algebra 1 lesson 8-8 warm-up algebra 1 “factoring by grouping” (8-8) how can you sometimes...

ALGEBRA 1

Lesson 8-8 Warm-Up

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


Example:

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.


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


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.


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


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.


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).


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

ALGEBRA 1

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)


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algebra 1 lesson 8-8 warm-up algebra 1 “factoring by grouping” (8-8) how can you sometimes...

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