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

### 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})
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