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ALGEBRA 1
“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, y2 + 4
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
4
6
8
6 x 8
6 x (4 + 8)
4 + 8
6
+ =
+ =24 48 24 4872
12
ALGEBRA 1
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).
= 6x3 – 4x + 3x2 – 2 Use FOIL.
Check: 6x3 + 3x2 – 4x – 2 (2x + 1)(3x2 – 2)
= 6x3 + 3x2 – 4x – 2 Write in standard form.
Factoring by GroupingLESSON 8-8
Additional Examples
ALGEBRA 1
Factor 8t4 + 12t3 + 16t2 + 24t.
8t4 + 12t3 + 16t2 + 24t = 4t(2t3 + 3t2 + 4t + 6) Factor out the common factor, 4t.
= 4t (2t3 + 3t2 + 4t + 6) Factor out 2t+3 from 2t3 + 3t2 and 4t + 6.
= 4t (t2 + 2) (2t + 3) Rewrite.
Factoring by GroupingLESSON 8-8
Additional Examples
8t4 = 222tt t t 12t3 = 223ttt 16t2 = 2222tt 24t = 2223t
2t3 = 2tt t 3t2 = 3tt
4t = 22t 6 = 23
= 4t [t2 (2t + 3) + 2(2t + 3)] Rewrite as the product of factors.
ALGEBRA 1
“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)
ALGEBRA 1
Factor 24h2 + 10h – 6.
Step 1: 24h2 + 10h – 6 = 2(12h2 + 5h – 3) Factor out the common factor, 2.
Step 2: 12 • –3 = –36 Find the product the a and c terms.
Step 4: 12h2 – 4h + 9h – 3 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.
Step 3: Factors Sum–2(18) = –36 –2 + 18 = 16–3(12) = –36 –3 + 12 = 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.
Factoring by GroupingLESSON 8-8
Additional Examples
Method 1: Group by Finding a Common Factor of Two Binomials
ALGEBRA 1
Example: Factor 2(12h2 + 5h – 3)
-36h2
9h
5h
-4h
12h2
-3-4h
9h3h
Answer: (4h + 3)(3h - 1)
1. Find two numbers whose product is ac and sum is b. These numbers will be the coefficients of the x terms.
“Factoring Trinomials of the Type ax2 + bx +c” (8-8)
-1
4h 3
2. 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.
3. 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.
Method 2: Use an Area Model to Group Two Binomials With a Common Factor (“X-Box”)
2
ALGEBRA 1
“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 80x3 + 224x2 + 60x
4x(10x + 3) (2x + 5)
ALGEBRA 1
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.
Step 1: 3x(12x2 + 17x + 6) Factor out the common factor, 3x.
Step 2: 12 • 6 = 72 Find the product of the a and c terms.
Step 3: Factors Sum 4 • 18 4 + 18 = 22 6 • 12 6 + 12 = 18 8 • 9 8 + 9 = 17
Find two factors of ac that have sum b. Use mental math to determine a good place to start.
Factoring by GroupingLESSON 8-8
Additional Examples
Method 1: Group by Finding a Common Factor of Two Binomials
ALGEBRA 1
(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).
Factoring by GroupingLESSON 8-8
Additional Examples
ALGEBRA 1
Factor 36x2 + 51x + 18 = 3x(12x2 + 17x + 6)
72x2
8x
17x
9x12x2
69x
8x4x
Answer: (3x + 2)(4x + 3)
Method 2: “X-Box Method”
“Factoring Trinomials of the Type ax2 + bx +c” (8-8)
3
3x 2
Factor 36x3 + 51x2 + 18x.
3x