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Question 1

Here’s a machine you can use for the questions below.

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It is just as easy to identify remainders in base xx division problems as it is in base 1010 arithmetic.

Look at

4x47x3+9x23x1x2x+1\dfrac {4x{4}-7x{3}+9x{2}-3x-1}{x{2}-x+1}
on the 1x1 \leftarrow x machine. Can you see that it equals 4x23x+24x^2-3x+2 with a remainder of 2x32x-3 yet to be divided by x2x+1x^2-x+1?

People typically write this answer as follows:
4x47x3+9x23x1x2x+1=4x23x+2+2x3x2x+1\dfrac {4x{4}-7x{3}+9x{2}-3x-1}{x{2}-x+1}=4x2-3x+2+\dfrac{2x-3}{{x{2}-x+1}}.

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

  1. Can you deduce what the answer to (2x2+7x+7)÷(x+2)\left( 2x^2+7x+7 \right) \div \left( x+2 \right) is going to be before doing it?
  2. Compute x4x23\dfrac{x4}{x2-3}.
  3. Try this crazy one: 5x52x4+x3x2+7x34x+1\dfrac{5x5-2x4+x3-x2+7}{x^3-4x+1}. (If you do it with paper and pencil, you will find yourself trying to draw 8484 dots at some point. Is it swift and easy just to write the number “8484”? In fact, how about just writing numbers and not bother drawing any dots at all?)

SOLUTIONS

Here are my answers.

  1. We know that (2x2+7x+6)÷(x+2)=2x+3\left( 2x^2+7x+6 \right) \div \left( x+2 \right)=2x+3 so I bet (2x2+7x+7)÷(x+2)\left( 2x^2+7x+7 \right) \div \left( x+2 \right) turns out to be 2x+3+1x+22x+3+\dfrac{1}{x+2}. Does it?
  2. x4x23=x2+3+9x23\dfrac {x4}{x2-3}=x2+3+\dfrac{9}{x2-3}.
  3. 5x22x+21+14x2+82x14x34x+15x^2-2x+21+\dfrac {-14x2+82x-14}{x3-4x+1}.
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