# How to Teach Factoring Trinomials (The Best Method!)

There are many, many different ways to factor trinomials and quadratic expressions. In fact, it almost seems like math teachers can’t quite agree when it comes to how to teach factoring trinomials.

This can make it very difficult for teachers to determine which factoring method they want to introduce to their class. It also makes it very difficult for students to learn this important skill!

I want to share an unbeatable strategy that I learned back when I was a student that will make learning how to factor a quadratic expression a breeze for any student! This is the *exact strategy* that I have taught throughout my teaching career that has proven over and over to be the most successful of the factoring methods.

Let’s dig into it!

## The Factoring Decision Tree

The key to this factoring method is using a decision tree. Students love a good decision tree because decision making is very hard in math!

There are so many different problem solving strategies taught that it can be hard for students to really understand when each should be applied.

Take the quadratic formula and factoring for example. These strategies are both helpful for finding the x-intercepts of a quadratic. But there is a time and place for each.

Giving students something visual that helps them see the decisions they have available to them is very helpful and makes the whole process much less stressful for them.

### Step 1: Check for a Greatest Common Factor

Finding the greatest common factor is a great starting point when you are thinking about how to teach factoring.

Finding a greatest common factor builds on the number sense knowledge that students already have. Because of this, I believe that the only way to start a factoring unit is to begin with a deep dive into common factoring. This will make it so much easier to factor quadratic equations later on.

The decision tree above uses checking for a greatest common factor as a starting point. If a student skips this step, the problem always feels so much harder!

### Step 2: Is the Quadratic Expression a Trinomial?

After students have checked for the greatest common factor, I have them check to see if they are working with a trinomial expression. If they aren’t, I direct them toward checking for differences of squares and factoring by grouping.

Factoring differences of squares always seems to confuse students. These expressions are kind of a special case when it comes to factoring. If students are expecting a factorable trinomial, seeing only two terms with a subtraction sign in between can be alarming.

Where is the middle term!? What’s with the minus sign! Where is the constant term!?Checking for differences of squares can get a rid of this confusion early on in the process.

If we are dealing with factoring by grouping, we have reached the end of the tree. Our only remaining choice would be to apply our knowledge of factoring by grouping problems at this point.

### Step 3: Check the Leading Coefficient

After checking for a greatest common factor, differences of squares, and grouping, the decision tree tells students to check the value of the leading coefficient next.

The leading coefficient is the number in front of the x term **that has an exponent of two** (if you are working with a quadratic expression).

**When the Leading Coefficient Is Equal to One**

If the leading coefficient is equal to one, we are working with a simple case of trinomial factoring. I normally start this process with decomposition, or the “splitting the middle term” process. This is where you take the second term and break it into a sum of two different numbers. This allows you to factor by grouping, which eventually ends up in a fully factored expression.

The first step of this process is taking the leading coefficient and multiplying it by the constant term (this is often the third term). Multiplying the coefficients of the first and last terms will result in some whole number. I then have my students write out a list of the possible numbers that will multiply to produce this number. Remember that negative numbers are important to consider as well!

It can be tough to find the right number combination in this step. Some teachers will use a factor tree to help students find the different factors that work for this step. Practice is key here, especially for students with weaker number sense.

Once students have this list of factor pairs, I have them select the pair that *also* adds to produce the middle term.

Breaking the middle term up into a sum of these two numbers will create an expression that can be factored by grouping.

**When the Leading Coefficient Is Not Equal to One**

If the leading coefficient is anything other than one, I direct my students to the right side of the decision tree. The same process as above can be applied to tackle a problem like this.

I have my own crazy trick for factoring these sorts of trinomials that you sort of have to see to believe. You can check out a video of my trinomial factoring process to see why it has become one of my most viewed videos on my YouTube channel!

### Step 4: Checking Your Work

After successfully factoring a quadratic polynomial, I always make sure to check if my answer makes sense. The only way to do this is to apply the distributive property for binomials (sometimes known as FOIL).

This process allows us to take a product of binomials and write them in standard form. If you see that your standard form expression is the same as the original trinomial, you know that your factored expression is correct!

## Why it is Important to Consider How to Teach Factoring

Factoring truly can be one of the most challenging skills that students learn in secondary mathematics. To make matters worse, it is used very often in more complex mathematics. This is why it is so important that teachers explore how to teach factoring in ways that support student learning.

There are so many different factoring methods out there that it is important to think about the best one for you. Whether you are a teacher or a student, finding a strategy that helps you feel comfortable is important.

I’ve never liked using ‘magic squares’ or the box method. I don’t like trying to memorize strategies that don’t seem to make sense to me. I also love a good decision tree and I know how useful they can be for students who struggle with decision making in mathematics. This is why I choose this method. But it may not be the best one for you!

I encourage you to try solving a series of factoring problems while using the decision tree above. With practice, you can master factoring trinomials!

After you’ve mastered factoring trinomials, check out my walkthrough of how to apply the quadratic formula to solve quadratic equations!

**Did you find this factoring walkthrough helpful? Share this post and subscribe to Math By The Pixel on YouTube for more helpful factoring content!**

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