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)