sometimes factor a four-term polynomial” by grouping? How do you factor by grouping?
Sometimes, two groups of terms have the same factor. If this is the case, you can use the Distributive Property to factor by grouping.
Example:
y + 3 is a common factor of each group of terms. Using Distributive Property, y can now be combined as the other factor.
To factor by grouping, look for a common factor of two pairs of terms. S
Example:
6 x 4 +
6 x 8 =
6 x (4 + 8) 6 24 + 6 48 = 6 24 72 48 4 8
4 + 8 12

3. “Factoring by Grouping” (8-8)
Example:

4. 6x3 + 3x2 – 4x – 2 = 3x2(2x + 1) – 2(2x + 1) Factor the common
Factoring by Grouping
LESSON 8-8
Additional Examples
Factor 6x3 + 3x2 – 4x – 2.
6x3 + 3x2 – 4x – 2 = 3x2(2x + 1) – 2(2x + 1) Factor the common
factor from each group of two terms.
= (3x2 – 2)(2x + 1) Factor out (2x + 1).
Check: 6x3 + 3x2 – 4x – (2x + 1)(3x2 – 2)
= 6x3 – 4x + 3x2 – 2 Use FOIL.
= 6x3 + 3x2 – 4x – 2 Write in standard form.

5. = 4t (2t3 + 3t2 + 4t + 6) Factor out 2t+3 from 2t3 + 3t2 and 4t + 6.
Factoring by Grouping
LESSON 8-8
Additional Examples
Factor 8t4 + 12t3 + 16t2 + 24t.
8t4 + 12t3 + 16t2 + 24t = 4t(2t3 + 3t2 + 4t + 6) Factor out the common factor, 4t.
8t4 = 222tt t t
12t3 = 223ttt
16t2 = 2222tt
24t = 2223t
= 4t (2t3 + 3t2 + 4t + 6) Factor out 2t+3 from 2t3 + 3t2
and 4t + 6.
= 4t [t2 (2t + 3) + 2(2t + 3)] Rewrite as the product of
factors.
2t3 = 2tt t
3t2 = 3tt
= 4t (t2 + 2) (2t + 3) Rewrite.
4t = 22t
6 = 23

6. “Factoring by Grouping” (8-8)
How do you factor a trinomial by grouping?
Sometimes, you can make a trinomial into a four-term polynomial (by splitting the middle term into two terms that add up to it) that you can factor by grouping Example: Factor 48x2 + 46x + 5 Answer: (6x + 5)(8x + 1)

7. Method 1: Group by Finding a Common Factor of Two Binomials
Factoring by Grouping
LESSON 8-8
Additional Examples
Factor 24h2 + 10h – 6.
Method 1: Group by Finding a Common Factor of Two Binomials
Step 1:  24h2 + 10h – 6 = 2(12h2 + 5h – 3) Factor out the common factor, 2.
Step 2:  12 • –3 = – Find the product the a and c terms.
Step 3: Factors Sum
–2(18) = –36 – = 16
–3(12) = –36 – = 9
–4(9) = –36 –4 + 9 = 5 
Find two factors of ac that have a sum b. Use mental math to determine a good place to start.
Step 4:  12h2 – 4h + 9h – Rewrite the trinomial.
Step 5:   4h(3h – 1) + 3(3h – 1) Factor by grouping.
  (4h + 3)(3h – 1) Factor again.
24h2 + 10h – 6 = 2(4h + 3)(3h – 1) Include the common factor in your final answer.

8. “Factoring Trinomials of the Type ax2 + bx +c” (8-8)
Method 2: Use an Area Model to Group Two Binomials With a Common Factor (“X-Box”)
Find two numbers whose product is ac and sum is b. These numbers will be the coefficients of the x terms.
Then, create a box divided into two columns and two rows. The top-left box will be the a term, the bottom right box will be the c term, and the middle two boxes will be the b terms.
Finally, find common factors of each column and row. The dimensions (length and width) of the box are factors (binomial times binomial) of the trinomials.
Example: Factor 2(12h2 + 5h – 3) 4h 3
-36h2 3h
12h2 9h 9h
-4h -1
-4h -3 5h
Answer: (4h + 3)(3h - 1) 2

9. “Factoring by Grouping” (8-8)
Sometimes, you need to “factor out” a common monomial of the three terms of a trinomial before you make a trinomial into a four-term polynomial that you can factor by grouping
Example: Factor 80x x2 + 60x
4x(10x + 3) (2x + 5)

10. Method 1: Group by Finding a Common Factor of Two Binomials
Factoring by Grouping
LESSON 8-8
Additional Examples
A rectangular prism has a volume of 36x3 + 51x2 + 18x. Factor to find the possible expressions for the length, width, and height of the prism.
Factor 36x3 + 51x2 + 18x.
Method 1: Group by Finding a Common Factor of Two Binomials
Step 1:  3x(12x2 + 17x + 6) Factor out the common factor, 3x.
Step 2:  12 • 6 = Find the product of the a and c terms.
Step 3:  Factors     Sum
4 • = 22
6 • = 18
8 • = 17
Find two factors of ac that have sum b. Use mental math to determine a good place to start.

11. Step 4: 3x (12x2 + 8x + 9x + 6) Rewrite the trinomial.
Factoring by Grouping
LESSON 8-8
Additional Examples
(continued)
Step 4:  3x (12x2 + 8x + 9x + 6) Rewrite the trinomial.
Step 5:  3x[4x(3x + 2) + 3(3x + 2)] Factor by grouping.
3x(4x + 3)(3x + 2) Factor again.
The possible dimensions of the prism are 3x, (4x + 3), and (3x + 2).

12. “Factoring Trinomials of the Type ax2 + bx +c” (8-8)
Factor 36x3 + 51x2 + 18x.
Method 2: “X-Box Method”
Factor 36x2 + 51x = 3x(12x2 + 17x + 6) 3x 2
72x2 4x
12x2 8x 8x 9x
17x 3 9x 6
Answer: (3x + 2)(4x + 3) 3x

13. Factor each expression.
Factoring by Grouping
LESSON 8-8
Lesson Quiz
Factor each expression.
1. 10p3 – 25p2 + 4p – 10
2. 36x4 – 48x3 + 9x2 – 12x
3. 16a3 – 24a2 + 12a – 18
(5p2 + 2)(2p – 5)
3x(4x2 + 1)(3x – 4)
2(4a2 + 3)(2a – 3)