Factor by Grouping Lesson

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)

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