Exploding Dots

6.5 (Optional) Remainders

It is just as easy to identify remainders in base \(x\) division problems as it is in base \(10\) arithmetic.


Play with  \(\dfrac{4x^{4}-7x^{3}+9x^2-3x+1}{x^2-x+1}\) in an \(1 \leftarrow x\) machine.

Can you see that it equals \(4x^2-3x+3\) with a remainder of \(2x-3\) yet to be divided by \(x^2-x+1\)?


People typically write this answer as follows:


\(\dfrac{4x^{4}-7x^{3}+9x^2-3x+1}{x^2=-x+1}=4x^2-3x+2+\dfrac{2x-3}{x^2-x+1}\) .



Here are some practice problems if you would like to play some more with this idea.


8. Can you deduce what the answer to \(\left(2x^2+7x+7\right) \div \left(x+2\right)\) is going to be before doing it?

9. Compute \(\dfrac{x^{4}}{x^2-3}\).

10. Try this crazy one: \(\dfrac{5x^{5}-2x^{4}+x^{3}-x^2+7}{x^{3}-4x+1}\).

If you do it with paper and pencil, you will find yourself trying to draw 84 dots at some point. Is it swift and easy just to write the number “84”? In fact, how about just writing numbers and not bother drawing any dots at all?




Please join the conversation on Facebook and Twitter and kindly share this page using the buttons below.

Share on FacebookTweet about this on Twitter




Take your understanding to the next level with easy to understand books by James Tanton.



Guides & Solutions

Dive deeper into key topics through detailed, easy to follow guides and solution sets.


light bulb


Consider supporting G'Day Math! with a donation, of any amount.

Your support is so much appreciated and enables the continued creation of great course content. Thanks!


Ready to Help?

Donations can be made via PayPal and major credit cards. A PayPal account is not required. Many thanks!