Lesson Objectives

- Demonstrate an understanding of how to find the GCF for a polynomial
- Demonstrate an understanding of how to factor out the GCF of a polynomial
- Learn how to factor a four-term polynomial using grouping

## Factoring by Grouping

In our last lesson, we learned how to factor out the GCF from a polynomial. In this lesson, we will expand on this concept and learn how to factor a four-term polynomial using the factoring by grouping method.

Example 1: Factor each polynomial using the grouping method.

20x

Step 1) Rearrange the terms into two groups, where each group has a common factor:

(20x

Step 2) Factor out the GCF or -(GCF) from each group

4x

Step 3) Factor out the common binomial factor:

4x

(5x - 2)(4x

Let's try an example where we need to rearrange the terms.

Example 2: Factor each polynomial using the grouping method.

24xy - 20x - 15x

Step 1) Rearrange the terms into two groups, where each group has a common factor:

Our first two terms (24xy and -20x) have a common factor of 4x, but the last two terms (-15x

(24xy + 32y) + (-20x - 15x

Now our first two terms (24xy and 32y) have a common factor of 8y and our last two terms (-20x and -15x) have a common factor of 5x or (-5x).

Step 2) Factor out the GCF or -(GCF) from each group. For the second group of terms, we want to factor out the -(GCF), which is (-5x). Let's look at what happens when we factor out 5x:

8y(3x + 4) + 5x(-4 - 3x)

Notice how (3x + 4) and (-4 - 3x) are opposites. We can simply factor out a (-1) in the case of (-4 - 3x):

8y(3x + 4) - 5x(3x + 4)

Step 3) Factor out the common binomial factor:

8y(3x + 4) - 5x(3x + 4) =

(3x + 4)(8y - 5x)

Example 3: Factor each polynomial using the grouping method.

16xy + x

Step 1) Rearrange the terms into two groups, where each group has a common factor:

(16xy + x

Step 2) Factor out the GCF or -(GCF) from each group

x(16y + x

Step 3 & 4) Factor out the common binomial factor: We don't have a common binomial factor, we need to try another grouping and repeat our steps:

Step 1) Rearrange the terms into two groups, where each group has a common factor:

(16xy - 2y) + (x

Step 2) Factor out the GCF or -(GCF) from each group

2y(8x - 1) + x

Again, (8x - 1) and (1 - 8x) are opposites. We can simply factor out a (-1) in the case of (1 - 8x):

2y(8x - 1) - x

Step 3) Factor out the common binomial factor:

2y(8x - 1) - x

(8x - 1) + (2y - x

In some cases, we may need to factor out the GCF before we begin our process. Let's look at an example.

Example 4: Factor each polynomial using the grouping method.

96x

Before we begin with our first step, notice that we have a common factor of 8. Let's factor this out:

8(12x

Now, we can continue using our procedure:

8[(12x

8[3x

8(3x

### Factoring by Grouping Method

- Rearrange the terms into two groups, where each group has a common factor
- In some cases, the common factor will be (1) or (-1)

- Factor out the GCF or -(GCF) from each group
- Factor out the common binomial factor when possible
- If no common binomial factor is found, we repeat the process with a different grouping

Example 1: Factor each polynomial using the grouping method.

20x

^{3}- 8x^{2}+ 25x - 10Step 1) Rearrange the terms into two groups, where each group has a common factor:

(20x

^{3}- 8x^{2}) + (25x - 10)Step 2) Factor out the GCF or -(GCF) from each group

4x

^{2}(5x - 2) + 5(5x - 2)Step 3) Factor out the common binomial factor:

4x

^{2}(5x - 2) + 5(5x - 2) =(5x - 2)(4x

^{2}+ 5)Let's try an example where we need to rearrange the terms.

Example 2: Factor each polynomial using the grouping method.

