Is There an Alternative to Polynomial Long Division?

For whatever reason, I've always been very familiar with the division algorithm. But much like my peers, I never really understood how it worked (until recently), I just knew it did work and I was able to use it in my arsenal of mathematical weapons. Polynomial long division is really no different either, but these algorithms don't really get to what is happening.

Division has a very simple definition: How many ways can I partition or group objects into equal amounts? Admittedly, with large quantity of numbers and abstract ideas such as polynomials, this definition is really hard to wrap your head around.

So how can we answer these abstract division problems in another fashion? Is there another way that promotes conceptual understanding? Well of course there is! There's actually quite a few ways to do so.

The easy answer is synthetic division but any maths educator knows that it only work with a linear divisor. The following are examples of many different ways we can look at division, including the algorithm.

Without further ado, ladies and gentlemen: The Standard Algorithm

Oh! And the synthetic division, of course!

Now the next one is a similar take as the standard algorithm. But rather than multiply and subtract, it's just a bunch of factoring. But I do believe it does grasp at the concept of how many times (x+2) can go into this quadratic:

Essentially the goal is to start with the quadratic term and make it look something like the denominator. And then once that is simplified, just go down to the linear term and do the same thing until there is no reducing left to do. This is also how I divide fractions in my head:

Just split up the numerator into something the denominator can divide into and work from there. It's the same idea!

Last but not least is the box method! If you know the box method for multiplying, it's the same concept except working backwards (well duh, Robo). I'll just leave my solution for the example above below:

If you would like to see how this is done in detail, my man James Tanton has a brilliant video that explains it in depth.

What I love about the box method is that, not only does it do well in explaining that multiplication is just the area of a rectangle, but it can also be reverse engineered to do the inverse and does it in such an elegant and easy way. Hence, this is my go-to option when it comes to division. Sure the algorithms are good and dandy, but they just do not have the same power and grace that this box method has.

Don't get me wrong, division is a very hard concept, especially when dealing with the abstract. There really is no out-and-out best way, but know that there are always other representations of it, and use those to your advantage.

Until next time, happy math!