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Factoring By Grouping: Steps, Verification & Examples

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  • 0:02 Factoring
  • 1:15 Grouping
  • 3:29 Practice
  • 5:02 Splitting the Middle Term
  • 7:06 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

How do you factor an expression if all the terms don't share a common factor? Grouping! In this lesson, we'll learn how to use grouping to help factor expressions.

Factoring

In algebra, we love to build and we love to take apart. It's like we're using building blocks to make a spaceship, only instead of blocks, we have terms. And once we build our spaceship, we love to smash it. Is that just me? I built and smashed a lot of spaceships as a kid. The smashing part is factoring. Factoring is the process of finding the factors. If our space ship looks like 5x + 10, we factor out a 5 and get 5(x + 2). But what if we encounter something like this?

3x^2 + 2x + 12x + 8

That's a pretty awesome spaceship. We have four terms here. Is there a common factor? Well, three of the terms have an x, but not the last one. The last three terms have a factor of 2, but not 3x^2. There's no common factor. It's like someone glued our spaceship together! That's not cool. What if we want to build something else? We need a way to break it apart. How do you factor an expression if you can't factor anything out of each term?

Grouping

The answer: grouping. With grouping, we break up the expression into smaller groups that can be factored. Then we do what we already know how to do. Let's take our expression and learn about the steps involved with grouping.

The first step is to group the first two terms and the last two terms. Think of these as separate sets of expressions. Here, let's do (3x^2 + 2x) and have that plus (12x + 8). We basically have two two-term expressions, or binomials. Think of this like delicately taking the spaceship apart and splitting up the pieces by shape or color. This is a less fun way of deconstructing your blocks but, hey, to each his own.

The second step is to factor the greatest common factor from each binomial. We got stuck trying to factor something from all four terms, but we can factor something from these binomials. With (3x^2 + 2x), we can factor out an x to get x(3x + 2). With (12x + 8), we can factor out a 4 to get 4(3x + 2). Okay, now we have x(3x + 2) + 4(3x + 2). That leads to our final step.

The third and final step is to factor out the common binomial. Do you see how we have two (3x + 2)'s? We can factor that out. That gets us (x + 4)(3x + 2). And that's it. It's a spaceship no more! But wait, let's check our work. (x + 4)(3x + 2) doesn't look anything like our original expression. How do we know if it's correct? Well, we rebuild it with FOIL. x(3x) is 3x^2, x(2) is 2x, 4(3x) is 12x, and 4(2) is 8. That's 3x^2 + 2x + 12x + 8. And that is our original expression. We did it!

Practice

Let's practice another one.

5y^3 - 3y^2 + 10y - 6

Wow, look at all those y's. This is a cool spaceship. Well, let's break it apart by following our steps.

Step 1: Group it as (5y^3 - 3y^2) + (10y - 6).

Okay, Step 2: What can we factor from (5y^3 - 3y^2)? Well, nothing from 5 and 3, but we can factor out a y. Not just that, but y^2. We get y^2(5y - 3). What about (10y - 6)? We can't factor out a y, but 10 and 6 have a common factor in 2. We get 2(5y - 3). Hey, look at that: 5y - 3 again! We found some common parts that go together.

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