24xy - 20x - 15x

^{2}+ 32yStep 1) Rearrange the terms into two groups, where each group has a common factor:

Our first two terms (24xy and -20x) have a common factor of 4x, but the last two terms (-15x

^{2}and 32y) have no common factor other than 1. We will rearrange the terms to:(24xy + 32y) + (-20x - 15x

^{2})Now our first two terms (24xy and 32y) have a common factor of 8y and our last two terms (-20x and -15x) have a common factor of 5x or (-5x).

Step 2) Factor out the GCF or -(GCF) from each group. For the second group of terms, we want to factor out the -(GCF), which is (-5x). Let's look at what happens when we factor out 5x:

8y(3x + 4) + 5x(-4 - 3x)

Notice how (3x + 4) and (-4 - 3x) are opposites. We can simply factor out a (-1) in the case of (-4 - 3x):

8y(3x + 4) - 5x(3x + 4)

Step 3) Factor out the common binomial factor:

8y(3x + 4) - 5x(3x + 4) =

(3x + 4)(8y - 5x)

Example 3: Factor each polynomial using the grouping method.

16xy + x

^{3}- 8x^{4}- 2yStep 1) Rearrange the terms into two groups, where each group has a common factor:

(16xy + x

^{3}) + (-8x^{4}- 2y)Step 2) Factor out the GCF or -(GCF) from each group

x(16y + x

^{2}) + (-2)(4x^{4}- y)Step 3 & 4) Factor out the common binomial factor: We don't have a common binomial factor, we need to try another grouping and repeat our steps:

Step 1) Rearrange the terms into two groups, where each group has a common factor:

(16xy - 2y) + (x

^{3}- 8x^{4})Step 2) Factor out the GCF or -(GCF) from each group

2y(8x - 1) + x

^{3}(1 - 8x)Again, (8x - 1) and (1 - 8x) are opposites. We can simply factor out a (-1) in the case of (1 - 8x):

2y(8x - 1) - x

^{3}(8x + 1)Step 3) Factor out the common binomial factor:

2y(8x - 1) - x

^{3}(8x - 1) =(8x - 1) + (2y - x

^{3})In some cases, we may need to factor out the GCF before we begin our process. Let's look at an example.

Example 4: Factor each polynomial using the grouping method.

96x

^{3}- 72x^{2}- 160x + 120Before we begin with our first step, notice that we have a common factor of 8. Let's factor this out:

8(12x

^{3}- 9x^{2}- 20x + 15)Now, we can continue using our procedure:

8[(12x

^{3}- 9x^{2}) + (-20x + 15)]8[3x

^{2}(4x - 3) - 5(4x - 3)]8(3x

^{2}- 5)(4x - 3)#### Skills Check:

Example #1

Factor each. $$40x^{3}+ 32x^{2}+ 15x + 12$$

Please choose the best answer.

A

$$4(5x + 3)(2x^{2}- 1)$$

B

$$4(5x + 3)(2x^{2}+ 1)$$

C

$$(2x^{2}+ 5)(x - 4)$$

D

$$(8x^{2}+ 3)(5x + 4)$$

E

$$(7x + 4)(5x^{2}+ 4)$$

Example #2

Factor each. $$4x^{3}- 8x^{2}+ 5x - 10$$

Please choose the best answer.

A

$$2(2x^{2}+ 5)(2x + 1)$$

B

$$(4x^{2}+ 5)(2x - 1)$$

C

$$(2x^{2}- 5)(5x^{2}+ 3)$$

D

$$(4x^{2}+ 5)(x - 2)$$

E

$$2(2x^{2}+ 1)(2x - 3)$$

Example #3

Factor each. $$14x^{4}- 28x^{3}- 8x^{2}+ 16x$$

Please choose the best answer.

A

$$2x(7x^{2}- 2)(x + 4)$$

B

$$2x(7x^{2}- 4)(x - 2)$$

C

$$(7x^{2}- 4)(4x^{2}- 7)$$

D

$$(8x^{2}- 6)(7x + 3)$$

E

$$2x(7x^{2}- 4)(x + 2)$$

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