Mathematical Synthesis

I know all of us know what long division is, but do you know what synthetic division is? Well it's a very fast method of calculation taught to me by my Additional Mathematics teacher in Secondary Three, Mr Michael Doyle. However, there are some limitations and downsides to using this method. I shall illustrate using an example:

Let's say I'd like to simplify the following fraction using long division first:


Notice that I haven't done it step by step because I'm assuming you, the reader, knows enough of long division already. Well notice the important things: we have a polynomial in x, being divided by a divisor D(x), which results in a quotient Q(x) and a remainder R(x).

We now introduce what we call the remainder theorem, that is, for any polynomial, we can write it as such:


If the remainder R(x) = 0, then we say that the divisor D(x) is a factor of the polynomial. So clearly for this particular polynomial we have it written as:

Now, that being done, we naturally have to ask ourselves: is there an easier method? Well, there is! And this method is known as synthetic division. We shall use the previous polynomial to illustrate. This method requires us to write down the following first:

Notice that I've written down the coefficients of the polynomial - these coefficients will represent our polynomial of interest. Now, to the left of these coefficients is the number -3, which you should recognise as representing the divisor (x + 3). We put -3 to represent (x + 3) because earlier we said that:


Now clearly when x = -3, notice that the divisor term (x + 3) goes to zero, and we're only left with the remainder term (x - 6). So to divide by (x + 3), a really simple method is to substitute in the value that makes (x + 3) = 0. Now, there is a basic limitation: clearly then the remainder is obtained no longer as a polynomial, but as a specific number. So this explains why long division is preferred in some instances.

No matter the case, we move on the next step, as shown below:


And of course, the next step:


And the last step:


Our final presentation is therefore:

Voila! Do you see how synthetic division has worked out the same exact answer as long division? Try it yourself today!