Synthetic Division

A while back I made a video covering the concept of Synthetic Division, which is an alternative to Long Division when dividing polynomials.

Here is the video.

Check it out for some examples.

This article will explore why synthetic division works, and the several advantages it provides over the traditional long division.

One fundamental mantra of mathematics is: If two methods accomplish the same correct solutions consistently, then they must be doing the same thing.

Long division and synthetic division are doing the same thing.

Let's have a look at an example:

Here are the workings for evaluating this example:

The first thing to note is that synthetic division leaves out the unknown terms, writing down only the coefficients. This can make it look simpler and may reduce working time. However the drawback is that positioning is very important.

In long division, each column does represent a different x term:

While there is a certain structure to keeping the columns tagged to each x^n term, writing it down explicitly removes any ambiguity and keeps it more straightforward when evaluating.

In synthetic division however, this isn't given much emphasis. We only really care about the operations of the coefficients. The x-terms still matter, but they're like this:

Whatever goes on in the middle yellow zone isn't important. We just have to write the numbers in the correct positions.

If we colour blot each term, we can see that these two methods are totally doing the same thing.

In fact, in synthetic division, we don't even fully evaluate the intermediate steps. Instead, all the addition steps are lumped together and evaluated in one step.

But long division makes it explicit: 7x - 20x = 13x then add +27x to get +14x.

Additionally, we see that the signs of the divisor have been reversed in synthetic division.

This reversal ensures that evaluation consists mostly of addition instead of subtraction, like in long division.

In synthetic division we add vertically, whereas in long division we have to perform a subtraction with each term of the quotient we obtain. This may introduce careless mistakes, and addition is arguably easier to perform.

However we must also take careful note of the signs. Signs are extremely important in synthetic division so I write them explicitly, which is probably a little unnecessary. As long as you're careful, it's fine.

So synthetic division is accomplishing the bare minimal amount of work that's required to produce the result. It's a hyper-optimised way for solving any polynomial division problem.

Footnote:

I've totally tried to generalise this with variables for coefficients but it gets really messy and unhelpful really quickly