1.
“Solve” the two equations above very carefully. Only use the operations above, but don’t just guess and check (even though that is a valid way to solve an equation in a finite set of numbers).
Worksheet Operation: Properties
In this activity, we will solve two simple equations:
However, the goal of this activity is really understand what properties of operations are needed to be able to solve these equations. To help us avoid taking things for granted, we will use a strange set of numbers and operations again. Here are addition and multiplication tables for the set of numbers we will use: \(\{0,1,2,3,4\}\)
| + | 0 | 1 | 2 | 3 | 4 |
| 0 | 0 | 1 | 2 | 3 | 4 |
| 1 | 1 | 2 | 3 | 4 | 0 |
| 2 | 2 | 3 | 4 | 0 | 1 |
| 3 | 3 | 4 | 0 | 1 | 2 |
| 4 | 4 | 0 | 1 | 2 | 3 |
| \(\cdot\) | 0 | 1 | 2 | 3 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 | 4 |
| 2 | 0 | 2 | 4 | 1 | 3 |
| 3 | 0 | 3 | 1 | 4 | 2 |
| 4 | 0 | 4 | 3 | 2 | 1 |
2.
Will we be able to solve any equation of the first type? There are 100 possible degree 1 polynomials here to check. Maybe don’t check them all, but still try to answer this question.
3.
There are 500 possible degree 2 polynomials. Will we be able to solve them all? Why or why not?
4.
Let’s try working with a different set of numbers: \(\{0, 1, 2, 3, 4, 5\}\text{.}\) Create operation tables for these that work the same way as they did for \(\{0, 1, 2, 3, 4\}\) on the last page. Then try to solve the same equations as before using these new tables. What goes wrong?
| + | 0 | 1 | 2 | 3 | 4 | 5 |
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| 5 |
| \(\cdot\) | 0 | 1 | 2 | 3 | 4 | 5 |
| 0 | ||||||
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| 5 |
