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.

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
Let's look at a few examples.
Example 1: Factor each polynomial using the grouping method.
20x3 - 8x2 + 25x - 10
Step 1) Rearrange the terms into two groups, where each group has a common factor:
(20x3 - 8x2) + (25x - 10)
Step 2) Factor out the GCF or -(GCF) from each group
4x2(5x - 2) + 5(5x - 2)
Step 3) Factor out the common binomial factor:
4x2(5x - 2) + 5(5x - 2) =
(5x - 2)(4x2 + 5)
Let's try an example where we need to rearrange the terms.
Example 2: Factor each polynomial using the grouping method.
24xy - 20x - 15x2 + 32y
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 (-15x2 and 32y) have no common factor other than 1. We will rearrange the terms to:
(24xy + 32y) + (-20x - 15x2)
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 + x3 - 8x4 - 2y
Step 1) Rearrange the terms into two groups, where each group has a common factor:
(16xy + x3) + (-8x4 - 2y)
Step 2) Factor out the GCF or -(GCF) from each group
x(16y + x2) + (-2)(4x4 - 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) + (x3 - 8x4)
Step 2) Factor out the GCF or -(GCF) from each group
2y(8x - 1) + x3(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) - x3(8x + 1)
Step 3) Factor out the common binomial factor:
2y(8x - 1) - x3(8x - 1) =
(8x - 1) + (2y - x3)
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.
96x3 - 72x2 - 160x + 120
Before we begin with our first step, notice that we have a common factor of 8. Let's factor this out:
8(12x3 - 9x2 - 20x + 15)
Now, we can continue using our procedure:
8[(12x3 - 9x2) + (-20x + 15)]
8[3x2(4x - 3) - 5(4x - 3)]
8(3x2 - 5)(4x - 3)

Skills Check:

Example #1

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

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

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

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