*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

**, that is, for any polynomial, we can write it as such:**

*remainder theorem*If the remainder

**, then we say that the divisor**

*R(x) = 0***is a**

*D(x)***of the polynomial. So clearly for this particular polynomial we have it written as:**

*factor*Now, that being done, we naturally have to ask ourselves: ** is there an easier method?** Well, there is! And this method is known as

**We shall use the previous polynomial to illustrate. This method requires us to write down the following first:**

*synthetic division.*

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

**, which you should recognise as representing the divisor**

*-3***. We put**

*(x + 3)***to represent**

*-3***because earlier we said that:**

*(x + 3)*Now clearly when

**, notice that the divisor term**

*x = -3***goes to zero, and we're only left with the remainder term**

*(x + 3)***. So to divide by**

*(x - 6)***a really simple method is to substitute in the value that makes**

*(x + 3),***. Now, there is a basic limitation:**

*(x + 3) = 0***. So this explains why long division is preferred in some instances.**

*clearly then the remainder is obtained no longer as a polynomial, but as a specific number*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:

**Do you see how synthetic division has worked out the same exact answer as long division?**

*Voila!**Try it yourself today!*

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