the trinomial 6x^2+11x+4

You Won’t Believe This Fast and Easy Trinomial Factoring Shortcut!

As a math teacher, I have seen thousands of students struggle with factoring trinomials. This is especially true when the leading coefficient is not 1. 

I have tried teaching traditional methods like guess-and-check, the AC method, the grouping method, or the box method, but these strategies often leave students confused, frustrated, and discouraged.

But what if I told you that there was a shortcut that streamlined the process? And what if this shortcut was so whacky that it not only simplified the process but also made factoring trinomials almost effortless?

A method that eliminates the need for long, confusing, and complex calculations and makes factoring trinomials with a leading coefficient easy, accurate…and dare I say it … fun? 

Surely it can’t be.

But it’s true!

My goal is to introduce you to the ultimate trinomial factoring shortcut. This is a top secret factoring strategy that my math teacher showed me way back when I was young and I have used reliably throughout my teaching career ever since!

By the end, you’ll be able to tackle any quadratic trinomial confidently and efficiently. 

This strategy almost feels like you are cheating the system because it sidesteps the complex common factor and grouping steps. Because of this, it is worth mentioning that I always recommend checking with your teacher before you factor trinomials using this shortcut on a quiz, test, or exam.

Let’s dig into this step-by-step approach to simplify the trinomial factoring process!

​Watch the following video for a walkthrough of my epic trinomial factoring shortcut!

Step 1: Understand the General Form of Trinomials

​The first step when learning this trinomial factoring shortcut is to make sure that you are comfortable with the general form of trinomials.

A quadratic trinomial typically takes the form \(ax^2+bx+c\), where:

  • \(a\) is the leading coefficient of the leading term (the coefficient of the \(x^2\) term)
  • \(b\) is the coefficient of the middle term
  • \(c\) is the constant term (often written as the last term without a variable)
standard form of a quadratic trinomial

The goal of factoring is to rewrite the given trinomial as the factored form of a product. In other words, we want to take \(ax^2+bx+c\) and write it in the form \((px+s)(qx+r)\).

Step 2: Finding the Right Numbers

Before I show you the epic trinomial factoring shortcut, it’s important that you have a good understanding of one of the most important steps in factoring trinomials.

The key to factoring successfully is identifying factor pairs. These are two numbers that:

  1. Multiply to give \(a \times c\), or the product of the leading coefficient and the constant term.
  2. Add to give \(b\), or the coefficient of the middle term.

For example, in the trinomial \(6x^2+11x+4\):

  • \(a=6\), \(b=11\), and \(c=4\)
  • Our goal is to find two numbers that multiply to get \(a \times c=6 \times 4 = 24\)
  • BUT, those two numbers must also add to \(b=11\) (ie. the b part of the middle term) 

Finding these two factor pairs is the part of the factoring process that most students struggle with. But I’m going to share a little secret that will make this step SO much easier.

The trick is to always start with multiplication of possible factor combinations. Write out the factors of \(a \times c\), then choose the ones that also add to get the b term.

In the example above, we know that the factors of 24 are: 

  • 1 and 24
  • 2 and 12
  • 3 and 8
  • 4 and 6

If you go through this list of numbers, you will see that the only integer pairs that add to get 11 are 8 and 3.

So let’s take those two numbers and write them down. Here is where things get wild!

Step 3: Start Roundhouse Kicking

​I think the formal name for this method is “the Australian Method”, but I like to call it the Roundhouse Kick Method. You’ll see why in a moment.

Remember those two numbers from the last step? Write them down anywhere on your page.

Next, we’re going to divide each of those numbers by the value of \(a\). Remember, this is the leading coefficient of your trinomial.

​We are going to end up with two fractions that we want to reduce to lowest terms.

In our example, we had 8 and 3. So we are going to divide each of those numbers by our \(a\)-value of 6. 

This results in:

\[\frac{8}{6} \quad \text{and} \quad \frac{3}{6}\]

which reduces to:

\[\frac{4}{3} \quad \text{and} \quad \frac{1}{2}\]

So we just made things worse, right? For a moment, maybe. But this next part is just so fun that it was worth it!

Next, you are going to roundhouse kick those fractions from left to right.

roundhouse kicking the numerator of a fraction

What this actually means is that you want to picture yourself tipping the fractions over by pushing the numerator (including its sign) to the right of the denominator.

What we get here is:

\[ 3 \quad +4 \quad 2 \quad +1 \]

Step 4: Finish It!

So we ended up with two sets of numbers. What is even happening right now?

​Well, it turns out we are basically done factoring this trinomial!

To finish it off, we place brackets around each set of numbers.

\[ 3 \quad +4 \quad 2 \quad +1 \]

\[ (3 \quad +4) \quad (2 \quad +1) \]

We place an \(x\) next to the first term in each set of brackets … and we have our product of binomial factors!

\[ (3x+4)(2x+1) \]

Step 5: Check Your Answer

Now I know what you’re thinking. There’s absolutely no way that works…. right?

The best part about any factoring problem is that you can always check your answer. To do this, we simply expand and simplify our factored form of the trinomial back into standard form by using the distributive property. If we have equivalent answers, we should end up with our original trinomial! 

For the result from the above example:

\[\begin{split} &(3x+4)(2x+1) \\  =&6x^2+3x+8x+4 \\  =&6x^2+11x+4 \end{split} \]

There’s your proof! And I can promise you that this strategy will work every time!

Don’t believe me? Try it with any trinomials of the form \(ax^2+bx+c\). It even works when you have a negative middle term in your quadratic polynomial!

Master This Trinomial Factoring Shortcut with Practice

Mastering any new math concept is a matter of practice. And this strategy is no different! 

Grab my free trinomial factoring worksheet and start practicing so that you can make this strategy your go-to for trinomial factoring problems!

I created this trinomial factoring worksheet to provide you with specific examples that will help you master this shortcut. I’ve also included an answer key and video solutions to each of the problems so that you can see the strategy in action!

Helpful Resources for Factoring Trinomials

Since I’ve been teaching this trinomial factoring strategy for years, I have a large collection of resources that you may find helpful while learning this technique.

If you are looking for more help, check out:

My hope is that this epic trinomial factoring shortcut simplifies what is often seen as one of the most difficult algebraic techniques to learn in secondary mathematics. Keep this post handy as a summary of the method!

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


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