Learning Objectives
- Determine the greatest common factor in a polynomial and factor it out.
- Factor by grouping.
Chapter 8 Common Factors
Section 8.1 You All Have This in Common
In the previous section, we learned how to multiply polynomials. In this section, we are going to start to learn how to undo everything we just did.
We will begin the process by looking at a basic grid product from the previous section. We started with values around the edges of the grid and used that information to tell us the values inside of the grid.
For this section, we are going to start with the numbers in the grid and try to figure out what values go around the edges.
As it turns out, there are several ways to do this. Here are four examples:
There is nothing mathematically wrong with any of these products. But we can see that some choices are better than others. The choice on the far left doesn’t accomplish anything at all. The one on the far right introduces fractions. So we really want to focus on the middle two.
Of the middle two options, the one on the right can be understood as being the "better" option. The reason for that is that we factored out the biggest amount possible. And that is the basic goal. Technically, we call this the greatest common factor.
Definition 8.1. Common Factor.
A common factor of the terms of a polynomial is a monomial that divides all of the terms. The greatest common factor is the monomial that has the largest degree and largest coefficient.
Activity 8.1. Greatest Common Factor (Part 1).
The end result of this manipulation is going to be an expression that is equivalent to the one that we started with. So for the sample grid we’ve been using, we’ve been trying to identify the greatest common factor of the polynomial \(4x + 24\text{.}\) Here’s what the presentation (including the grid) would look like.
Try it!
Identify the greatest common factor of \(5x^2 + 15\text{,}\) then factor it out of the polynomial. Draw the grid and write the corresponding equation.
Solution.
Activity 8.2. Greatest Common Factor (Part 2).
Conceptually, there is nothing different about this process when there are more variables and more terms involved. The main challenge is to avoid simple arithmetic errors by miscounting the variables in the expressions when we factor them out.
Try it!
Identify the greatest common factor of \(6x^2y - 8xy + 14y\text{,}\) then factor it out of the polynomial. Draw the grid and write the corresponding equation.
Solution.
A slightly more complex version of this concept is known as factoring by grouping. The idea here is that we’re going to factor out the greatest common factor of two pairs of terms, then we’re going to look at the new terms you’ve created and see if there’s something to factor out there. But this requires us to expand our concept of what a common factor can be.
To visualize this, we will look at a specific example through a substitution.
The key to understanding this is that we can treat terms inside of parentheses as if they were a single object when we’re factoring. As long as the objects in the parentheses are identical, we can factor them out like any other variable.
Activity 8.3. Factor by Grouping.
Let’s take a look at a full example of factoring by grouping. Consider the expression \(x^3 + 3x^2 + 2x + 6\text{.}\) We are going to think of this as two pairs of expressions:
Each of these grids gives rise to a factorization.
We can then take these results and put that into another grid and look for another factorization.
The presentation that we’ve given above is much more focused on understanding the process. The presentation was broken into pieces that facilitate explanation, and a final presentation can be much more compact.
Try it!
Factor \(x^3 - 2x^2 + 4x - 8\) by grouping. Draw the grids and use a complete presentation.
Solution.
Activity 8.4. Factor by Grouping with Negatives.
It is very important to be careful with negative signs, especially when the negative sign is the third term of the polynomial. Students will sometimes make unfortunate groupings that are incorrect.
\begin{equation*}
x^2 + 4x - 2x - 8 \overset{\times}{=} (x^2 + 4x) - (2x - 8)
\end{equation*}
When thinking about the factorization, it is important to keep the signs with the corresponding terms when you visualize the grid.
Try it!
Factor \(x^2 - 3x - 4x + 12\) by grouping. Draw the grids and use a complete presentation.
Solution.
Section 8.2 Worksheets
PDF Version of these Worksheets
1
external/worksheets/08-Worksheets.pdfWorksheet Worksheet 1
1.
Identify the greatest common factor of \(6x + 9\text{,}\) then factor it out of the polynomial. Draw the grid and write the corresponding equation.
2.
Identify the greatest common factor of \(4y - 10\text{,}\) then factor it out of the polynomial. Draw the grid and write the corresponding equation.
3.
Identify the greatest common factor of \(3a^2b + 9ab - 15b\text{,}\) then factor it out of the polynomial. Draw the grid and write the corresponding equation.
4.
Identify the greatest common factor of \(8n + 4\text{,}\) then factor it out of the polynomial. Write the corresponding equation, but do not draw a grid.
