"Trinomial Method"

From the looks of Mr. Felix’s classroom, you might think he was a narcissist. Through his genealogy, he was graced with the same name as the 1919 silent-film-star-turned-cartoon, Felix the Cat, and he was unashamed to fill the majority of his walls with posters of Felix.

In addition, he was affectionately known by the student body as “that guy with the flat-top hair cut that coaches the basketball team,” but even more than that, he taught the all-so-frightening Algebra II.

I can remember his words clearly: “If you miss tomorrow’s class on factoring, you are going to fail my class.” My fear drove me to class the next day, and I sat through what was, in retrospect, one of the strangest and most unique lectures of my life. I did not realize how strange this lecture was until, well, today when a few of my classmates looked at a paper I had on factoring and asked why I had T-charts all over it. 

You might be thinking to yourself, “T-charts… factoring… what do these even have to do with each other?” Well, sit back, enjoy the show, and let me share with you Mr. Felix’s “Trinomial Method,” because somewhere behind the flat-top and basketballs is a pretty smart guy.

Polynomials may look like big scary monsters, but they are really easy to tackle with the right method.

TRINOMIAL METHOD

Let’s say we’re given the snazzy polynomial

3x² + 12x + 9,

and asked to factor it. First, we’re going to draw a cute little T-chart

that will also double as abstract art. Now, we’re going to take the first and last coefficient of the polynomial and multiply them together, and place this number in the upper left of the T-chart. Then we will take the middle coefficient and place it in the upper right of the T-chart.

Next, we will list all of the ways the leftmost number can be factored. We’ll get something like this:

Don't forget all possible ways to factor! 

So many factors...

Now, we will take each of these numbers and add them together (and also realize it’s easier to type it than to write it by hand), to get something like…

You might notice that 3 + 9 = 12 is in the 12 column… hmmm… I wonder if that means anything.

Let’s use these numbers to split up the middle term in the polynomial. We’ll get something like

3x² + 12x + 9 = 3x² + 3x + 9x + 9.

Now we’ll factor this, and we get

3x² + 3x + 9x + 9

3x (x + 1) + 9 (x + 1)

(x + 1) (3x + 9),

thus factoring our polynomial.

“How does this work, exactly?” you might ask. 

Let's flush this out.

Well, the general thought process behind it is that you need to find two numbers that when multiplied together equal the first and last coefficient multiplied together, but when added together equal the middle coefficient. Obviously, this does not work easily for all cases. You can try a few on your own if you’d like. But, in essence, this is it:

There are a few cool things about the “Trinomial Method.” If your leading coefficient is 1, s and t are how you factor your polynomial. As an example:

Math is trippy sometimes.

Obviously, the more time you spend using this method, the better you will get at it. Eventually you won't need to list out all of the factors and it will come more naturally to you.

Destroy all the polynomials with factoring! 

This can greatly reduce time spent factoring. Although it's important to learn how to factor the traditional way, Mr. Felix's "Trinomial Method" can be very helpful and is definitely worth looking at. 

So, thank you Mr. Felix! You keep doin' you.