## 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 $$2x-1$$ yet to be divided by $$x^2-x+1$$?

People typically write this answer as follows:

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

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?

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