Worksheet Worksheet 2
1.
Identify the greatest common factor of \(4x^5 - 10x^2 + 12x\text{,}\) then factor it out of the polynomial. Draw the grid and write the corresponding equation.
2.
Identify the greatest common factor of \(12p^2 - 18p\text{,}\) then factor it out of the polynomial. Write the corresponding equation, but do not draw a grid.
3.
Identify the greatest common factor of \(3y^3 + 15y^2 - 6y\text{,}\) then factor it out of the polynomial. Write the corresponding equation, but do not draw a grid.
4.
Identify the greatest common factor of \(x(x - 4) + 3(x - 4)\text{,}\) then factor it out of the polynomial. Draw the grid and write the corresponding equation.
Worksheet Worksheet 3
1.
Identify the greatest common factor of \(x(3y - 2) - 5(3y - 2)\text{,}\) then factor it out of the polynomial. Write the corresponding equation, but do not draw a grid.
2.
Factor \(x^3 + 4x^2 + 3x + 12\) by grouping. Draw the grids and use a complete presentation.
3.
Factor \(x^2 - 6x - 3x + 18\) by grouping. Draw the grids and use a complete presentation.
Worksheet Worksheet 4
1.
Factor \(2x^2 - 3x - 6x + 9\) by grouping. Draw the grids and use a complete presentation.
2.
Factor \(3xy + 6x - 2y - 4\) by grouping. Draw the grids and use a complete presentation.
3.
Factor \(x^3 - 3x^2 + 4x - 12\) by grouping. Use a complete presentation, but do not draw the grids.
Worksheet Worksheet 5
1.
Factor \(x^2 - 4x - 4x + 16\) by grouping. Use a complete presentation, but do not draw the grids.
2.
Factor \(2x^2 + 5x - 4x - 10\) by grouping. Use a complete presentation, but do not draw the grids.
3.
Factor \(x^2 - 3x + 5x - 15\) by grouping. Then factor \(x^2 + 5x - 3x - 15\) by grouping. Use a complete presentation for both, but do not draw the grids. Was one easier than the other? Explain your perspective.
Section 8.3 Deliberate Practice: Factor by Grouping
Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.
Focus on these skills:
- Write the original expression.
- Visualize the grids, but try to do the calculations without drawing them.
- Present your work legibly.
Worksheet Worksheet
Instructions: Factor by grouping.
1.
Factor \(x^2 + 3x - 2x - 6\) by grouping.
2.
Factor \(x^2 - 5x + 3x - 15\) by grouping.
3.
Factor \(x^2 - 2x - 4x + 8\) by grouping.
4.
Factor \(x^2 + 4x + 3x + 12\) by grouping.
5.
Factor \(2x^2 - 8x - 3x + 12\) by grouping.
6.
Factor \(2x^2 + 5x - 4x - 10\) by grouping.
7.
Factor \(3x^2 - 6x - 7x + 14\) by grouping.
8.
Factor \(x^3 + 3x^2 - 2x - 6\) by grouping.
9.
Factor \(2x^3 - 3x^2 - 6x + 9\) by grouping.
10.
Factor \(x^3 - 4x^2 + 4x - 16\) by grouping.
Section 8.4 Closing Ideas
Factoring is a basic but important algebraic manipulation. It’s nothing more than the distributive property (Definition 3.4 ) applied backwards.
\begin{equation*}
\begin{aligned}
3x + 4x \amp = (3 + 4)x = 7x \amp \eqnspacer \amp \text{Combine like terms} \\
x^2 + 4x \amp = (x + 4)x \amp \amp \text{Factor out the common factor}
\end{aligned}
\end{equation*}
And if there were some aspects of this section that feels familiar, it’s because this is the exact same algebra that is used with combining like terms.
\begin{equation*}
\begin{aligned}
3x + 4x \amp = (3 + 4)x = 7x \amp \amp \text{Combine like terms} \\
x^2 + 4x \amp = (x + 4)x \amp \amp \text{Factor out the common factor}
\end{aligned}
\end{equation*}
The difference between combining like terms and factoring out the common factor is that there’s an extra step of arithmetic that we can do with numbers that we can’t do with algebraic expressions.
Math is an extremely scaffolded subject. What this means is that new ideas are very often built on older ones. This also means that weaknesses in the foundation make the higher levels of mathematical thinking less stable. Hopefully, as you’ve been working your way through these sections, you have been taking the time to think through the ideas and solidify those core concepts.
